three essays in financial economics
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Three Essays in Financial Economics
Isakin, Maksim
Isakin, M. (2016). Three Essays in Financial Economics (Unpublished doctoral thesis). University
of Calgary, Calgary, AB. doi:10.11575/PRISM/28438
http://hdl.handle.net/11023/3044
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UNIVERSITY OF CALGARY
Three Essays in Financial Economics
by
Maksim Isakin
A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE INTERDISCIPLINARY DEGREE OF DOCTOR OF PHILOSOPHY
GRADUATE PROGRAM IN ECONOMICS
and
GRADUATE PROGRAM IN MANAGEMENT
CALGARY, ALBERTA
June, 2016
c� Maksim Isakin 2016
Abstract
This dissertation consists of three essays on financial economics. In the first essay, I build a theo-
retical model for pricing collateralized debt obligations (CDO) based on varying precision of credit
ratings. A credit rating agency (CRA) produces noisy ratings to maximize the proportion of firms
with high ratings but still ensures that firms choose lower risk projects. Increased fundamental
volatility in bad times makes high-risk choices more appealing to firms, which the CRA responds
to by increasing the precision of ratings. Only firms that can call existing bonds and issue new
ones will choose low risk projects at such times. Therefore, the resulting high risk strategy for
constrained firms in such periods implies that junior tranches get seriously impacted. In contrast,
senior tranches are more exposed to growth shocks, which increase the risk of all firms’ projects.
I structurally estimate the parameters of the model and show that the model is able to explain the
levels and a significant fraction of the volatilities of CDO tranche spreads. In the second essay, I
take the user cost approach to modelling a banking firm and analyze banks’ optimal response to
monetary and regulatory changes. The bank maximizes its profit choosing the quantities of finan-
cial goods such as deposits, loans, and investments based on their user costs. I estimate the system
of demand and supply of financial goods using data on U.S. banks over the 1992-2013 period. The
policy tools change the user costs of the financial goods and, therefore, bank’s demand and supply
of financial goods. I report the effects of an increase in interest paid on reserves, federal funds
rate and others. In the third essay, I develop a framework for estimating demand systems with
autoregressive conditional heteroscedasticity (ARCH). In this setup, the conditional variance is a
random variable depending on current and past information. Since most economic and financial
time series are nonlinear, using parametric nonlinear demand systems with an ARCH-component
can significantly improve the quality of a model. I prove the invariance of the maximum likelihood
estimator with respect to the choice of an estimated demand subsystem.
ii
Acknowledgements
I would like to express my gratitude to my supervisor, Professor Apostolos Serletis, who has helped
me throughout this entire venture with his guidance and insightful remarks. I thank him for his
continuous willingness to help with every step. It was my pleasure to work with him.
I would like to thank my co-supervisor, Professor Alexander David, for whom my admiration
and respect are enormous. Without him this research would have been most difficult. His constant
advice and support have been essential in my development as a researcher.
I also wish to thank Joanne Roberts, Kenneth James McKenzie, Kunio Tsuyuhara, Curtis Eaton,
Jean-Francois Wen for their guidance and helpful discussions.
I would like to thank my mother Tatiana Isakina and my father Aleksandr Isakin for encourag-
ing my interest in economics, for their patience and sense of humour. I thank my sister Ekaterina
Isakina for her support and long discussions of my papers.
Finally, I would like to thank my wife Alena Isakina for her belief in me. She has shared all
difficult and happy moments in my research.
iii
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii1 Credit Ratings, Credit Crunches,
and the Pricing of Collateralized Debt Obligations . . . . . . . . . . . . . . . . . . 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 Belief Updating From Learning and Bayesian Persuasion . . . . . . . . . . 161.4 Securitized Debt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.5 Empirical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5.1 First Stage Maximum Likelihood Estimation of Regime Switching Model . 191.5.2 Second Stage SMM Estimation of Firms’ Project Return Parameters . . . . 211.5.3 Risk-Adjustment Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 221.5.4 The Credit Spreads Puzzle, Spread Dynamics, and the Convexity Effect . . 22
1.6 Data Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 User Costs, the Financial Firm, and Monetary
and Regulatory Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.2 The User Cost Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.3 The Variable Profit Function Approach . . . . . . . . . . . . . . . . . . . . . . . . 472.4 Flexible Functional Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.4.1 The Translog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.4.2 The Normalized Quadratic . . . . . . . . . . . . . . . . . . . . . . . . . . 512.4.3 The Generalized Symmetric Barnett . . . . . . . . . . . . . . . . . . . . . 55
2.5 Data and Measurement Matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.6 Econometric Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.7 Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.7.1 Theoretical Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.7.2 Elasticities of Transformation . . . . . . . . . . . . . . . . . . . . . . . . 662.7.3 Compensated Price Elasticities . . . . . . . . . . . . . . . . . . . . . . . . 67
2.8 Monetary and Regulatory Policy Analysis . . . . . . . . . . . . . . . . . . . . . . 692.8.1 Interest on Reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.8.2 Reserve Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.8.3 Changes in the Federal Funds Rate . . . . . . . . . . . . . . . . . . . . . . 702.8.4 Changes in the Return on Investments . . . . . . . . . . . . . . . . . . . . 71
2.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
iv
3 Stochastic Volatility Demand Systems . . . . . . . . . . . . . . . . . . . . . . . . 1073.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.2 Neoclassical Demand Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083.3 Stochastic Volatility Demand Systems . . . . . . . . . . . . . . . . . . . . . . . . 1103.4 A Specific Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.5 Empirical Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1153.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
v
List of Tables
1.1 What Explains CDO Tranche Spreads? . . . . . . . . . . . . . . . . . . . . . . . 271.2 Maximum Likelihood Estimates of 4-Regime Markov Switching Model for Ratio
of Credit Growth at Nonfinancial Firms to GDP and Real GDP Growth . . . . . . 281.3 Second Stage SMM Estimation of Firms’ Project and Risk Adjustment Parameters 291.4 Implied Spreads (In Basis Points) From SMM Parameter Estimates . . . . . . . . 29
2.1 Size distribution of U.S. banks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.2 Assets and liabilities of U.S. banks (in billions) . . . . . . . . . . . . . . . . . . . 762.3 Percentage of Observations When Financial Goods are Outputs . . . . . . . . . . . 772.4 User Costs Averaged Across Banks (and the Discount Rate) . . . . . . . . . . . . 782.5 Translog Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.6 Normalized Quadratic Parameter Estimates . . . . . . . . . . . . . . . . . . . . . 802.7 Generalized Barnett Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . 812.8 Elasticities of Transformation, All Banks . . . . . . . . . . . . . . . . . . . . . . . 832.9 Elasticities of Transformation, Banks With Assets Less Than $100 Million . . . . . 842.10 Elasticities of Transformation, Banks With Assets Between $100 Million and $1
Billion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852.11 Elasticities of Transformation, Banks With Assets Between $1 Billion and $10
Billion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862.12 Elasticities of Transformation, Banks With Assets More Than $10 Billion . . . . . 872.13 Price Elasticities, All Banks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882.14 Price Elasticities, Banks With Assets Less Than $100 Million . . . . . . . . . . . . 892.15 Price Elasticities, Banks With Assets Between $100 Million and $1 Billion . . . . 902.16 Price Elasticities, Banks With Assets Between $1 Billion and $10 Billion . . . . . 912.17 Price Elasticities, Banks With Assets More Than $10 Billion . . . . . . . . . . . . 92
vi
List of Figures and Illustrations
1.1 Tranche Spreads, Economic Growth, and Credit Availability . . . . . . . . . . . . 301.2 Sequence of Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.3 Belief Updating From Learning and Bayesian Persuasion . . . . . . . . . . . . . . 321.4 Probabilities of the States From Regime Switching Model (1950:Q1 – 2014:Q4) . 331.5 Fundamentals: Data and Fitted From Regime Switching Model (1950:Q1 - 2014:Q4)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.6 Model and Actual Spreads on Senior and Equity Tranches . . . . . . . . . . . . . 351.7 Distribution of Firms’ Capital Stocks . . . . . . . . . . . . . . . . . . . . . . . . 361.8 Equity Value under the LR and HR Projects . . . . . . . . . . . . . . . . . . . . . 37
2.1 Dynamics of Bank Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932.2 Dynamics of Bank Liabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942.3 Average Reference Discount Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 952.4 Own-price elasticities for debt securities, hSS . . . . . . . . . . . . . . . . . . . . 962.5 Own-price elasticities for loans and leases, hLL . . . . . . . . . . . . . . . . . . . 972.6 Own-price elasticities for deposits, hDD . . . . . . . . . . . . . . . . . . . . . . . 982.7 Own-price elasticities for other debt, hOO . . . . . . . . . . . . . . . . . . . . . . 992.8 Own-price elasticities for equity, hEE . . . . . . . . . . . . . . . . . . . . . . . . 1002.9 Effects of a 25 Basis Points Increase in the Federal Funds Rate . . . . . . . . . . . 1012.10 Effects of a 25 Basis Points Increase in the Return to Investment . . . . . . . . . . 102
3.1 Squared residuals of equation (22), e21 . . . . . . . . . . . . . . . . . . . . . . . . 119
3.2 Squared residuals of equation (23), e22 . . . . . . . . . . . . . . . . . . . . . . . . 120
vii
Overview
After the 2008 financial crisis, the riskiness of debt markets and efficiency of monetary regulation
have become a focus of attention among academics, policy makers, and practitioners. A large part
of this discussion centres on the role of imperfect credit ratings for debt markets, in particular mar-
kets of structured debt products. An important question is whether information frictions brought
by imperfect credit ratings create additional risk for structured debt securities. Another subject of
ongoing debate is the efficiency of the monetary policy conducted by the Federal Reserve before
and after the crisis. A natural question that arises in this analysis is how monetary and regulatory
changes affect the U.S. banking sector. In this dissertation, I address these questions.
In the first chapter, I develop an equilibrium model with imperfect information to analyze
the pricing and risk of structured finance products. In particular, I explain why senior tranche
spreads are relatively more exposed to macroeconomic growth shocks, while junior spreads are
more exposed to credit availability shocks. In the model, a credit rating agency optimally changes
precision of its credit ratings. The credit ratings affect firm’s cost of capital and create dynamic
incentives for firms to raise or refinance their debt. If the credit market is disrupted, the firms cannot
refinance their debt they may increase their risk-taking according to asset substitution mechanism.
The increased risk of constrained firms in such periods implies that junior tranches get seriously
impacted. In contrast, senior tranches are more exposed to macroeconomic growth shocks, which
increase the risk of all firms’ projects. I structurally estimate the parameters of the model and show
that an endogenously generated ”convexity effect”, in large part due to the time varying precision
of credit ratings, is much more important in understanding CDO tranche spreads than the spread
on the entire pool of firms, the subject of past studies.
The second chapter investigates how monetary policy instruments and financial regulation af-
fect a banking firm. In doing so, I take the user cost approach to determine the value of bank’s
financial goods such as loans, deposits and equity capital. The user cost approach explicitly takes
viii
into account bank’s cost of funds and make bank’s decision problems aligned across financial insti-
tutions. Then I assume that the banking firm is a profit-maximizing entity producing intermediation
services between lenders and borrowers. Since the form of banking technology and profit function
is unknown, I approximate the profit function using three flexible functional forms: translog, nor-
malized quadratic and generalized Barnett. With a particular functional form I derive the system
of demand and supply equations and estimate this system using bank level data.
In the third chapter, I address the estimation of stochastic volatility demand systems. Since the
homoscedasticity assumption is unrealistic for most economic and financial time series, I assume
that the covariance matrix of the errors of demand systems is time-varying and derive the maximum
likelihood estimator. In the model, unconditional variance is constant but the conditional variance,
like the conditional mean, is also a random variable depending on current and past information. I
also prove an important practical result of invariance of the maximum likelihood estimator with
respect to the choice of an equation eliminated from a singular demand system. An empirical
application is provided, using the BEKK specification to model the conditional covariance matrix
of the errors of the basic translog demand system.
Statement of contribution. The paper presented in the first chapter is co-authored with
Alexander David who conducted the first stage of the estimation and described the results in the
introduction and section 1.5 and edited other sections. The papers in the second and third chap-
ters are co-authored with Apostolos Serletis who edited the text and described the results in the
introductions of both papers.
1
Chapter 1
Credit Ratings, Credit Crunches,
and the Pricing of Collateralized Debt Obligations
Joint paper with Alexander David
1.1 Introduction
A typical structured debt product such as a collateralized debt obligation (CDO) is a large pool
of economic assets with a prioritized structure of claims (tranches) against this collateral. These
instruments have made it possible to repackage credit risks and produce claims with significantly
lower default probabilities and higher credit ratings than the average asset in the underlying pool.
The structured finance market demonstrated spectacular growth during the decade before the fi-
nancial crisis of 2007/08 but almost dried up following massive downgrades and defaults of highly
rated structured products during the crisis (see Coval, Jurek and Stafford (2009b)). In an influential
paper, Coval, Jurek and Stafford (2009a) argue that investors did not adequately price the risk in
senior CDO tranches prior to the financial crisis (see also Collin-Dufresne, Goldstein and Yang
(2012) and Wojtowicz (2014)). In this paper, we do not focus on mispricing at particular points
of time, but provide a new theoretical model, which is based on the dynamic information content
of credit ratings through macroeconomic and credit cycles. We then structurally estimate the pa-
rameters of this model, and show that it is able to explain a substantial proportion of the historical
variation in CDO tranche spreads.
Following the work by these above authors, we study the time series of spreads on tranches on
the Dow Jones North American Investment Grade Index of credit default swaps, which are shown
in Figure 1.1. The “equity” tranche (top-left panel) represents the 0 to 3 percent loss attachment
2
points (these securities suffer losses if the loss on the entire collateral pool is between 0 and 3
percent of the underlying capital, are wiped out if the losses exceed 3 percent), while the “senior”
tranche (top-right panel) represents the 15 to 100 loss attachment points. While both spread series
rose rapidly during the financial crisis, the rise in the senior tranche spread was more spectacular,
from only about 10 basis points (b.p.) before the crisis, to above 230 b.p. at its peak. The equity
tranche by comparison, only roughly doubled from its pre-crisis level of 1175 b.p to 2700 b.p. at its
highest point. Post-crisis (2012-2014), the senior tranche spread was still 27 b.p., while the equity
tranche spread returned to its pre-crisis level.
The bottom-left panel of Figure 1.1 shows the quarterly growth in real GDP between 2004 and
2014. As seen, GDP growth bottomed out in the middle of the great recession, and resumed at a
more normal pace soon after the recession. The bottom-right panel shows that the ratio of credit
growth at nonfinancial companies to nominal GDP fell through the recession, and only bottomed
out after 2-3 quarters of the end of the recession. Matching up with the tranche spreads in the top
panels, the figures suggest that the senior tranche was more affected by the fluctuations in growth,
while the equity tranche was more affected by credit growth fluctuations. We examine if this is
true with some simple regressions.
In Table 1.1, we regress the spread on the entire pool (CDX) as well as different tranche spreads
on the two fundamentals. For each of the spread series, it is noteworthy that despite the presence of
a macroeconomic factor, credit growth additionally impacts tranche spreads. However, the relative
importance of the two fundamentals for junior and senior spreads is quite different. In lines 4 to
6, we see that GDP growth only explains only about 14.5 percent of the variation in the equity
tranche spread, while credit growth explains nearly 51 percent of its variation. Both variables are
significant in a joint regression. In contrast, in lines 13 to 15, we see that GDP growth explains
56 percent of the variation in the senior tranche spread, but credit growth explains only about 18
percent. In this paper, we ask why the relative exposure of the junior and senior tranches to the
alternative shocks is so different, and provide a new model to explain it.
3
There are three crucial ingredients in our model. First, we endogenize the risk of the firms
using an asset substitution mechanism. In particular, firms optimally choose their risk based on the
amount of debt that they need to service. Second, we introduce imperfect credit ratings using the
Bayesian persuasion concept, which we discuss more completely below. This concept implies that
the rating agency changes the intensity of its investigation of firms’ credit quality with the goal
of maximizing the proportion of firms with high credit ratings. Finally, credit availability in the
model can be in “available” or “nonavailable” states.
These features generate a mechanism that amplifies and propagates macroeconomic shocks
and can create catastrophic risk observed in the prices of CDO tranches (see Collin-Dufresne et al.
(2012)). According to this mechanism the rating agency produces a noisy signal (ratings) that al-
lows the firms to borrow at the cost compatible with low-risk behaviour, i.e. the credit ratings abate
the moral hazard problem just enough to induce low-risk behavior in current economic conditions.
In a sense, this puts the firms on the edge of low-risk and high-risk technologies and if economic
conditions change the firms could switch to risky behavior. To prevent this switching the rating
agency steps in and produces more precise signal (ratings). The new ratings can decrease the cost
of borrowing for the firms to maintain low-risk behavior, if they can call existing debt. However, if
credit availability is off, then, firms cannot call existing debt and will continue to choose high risk
projects.
We incorporate these features into a model of CDO tranches, where firms’ bonds are pooled
each period, and provide returns over 5 years, broken up in a short initial period of 1 year, after
which the bonds can be called, and a longer period of 4 years, at the end of which the returns are
distributed. In our model, we study how the information in credit ratings evolve over the business
cycle and the pricing consequences of these dynamics. We apply our model to explain risk and
pricing dynamics of the different CDO tranches. In particular we shed light on the difference in
relative exposure of junior and senior tranches to macro and credit availability shocks.
Our paper builds on the coordinating role of rating agencies in driving better investment deci-
4
sions by firms as in Boot, Milbourn and Schmeits (2006) and Manso (2013).1 In both papers the
models exhibit multiplicity of equilibria and the credit rating agency plays a coordinating role. In
their work, ratings lower the cost of finance specially since certain classes of investors are forced
by institutional rigidities to invest in highly rated securities. We instead build on the concept of
Bayesian persuasion (exemplified in a litigation context in Kamenica and Gentzkow (2011)), in
which the precisions of the ratings are controlled by the rating agencies investigation process. In
good times, the agencies allow some degree of contamination of the good ratings class by con-
ducting a less thorough examination of firms credit quality, but still ensuring that the overall cost
of capital of the mix of firms is low enough to induce the low risk project choice by high quality
firms. In periods of deteriorating fundamentals, the quality of the ratings are improved to weed
out bad firms from the high rating class, so that once again good quality firms still purse low risk
projects. Overall, the procedure maximizes the amount of debt with high ratings. It is important
to note that the time varying quality of ratings is distinct from alternative rating agency behaviors
such as misreporting and ratings inflation (see e.g. Fulghieri, Strobl and Xia (2014)), which might
also have played a role in financial crisis.
A significant contribution of our paper is a structural estimation of our Bayesian persuasion
model. Our estimation proceeds in two stages. At the first stage we use standard maximum likeli-
hood of regime switching models (see Hamilton (1994)) to estimate the cycles in credit availability
and macroeconomic growth. The regimes are observed by the agents in the model, but are unob-
served by the econometrician. In the second stage, we use the simulation method of moments
(SMM) to estimate the parameters of firms’ projects that fit tranche spreads. Our estimated model
provides several insights. In the model, during credit nonavailability states, several firms cannot
refinance their existing debt, and hence, they choose HR projects. Therefore an increase in risk of
some firms (relative to credit availability states) implies that the chance of the equity tranches ex-
periencing significant losses increases. But, since all firms do not increase their risk, the chance of1We therefore have a coordination game with strategic complementarity as for example in Milgrom and Roberts
(1990) and Cooper (1999).
5
all of them defaulting, an event that triggers losses in the senior tranche, does not increase. Instead,
the spreads for senior tranches, are higher in low growth (R) states, where all firms’ volatility in-
creases. This differential impact on senior and junior tranches helps our model match the different
dynamics of these tranches, and in particular why junior tranches are relatively more exposed to
credit shocks, and senior tranches to growth shocks. It is worth mentioning that as part of our
specification of our model, investors require risk adjustments to the transition probabilities across
different growth and credit availability states, which raise Q-measure or risk-adjusted default losses
(credit spreads) even though we constrain project parameters for firms to match the historical low
levels of default probability under the objective measure.
One of the key aspects of our model is the endogenously generated convexity effect of credit
spreads. As was pointed out by David (2008), in structural form models of credit risk (such as
the one presented here), credit spreads are convex function of firms’ asset values (capital stocks).2
Due to heterogeneity in firms’ capital accumulation, spreads for firms with low realized capital
rise more dramatically, then for the fall of spreads of firms that have high realized capital. The
greater the dispersion in capital stocks across firms, the greater is the difference in average spreads
across firms, and the spread calculated for a representative firm with an average capital stock. In
the model, heterogeneity increases in low growth states, but also to some extent when credit is
unavailable. Therefore spreads increase in such states. The convexity effect not only implies an
increase in the average spread generated by the model, but also the dynamics of spreads, as spreads
increases in states with higher dispersion, which endogenously varies as the economy transitions
through the macro and credit states. The convexity effect arises endogenously in our model as
the credit rating agency changes the precision of its rating over time. By doing so, it affects the
dispersion in borrowing costs across firms, which in turn affects their project choices, and the
dispersion in their capital stocks. This is a feature not present in prior work on the convexity
effect, such as in David (2008).2Bhamra, Kuehn and Strebulaev (2009), Kuehn and Schmid (2014), Feldhutter and Schaefer (2015), Chen,Cui, He and Milbrandt (2015), Christoffersen, Du and Elkamhi (2013), and Culp, Nozawa and Veronesi(2015) have also used this convexity effect to understand empirical properties of credit spreads.
6
The remainder of the paper follows the following plan: Section 2 introduces the model. Section
3 analyzes the equilibria in the model with two different credit rating standards. Section 4 provides
results on the pricing of CDO tranches. Section 5 presents empirical results. Section 6 provides
the data description. Section 7 concludes.
1.2 Model
The economy has a continuum of firms and investors and a monopolistic credit rating agency
(CRA). The economy goes through macroeconomic cycles with two states, booms (B) and reces-
sions (R), and credit cycles where either credit is available (state A) or not available (state N).
The macro states are identified by GDP growth, which in a given period is distributed N(µ ig,sg),
for i 2 {B,R}. Credit availability states are identified by the ratio of credit growth of nonfinancial
firms to GDP, which in a given period is distributed N(µ ic,sc), for i2 {A,N} Overall, the composite
states are S ⌘ {BA,BN,RA,RN}, and are driven by a stationary Markov process with a 4⇥4 tran-
sition matrix under the objective measure L ⌘ (lss0). Under the risk-neutral measure, the Markov
transition matrix is LQ, with elements l Qss0 = lss0 · eb1(µs
g�µs0g )+b2(µs
c�µs0c ), where b1 and b2, are the
risk adjustment factor loadings for macroeconomic and credit state transitions, respectively.3
Firms: There are two types of firms: good and bad. In each period a new pool of N bonds is
created, with a constant proportion a0 of good firms. Each firm in the pool provides returns over
three periods. In each period, every good firm chooses between two one-period projects: low risk,
LR, and high risk, HR. A bad firm can only implement the HR project. There are no switching
costs and a good firm could choose different projects in the first and second periods. Each project
returns r, which has a lognormal distribution with parameters µsp and s s
p for p 2 {LR,HR} and
s 2 S. The parameters depend only on the macro state, i.e. µBAp = µBN
p , µRAp = µRN
p , sBAp = sBN
p ,
sRAp = sRN
p for p 2 {LR,HR}. Conditional on the state of the economy, the project returns across
3Such risk adjustments are required for models to simultaneously match the low average historical default ratesof investment grade firms and the high level of their credit spreads in the “credit spreads puzzle” literature(see David (2008), Chen, Collin-Dufresne and Goldstein (2009), Bhamra et al. (2009), and Chen (2010)).
7
firms are independent, that is conditional firms’ risk is idiosyncratic. Each firm has capital in place
kt , which evolves as kt+1 = rt+1kt . We assume that firms pay no dividends, and that each unit
of capital is freely convertible into a unit of the numeraire good, i.e. the price of capital is one.
Further, we assume that the choice of the project is not contractable, even though returns of the
projects are observable ex-post. Since the returns have full support, the investors and the CRA
cannot ex-post infer the true type of the project even though they update their beliefs about the
type of the firm as described below.
We assume that at t = 0, each firm has capital in place K and raises debt D by issuing a two-
period zero-coupon callable bond with call price H. Therefore, total capital at t = 0 is K0 = K+ D.
We assume that the call price H = D, i.e. the bond can be called at par value. If at t = 1 credit is
available, each firm can refinance its debt. In this case, the firm redeems the existing two-period
bond and issues a new one-period bond in order to finance the call price of its existing bond. In
case of default, the debt holders incur a proportional dead weight cost, d , of the existing capital.
CRA: Firms’ type is not observable by either investors or the CRA. The CRA however can
conduct an investigation procedure whose results it reports truthfully to investors, and hence it
can influence the beliefs of investors. Even though the investigation process is costless, the CRA
can control its precision. In particular the G-rating could be assigned to a bad firm or the B-rating
could be assigned to a good firm (although as we show below, the latter is never optimal). The
type I and II errors associated with the ratings are n ⌘ P[B|good] and p ⌘ P[G|bad], respectively.
As in Lizzeri (1999) and Kartasheva and Yilmaz (2012), we assume that the CRA commits to
this structure of ratings. Investors’ beliefs about the type of a firm affects its cost of capital, and
ultimately its project choice. For example, in periods when investors’ assess that the firm is less
likely to be good, they charge a higher cost of capital, which leads even a good firm to choose the
HR project. In this case, the CRA can influence the investors’ beliefs by changing the precision
of ratings, and based on the new ratings standards that it announces, investors update their beliefs
about firms’ quality. Under the new beliefs, G-rated firms may refinance their debt at lower cost,
8
and subsequently chose the LR project.
We assume that the CRA attempts to issue as many G-ratings as possible. This preference
for high ratings can result from institutional investment constraints, as is assumed in Boot et al.
(2006).4 In particular, given a prior probability a that the firm is a good type, the CRA chooses n
and p to maximize the unconditional probability of assigning the good rating P[G] = a(1�n)+
(1�a)p . The process of changing the precision of the investigation process to induce a particular
outcome has been called “Bayesian persuasion” by Kamenica and Gentzkow (2011). The posterior
investors’ beliefs are represented by the probabilities that the firm is good conditional on observing
a G or a B rating, P[good|G] and P[good|B] respectively. There are two extreme cases. First, the
CRA can perfectly separate good and bad firms choosing n = 0 and p = 0. Second, the CRA can
produce completely uninformative ratings assigning G-rating to all firms, i.e. choosing n = 0 and
p = 1. If what follows, we denote a0 and a1 posterior beliefs that a firm is good if it gets the
G-rating at t = 0 and if it retains the G-rating at t = 1 respectively.
Investors: Let a0 be investors’ prior belief that any particular firm is good at t = 0. After observ-
ing the ratings, investors update their beliefs using Bayes’ law. In particular, their posterior belief
satisfies
a0 ⌘ P[good|G] =a0(1�n)
a0(1�n)+(1� a0)p(1.1)
c0 ⌘ P[good|B] =a0n
a0n +(1� a0)(1�p). (1.2)
As we will see, given the face value of debt, F , investors are able to tell if G-rated firms opti-
mally choose LR projects. Since the return distributions of the LR and HR projects are different,
they are able to partially learn about the type of the firm by observing the realized returns of its
project. Their updated belief satisfies
a1(r) =a0f s
LR(r)a0f s
LR(r1)+(1�a0)f sHR(r1)
, (1.3)
4More generally, this assumption is consistent with the widespread view that the issuer-pays business modeladopted by credit rating agencies leads to rating inflation (see e.g. Bar-Isaac and Shapiro (2011), Bolton,Freixas and Shapiro (2012), Fulghieri et al. (2012), Kartasheva and Yilmaz (2012), Harris, Opp and Opp(2013), and Cohn, Rajan and Strobl (2013)).
9
where f sp(r) is the probability density function of the return on p-type project in state s. If at t = 0,
good firms choose the HR project, the outcome of the project contains no information about the
type of the firm and, therefore, a1 = a0.
We assume that credit markets are perfectly competitive. Thus, in equilibrium the investors
require return that yields them zero expected profit.
Sequence of Events At t = 0 the investors have prior beliefs a0. The CRA chooses rating standard
parameters and issues G- and B-ratings for all firms. After observing the ratings, the investors
update their beliefs. If n � p , the investors’ beliefs that a firm is good increases for G-rated firms
and decreases for B-rated firms. In what follows, we focus on the firms that obtain the G-rating.
The investors’ beliefs for these firms increase from prior a0 to a0. After obtaining a rating, each
firm issues a two-period bond and starts a project. If a good firm chooses the LR project at t = 0,
the investors update their beliefs based on ((1.3)).
At t = 1, the CRA may adjust its ratings precisions, and investors update their beliefs according
to (1.1) and (1.2). Since the bond is callable, if at t = 1 credit is available, firms can refinance their
debt. The refinancing decision will be discussed in the next section. At t = 1 firms initiate new
projects as at t = 0. At t = 2, firms repay their debt. Figure 1.2 summarizes the sequence of events.
1.3 Equilibrium
This section describes a rational expectations equilibrium that arises in the model.
Definition 1 An equilibrium is a set of strategies of the CRA, firms, and investors such that:
1. Good firms choose optimally between LR and HR projects at t = 0 and t = 1 in each
state and decide whether to refinance their debt at t = 1 in the states when credit is
available.
2. Investors earn zero expected profits under rational expectations about the type of
the firm at t = 0 and t = 1, firms’ projects and refinancing decisions, and the CRA’s
10
rating precision.
3. The CRA follows the Bayesian persuasion strategy.
Equity value of firms known to be good: To determine the project choices of good firms, we
need to find the value to equity holders from each alternative. We use dynamic programming to
determine a good firm’s equity value. At t = 2, the good firm’s equity value is E2(K2) = max(K2�
F,0) where K2 the accumulated capital, and F is the face value of debt to be repaid. The following
lemma establishes the equity value at t = 1 given the project choice.
Lemma 1 Suppose a good firm with capital K1 chooses project p at t = 1 in state s. Then the
value its equity at t = 1 is
Es1(K1, p) = Â
s02Sl Q
ss0
h
K1 eµs0p +0.5(s s0
p )2N(�ds0
p )�F N(�ds0p �s s0
p )i
, (1.4)
where ds0p = (ln(F/K1)�µs0
p � (s s0p )
2)/s s0p and N(x) is the standard normal CDF.
Proof The value of the equity is the expected value of the firm after debt repayment under limited
liability, i.e.
E1(K1) = Âs02S
l Qss0
Z •
0(rK1 �F)+ f (r|µs0
p ,s s0p )dr
= Âs02S
l Qss0
Z •
F02/K1(rK1 �F) f (r|µs0
p ,s s0p )dr
= Âs02S
l Qss0
h
K1E⇥
r|r > F/K1,µs0p ,s s0
p⇤
�F P⇥
r > F02/K1|µs0p ,s s0
p⇤
i
,
where f (r|µ,s) is the PDF of the log normal distribution with parameters µ and s . Now using
standard results on the log normal distribution implies that (1.4) holds. ⌅
At t = 1 in state s, the good firm chooses between the low and high risk projects
Es1(K1) = max
P2{LR,HR}Es
1(K1,P). (1.5)
At t = 0, the good firm chooses the project to maximize the equity value, i.e.
Es0(K0) = max
P2{LR,HR}
n
Âs02S
l Qss0E⇥
Es01�
r1K0�
|P⇤
o
. (1.6)
Bad firms including those with the G-rating always implement the HR project.
11
Credit ratings At t = 0, the CRA issues ratings to every firm. After observing the ratings, the
investors update their beliefs about the type of each firm. At t = 1 the investors further update their
beliefs based on the outcomes of the projects. If the state changes, the CRA may adjust the ratings
and induce another update of investors’ beliefs. The following lemma shows the CRA’s optimal
choice of the rating standard in each period.
Lemma 2 Suppose that prior to observing ratings investors have beliefs at that a firm is good.
Then to induce target level of beliefs at at either t = 0 or t = 1, the CRA chooses the following
parameters of rating standard:
n = P[B|good] = 0 (1.7)
p = P[G|bad] =at(1�at)
at(1� at). (1.8)
Proof The CRA’s problem is
maxd1,d2
P[G] (1.9)
such that P[g|G]� q 0, (1.10)
where q 0 is the target level of beliefs. By the law of total probability
P[G] = P[G|g]P[g]+P[G|b]P[b] = qd1 +(1�q)d2. (1.11)
Given q , to maximize unconditional probability P[G] the CRA chooses probabilities d1 and d2 as
large as possible (but not greater than one). The optimal solution follows from the fact that these
variables are related by (1.10), i.e.
P[g|G] =P[G|g]P[g]
P[G|g]P[g]+P[G|b]P[b] =qd1
qd1 +(1�q)d2� q 0, (1.12)
or, equivalently,
d2 q(1�q 0)
q 0(1�q)d1. (1.13)
12
Conditions (1.13), d1 1 and d2 1 imply that maximum values of d1 and d2 are given by (1.7)
and (1.8). ⌅
Solution (1.7) and (1.8) results in posterior beliefs such that the investors are certain that a
firm is bad if it has the B-rating. Similar to Proposition 4 in Kamenica and Gentzkow (2011), it is
optimal for the CRA to assign all good firms G ratings but mix some bad firms into the G rating.
Expression (1.8) shows that conditional probability p is decreasing in at , that is higher posterior
beliefs require less noisy ratings. At the same time, p is increasing in at meaning that higher prior
beliefs allow the CRA to choose a looser rating standard. Since the firms with the B-ratings are all
bad, the CRA never changes their ratings at t = 1.
The CRA adopts the following logic at t = 1 in state s. If under prior beliefs (and possible
refinancing of its 2-period bond, to be discussed) the G-rated firms with capital K1 chooses the LR
project, the CRA keeps the ratings unchanged. Otherwise, if credit is available at t = 1, the CRA
reevaluates firms with the G-ratings such that under updated beliefs
Es1(K1,LR) � Es
1(K1,HR), (1.14)
i.e. they prefer the LR project. We assume that if G-rated firms choose the HR project under the
highest level of beliefs, i.e. P[good|G] = 1, or credit is unavailable, the CRA perfectly separates
good and bad firms and, therefore, increases the level of beliefs to unity. In this case, the rating
parameters are ns1 = 0 and ps
1 = 1. The rating accuracy implicitly depends on the level of firm’s
capital K1 and, thus, on firm’s leverage, since firms with higher leverage, are more likely to choose
the HR project. Therefore, at t = 1 there is a continuum of ratings indexed by firms’ level of capital
and letter G(K1) or B(K1).5
Similarly, at t = 0 in state s, the CRA chooses rating parameters ns0 and ps
0 as in Lemma 2 to
induce beliefs a0, which is the minimum level of beliefs such that
Es0(K0,LR)� Es
0(K0,HR). (1.15)5Since the debt of the firm is fixed, the capital is a measure of the leverage of the firm. The leverage dependenceof credit ratings is consistent with the ratings procedures used by most CRAs.
13
Debt value: Due to limited liability if at maturity the value of firm’s capital is less than the face
value of the bond, the value of the debt is the value of capital less bankruptcy costs. Therefore, at
t = 2 the value of a bond belonging to a firm with capital K2 is
D2(K2) =
8
>
>
<
>
>
:
F if F K2
(1�d )K2 if F > K2,
(1.16)
where F is the bond’s face value. The following lemma gives the value of a bond at t = 1 in state
s if investors know which project is going to be chosen.
Lemma 3 Suppose a good firm with capital K1 chooses project p at t = 1. Then, its value at t = 1
is
Ds1(K1, p) = Â
s02Sl Q
ss0
h
(1�d )K1 eµs0p +0.5(s s0
p )2N(ds0
p ) + F N(�ds0p �s s0
p )i
, (1.17)
where dsp is defined in the statement of Lemma 1.
Proof The value of the debt is the expected value of repayment, i.e.
D1(K1) = Âs02S
l Qss0
Z •
0D2(rK1) f (r|µs0
p ,s s0p )dr, (1.18)
where f (r|µ,s) is the PDF of the log normal distribution with parameters µ and s . Since
D2(K2) =
8
>
>
<
>
>
:
F02, if K2 � F02
(1�d )K2, if K2 < F02,
(1.19)
the expected repayment can be written as
D1(K1) = Âs02S
l Qss0
h
Z F02/K1
0(1�d )rK1 f (r|µs0
p ,s s0p )dr+
Z •
F02/K1F02 f (r|µs0
p ,s s0p )dr
i
= Âs02S
l Qss0
h
(1�d )K1E⇥
r|r F02/K1,µs0p ,s s0
p⇤
P⇥
r F02/K1|µs0p ,s s0
p⇤
+F02P⇥
r > F02/K1|µs0p ,s s0
p⇤
i
, (1.20)
14
where r is log normally distributed random variable. Using the fact that the expected value of the
truncated from above log normal distribution is
E[x|x u] = exp(µ +s2/2)F(u�s)/F(u), (1.21)
where u = (lnu�µ)/s and F(x) is the standard normal CDF, we have the result. ⌅
Let Ps1(K1) 2 {LR,HR} be a good firm’s optimal project choice at t = 1 in state s. Then
investors value of a G-rated firm’s bond is
Ds1(K1,G,as
1) = as1Ds
1�
K1,Ps1(K1)
�
+(1�as1)D
s1�
K1,HR�
. (1.22)
Since investors do not observe the firm’s type, they take expectations of the value of bond condi-
tional on its type. The value of a bond Ds1(K1,B) belonging to a B-rated firm is given by (1.22)
when as1 = 0.
The refinancing decision: A firm with capital K1 and rating Q1 2 {G,B}, can refinance its debt
at t = 1 in state s if credit is available. It can issues a one-period bond with face value Fs12 such
that Ds1(K1,Q1,as
1) = H. If Fs12 < F02, then the firm can lower its borrowing costs. It is worth
mentioning that a firm may be unable to refinance its debt if H is greater than its debt capacity in
that state. Given the log-normality of returns, the probability of the debt being repaid declines to
zero as the face value increases. In particular, since we model bankruptcy costs, the value of the
firm’s bond has a maximum value (its debt capacity) as we increase its face value. For a G-rated
firm, the face value is decreasing in as1, as investors believe it is more likely to be a good type.
The value of the two-period bond at t = 0 depends on firms’ optimal decision on refinancing
at t = 1. Let Rs(K1,Q1,as1) be the payment made by the firm with capital K1 and rating Q1 to
bondholders.
Rs(K1,Q1,as1) =
8
>
>
<
>
>
:
H if Fs12 < F02
Ds1(K1,Q1,as
1) otherwise,(1.23)
15
Then if Ps0 2 {LR,HR} is the firm’s optimal project choice at t = 0 in state s, the value of the
two-period bond at t = 0 in state s is
Ds0(K0) = Â
s02Sl Q
ss0
⇣
as0 Eh
Rs0�r1K0,G,as1�
| Ps0 ,s
0i
(1.24)
+(1�as0) E
h
ps01�
r1K0�
Rs0�r1K0,G,as1�
| HR,s0i
+(1�as0) E
h
�
1�ps01�
r1K0��
Rs0�r1K0,B,0�
| HR,s0i⌘
,
where ps01�
K1�
is the probability that a bad firm with capital K1 retains the G-rating at t = 1 in state
s0. The first expectation in (1.24) corresponds to the value of the two-period bond of a good firm.
A bad firm rated G at t = 0 gets either G or B rating at t = 1. The bond of a bad firm that retains
the G-rating at t = 1 has the same value as the bond of a good firm. The bond of a bad firm that
gets downgraded at t = 1 has the value when investors are certain that the firm is bad. The second
and third expectations in (1.24) provide the values for these two mutually exclusive events.
1.3.1 Belief Updating From Learning and Bayesian Persuasion
At t = 0 all good firms start with the same level of capital and debt, and hence choose the same
project. At t = 1, updated beliefs of firms rated G being good, depend on the project return as
shown in (1.3). Figure 1.3 shows the level of beliefs, P(good|G), before and after observing ratings
at t = 1. As can be seen, the posterior of the firm being good, increases in the level of capital upto
a range, since the mean return of the LR project exceeds that of the HR project. For higher levels
of capital, the belief falls, as the relative likelihood of very high returns (capital) increase for the
HR project, which has a higher variance.
If K1 is sufficiently low (leverage of the firm is high), the asset substitution problem urges the
firm to choose the HR project. In particular, if K1 < K1, the firm chooses the HR project even
if investors are certain that the firm is good, i.e. P[good—G]=1, and the firm is able to refinance
its debt. In this case, we assume that the CRA chooses perfectly precise ratings and therefore
posterior beliefs, as1 ⌘ P[good|G] = 1. On the contrary, if the level of capital is sufficiently high,
i.e. K1 � K1, the firm chooses the LR project under any level of as1. In this case, the CRA does
16
not adjust the ratings and as1 = as
1. Finally, if K1 2 [K1,K1] (the shaded are in the figure), the
CRA is able to influence firms’ project choice at t = 1. With beliefs updated only after observing
project returns, as1, the firm chooses the HR project. In this case, the CRA increases the precision
of ratings such that under updated beliefs, as1, the G-rated firms would choose the LR project if
they could refinance their debt to lower their cost of capital. If credit is unavailable however, the
firms would continue to choose HR projects Therefore, the lack of credit is an additional factor of
systematic risk that coupled with a business downturn leads to the simultaneous increase of firms
risk. We use this mechanism to explain the dynamics of spreads on the tranches of a collateralized
debt obligation.
1.4 Securitized Debt
In this section we apply the model to price the tranches of a collateralized debt obligation (CDO).
The pricing of the tranches of a synthetic CDO with a large homogeneous collateral pool is similar
to that in Coval et al. (2009a) and Gibson (2005). We assume that at t = 0, the collateral pool
consists of a large number of callable two-period bonds issued by G-rated firms, each with capital
K0. The project choices by these firms at t = 0, and the realized returns on these projects implies
that at t = 1 their capital stocks differ, as do their leverage ratios. Moreover, the rating precision at
t = 1 depends on firms’ capital, contributing to an additional dispersion in investors’ beliefs about
these firms. As before, firms’ project returns bear purely idiosyncratic risk conditional on the state
of the economy. At t = 1 if credit is available the firms decide whether to refinance their debt. If a
firm refinances its bond, the newly issued bond replaces the old bond in the pool. Then the firms
again choose a project and end up with capital K2 at t = 2.
Since at t = 1 the pool is heterogeneous we cannot obtain the distribution of losses in a pool at
t = 2 in closed form as in Gibson (2005). Instead, to estimate the losses and determine the tranche
spreads at t = 1 we resort to a simulation technique. First, we simulate the levels of capital of
each firm in the pool at t = 1. In doing so, we fix state s at t = 0 and form a pool of N firms
17
with capital K0 and randomly assigned type such that probability of the good type is a0. For each
firm in the pool randomly determine the level of capital at t = 1 according to equation Ki1 = K0ri,
where random return ri is drawn from the log-normal distributions with parameters (µsP, s s
P) with
P project choice of the good firms at t = 0 and s state at t = 1 for a good firm and (µsHR, s s
HR) for
a bad firm.
Second, for each state s at t = 1 simulate the distribution of losses in the pool and calculate
average losses L[AL,AU ] on each CDO tranche with attachment points AL and AU . In particular,
we conduct T Monte-Carlo simulations and in each trial:
1. Given state s at t = 1 and transition probabilities l Qss0 randomly choose state s0 at
t = 2.
2. For each firm in the pool randomly determine the level of capital at t = 2 according
to equation Ki2 = Ki
1ri, where random return ri is drawn from the log-normal distri-
butions with parameters (µs0P , s s0
P ) with P project choice of the good firms at t = 1
and s0 state at t = 2 for a good firm and (µs0HR, s s0
HR) for a bad firm.
3. Calculate the value Di2(K2) of ith bond according to (1.16) and the total payoff of
the portfolio of N bonds
D⇤2 =
ÂNi=1 Di
2(K2)
ÂNi=1 Fi
⇤2,
where Fi⇤2 is equal Fi
02 if the ith bond has not been refinanced or Fi12 otherwise.
4. Calculate expected loss on each tranche with lower and upper attachment points AL
and AU
L[AL,AU ] = max(L⇤2 �AL,0)�max(L⇤
2 �AU ,0),
where L⇤2 = 1�D⇤
2.
Finally, given states at t = 0 and t = 1 we use the average losses on each tranche to calculate
18
the spreads for each tranche using equation
Sss[AL,AU ] =AU �AL
AU �AL � L[AL,AU ]�1.
1.5 Empirical Analysis
In this section, we structurally estimate our model and evaluate its implications for the pricing of
CDO tranches. Our empirical estimation is implemented in two stages. At the first stage we use
standard maximum likelihood of regime switching models (see Hamilton (1994)) to estimate the
cycles in credit availability and macroeconomic growth. The regimes are observed by the agents in
the model, but are unobserved by the econometrician. In the second stage, we use the simulation
method of moments (SMM) to estimate the parameters of firms’ projects that fit tranche spreads.
1.5.1 First Stage Maximum Likelihood Estimation of Regime Switching Model
The specification of the regime model is at the beginning of Section 1.2. Macro cycles are identified
as regimes of real GDP growth (states B and R), while credit cycles are identified as regimes in the
ratio of credit growth at nonfinancial companies to nominal GDP (A and N). We then form the four
composite states (BA), (R,A), (BN), and (R,N). The specification has homoskedastic fundamentals,
so that the volatility of each process is the same in each regime. We maximize the likelihood of
the econometrician observing these four composite regimes. It is useful to note, that we estimate
the model from 1951 to 2004, before the start of the CDO tranche data. Using these estimates, we
filter the data to provide the econometrician’s filtered probability of the underlying states in-sample
(1951 – 2004:Q2) and then out-of-sample (2004:Q3 – 2014). By doing so, we attempt to mitigate
over-fitting of tranche spreads in the second subsample. The time series of the ecoometrician’s
filtered probabilities are denoted as {wmc(t)}.
Parameter estimates of the model are in Table 1.2. As seen in the top panel, the ratio of credit
growth to GDP is about 2.5 times as high in A states relative to N states, although even the latter is
positive. This is consistent with our model in which firms can obtain credit with some probability
19
even in N states. Real GDP growth is about 4.5 percent (at an annual rate) in B states, and shrinks
at nearly 1 percent in R states. GDP growth is significantly more volatile than credit growth. The
quarterly transition matrix (and its standard errors) under the objective measure are in the two
subsequent panels. We will discuss the risk-adjusted transition matrix (under the Q-measure) in
a subsequent subsection. As in several other estimates of growth regimes, booms are far more
persistent than recessions. In addition, booms are more persistent in credit availability states.
Indeed, the RN state is the least persistent.
The econometrician’s filtered probabilities of the four composite regimes are in Figure 1.4. As
seen, each of the four regime probabilities become quite likely in different stages of the cycles.
Quite notably, the probabilities of BA regimes increase significantly in the middle of most NBER
recessions (shaded areas) in the sample. However, the probabilities of N (credit unavailability)
regimes, remain quite high even after the end of recessions. Therefore, credit availability lags
GDP growth, and a simple linear regression of credit growth on lagged GDP growth verifies this
intuition.
Ratio of Credit Growth(t)/GDP(t) = 0.219+0.451GDP Growth(t �4)+ e(t) (1.25)
[1.431] [3.306] (1.26)
where t-stats adjusted for autocorrelation and heteroskedasticity are in parenthesis.
Figure 1.5 shows that the expected growth rates from the model, fit the data quite well. In fact,
the regression of each of the realized data series on its expected value calculated using the regime
parameters and the filtered probabilities explains close to 57 percent of the variation in each series.
The plots also show that expected GDP growth troughs in recessions, while expected credit growth
troughs 2 to 4 quarters after the end of each recession. This was specially true for the last three
recessions.
While we have a reduced form specification of macro and credit regimes, our estimates are
consistent with the view that credit availability shrinks at the onset of weak growth, and persists
for several quarters even after growth resumes, perhaps because lenders become more cautious.
20
1.5.2 Second Stage SMM Estimation of Firms’ Project Return Parameters
We now provide a description of the SMM estimation of the parameters of firms’ projects and the
risk adjustment demanded by investors, which are estimated at the second stage. These parameters
are chosen to match the time series of spreads using the econometrician’s filtered probabilities of
the states (regimes) at the first stage. We recall, that the probabilities are out-of-sample from the
first stage estimation. To fit spreads, we use the pricing formulae of tranche and the entire CDX
spread developed in Section 1.4. To implement the 3-period model, we use time periods of unequal
length. The physical time between periods 0 and 1, is 1-year, and the time between periods 1 and 2,
is 4-years. Recall that the bond is callable after the first period, which is similar to actual callability
restrictions on bonds, which can be called for only a fraction of their maturities.
Using the filtered probabilities, we have
St [AL,AU ] = Âs2S
Âs2S
l Qss
Âr2S l Qrs
wst Ss,s[AL,AU ]. (1.27)
As in the Section 1.4, tranche spreads at t depend on the face value of debt issued at t � 1, and
hence the expected spread at t depends not only on the probabilities of states at t, but in addition,
states at t �1.
The parameters that we need to estimate are a) µsp and s s
p, for p 2 {LR,HR}, and s 2 {B,R},
resulting in 8 parameters. We also estimate b1 and b2, which are the parameters to risk-adjust the
transition probability matrix, overall resulting in 10 parameters.
For the SMM procedure we use the time series of the five spreads (one for the full pool, CDX,
and four tranches). We also calculate the conditional volatility of spreads at each date using the
filtered probabilities, wmc(t), and match these to the unconditional sample volatility of each spread.
In addition, we calculate the P-measure probability conditional probability of default, using the
simulated beliefs of investors of each firm being good states. The average of this time series is used
to match the historical 4-year default probability of BBB-rated firms by S&P. Finally, we target the
endogenously determined leverage ratio in the model at each date, to match the unconditional
average of leverage of BBB-rated firms in the data. This gives us 11 moments to match, overall
21
leading to an overidentified identified SMM estimator.
The parameter estimates are given in Table 1.3. As seen, both type of projects are riskier in
recession states. In addition, HR projects have higher risk and lower returns than LR projects in
each state.
1.5.3 Risk-Adjustment Parameters
The signs of the risk-adjustment for the growth and credit growth transitions in Table 1.3 are
quite compelling. As in common parlance, the price of risk of a shock is positive (negative) if it
over (under) weights the transition probability from a good to a bad state for investors. For our
estimated parameters, we find b1, the adjustment for the growth transition probability is positive,
as is consistent with several other empirical studies. Quite interestingly, our estimate of b2, the
credit growth transition, is negative. Above, we showed that credit growth remains strong at the
start of a recession, but then weakens, and remains weak after the end of the recession. Therefore,
credit growth shocks have a slightly countercyclical property, and hence has a negative price of
risk. The overall risk-adjustment across the composite states in our model does deliver us the
increase in credit spreads, which measure expected default losses under the Q-measure, to match
the historical spread level.
1.5.4 The Credit Spreads Puzzle, Spread Dynamics, and the Convexity Effect
Using the parameters estimated from the SMM procedure, we calculated the spreads for each of
the tranches in period t = 1 of the model for each state at t = 0. The state at t = 0 is relevant for the
spreads at t = 1, since it determines the proportion of firms choosing HR projects, and hence the
face values of debt. These implied spreads are in Table 1.4. As seen senior spreads S(15,100) are
zero in credit availability (A) states, while equity tranche spreads are close to their values in credit
unavailability (N) states. In the model, during N states, several firms cannot refinance their existing
debt, and hence, they choose HR projects. Therefore an increase in risk of some firms (relative to
A states) implies that the chance of the equity tranches experiencing significant losses increases.
22
But, since all firms do not increase their risk, the chance of all of them defaulting, an event that
triggers losses in the senior tranche, does not increase. Instead, the spreads for senior tranches, are
higher in low growth (R) states, where all firms’ volatility increases. This differential impact on
senior and junior tranches helps our model match the different dynamics of these tranches.
Using these state-dependent spreads, we calculate the fitted spreads at each date, which are
shown in Figure 1.6. Due to high risk-adjusted transition probabilities, the model is able to pro-
vide average spread levels fairly close to their historical averages, even as we match the objective-
measure average default probability. As in the credit spreads puzzle literature, this happens, be-
cause more defaults happen in recessions, which have boosted transition probabilities under the Q-
measure. In particular, the model’s spreads rose close to their historical values for junior tranches
in the great recession, but fell a bit short for senior tranches. Also, significantly, the model’s senior
tranche spreads, fell once economic growth picked up at the end of the recession, but the equity
tranche in particular remained at high levels until nearly 2010, when credit growth resumed. This
is in line with our motivating regressions in the introduction, where a much larger proportion of
the equity tranche is explained by credit growth rather than economic growth, while the reverse is
true for the senior tranche. Overall, the model-fitted spreads explain between 51 percent (equity
tranche S(0,3)) and 71 percent (S(15,100) tranche), with better for senior tranches.
One of the key aspects of our model is the endogenously generated convexity effect of credit
spreads. As was pointed out by David (2008), essentially in structural form models of credit
risk (such as this one), credit spreads are convex function of firms’ asset values (capital stocks).
Due to heterogeneity in firms’ capital accumulation, spreads for firms with low realized capital
rise more dramatically, then for the fall of spreads of firms that have high realized capital. The
greater the dispersion in capital stocks across firms, the greater is the difference in average spreads
across firms, and the spread calculated for a representative firm with an average capital stock. In
the model, heterogeneity increases in low growth states, but also to some extent when credit is
unavailable. Therefore spreads increase in such states. The convexity effect not only implies an
23
increase in the average spread generated by the model, but also the dynamics of spreads, as spreads
increases in states with higher dispersion, which endogenously varies as the economy transitions
through the macro and credit states. Figure 1.7 shows the distribution of firm’s capital stocks at
t=1 in four possible states of the economy.
As mentioned above, the convexity effect arises endogenously in our model. In particular, as
the CRA changes the precision of its rating over time, it affects the dispersion in borrowing costs
across firms, which in turn affects their project choices, and the dispersion in their capital stocks.
This is a feature not present in prior work on the convexity effect, such as in David (2008).
1.6 Data Description
We obtain monthly time series of tranche spreads on synthetic CDOs based on the DJ CDX North
American Investment Grade Index (CDX.NA.IG). This index consists of an equally weighted port-
folio of 125 credit default swap (CDS) contracts on US firms with investment grade debt. Our sam-
ple covers the eleven year period from September 2004 to October 2014. The data from September
2007 to October 2014 is provided by Bloomberg (CMA New York). The data from September 2004
to August 2007 is from Coval et al. (2009a).
The CDX indices roll every six months. In particular, on September 20 and March 20 new
series of the index with updated constituents are introduced. After a new series is created, the
previous series continue trading though liquidity is usually concentrated on the on-the-run series.
An exception is series 9 introduced in September 2007 and traded till the end of 2012 together
with less liquid on-the-run series. The CDX indices have 3, 5, 7 and 10 year tenors. We use 5 year
CDX indices which are most liquid for most series.
We build our sample from on-the-run series except period from March 2008 to September 2010
where we use most liquid series 9. Before series 15 introduced in September 2010 the CDX index
has been traded with tranches 0-3%, 3-7%, 7-10%, 10-15%, 15-30% and 30-100%.6 Starting from6The tickers of the tranches of series 9 are CT753589 Curncy, CT753593 Curncy, CT753597 Curncy, CT753601
24
series 15 and onward, only odd series of the index are traded with tranches and the structure of
tranches changes to 0-3%, 3-7%, 7-15% and 15-100%.7
We focus our analysis on the equity and the most senior tranches. Since the equity 0-3%
tranche is quoted as an upfront payment, we calculate the par spread using the formula S0�3% =
500b.p.+U/D where U is the upfront fees and D is the time to maturity of the tranche. While
earlier series (before 15) have tranches 15-30% and 30-100%, there is only one tranche 15-100%
for later series. To make the series consistent we create a tranche 15-100% for earlier series as the
sum of tranches 15-30% and 30-100%.
We obtain credit growth at nonfinancial corporate businesses from the Federal Reserve Board’s
flow of funds accounts (series FA104104005.Q), and nominal and real GDP from the St. Louis
Fed FRED database.
1.7 Conclusion
In this paper, we provide a new model to show how imperfect credit ratings and occurrence of
credit crunches can create catastrophic risk observed in the prices of CDO tranches. There are
three crucial ingredients in our model. First, we endogenize firms’ risk-taking using the asset
substitution mechanism. In particular, the firms choose the riskiness of their projects based on the
amount of debt that they need to service. Second, a credit rating agency changes the intensity of the
investigation of firms’ credit quality to maximize the proportion of firms with high credit ratings.
Finally, the credit shortage can trigger firms’ risky behaviour if they are unable to refinance their
debt under a more precise rating standard. This increases the risk of senior tranches of structured
finance products.
We structurally estimate the parameters of our dynamic Bayesian persuasion model and show
Curncy, CT753605 Curncy, CT753609 Curncy.7The tickers of the tranches of series 15, 17, 19 and 21 are CY071225 Curncy, CY071229 Curncy, CY071233
Curncy, CY071237 Curncy, CY087579 Curncy, CY087583 Curncy, CY087587 Curncy, CY087591 Curncy,CY125375 Curncy, CY125380 Curncy, CY125385 Curncy, CY125390 Curncy, CY181667 Curncy, CY181672Curncy, CY181677 Curncy and CY181682 Curncy.
25
that it can shed light on the puzzling phenomenon that senior tranche spreads are relatively more
exposed to growth shocks, while junior spreads are more exposed to credit availability shocks.
In particular, refinancing of existing debt may not be possible during a credit crunch, and hence
the resulting high risk strategy for some firms in such periods implies that junior tranches, get
seriously impacted. In contrast senior tranches get affected by growth shocks, which increase
the risk of all firms’ projects. A crucial aspect of our model is that an endogenously generated
“convexity effect”, in large part due to the time varying precision of credit ratings, is much more
important in understanding CDO tranche spreads than the spread on the entire pool of firms, the
subject of past studies.
26
Table 1.1: What Explains CDO Tranche Spreads?
No. a b1 b2 R2
CDX Spread1. 111.61 -49.31 0.433
[8.71] [-3.88]2. 113.69 -51.57 0.344
[7.30] [-3.08]3. 126.84 -43.036 -42.89 0.669
[14.29] [-4.96] [-6.31]Spread (0-3)4. 1564.69 10.73 0.145
[10.73] [-2.50]5. 1717.56 -646.36 0.508
[15.11] [-4.73]6. 1779.44 -205.87 -604.85 0.577
[17.01] [-3.58] [-4.42]Spread (3-7)7. 627.11 -362.28 0.362
[5.48] [-3.04]8. 686.00 -484.66 0.470
[114.18] [126.75]9. 777.69 -300.23 -424.13 0.711
[10.30] [-4.22] [-7.76]Spread (7-15)10. 406.83 -343.59 0.498
[4.53] [-3.32]11. 410.68 -333.43 0.341
[3.57] [-2.60]12. 503.47 -303.78 -272.17 0.718
[6.898] [-4.17] [-5.49]Spread (15-100)13. 66.67 -53.30 0.564
[6.06] [-4.88]14. 60.48 -35.26 0.179
[3.53] [-1.94]15. 75.64 -49.61 -25.26 0.653
[-5.28] [-3.17]Tranche spreads are on the Dow Jones North American Investment Grade Index, which are reported by CreditMarket Analysis (CMA) and obtained from Bloomberg (see Data Appendix for construction of our time series).CDX represents the full CDO. Spread (AL,AU) represents the spread on a trance with loss attachment points ALand AU in percentage points. For example, the “senior” spread represents the 15 to 100 percent loss attachmentpoints, while the “equity” tranche represents the 0 to 3 loss attachment points. We report the coefficients of thefitted regression:
Tranche Spread(t) = a + b1 Real GDP Growth + b2 Credit Growth(t)/GDP(t)+ e(t)
for alternative tranches T-statistics are in parenthesis and are adjusted by White’s procedure for heteroskedasticity.
27
Table 1.2: Maximum Likelihood Estimates of 4-Regime Markov Switching Model for Ratio ofCredit Growth at Nonfinancial Firms to GDP and Real GDP Growth
Ratio of Credit Growth to GDP (%)µ1
c µ2c
0.901 0.366(0.185) (0.005)
Quarterly Real GDP Growth (%)µ1
g µ2g
1.109 -0.237(0.031) (0.002)
Volatilities (%)sg sc
0.251 0.755(7.432) (0.019)
Standard errors are in parenthesis.
Quarterly Transition Probability Matrix (Estimates)(BA) (RA) (BN) (RN)
(BA) 0.930 0.000 0.039 0.030(RA) 0.168 0.832 0.000 0.000(BN) 0.000 0.000 0.900 0.099(RN) 0.102 0.119 0.133 0.644
Quarterly Transition Probability Matrix (Asymptotic Standard Errors)(BA) (RA) (BN) (RN)
(BA) 3.624⇥10�04 1.338⇥10�4 1.363⇥10�04
(RA) 3.96⇥10�04 1.652⇥10�04 6.081⇥10�04
(BN) 8.322⇥10�04 3.713⇥10�04 2.199⇥10�04
(RN) 5.054⇥10�04 4.482⇥10�04 2.55⇥10�04
Log Likelihood =-401.725
28
Table 1.3: Second Stage SMM Estimation of Firms’ Project and Risk Adjustment Parameters
Firms’ Project ParametersµB
LR 0.132 sBLR 0.019
µRLR 0.047 sB
LR 0.035µB
HR 0.022 sBHR 0.036
µRHR 0.002 sR
HR 0.152
Risk Adjustment Parametersb1 0.129 b2 -0.220
J-Statistic = 0.988
Table 1.4: Implied Spreads (In Basis Points) From SMM Parameter Estimates
CDX S(0,3) S(3,7) S(7,15) S(15,100)
State at t=0 is (BA)(BA) 27 559 258 36 0(RA) 290 1344 1067 1067 173(BN) 37 1031 278 29 0(RN) 243 5729 873 574 119State at t=0 is (BN)(BA) 27 559 258 36 0(RA) 234 1067 1067 1067 116(BN) 37 1031 278 29 0(RN) 159 1422 574 574 74State at t=0 is (RA)(BA) 27 559 258 36 0(RA) 290 1344 1067 1067 173(BN) 37 1031 278 29 0(RN) 243 5729 873 574 119State at t=0 is (RN)(BA) 27 559 258 36 0(RA) 253 1104 1067 1067 136(BN) 37 1031 278 29 0(RN) 198 2529 627 574 99
29
Figure 1.1: Tranche Spreads, Economic Growth, and Credit Availability
0
40
80
120
160
200
04 05 06 07 08 09 10 11 12 13 14
CDO Senior Tranche: S(15,100)
Ba
sis
Po
ints
400
800
1,200
1,600
2,000
2,400
2,800
04 05 06 07 08 09 10 11 12 13 14
CDO Equity Tranche (0,3)
Ba
sis
Po
ints
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
04 05 06 07 08 09 10 11 12 13 14
Quarterly Real GDP Growth
Pe
rce
nta
ge
Po
ints
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
04 05 06 07 08 09 10 11 12 13 14
Ratio of Credit Growth at Nonfinancial Firms to GDP
Pe
rce
nta
ge
Po
ints
Tranche spreads are on the Dow Jones North American Investment Grade Index, which are reported by Credit MarketAnalysis (CMA) and obtained from Bloomberg (see Data Appendix for construction of our time series). The “senior”spread represents the 15 to 100 percent loss attachment points, while the “equity” tranche represents the 0 to 3 lossattachment points.
30
Figure 1.2: Sequence of Events
Good firms choose LR or HRand invests all its capital.
The firm issues a two-periodbond with face value F02 andcall price H.
If the credit is available, thefirm may refinance its debt, i.e.pay call price H and issue aone-period bond with face valueF12 to finance the repayment.
If refinanced at t = 1 thefirm repays F12; other-wise the firm repays F02.
t=2BA/BN/RA/RN
t=1BA/BN/RA/RN
t=0BA/RA
(RE-)RATING
FINANCING
INVESTMENT
The CRA produces ratings andmoves investors’ belief from ↵0
to ↵0.
Investors observe project out-come and update their beliefs to↵1. If necessary, the CRA ad-justs ratings so that investorsupdate their beliefs from ↵1 to↵1.
Good firms choose LR/HRand invests all its capital.
31
Figure 1.3: Belief Updating From Learning and Bayesian Persuasion
0 1 2 3 4
0.0
0.2
0.4
0.6
0.8
1.0
Capital K1
Post
erio
r bel
ief P
[goo
d|G
]
After observing project returnAfter observing project return and rating
K1 K1
32
Figure 1.4: Probabilities of the States From Regime Switching Model (1950:Q1 – 2014:Q4)
0.0
0.2
0.4
0.6
0.8
1.0
55 60 65 70 75 80 85 90 95 00 05 10
Probability of High Credit and High GDP Growth
0.0
0.2
0.4
0.6
0.8
1.0
55 60 65 70 75 80 85 90 95 00 05 10
Probability of High Credit and Low GDP Growth
0.0
0.2
0.4
0.6
0.8
1.0
55 60 65 70 75 80 85 90 95 00 05 10
Probability of Low Credit and High GDP Growth
0.0
0.2
0.4
0.6
0.8
1.0
55 60 65 70 75 80 85 90 95 00 05 10
Probability of Low Credit and Low GDP Growth
33
Figure 1.5: Fundamentals: Data and Fitted From Regime Switching Model (1950:Q1 - 2014:Q4)
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
45 50 55 60 65 70 75 80 85 90 95 00 05 10
Credit Growth to GDP (Data)Credit Growth to GDP (Model)
-3
-2
-1
0
1
2
3
4
45 50 55 60 65 70 75 80 85 90 95 00 05 10
GDP Growth (Data)GDP Growth (Model)
R-square = 56.8%
R-square=57.8%
34
Figure 1.6: Model and Actual Spreads on Senior and Equity Tranches
R2 = 0.603
50
100
150
200
2006 2008 2010 2012 2014
Actual
Model
CDX
R2 = 0.514
1000
1500
2000
2500
2006 2008 2010 2012 2014
Actual
Model
0−3% Tranche
R2 = 0.601
500
1000
1500
2006 2008 2010 2012 2014
Actual
Model
3−7% Tranche
R2 = 0.602
500
1000
2006 2008 2010 2012 2014
Actual
Model
7−15% Tranche
R2 = 0.712
50
100
150
2006 2008 2010 2012 2014
Actual
Model
15−100% Tranche
35
Figure 1.7: Distribution of Firms’ Capital Stocks
0
1
2
3
0 1 2 3Capital Stock K1
Den
sity
Bad firms
Good firms
High Credit and High GDP Growth
0
1
2
3
0 1 2 3Capital Stock K1
Den
sity
Bad firms
Good firms
Low Credit and High GDP Growth
0
1
2
3
4
0 1 2 3Capital Stock K1
Den
sity
Bad firms
Good firms
High Credit and Low GDP Growth
0
1
2
3
4
0 1 2 3Capital Stock K1
Den
sity
Bad firms
Good firms
Low Credit and Low GDP Growth
The figure plots the probability densities of firms’ capital stock at t=1 in each of the four states. Solid/dashed linecorresponds to good/bad firms. Since bad firms always undertake the HR project with high volatility, bad firms’capital stocks have greater dispersion. In the low credit and low GDP growth state, good firms with capital stock lessthan 0.7 (shaded area) choose the HR project at t=1 due to inability to refinance their debt. In other states, all goodfirms choose the LR project.
36
Figure 1.8: Equity Value under the LR and HR Projects
0.0
0.5
1.0
1.5
0.0 0.5 1.0 1.5Capital Stock K1
Equi
ty V
alue
LR
HR
The figure plots the equity value of a good firm as a function of capital stock at t=1 in the high credit and low GDPgrowth state. The solid line is the equity value if the firm undertakes the LR project. The dashed line is the equityvalue under the HR project. If a firm has capital stock less than 0.56, the equity value is greater under the HR project.For the firms with capital stock between 0.56 and 0.71 (shaded area), the credit rating agency delivers more precisecredit ratings. The re-rating reduces the cost of borrowing for good firms and they refinance their debt. The creditrating agency chooses the precision of the ratings such that after the refinancing the equity values under the LR andHR projects become the same. In this case the firms choose the LR project. Firms with capital stock greater than 0.71optimally choose the LR project.
37
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Chapter 2
User Costs, the Financial Firm, and Monetary
and Regulatory Policy
Joint paper with Apostolos Serletis
2.1 Introduction
The current mainstream approach to monetary policy and business cycle analysis is based on the
new Keynesian model and is expressed in terms of the interest rate on overnight loans between
banks, such as the federal funds rate in the United States. This approach ignores the financial
intermediary sector. As Adrian and Shin (2011, p. 602) put it, “in conventional models of mon-
etary economics commonly used in central banks, the banking sector has not played a prominent
role. The primary friction in such models is the price stickeness of goods and services. Finan-
cial intermediaries do not play a role, except as a passive player that the central bank uses as a
channel to implement monetary policy.” However, banks and other financial intermediaries have
been at the center of the global financial crisis, and there is almost universal agreement that the
crisis originated in the financial intermediary sector. There is also a number of recent theories that
give financial intermediaries a central role in economics and finance. See, for example, Fostel
and Geanakoplos (2008), Geanakoplos (2010), He and Krishnamurthy (2013), and Adrian et al.
(2014).
However, it is not uniform view in the literature as to what financial intermediaries do. As
Diewert et al. (2012) put it, “one of the most controversial areas in the field of economic mea-
surement is the measurement of the real and nominal output of the banking sector. There is little
consensus on all aspects of this topic: even the measurement of banking sector nominal outputs
41
and inputs is controversial and there is little agreement on how to measure the corresponding real
outputs and inputs.” There is also a broader aspect to what banks do than just being financial inter-
mediaries. Banks are allowed to create money and play an important role in the monetary policy
transmission process. In this regard, the current approach to monetary policy also ignores the role
of money, as the short-term nominal interest rate is the sole monetary variable and there is no
reference to any monetary aggregate.
In this paper we follow Hancock (1985) and take a microeconomic theory approach to the fi-
nancial firm in an attempt to elevate financial intermediaries to the center stage of business cycles
and economic growth. We assume that the banking firm is a profit-maximizing entity producing
intermediation services between lenders and borrowers. We take the user cost approach to the con-
struction of prices for monetary and nonmonetary goods. This approach to modelling the banking
firm has rarely been implemented in practice and permits monetary goods (such as cash and de-
posits of various types), other financial goods (such as loans), and physical goods (such as labor
and materials) to be classified as inputs or outputs. Those items with a positive user cost are
classified as inputs and those with a negative user cost are classified as outputs. More importantly,
the user cost approach explicitly takes into account each bank’s cost of funds, as user costs are
calculated based on each bank’s cost of capital. This makes decision problem comparable across
financial institutions. For example, the same investment opportunity could be profitable for a bank
with a low cost of capital but unprofitable for a bank with a high cost of capital. Thus, the optimal
demands for and supplies of monetary and nonmonetary goods vary across institutions, an aspect
of banking that is often ignored in the literature.
The technology of the financial firm is relevant in the investigation of the demand for and supply
of monetary and nonmonetary goods and their substitutability/complementarity relationship. It is
also relevant in the conduct of monetary policy and the investigation of the effects of financial
regulation. One approach to financial technology is to estimate cost functions in which input costs
and output quantities are the explanatory variables. In this approach to bank technology, however,
42
output is not a predetermined variable and it is difficult to distinguish between inputs and outputs.
In this paper we take an alternative (and more promising) approach based on the variable profit
function, which depends on prices of outputs and inputs and the quantities of fixed inputs.
In doing so, we also provide a comparison among three flexible functional forms for the
variable profit function — the translog, the symmetric generalized Barnett, and the Normalized
Quadratic (NQ), the latter also known in the literature as symmetric generalized McFadden. We
use recent, state-of-the-art advances in microeconometrics to produce inference consistent with
theoretical regularity — see, for example, Barnett and Serletis (2008) and Feng and Serletis (2008).
In particular, motivated by the wide-spread practice of ignoring theoretical regularity, we estimate
the translog, NQ, and symmetric generalized Barnett variable profit functions subject to the con-
vexity conditions and produce inference consistent with neoclassical microeconomic theory. Our
objective is to identify the channels through which financial intermediaries affect the real economy,
and investigate the implications for regulatory and monetary policies.
The paper is organized as follows. In Section 2, we provide a brief review of two fundamen-
tally different approaches to modeling the banking firm — the accounting balance-sheet approach
(or intermediation approach) of Sealey and Lindley (1977) and the user-cost approach. In Section
3 we discuss the variable profit function approach and in Section 4 we discuss the three flexible
functional forms for the variable profit function as well as the procedures for imposing convexity
on each of these functions in order to achieve theoretical regularity. In Section 5 we deal with
data issues. Our primary focus is on empirical application, specifically to annual data on the U.S.
commercial banking sector, over the period from 1992 to 2013, obtained from the quarterly Uni-
form Bank Performance Reports (UBPR) provided by the Federal Deposit Insurance Corporation.
Section 6 discusses related econometric issues and in Section 7 we estimate the models, present
the empirical results, and provide a comparison among the three flexible functional forms for the
variable profit function. In Section 8, we conduct a monetary and regulatory policy analysis. The
final section concludes the paper.
43
2.2 The User Cost Approach
A long-standing debate over the inputs and outputs of the banking technology has shaped sev-
eral approaches to modelling the banking firm. We briefly discuss two of them: the Sealey and
Lindley’s (1977) intermediation approach and the Barnett (1978) and Hancock (1985) user cost
approach. Although these two approaches have many implications in common, they stem from
conceptually different premises. The intermediation approach posits that, by nature, banks are fi-
nancial liaisons between liability holders and those who receive bank funds. This premise suggests
that all bank assets including loans and leases, investments in securities, and reserves are outputs
for the banking technology. At the same time, all bank liabilities such as deposits, other debt,
and equity capital are inputs. Sealey and Lindley (1977, p. 1253) claim that deposits, the most
controversial part of liabilities, are an economic input since “these services require the financial
firm to incur positive costs without yielding any direct revenue.” Classifying deposits as inputs
has exposed the intermediation approach to criticism for neglecting substantial services that banks
provide to their depositors. Later, the literature has attempted to disentangle the intermediation
and deposit services of banks. See Berger and Humphrey (1992) for discussion.
The user cost approach arises from an intertemporal model of the banking firm which chooses
the quantities of different assets and liabilities to maximize profit. This approach renounces the
ex-ante classification of assets and liabilities and instead derives the nature of the financial goods
based on their contribution to the bank profit. In particular, an asset is considered to be an output
if the return on investment into this asset exceeds the opportunity cost of funds. Similarly, a
liability is classified as an output if the financial cost of this liability is less than the opportunity
cost of funds. According to this classification scheme, deposits are likely to be labeled as outputs,
especially, if only interest expenses are taken into consideration. Interestingly, because of different
cost of funds, the same asset or liability can be classified as input in one bank and as output in
another. For this same reason, the nature of a financial good can change over time.
A formal representation of the user cost approach can be found in Barnett (1978, 1987), Barnett
44
and Hahm (1994), Barnett and Zhou (1994), Barnett et al. (1995), and Hancock (1985, 1991).
Here, we begin with Barnett’s (1978) definition of user costs and Hancock’s (1985) application of
the user cost approach to the banking firm. In doing so we construct user costs (per unit, with the
unit taken to be one dollar per period) for the services from all assets and liabilities in a financial
firm’s balance sheet. Following Hancock (1985), we model banks that maximize the capitalized
value of variable profit over a certain period of time. Since we estimate the model using annual
data, we assume that banks can fully adjust the quantities of the goods to optimal values every year.
The absence of adjustment costs effectively reduces our setup to a one-period profit maximization
problem.
We assume that there are N banks in the economy (indexed by i) and each bank operates with
n financial goods indexed by k. The first A financial goods, i.e. k = 1, ...,A, are assets and the
following L, i.e. k = A+ 1, ...,A+L are liabilities, so that n = A+L. We let Pt denote a general
price index in period t, ykit the real balance and hk
it the holding cost/revenue (per unit) of the kth
financial good of bank i in period t. Besides, we let yit to be the quantity of a fixed in the short run
good of bank i in period t. Then the variable profit of bank i during period t is
git =A
Âk=1
h⇣
1+hki,t�1
⌘
yki,t�1Pt�1 � yk
itPt
i
+A+L
Âk=A+1
h
ykitPt �
⇣
1+hki,t�1
⌘
yki,t�1Pt�1
i
. (2.1)
In equation (2.1), for a liability k (such as a deposit), ykitPt �
⇣
1+hki,t�1
⌘
yki,t�1Pt�1 represents
the total net cost of services to the firm and equals the total nominal liability to depositors at the
end of the period, ykitPt , minus the initial nominal liability, yk
i,t�1Pt�1, and holding costs or revenues
incurred at the rate hki,t�1, hk
i,t�1yLi,t�1Pt�1. For an asset k (such as a loan),
⇣
1+hki,t�1
⌘
yki,t�1Pt�1 �
ykitPt represents the total net revenue to the firm during period t and equals the initial nominal asset,
yki,t�1Pt�1, plus holding revenue incurred at the rate hk
i,t�1, hki,t�1yk
i,t�1Pt�1, minus the total nominal
asset at the end of the period, ykitPt .
If Ris is the ith bank discount rate in period s, then the capitalized value of variable profit over
45
T periods is
T
Ât=2
t
’s=1
1(1+Ris)
git =T
Ât=2
t
’s=1
1(1+Ris)
A
Âk=1
h⇣
1+hki,t�1
⌘
yki,t�1Pt�1 � yk
itPt
i
+T
Ât=2
t
’s=1
1(1+Ris)
A+L
Âk=A+1
h
ykitPt �
⇣
1+hki,t�1
⌘
yki,t�1Pt�1
i
. (2.2)
The negative of the coefficients of real balances, ykit , in equation (2.2) are beginning of the
period nominal user costs — see also Barnett (1978). Thus, the beginning of period nominal user
cost for assets is
ukit =
✓
Rit �hkit
1+Rit
◆
Pt , k = 1, ...,A (2.3)
and that for liabilities is
ukit =
✓
hkit �Rit
1+Rit
◆
Pt , j = A+1, ...,A+L. (2.4)
Equations (2.3) and (2.4) imply that the user costs may be positive or negative. The sign of the
user cost permits monetary goods (such as cash and deposits of various types) and other financial
goods (such as loans) to be classified as inputs or outputs. Those items with a positive user cost are
classified as inputs (because variable profit is reduced when the quantity is increased) and those
with a negative user cost are classified as outputs (because variable profit is increased when the
quantity is increased). With this classification of goods, we can also perform a change of variables,
transferring the sign from the user costs to the quantities (so that the variable profit function in the
next section has only nonnegative arguments), as follows
vkit =
�
�
�
ukit
�
�
�
, k = 1, ...,A+L
and
xkit =�sign
⇣
ukit
⌘
⇥ ykit , k = 1, ...,A+L.
We let vit =(v1it , ...,v
nit) be the vector of positive user costs of all financial goods and xit =(x1
it , ...,xnit)
the vector of quantities, with xkit < 0 for inputs and xk
it � 0 for outputs. This setup is consistent with
the neoclassical microeconomic theory of the firm — see, for example, Mas-Colell et al. (1995).
46
In constructing the user costs of financial goods (in Section 5 below) we deviate from the Han-
cock (1985) and Barnett and Hahm (1994) approach, and follow Diewert et al. (2012). Hancock
(1985) uses longitudinal observations on the balance sheet of 18 New York-New Jersey banks (all
members of Federal Reserve District 2), over the period from 1973 to 1978. Barnett and Hahm
(1994) also use longitudinal observations on 41 Chicago (Federal Reserve District 7) banks, over
the period from 1979 to 1983. Both Hancock (1985) and Barnett and Hahm (1994) calculate
holding and user costs using data on interest rates, deposit insurance premium rates, reserve re-
quirement rates, and service charges. Here we follow Diewert et al. (2012) and construct holding
and user costs using data on realized bank interest income and expenses. In particular, the holding
cost (revenue) is the ratio of interest expenses (income) to the value of the corresponding asset or
liability. To transform the holding costs into user costs we choose a time-varying bank specific dis-
count rate, in particular, the weighted average cost of raising capital via deposits, debt and equity.
With this discount rate bank deposits are mostly classified as outputs because typically deposits
are the cheapest source of funds [see Basu et al. (2011) for discussion]. We discuss data and
measurement matters in detail in Section 5.
2.3 The Variable Profit Function Approach
As in Hancock (1985), we use the profit function to obtain the functional forms for the estimating
equations. In this section and sections 4 and 5 we omit time and bank indexes for simpler expo-
sition. We let x = y denote the quantity of a fixed in the short run input good, and x = (x1, ...,xn)
the vector of quantities of n variable financial goods — the services of assets, liabilities, and items
off the balance sheet. Outputs are measured positively and inputs are measured negatively, both
with positive user costs v = (v1, ...,vn). As discussed in the previous section, whether a good is an
input or an output is not known a priori.
The variable profit of the bank (i.e., gross returns minus variable costs) is v0x and the bank’s
47
profit maximization problem is
p (v, x) = maxx2S(x)
v0x. (2.5)
where p (v, x) denotes the variable profit function and S(x) is the production possibility set. The
variable profit function is: (i) nondecreasing in output prices and nonincreasing in input prices; (ii)
homogeneous of degree one in v; (iii) continuous in v; and (iv) convex in v.
In principle, assuming an explicit functional form for the variable profit function and having
data on prices and observed profit, one could estimate equation (2.5) directly. However, we can
substantially improve the accuracy of the estimation if we simultaneously estimate the system of
supply and demand functions induced by the variable profit function. Since U.S. financial markets
are relatively competitive and most of the banks have little market power (the number of banks
in our sample ranges between approximately eleven and six thousands), we model the bank as a
price taker. We obtain the system of supplies of outputs and demands for inputs using Hotelling’s
lemma, differentiating (2.5) with respect to prices
xi (v, x) =∂p (v, x)
∂vi , i = 1, ...,n. (2.6)
Estimation of (2.6) allows the calculation of own- and cross-price elasticities of supply and
demand for the financial goods. These elasticities can then be used to investigate the effects of
interest rate and user cost changes on the production of financial goods (including both inputs and
outputs). In particular, the elasticity of transformation can be calculated from the Hessian matrix,
H, as follows
si j = p ∂ 2p∂vi∂v j
∂p∂vi
∂p∂v j
��1=
pHi j
xix j , i, j = 1, ...,n (2.7)
and the compensated price elasticities of supply and demand as
hi j = si jvixi
p, i, j = 1, ...,n. (2.8)
48
2.4 Flexible Functional Forms
In this section we discuss three flexible functional forms that we use to approximate the unknown
underlying variable profit function (2.5). They are the translog of Christensen et al. (1975),
the Normalized Quadratic (NQ), also known as symmetric generalized McFadden, of Diewert and
Wales (1987), and the symmetric generalized Barnett functional form of Barnett and Hahm (1994).
The translog and NQ models are both locally flexible, capable of approximating an arbitrary twice
continuously differentiable function to the second order at an arbitrary point in the domain.
Regarding the symmetric generalized Barnett functional form, Diewert and Wales (1987) prove
that it is quasi-flexible. They define that to mean that it can locally attain all first derivatives, all
levels, and all second derivatives, except for those in one column of the Hessian (and its corre-
sponding identical row, since symmetric). So, it also is local, but not fully flexible as the translog
and NQ are. But the number of derivatives potentially missed by the symmetric generalized Bar-
nett functional form is linear in the number of goods, while the number attained is quadratic in the
number of goods, so many more than the ones possibly missed. Hence, while the generalized Bar-
nett cannot exactly attain all possible elasticities at a point, it comes very close. The big advantage
is that the generalized Barnett has much better global regularity properties than the translog and
the NQ.
2.4.1 The Translog
We start with the translog variable profit function used by Hancock (1985). Assuming constant
returns to scale and n financial goods, we have
lnp (v, x) = a0 +n
Âi=1
ai lnvi +12
n
Âi=1
n
Âj=1
bi j lnvi lnv j + a ln x (2.9)
where bi j = b ji. Linear homogeneity in prices implies Âni=1 ai = 1 and Ân
j=1 bi j = 0, i = 1, ...,n.
Applying Hotelling’s lemma to (2.9) yields the expenditure shares on each good with respect to
49
profit
si =∂ lnp∂ lnvi =
vixi
p= ai +
n
Âj=1
bi j lnv j, i = 1, ...,n. (2.10)
As it is well-known, the system (2.10) is singular and only n�1 equations in (2.10) can be used
for estimation purposes — see Barten (1969) for more details. With n goods, the translog system
(2.10) contains n(n� 1)/2+(n�1) free parameters (that is, parameters estimated directly). For
n = 5 (as in our case), the number of free parameters is 14.
For the translog specification the elasticities of transformation at the geometric sample mean
are
si j = 1+bi j
aia j�
1{i= j}a j
, i, j = 1, ...,n
where 1{x} is an indicator function of x. The compensated price elasticities of supply and demand
at the geometric sample mean are
hi j = a j +bi j
ai�1{i= j}, i, j = 1, ...,n.
The translog model provides the capability to approximate systems resulting from a broad class
of generating functions and also to attain arbitrary elasticities of substitutions, although only at one
point (that is, locally). However, although the translog flexible functional form provides arbitrary
elasticity estimates at the point of approximation, there is evidence that this model fails to meet the
regularity conditions of neoclassical microeconomic theory in large regions even when convexity
of the variable profit function is locally imposed. See, for example, Barnett and Serletis (2008).
As we shall discuss in Section 7, it is exactly for this reason that in this paper we do not use
the translog and instead use the NQ and symmetric generalized Barnett flexible functional forms,
because of their ability to impose convexity globally without losing the flexibility property.
50
2.4.2 The Normalized Quadratic
Following Diewert and Wales (1987), the NQ variable profit function can be written for n financial
goods as
p (v, x) =✓
b0v+v0Bv2a 0v
◆
x
=
n
Âi=1
bivi +12
Âni=1 Ân
j=1 bi jviv j
Âni=1 aivi
!
x (2.11)
where b = [b1, ...,bn], a = [a1, ...,an], and the elements of the n⇥ n matrix B ⌘ [bi j] are the
unknown parameters to be estimated — usually the a vector (a > 0) is predetermined (in the
empircal section we assume that it is a vector of ones). As can be seen, p (v, x) in (2.11) is linearly
homogeneous in v. Since B is only used in quadratic form, v0Bv, it is reasonable to assume that B
is symmetric, bi j = b ji, for all i, j. Further, we impose the following restriction on the B matrix
Bv⇤ = 0, for some v⇤ > 0.
A natural choice of v⇤ = 1 allows us to express the main diagonal elements of B in terms of its
off-diagonal elements as follows
bii =�Âj 6=i
bi j, i = 1, ...,n. (2.12)
It is easy to see that the variable profit function (2.11) is convex if and only if B is positive semidef-
inite.
Differentiating (2.11) with respect to prices yields the following system of supplies of outputs
and demands for inputs
xi =
0
@bi +Ân
j=1 bi jv j
Âni=1 aivi � 1
2
ai
⇣
Âni=1 Ân
j=1 bi jviv j⌘
(Âni=1 aivi)2
1
A x (2.13)
or, equivalently,xi
x= bi +
n
Âj=1
bi jz j � 12
ai
n
Âi=1
n
Âj=1
bi jziz j
!
, i = 1, ...,n (2.14)
51
where z j = v j/Âni=1 aivi for j = 1, ...,n denotes normalized prices. Finally, equation (2.14) can be
written [after imposing (2.12)] as
xi
x= bi +
i�1
Âj=1
b jiw ji �n
Âj=i+1
bi jwi j �12
ai
n�1
Âj=1
n
Âk= j+1
b jkw2jk
!
, i = 1, ...,n (2.15)
where wi j = zi � z j, denoting differences in normalized prices.
The elasticities of transformation and compensated price elasticities for the NQ model can be
obtained using (2.7) and (2.8) with the Hessian of the NQ profit function
H =B�Bza 0 �az0B+ z0Bzaa 0
a 0vx, (2.16)
where z = [z1, ...,zn], or equivalently,
hi j =
bi j �n
Âk=1
bikzk �n
Âk=1
bk jzk +n
Âk=1
n
Âl=1
bklzkzl
!
n
Âk=1
akvk
!�1
x.
With n goods, the normalized quadratics system (2.15) contains n(n�1)/2+n free parameters.
For n = 5 (as in our case), the number of free parameters is 15 and the estimating demand and
supply equations are as follows
x1/x = b1 �b12w12 �b13w13 �b14w14 �b15w15 �12
a1z0Bz
x2/x = b2 +b12w12 �b23w13 �b24w14 �b25w25 �12
a2z0Bz
x3/x = b3 +b13w13 +b23w23 �b34w34 �b35w35 �12
a3z0Bz (2.17)
x4/x = b4 +b14w14 +b24w23 +b34w34 �b45w45 �12
a4z0Bz
x5/x = b5 +b15w15 +b25w25 +b35w35 +b45w45 �12
a5z0Bz
where
z0Bz =n�1
Âj=1
n
Âk= j+1
b jkw2jk
= b12w212 +b13w2
13 +b14w214 +b15w2
15 +b23w223
+b24w224 +b25w2
25 +b34w234 +b35w2
35 +b45w245.
52
In practice, the convexity of the NQ variable profit function p (v, x) may not be satisfied, in
the sense that the estimated B matrix may not be positive semidefinite.1 In this paper, to ensure
convexity of p�
v,x f�
, we follow Wiley et al. (1973) and impose B = KK0, where K = [ki j] is a
lower triangular matrix which also satisfies K0v⇤ = 0. For example, choosing v⇤ = 1, we can also
express the diagonal elements of K as
kii =�n
Âj=i+1
k ji, i = 1, ...,n�1
and knn = 0. In our case with n = 5, convexity of the NQ variable profit function (2.11) can be1Convexity is checked by performing a Cholesky factorization of B and checking whether the Cholesky values are
nonnegative [since a matrix is positive semidefinite if its Cholesky factors are nonnegative — see Lau (1978, Theorem3.2)]. This can be checked by evaluating the eigenvalues of the Hessian — all eigenvalues of a positive semidefinitematrix should be nonnegative.
53
imposed by replacing the elements of B in (2.17) by the elements of K, as follows
b11 = (k21 + k31 + k41 + k51)2
b12 =�(k21 + k31 + k41 + k51)k21
b13 =�(k21 + k31 + k41 + k51)k31
b14 =�(k21 + k31 + k41 + k51)k41
b15 =�(k21 + k31 + k41 + k51)k51
b22 = k221 +(k32 + k42 + k52)
2
b23 = k21k31 � (k32 + k42 + k52)k32
b24 = k21k41 � (k32 + k42 + k52)k42
b25 = k21k51 � (k32 + k42 + k52)k52
b33 = k231 + k2
32 +(k43 + k53)2
b34 = k31k41 + k32k42 � (k43 + k53)k43
b35 = k31k51 + k32k52 � (k43 + k53)k53
b44 = k241 + k2
42 + k243 + k2
54
b45 = k41k51 + k42k52 + k43k53 � k254
b55 = k251 + k2
52 + k253 + k2
54.
The estimation of the NQ profit function with the full sample (from 1992 to 2013) could be
computationally intensive due to the nonlinearity of the system, especially if dimensionality is
high. To obtain robust convergence we apply an iterative algorithm by varying the rank of the
quadratic form in the NQ representation. Diewert and Wales (1988) use this procedure to deal with
lack of degrees of freedom and computational difficulties in estimating a normalized quadratic
semiflexible expenditure function. In particular, we construct the matrix B using the rank deficient
matrix K. At the first step, we set the rank of K to one and let only the first column of K have
non-zero elements. Next, we let the first and second columns have non-zero elements (preserving
54
the triangular form of K), and so on. At the lth step, si, j = 0 for 1 i < j N � 1 and j =
l +1, ...,N �1.
2.4.3 The Generalized Symmetric Barnett
In this section we introduce the generalized symmetric Barnett variable profit function based on
Barnett and Hahm (1994). The variable profit function is
p(v, x) =
n
Âi=1
aiivi �2n
Âi=1
n
Âj=i+1
ai j�
viv j�1/2
+n
Âi=1
n
Âj=1
j 6=i
n
Âk= j+1
k 6=i
ai jk�
vi�2⇣
v jvk⌘�1/2
!
x (2.18)
where ai j � 0 and ai jk � 0. With this specification the conditional variable profit function (2.18)
is linearly homogeneous and globally convex in prices v. The convexity of the function follows
from convexity of the summands in (2.18) and nonnegativity of the coefficients ai j and ai jk. Using
duality and the envelope theorem we obtain the demand and supply system induced by (2.18):
xi/x = aii � Âj: j>i
ai j�
v j/vi�1/2 � Âj: j<i
a ji�
v j/vi�1/2
+2 Âj: j 6=i
Âk:k> j,k 6=i
ai jkvi⇣
v jvk⌘�1/2
� 12 Â
j: j 6=iÂ
k:k>i,k 6= ja jik�
v j/vi�2⇣
vi/vk⌘1/2
� 12 Â
j: j 6=iÂ
k:k<i,i 6= ja jki�
v j/vi�2⇣
vi/vk⌘1/2
. (2.19)
With n goods, the generalized Barnett profit function (2.18) contains n�
n2 �2n+3�
/2 free
parameters. For n = 5 (as in our case), the number of free parameters is 45 and the estimating
55
demand and supply equations are as follows
x1/x = a11 �⇣
a12(v1)�1/2
(v2)1/2
+a13(v1)�1/2
(v3)1/2
+a14(v1)�1/2
(v4)1/2
+a15(v1)�1/2
(v5)1/2⌘
+2
"
a123v1
(v2v3)1/2 +a124v1
(v2v4)1/2 +a125v1
(v2v5)1/2 +a134v1
(v3v4)1/2 +a135v1
(v3v5)1/2 +a145v1
(v4v5)1/2
#
� 12
"
a213(v2)2
(v1)3/2(v3)1/2 +a312(v3)
2
(v1)3/2(v2)1/2 +a214(v2)
2
(v1)3/2(v4)1/2 +a412(v4)
2
(v1)3/2(v2)1/2
+a215(v2)
2
(v1)3/2(v5)1/2 +a314(v3)
2
(v1)3/2(v4)1/2 +a512(v5)
2
(v1)3/2(v2)1/2 +a413(v4)
2
(v1)3/2(v3)1/2
+a315(v3)
2
(v1)3/2(v5)1/2 +a513(v5)
2
(v1)3/2(v3)1/2 +a415(v4)
2
(v1)3/2(v5)1/2 +a514(v5)
2
(v1)3/2(v4)1/2
#
x2/x = a22 �⇣
a23(v2)�1/2
(v3)1/2
+a24(v2)�1/2
(v4)1/2
+a25(v2)�1/2
(v5)1/2⌘
+2
"
a213v2
(v1v3)1/2 +a214v2
(v1v4)1/2 +a215v2
(v1v5)1/2 +a234v2
(v3v4)1/2 +a235v2
(v3v5)1/2 +a245v2
(v4v5)1/2
#
� 12
"
a123(v1)2
(v2)3/2(v3)1/2 +a312(v3)
2
(v2)3/2(v1)1/2 +a124(v1)
2
(v2)3/2(v4)1/2 +a412(v4)
2
(v2)3/2(v1)1/2
+a125(v1)
2
(v2)3/2(v5)1/2 +a512(v5)
2
(v2)3/2(v1)1/2 +a324(v3)
2
(v2)3/2(v4)1/2 +a423(v4)
2
(v2)3/2(v3)1/2
+a325(v3)
2
(v2)3/2(v5)1/2 +a523(v5)
2
(v2)3/2(v3)1/2 +a425(v4)
2
(v2)3/2(v5)1/2 +a524(v5)
2
(v2)3/2(v4)1/2
#
x3/x = a33 �⇣
a34(v3)�1/2
(v4)1/2
+a35(v3)�1/2
(v5)1/2⌘
+2
"
a312v3
(v1v2)1/2 +a314v3
(v1v4)1/2 +a315v3
(v1v5)1/2 +a324v3
(v2v4)1/2 +a325v3
(v2v5)1/2 +a345v3
(v4v5)1/2
#
� 12
"
a123(v1)2
(v2)1/2(v3)3/2 +a213(v2)
2
(v1)1/2(v3)3/2 +a134(v1)
2
(v3)3/2(v4)1/2 +a413(v4)
2
(v1)1/2(v3)3/2
+a135(v1)
2
(v3)3/2(v5)1/2 +a234(v2)
2
(v3)3/2(v4)1/2 +a513(v5)
2
(v1)1/2(v3)3/2 +a423(v4)
2
(v2)1/2(v3)3/2
+a235(v2)
2
(v3)3/2(v5)1/2 +a523(v5)
2
(v2)1/2(v3)3/2 +a435(v4)
2
(v3)3/2(v5)1/2 +a534(v5)
2
(v3)3/2(v4)1/2
#
56
x4/x = a44 �a45(v4)�1/2
(v5)1/2
+2
"
a412v4
(v1v2)1/2 +a413v4
(v1v3)1/2 +a423v4
(v2v3)1/2 +a415v4
(v1v5)1/2 +a425v4
(v2v5)1/2 +a435v4
(v3v5)1/2
#
� 12
"
a124(v1)2
(v2)1/2(v4)3/2 +a214(v2)
2
(v1)1/2(v4)3/2 +a134(v1)
2
(v3)1/2(v4)3/2 +a314(v3)
2
(v1)1/2(v4)3/2
+a234(v2)
2
(v3)1/2(v4)3/2 +a324(v3)
2
(v2)1/2(v4)3/2 +a145(v1)
2
(v4)3/2(v5)1/2 +a514(v5)
2
(v1)1/2(v4)3/2
+a245(v2)
2
(v4)3/2(v5)1/2 +a524(v5)
2
(v2)1/2(v4)3/2 +a345(v3)
2
(v4)3/2(v5)1/2 +a534(v5)
2
(v3)1/2(v4)3/2
#
x5/x = a55
+2
"
a512v5
(v1v2)1/2 +a513v5
(v1v3)1/2 +a514v5
(v1v4)1/2 +a523v5
(v2v3)1/2 +a524v5
(v2v4)1/2 +a534v5
(v3v4)1/2
#
� 12
"
a125(v1)2
(v2)1/2(v5)3/2 +a215(v2)
2
(v1)1/2(v5)3/2 +a135(v1)
2
(v3)1/2(v5)3/2 +a315(v3)
2
(v1)1/2(v5)3/2
+a145(v1)
2
(v4)1/2(v5)3/2 +a235(v2)
2
(v3)1/2(v5)3/2 +a415(v4)
2
(v1)1/2(v5)3/2 +a325(v3)
2
(v2)1/2(v5)3/2
+a245(v2)
2
(v4)1/2(v5)3/2 +a425(v4)
2
(v2)1/2(v5)3/2 +a345(v3)
2
(v4)1/2(v5)3/2 +a435(v4)
2
(v3)1/2(v5)3/2
#
The convexity of the generalized symmetric Barnett variable profit function is ensured by im-
posing nonnegativity constraints on the parameters ai j and ai jk. In particular, we reparameterize
the model such that these coefficients are squared new parameters. See Barnett (1976, 1983) for
a detailed discussion of squaring techniques and the asymptotic properties of such nonnegative
estimators.
2.5 Data and Measurement Matters
We use annual data on the U.S. commercial banking sector, over the period from 1992 to 2013,
obtained from the quarterly Uniform Bank Performance Reports (UBPR) provided by the Federal
Deposit Insurance Corporation. In a sense, the UBPR are a refined version of the data from the Call
57
Reports. The sample of the UBPR covers federal and state chartered commercial banks, savings
banks, savings associations (and as of July 21, 2011 thrifts), and insured U.S. branches of foreign
chartered institutions. Although the range of activity of the savings associations has substantially
expanded over the period under consideration, banks and savings associations are still distinct
institutions subject to different regulation. In our analysis, we exclude from the sample savings
associations and thrifts which account on average for 10.6% of total assets.
The resulting panel which contains about 184,888 observations is unbalanced: only 32.4 per-
cent of the participation patterns cover the entire twenty-two year period. During the period from
1992 to 2013 the number of banks declined from 11,725 to 6,262 because of the intense consoli-
dation in the U.S. banking industry. At the same time the assets became more concentrated in the
largest financial institutions, as can be seen in Table 1, which shows the distribution of assets over
four groups of banks in 1992, 2002 and 2012. Our use of the unbalanced panel reduces the sur-
vival bias in parameter estimates. Olley and Pakes (1996) show that using an artificially balanced
sample can lead to significant bias in parameter estimates due to the survival effect. They also
demonstrate that an explicit selection correction has insignificant effect if the estimation is based
on the unbalanced sample.
We do not consider all the profit of banks and abstract from the interaction between commercial
and investment banking. In particular, we focus only on commercial banking activity, and investi-
gate how changes in the user costs of certain financial goods (which are based on interest income
and interest expenses) affect the supplies of outputs and demands for inputs. In doing so, we ig-
nore non-interest income and expenses. For example, in the case of JP Morgan Chase, according to
its 2012 financial statements, we do not consider non-interest expenses ($53 billion) and consider
only interest expenses ($6 billion). The reason is that we cannot associate the non-interest ex-
penses (more than half of which is remuneration) with particular financial goods. Similarly, we do
not consider non-interest revenue ($37 billion) compared to interest income ($40 billion). Again,
there is no information to associate non-interest revenue with particular financial goods. In fact,
58
we analyze the whole sample of banks and more homogeneous subsamples of banks categorized
by the total amount of assets held. For instance, we separate the largest banks, those with assets in
excess of $100 billion in 2012. These big banks are really bank holding companies (and financial
holding companies). They have offices and branches nationwide and their activity is very diverse
across types and geography. For example, Morgan Stanley, a bank holding company since 2008, is
a major generator of electricity, JP Morgan Chase controls the U.S. copper warehouse market, and
Goldman Sachs the U.S. aluminum warehouse business.
We consider five variable financial goods, two assets and three liabilities. They are debt se-
curities and trading accounts (y1) and loans and leases (y2), and deposits (y3), other debt (that is,
debt other than deposits) (y4), and equity (y5). We also use bank premises and fixed assets as a
quasi-fixed good and denote it by x. The level of aggregation for these financial goods is mainly
due to data limitation. Aggregation bias may be present, for example, because we combine de-
mand deposits and term deposits with different maturities. In Table 2 we list asset and liability
values (in billions of dollars) for each of the 22 years in the sample for each of the five financial
goods. Figure 1 shows the changes in the U.S. banks’ asset structure over the sample period. The
loans and leases on average account for over 60% of assets and debt securities and trading accounts
account for about 26% of assets. Together with bank premises and fixed assets, these two assets on
average account for about 88% of all assets. Similarly, Figure 2 shows the changes in the liabilities
structure over the 1992 to 2013 period. Deposits on average account for about 84%, other debt for
about 5%, and equity capital for close to 11% of total liabilities.
Next we calculate the user costs of the financial goods based on realized bank interest income
and expenses from holding these goods. Given available data, these ex post average realized user
costs provide a reasonable approximation for the ex ante marginal user costs defined by (2.3) and
(2.4). We let ri, i = 1, ...,5, denote the interest income on the corresponding asset or the interest
expense on the corresponding liability. We use the net income r5 as a proxy to calculate the return
to equity capital. Then, using the values of interest and non-interest income and expenses we
59
calculate the holding costs and revenues for the assets and liabilities, as follows
hi =ri
yi , i = 1, ...,5.
According to equations (2.3) and (2.4), user costs are linear transformation of holding costs
parameterized by the discount rate. By nature, the discount rate is specific for each bank and could
vary over time reflecting a bank’s riskiness. In general, there are several recognized methods of
choosing the discount rate, such as, for example, the weighted average cost of capital (WACC)
for mixed capital and the capital asset pricing model (CAPM) for equity capital. However, the
literature shows no consensus on how to determine the discount rate for a bank. For example, a
contentious issue in calculating a bank’s WACC is accounting for the cost of deposits which on
average account for more than 80% of the total liabilities of a commercial bank. At the same
time, the interest rates paid by banks on deposit balances are usually quite low and do not reflect
non-interest expenses associated with attracting and servicing deposits. In this regard, Diewert et
al. (2012) discuss three options for the choice of the discount rate: (i) the average cost of raising
financial capital via debt other than deposits; (ii) the weighted average cost of raising capital via
deposits and debt; and (iii) the weighted average cost of raising capital via deposits, debt, and
equity.
Although each of these methods can provide a reasonable proxy for the discount rate, the
choice depends on the composition of bank liabilities. The first method might provide a biased
estimate of the discount rate if the share of debt other than deposits in all liabilities is small. In
our sample the average share of debt other than deposits is about five percent of total liabilities
and equity capital. At the same time, the second method can produce a significantly downward
biased estimate because (1) the interest rate on deposits underestimates the cost of this source of
funds [see Basu and Wang (2013) for a discussion] and (2) the method excludes equity capital
which is, typically, the most expensive source of bank capital. Figure 3 shows the dynamics of the
(across banks) average reference discount rates calculated using each of the three methods. In what
follows, we choose the third method and calculate the weighted average cost of deposits, debt, and
60
equity for each bank in every period.
Although there is a prior belief about which financial goods are inputs and which are outputs,
the actual nature of these goods is determined by the sign of the corresponding user costs. More-
over, in our framework the sign of a particular user cost for any particular bank could change over
time because both holding costs and discount rates vary over time. In particular, the sign of the
user costs depends on the sign of the numerator in (2.3) and (2.4). For each year in the sample,
we calculate the percentage of observations in the pooled sample of all 184,888 observations when
each financial good is an output (i.e. the user cost is negative) and report the results in Table 3. On
average, the most stable output is loans and leases, with a negative user costs in 99.8 percent of
the observations. Another asset, debt securities and trading accounts, is an output in 92 percent of
the observations. Deposits have a negative user cost in about 88 percent of the observations. Inter-
estingly, because of low or even negative return on equity capital during (and around) the period
of the global financial crisis, 2007 to 2011, deposits became a relatively expensive source of funds
and the observations with a negative user cost during this period dropped to 75.5 percent. Debt
other than deposits is an output in about 67 percent of the observations. Finally, equity is an output
only in 12.3 percent of the observations. In Table 4 we report the (across banks) average nominal
user costs corresponding to the assets, y1 and y2, and the liabilities, y3, y4, and y5, for each year
from 1992 to 2013. Finally, to obtain real quantities, we use the GDP deflator (from the St. Louis
Fed FRED database) to proxy the general price index.
2.6 Econometric Issues
We estimate the translog share equation system and the systems of input demands and output
supplies for the NQ and generalized symmetric Barnett models using the maximum likelihood
method. We do not estimate the variable profit function itself, because it contains no additional
information. In order to estimate equation systems (2.10), (2.15), and (2.19), we add a stochastic
61
component, e it , as follows
w it = y (vit ,q)+ e it (2.20)
where w it = (w1it , ...,wn
it) is the vector of expenditure shares on each good with respect to profit
in the case of the translog model and the vector of input demands and output supplies in the case
of the NQ and generalized symmetric Barnett models. e it is a vector of stochastic errors and we
assume that e it ⇠ N (0, ) where 0 is a null vector and is the n⇥ n symmetric positive definite
error covariance matrix. y(vit ,q) =�
y1 (vit ,q) , ...,yn (vit ,q)�0, and yk (vit ,q) is given by the
right-hand side of each of (2.10), (2.15), and (2.19). Recall that in the case of the translog model
we estimate only n�1 share equations, because of the singularity problem.
Under the assumption that the additive stochastic term in (2.20) is normally distributed with 0
mean and constant covariance matrix , the full log likelihood function for the system (2.20) over
the pooled panel is
L⇣
q |{vit} ,{w it} ,{xit}⌘
=�T
Ât=1
nNt
2log(2p)� Nt
2log(| |)
�
� 12
T
Ât=1
Nt
Âi=1
(w it � bw it)0 �1 (w it � bw it) (2.21)
where Nt denotes the number of banks (a function of t). The coefficients of the approximating
form in bw it must be estimated together with the covariance matrix . The log likelihood function in
(2.21) is computationally cumbersome, especially if the dimensionality is high and/or the sample is
large. In our estimation we ignore the possibility of autocorrelation for empirical implementability
purposes and use the concentrated log likelihood function [see Greene (2008) for more details]
Lc (vt , xt ,q) =�T
Ât=1
Nt
2
h
n(1+ log(2p))+ log |W |i
,
where
W =1T
T
Ât=1
1Nt
Nt
Âi=1
(w it � bw it)0 (w it � bw it) .
We admit that the errors in (2.20) can be conditionally heteroskedastic and use the Huber-White
62
estimator for the asymptotic covariance matrix of the maximum likelihood estimates
bV⇣
bq⌘
= A�1⇣
bq⌘
I⇣
bq⌘
A�1⇣
bq⌘
(2.22)
where A⇣
bq⌘
is the mean of the Hessian of the log likelihood function
A⇣
bq⌘
=T
Âi=1
Nt
Âi=1
∂ 2Lc (vit , xit ,q)∂q∂q 0
�
�
�
�
bq
and I⇣
bq⌘
is the mean of the outer product of the gradient of the log likelihood function
I⇣
bq⌘
=T
Âi=1
Nt
Âi=1
∂Lc (vit , xit ,q)∂q
∂Lc (vit , xit ,q)∂q 0
�
�
�
�
bq.
The estimation is performed in C++ using the concentrated log likelihood function. The regu-
larity conditions of the variable profit function are checked as follows:
• Monotonicity requires that the variable profit function is nondecreasing in output
prices and nonincreasing in input prices. Since vi > 0 (by definition) and p > 0,
monotonicity is checked by direct computation of the estimated expenditure on
each good relative to variable profit, since sign(∂p/∂vi) = sign(vixi/p).
• Convexity requires the Hessian matrix of the variable profit function, H, to be pos-
itive semidefinite. It is checked by performing a Cholesky factorization of that
matrix and checking whether the Cholesky values are nonnegative [since a matrix
is positive semidefinite if its Cholesky factors are nonnegative — see Lau (1978,
Theorem 3.2)].
Finally, using the parameter estimates bq we calculate elasticities of transformation and com-
pensated price elasticities using equations (2.7) and (2.8), respectively. We also apply the Delta
method to obtain the standard errors for these elasticities. For example, the standard errors of the
price elasticities are
s.e.�
hi j�
=q
—0q hi j(bq)V (bq)—q hi j(bq)
where —q hi j(bq) is the gradient of the elasticity with respect to q .
63
2.7 Empirical Evidence
2.7.1 Theoretical Regularity
In this section we estimate the translog, NQ, and generalized Barnett systems using the maximum
likelihood method. Tables 5, 6, and 7 contain a summary of results from each of the models in
terms of parameters estimates and theoretical regularity (convexity and monotonicity) violations
when the models are estimated with the convexity conditions imposed (locally for the translog, and
globally for each of the NQ and generalized Barnett models). We estimate the models for five bank
groups, based on asset size, as follows: all banks (184,888 observations), banks with assets less
than $100 million (96,340 observations), banks with assets between $100 million and $1 billion
(77,945 observations), banks with assets between $1 billion and $10 billion (8,874 observations),
and banks with assets in excess of $10 billion (1,729 oservations).
We make several premises about bank heterogeneity. First, we allow for the heterogeneity in
the level of bank profit captured by the intercept of the profit function. However, since we estimate
the demand and supply system (2.20) derived from the profit function, but not the profit func-
tion per se, the Hotelling’s lemma transformation eliminates a potentially heterogeneous intercept.
Second, we control for bank heterogeneity by estimating models for five size classes, acknowl-
edging differences in the degree of competition and institutional structure among small and large
banks. Third, bank specific dummy variables and the within estimator cannot be used to address
the heterogeneity in the levels of the demand and supply equations in our model, because three
of the five samples contain more than a thousand observations each, and the inclusion of dummy
variables introduces a dimensionality problem. The nonlinearity of the NQ and the generalized
Barnett demand and supply systems impedes direct application of the within or the first difference
transformations. In this regard, Isakin and Serletis (2015) address the heterogeneity in the levels
of the NQ demand and supply system, and propose a two-step estimation procedure based on the
first difference transformation and test it on a sample of small banks. Since they find no dramatic
changes in the elasticity estimates, in this paper we focus on the comparison of the estimation
64
results based on the NQ and the generalized Barnett functional forms without fixed effects.
As can be seen in Tables 5, 6, and 7, the NQ and generalized Barnett models satisfy convexity
of the variable profit function at every data point in each of the samples. The translog, however,
violates convexity at a very large number of observations in each of the samples: at 69% of the
observations in the case of all banks, 97% in the case of banks with assets less than $100 million,
98% in the case of banks with assets between $100 million and $1 billion, 99% in the case of banks
with assets between $1 billion and $10 billion, and 81% in the case of banks with assets in excess
of $10 billion. We also find that the imposition of convexity does not always assure economic
regularity, as there are monotonicity violations at many data points in each of the samples even
with the NQ and generalized Barnett models that satisfy convexity at every data point; only in the
case of banks with assets less than $100 million, and with the generalized Barnett model, we report
zero monotonicity and zero convexity violations.
Our evidence regarding economic regularity supports Barnett’s (2002, p. 199) argument that
“although unconstrained specifications of technology are more likely to produce violations of cur-
vature than monotonicity, I believe that induced violations of monotonicity become common, when
curvature alone is imposed. Hence, the now common practice of equating regularity with curvature
is not justified.” Although convexity of the variable profit function is not sufficient for regularity,
we believe that in the context of our models the convexity condition is the most crucial. That is,
although monotonicity is a desirable property, in our framework, since each financial good may
be an input in one period and an output in another period, the monotonicity condition indicates
whether a particular model can predict the sign of a financial good at a point in the sample.
In what follows, because of the disappointing results with the translog, we only use the NQ and
generalized Barnett models that are able to produce inferences about the elasticities of transfor-
mation, si j, and the compensated price elasticities, hi j, that are more consistent with neoclassical
microeconomic theory. In this regard, according to equations (2.7) and (2.8), the Hessian matrix of
the variable profit function, H, is the basis for the calculation of the elasticities of transformation
65
and the price elasticities of supply and demand. The satisfaction of the convexity condition with
each of the NQ and generalized Barnett models suggests that our estimates of the elasticities of
transformation and the own- and cross-compensated price elasticities (reported in what follows)
are well behaved. Regarding the calculation of the elasticities, it is to be noted that in the case of
the NQ we use analytic formulas whereas in the case of the generalized Barnett we use numerical
approximations due to its complexity.
2.7.2 Elasticities of Transformation
The estimated (symmetrical) elasticities of transformation, si j, calculated using equation (2.7), are
shown in Tables 8.1 to 8.5 for each of the five bank samples and for each of the NQ and generalized
Barnett models. We expect the on-diagonal elements for all five goods to be positive and this
expectation is clearly achieved. The off-diagonal elements indicate the degree of substitutability
or complementarity between financial goods. In particular, between two inputs or two outputs, if
si j > 0, they are substitutes, and if si j 0, they are complements. In the case of one output and
one input, if si j > 0, they are complements, and if si j 0, they are substitutes.
According to the generalized Barnett model, equity is a complement for each of the outputs
and in each of the five bank samples. However, with the NQ functional form for the variable
profit function, equity is a complement for loans and leases and deposists, but a substitute for
debt other than deposits, with an estimate of sEO = �13.166 (and a standard error of 0.290) in
the case of all banks, sEO = �2.676 (with a standard error of 0.215) in the case of banks with
assets less than $100 million, sEO =�5.826 (with a standard error of 5.334) in the case of banks
with assets between $100 million and $1 billion, sEO = �3.268 (with a standard error of 0.271)
in the case of banks with assets between $1 billion and $10 billion, and sEO = �13.166 (with a
standard error of 0.290) in the case of banks with assets in excess of $10 billion. With the NQ,
equity is also a substitute for debt securities in the case of banks with assets less than $100 million
(sES = �0.226 with a standard error of 0.012), banks with assets between $100 million and $1
billion (sES = �0.607 with a standard error of 0.124), and banks with assets between $1 billion
66
and $10 billion (sES =�0.062 with a standard error of 0.133).
2.7.3 Compensated Price Elasticities
We report the own- and cross-compensated price elasticities in Tables 9.1 to 9.5 for each of the
five banks groups and for each of the NQ and generalized Barnett models. The elasticities are
evaluated at the arithmetic sample mean of the data, and numbers in parentheses are standard
errors. According to the results, the financial technology is relatively inflexible, consistent with
Hancock’s (1985) conclusion.
The signs of the own-price elasticities of supply and demand, hii, appear reasonable for both
models and for each of the five bank samples. They are, however, less than unity, with both models,
for both outputs and inputs, except for debt other than deposits and only with the NQ functional
form in the case of all banks (hOO = 1.168 with a standard error of 0.027) and banks with assets
in excess of $10 billion (hOO = 2.193 with a standard error of 0.290). The own-price elasticity
of supply for deposits based on the generalized Barnett model is hDD = 0.220 (with a standard
error of 0.004) in the case of all banks, hDD = 0.356 (with a standard error of 0.030) in the case of
banks with assets less than $100 million, hDD = 0.323 (with a standard error of 0.009) in the case
of banks with assets between $100 million and $1 billion, hDD = 0.388 (with a standard error of
0.057) in the case of banks with assets between $1 billion and $10 billion, and hDD = 0.192 (with
a standard error of 0.035) in the case of banks with assets in excess of $10 billion. The own-price
elasticity of supply for deposits based on the NQ functional form is in general lower than that based
on the generalized Barnett model.
The demand for equity is also relatively inelastic. In particular, the own-price elasticity of
demand for equity is statistically significant and around �0.500 with the generalized Barnett model
for each of the five bank groups. The own-price elasticity of demand for equity is a bit higher with
the NQ model with an estimate of hEE = �0.154 (with a standard error of 0.041) in the case of
all banks, hEE = �0.041 (with a standard error of 0.004) in the case of banks with assets less
than $100 million, hEE =�0.331 (with a standard error of 0.069) in the case of banks with assets
67
between $100 million and $1 billion, hEE =�0.286 (with a standard error of 0.033) in the case of
banks with assets between $1 billion and $10 billion, and hEE =�0.327 (with a standard error of
0.075) in the case of banks with assets in excess of $10 billion. In Figures 4-8 we show the own-
price elasticities for all five financial goods, and we also include the user cost of each good (which
is the price that is being varied to produce these elasticities). The graphs reveal the business cycle
pattern of the user costs. As can be seen, for example, the global financial crisis of 2007-2009
dramatically reduced the return on equity capital, and consequently the user costs of equity, loans,
and debt securities. On the other hand, the user cost of deposits increased during the financial
crisis, together with the own-price elasticity of deposits, as indicated with the generalized Barnett
model (but not with the NQ model).
The entries off the principal diagonals in Tables 9.1 to 9.5 are the cross-price elasticities of
supply and demand, hi j, representing the percentage change in quantity of a given good with unit
percentage change in the user cost for each of the other goods. As with the own-price elasticities,
the cross-price elasticities of supply and demand do not differ significantly between the two mod-
els. They all are less than unity in absolute value, except with the NQ model for the supply of debt
other than deposits when the user cost of equity changes in the case of banks with assets between
$100 million and $1 billion (hOE =�1.308 with a standard error of 1.211) and banks with assets
in excess of $10 billion (hOE = 4.129 with a standard error of 0.006). According to the generalized
Barnett model, equity demand is increasing in the user cost of all outputs — debt securities, loans
and leases, deposits, and debt other than deposits — in each of the five bank samples. Moreover,
with the generalized Barnett model and the bank sample that satisfies full economic regularity (that
is, banks with assets less than $100 million), debt securities, loans and leases, deposits, and debt
other than deposits are each decreasing in the user cost of all financial goods.
68
2.8 Monetary and Regulatory Policy Analysis
Our estimates of the compensated price elasticities of banking technology are, in general, moderate
or small in magnitude, a result consistent with Hancock (1985). However, since the prices of the
financial goods are user costs, the interpretation of the elasticities requires taking into account the
relation between holding costs and user costs. In some cases, small changes in holding costs may
have significant effects on quantities even when the price elasticities are small. The reason for this
is that the user costs are centred around the discount rates and the user costs are rates (percentages)
by nature. In fact, if the holding cost of a financial good is close to the discount rate, then even
small changes in the holding cost result in significant swings in the user cost.
2.8.1 Interest on Reserves
For years, the Federal Reserve did not pay interest on bank reserves. However, during the global
financial crisis, legislation was passed and authorized the Fed to remunerate bank reserve holdings.
Thus, since October 2008, the Federal Reserve operates a channel system of monetary control by
paying interest on bank reserves.
We analyze the effect of a 25 basis points increase in the interest rate paid by the Fed on reserves
on the demand and supply of financial goods, using compensated price elasticities. Beginning of
period t nominal user cost of deposits of bank i is given by
u⇤it =hit �Rit + k (Rit �q)� s+d
1+RitPt (2.23)
where k is the reserve requirement on deposits (a flat rate), s is the service charge earned per dollar,
d is the deposit insurance premium rate, and q is the interest rate on required and excess reserves
(we assume equal rates). It follows that the effect of a 1 percent increase in the interest rate on
reserves on the user cost of deposits is �k/(1+Rit). Assuming a reserve requirement ratio of 10%
(the actual ratio depends on the amount of deposits and it is lower for small banks), a 25 basis point
increase in this rate results in, on average, 83 basis points increase in the amount of deposits and
approximately only 2 basis points increase in loans and leases.
69
2.8.2 Reserve Requirements
Now we consider a decrease in the reserve ratio by one percent. According to equation (2.23),
if the reserve ratio rises by one percent, the user cost of deposits increase by (Rit �q)/(1+Rit).
Assuming a 0.5 percent interest rate on reserve balances (effective as of December 17, 2015), then
a one percent decrease in the reserve ratio leads to approximately 82 basis point increase in the
amount of deposits and 141 basis point increase in investment in debt securities.
2.8.3 Changes in the Federal Funds Rate
With the economy showing sings of recovery recently, the Federal Reserve has moved its focus
back to conventional monetary policy instruments. We consider the effects of an increase in the
federal funds rate on the demand for and supply of financial goods in our model. We assume that
the primary channel through which the federal funds rate affects the production of financial goods
by banks is the interest rate on debt other than deposits (“other debt” in our model), specifically,
interbank loans and money markets.
An increase in the federal funds rate has two opposing effects on the user cost of other debt.
First, it increases the holding cost of other debt. Second, it raises the discount rate. However, since
other debt is a relatively small component in the discount rate (average weight is about 5 percent),
the magnitude of the second effect is economically insignificant. Therefore, the overall effect on
the user cost of other debt is positive. We ignore the effect of the discount rate on the user costs of
other financial goods.
Figure 9 shows the effect of a 25 basis point increase in the federal funds rate on the quantities
of financial goods over the sample period. Consistent with our previous discussion, a 25 basis
point increase in the fed funds rate results in significant fluctuations in the user cost of other debt,
ranging between 35 and 55 percent (in absolute value). Given the compensated price elasticity
of other debt (see Table 9.1), the expected percentage decrease in the supply of other debt ranges
between 37 and 73 percent.
70
Also, using the cross-price elasticities, we find that a 25 basis point increase in the fed funds rate
results approximately in a 15 percent increase in the amount of deposits. Moreover, our estimates
suggest that the average effect on the supply of loans and leases is only 2 percent. This finding
gains support from the recent evidence of large excess reserves accumulated by banks when the
cost of capital has approached zero.
2.8.4 Changes in the Return on Investments
It is also to be noted that in the aftermath of the global financial crisis, the Federal Reserve and
many central banks around the world have departed from the traditional interest-rate targeting
approach to monetary policy and are now focusing on their balance sheet instead, using noncon-
ventional monetary policy, such as quantitative easing and credit easing. There has also been a
move towards tougher standards in prudential regulation for banks, mostly in the form of higher
regulatory capital requirements. Financial firm production can be influenced by nonconventional
monetary policy, as well as by regulatory policy requirements, even when the policy rate is at the
zero lower bound. The transmission mechanisms of such policies include traditional interest-rate
channels that operate through the cost of borrowing and lending, other asset price channels, as well
as the bank lending channel.
In general, the user cost approach has no internal constraints related to the zero lower bound
constraint on the policy rate and can be helpful in analyzing nonconventional monetary policy.
Effectively, the unconventional monetary policy used by the Federal Reserve has also been reduc-
ing (long term) interest rates. For example, the Federal Reserve’s quantitative easing programme
consisted in monthly purchases of government and mortgage bonds has been raising the prices of
these financial assets and lowering long term yields, while simultaneously increasing the monetary
base. Here the changes in the interest rate affect banks mainly through user costs of loans and
leases and debt other than deposits. The analysis of the effects of these changes is similar to that
for conventional monetary policy.
Since in our model banks optimally choose the quantities of financial goods, we cannot model
71
the direct purchases of bank (problematic) assets. Instead, we can assume that asset purchases
increase the return on bank investments (“debt securities and trading accounts” in our model) and,
therefore, user revenues of these assets. Figure 10 shows the effects of a 25 basis point increase of
return on bank investments on the supply and demand of financial goods. In particular, this increase
implies the growth of the investments in debt securities and trading accounts by approximately 5
percent and growth of deposits by 2 percent. It also has a pronounced negative effect of about 9
percent on debt other than deposits.
2.9 Conclusion
In this article, we build on the path-breaking work by Hancock (1985) and Barnett and Hahm
(1994) and develop an estimable model of the microeconomics of the financial firm, using recent
state-of-the-art advances in microeconometrics. We assume that the deposit-taking financial firm
produces intermediation services between lenders and borrowers and maximizes variable profit
(total revenue less variable cost). We also follow the user cost approach and define outputs as
those assets or liabilities that contribute to a bank’s revenue and inputs as those assets or liabilities
that contribute to a bank’s cost of production. With the calculation of user costs for financial goods,
we use three flexible specifications for the variable profit function in order to derive demands for
and supplies of financial goods.
In constructing user costs, we deviate from Hancock (1985) and Barnett and Hahm (1994) and
follow Diewert et al. (2012) using data on realized bank interest income and expenses in order
to classify financial goods (debt securities and trading accounts, loans and leases, deposits, other
debt, and equity) as inputs or outputs. We estimate the translog, NQ, and generalized symmetric
Barnett variable profit functions with the convexity conditions imposed (locally in the case of the
translog and globally in the case of the NQ and generalized symmetric Barnett models) in order to
produce inference consistent with neoclassical microeconomic theory. We also take the literature
to a new level by using annual panel data on all commercial banks in the United States, over the
72
period from 1992 to 2013 (a total of 184,888 observations).
In providing a comparison among the three flexible functional forms for the variable profit
function — translog, NQ, and generalized symmetric Barnett — our finding in terms of convexity
violations is disappointing in the case of the translog, even when convexity is locally imposed. We
find that the locally flexible NQ and the quasiflexible generalized symmetric Barnett models are
able to provide inferences about the microeconomics of financial firm production consistent with
neoclassical microeconomic theory, although the imposition of convexity globally does not always
assure full theoretical regularity, as pointed out by Barnett (2002). We provide a comparison
between the convexity-constrained NQ and generalized symmetric Barnett models and find that
both models are able to produce well-behaved elasticities of transformation, supply, and demand.
We consider the estimation results based on the NQ functional form relatively more robust.
Although the size of our sample is sufficiently large, the generalized Barnett system can possibly
be over-parameterized. Specifically, there are two potential problems. First, a large number of
interaction terms could cause multicollinearity in the demand and supply equations of the gener-
alized Barnett system. Second, a set of fractional polynomials in the generalized Barnett system
leads to a likelihood function with multiple local maxima. The maximization of such a function in
a multidimensional space cannot always guarantee that the global maximum is reached.
We find that supplies of outputs and demands for inputs of financial goods are relatively in-
elastic, that the degree of substitutability among financial goods is low and variable over time, and
that production of financial firms is relatively insensitive to changes in user costs — consistent
with Hancock (1985) and Barnett and Hahm (1994). These results are robust across four different
bank samples, based on asset size — banks with assets less than $100 million, banks with assets
between $100 million and $1 billion, banks with assets between $1 billion and $10 billion, and
banks with assets in excess of $10 billion. However, we conduct a monetary and regulatory policy
analysis and find that small changes in holding costs may have significant effects on user costs
and the demands for and supplies of financial goods, even when the user cost elasticities are small
73
in magnitude. This has significant implications for the conduct of monetary policy, as it suggests
that the central bank can signal policy changes via changes in interest-rate policy instruments even
when the zero lower bound constraint on the policy rate is binding.
74
Tabl
e2.
1:Si
zedi
strib
utio
nof
U.S
.ban
ks
1992
2002
2012
Shar
eSh
are
Shar
eN
umbe
rA
sset
sof
asse
tsN
umbe
rA
sset
sof
asse
tsN
umbe
rA
sset
sof
asse
tsA
sset
sof
bank
she
ldhe
ld(%
)of
bank
she
ldhe
ld(%
)of
bank
she
ldhe
ld(%
)
Less
than
$100
mill
ion
8,28
235
09.
64,
125
213
2.9
1,88
611
20.
8$1
00m
illio
n–
$1bi
llion
3,02
474
920
.53,
583
963
13.1
3,80
41,
152
8.5
$1bi
llion
–10
billi
on36
81,
117
30.5
359
1,02
213
.948
41,
275
9.4
Mor
eth
an$1
0bi
llion
511,
445
39.5
875,
175
70.2
8811
,042
81.3
Tota
l11
,725
3,66
210
08,
154
7,37
410
06,
262
13,5
8010
0
Not
e:B
ank
asse
tsar
ein
billi
ons
ofdo
llars
.
75
Table 2.2: Assets and liabilities of U.S. banks (in billions)
Debt Loans Premises & TotalYear securities & leases fixed assets Deposits Other debt Equity liabilities
1992 893 2,299 99 2,853 535 274 3,6621993 1,003 2,449 94 2,925 650 311 3,8851994 1,069 2,671 96 3,048 809 326 4,1831995 1,082 2,953 102 3,213 923 365 4,5011996 1,088 3,185 117 3,372 969 385 4,7261997 1,206 3,354 136 3,598 1,181 433 5,2121998 1,299 3,615 152 3,849 1,294 473 5,6171999 1,339 3,904 178 4,015 1,466 499 5,9802000 1,410 4,221 186 4,362 1,560 547 6,4692001 1,538 4,297 207 4,595 1,610 613 6,8192002 1,805 4,570 217 4,921 1,779 673 7,3742003 1,988 4,880 256 5,281 1,917 719 7,9182004 2,129 5,365 380 5,801 1,994 878 8,6732005 2,138 5,850 407 6,279 2,067 937 9,2842006 2,338 6,514 467 6,906 2,341 1,050 10,2972007 2,499 7,174 546 7,474 2,678 1,159 11,3112008 2,725 7,201 532 8,244 3,035 1,168 12,4482009 2,949 6,722 541 8,500 2,152 1,324 11,9762010 3,125 6,852 526 8,651 2,097 1,359 12,1072011 3,306 7,121 496 9,369 1,922 1,418 12,7092012 3,519 7,622 508 10,162 1,893 1,524 13,5802013 3,383 7,830 490 10,531 1,727 1,532 13,790
76
Table 2.3: Percentage of Observations When Financial Goods are Outputs
Year Debt securities Loans & leases Deposits Other debt Equity
1992 96.86 99.78 88.58 88.17 10.751993 96.50 99.86 92.32 87.51 7.291994 97.56 99.83 93.73 77.67 5.951995 95.66 99.81 93.23 71.12 6.631996 96.67 99.80 91.35 79.47 8.101997 96.64 99.80 91.69 76.90 7.871998 92.10 99.82 89.41 76.41 10.191999 95.27 99.79 87.71 76.23 11.372000 96.32 99.77 88.09 55.64 12.832001 89.36 99.86 84.95 68.53 14.482002 92.73 99.83 90.86 65.73 9.082003 85.61 99.83 93.26 63.66 7.062004 90.86 99.80 93.48 60.44 6.942005 87.23 99.78 93.04 52.15 7.422006 82.26 99.77 90.04 46.85 11.092007 83.72 99.72 81.52 61.46 18.532008 94.12 99.90 69.16 54.80 32.172009 94.79 99.83 66.56 43.59 36.112010 89.73 99.79 76.61 46.50 25.482011 88.62 99.77 83.86 53.33 17.482012 84.24 99.81 89.78 58.37 11.082013 82.34 99.80 92.39 64.42 8.19
Total 92.04 99.81 87.95 66.99 12.28
Note: Sample period, annual data 1992-2013 (184,888 observations).
77
Tabl
e2.
4:U
serC
osts
Ave
rage
dA
cros
sB
anks
(and
the
Dis
coun
tRat
e)
Year
Deb
tsec
uriti
esLo
ans
&le
ases
Dep
osits
Oth
erde
btEq
uity
Dis
coun
trat
e
1992
�0.
0182
3�
0.03
693
�0.
0039
5�
0.01
695
0.03
450
0.04
329
1993
�0.
0160
5�
0.03
564
�0.
0053
8�
0.01
425
0.05
032
0.03
730
1994
�0.
0155
4�
0.03
512
�0.
0054
5�
0.00
768
0.05
010
0.03
655
1995
�0.
0143
9�
0.03
596
�0.
0053
6�
0.00
579
0.04
603
0.04
274
1996
�0.
0149
3�
0.03
511
�0.
0050
8�
0.01
215
0.04
567
0.04
282
1997
�0.
0160
9�
0.03
484
�0.
0052
5�
0.01
018
0.04
446
0.04
319
1998
�0.
0140
8�
0.03
611
�0.
0048
6�
0.01
078
0.04
665
0.04
219
1999
�0.
0154
8�
0.03
409
�0.
0046
4�
0.00
958
0.05
045
0.04
007
2000
�0.
0172
6�
0.03
471
�0.
0049
60.
0021
10.
0447
70.
0436
420
01�
0.01
524
�0.
0363
4�
0.00
428
�0.
0067
30.
0405
90.
0399
020
02�
0.01
634
�0.
0365
4�
0.00
660
�0.
0052
70.
0567
60.
0304
620
03�
0.01
067
�0.
0367
4�
0.00
732
�0.
0026
50.
0605
40.
0251
420
04�
0.01
245
�0.
0355
7�
0.00
793
�0.
0010
00.
0679
00.
0230
020
05�
0.01
060
�0.
0364
7�
0.00
818
0.00
379
0.07
049
0.02
682
2006
�0.
0099
7�
0.03
769
�0.
0071
70.
0085
60.
0608
20.
0331
120
07�
0.01
234
�0.
0385
5�
0.00
457
�0.
0005
90.
0415
30.
0350
320
08�
0.02
333
�0.
0427
2�
0.00
005
0.00
267
�0.
0177
30.
0242
020
09�
0.02
636
�0.
0480
50.
0015
90.
0088
0�
0.25
790
0.01
559
2010
�0.
0186
1�
0.04
904
�0.
0021
80.
0083
2�
0.05
807
0.01
465
2011
�0.
0149
8�
0.04
823
�0.
0051
90.
0042
00.
0001
90.
0142
220
12�
0.01
152
�0.
0462
3�
0.00
765
0.00
231
0.04
677
0.01
397
2013
�0.
0089
8�
0.04
376
�0.
0084
1�
0.00
004
0.05
340
0.01
309
78
Tabl
e2.
5:Tr
ansl
ogPa
ram
eter
Estim
ates
Ass
ets
Less
than
$100
mill
ion
$1bi
llion
toM
ore
than
Para
met
erA
llba
nks
$100
mill
ion
to$1
billi
on$1
0bi
llion
$10
billi
on
a 1�
0.11
0(0.7
57)
�0.
107(0.0
13)
0.94
9(2.7
33)
�0.
460(2.2
84)
0.49
8(0.0
04)
a 20.
038(0.5
62)
0.74
9(0.1
35)
�0.
588(1.2
65)
1.11
0(0.2
46)
0.17
2(0.5
54)
a 30.
571(0.2
12)
0.03
6(0.0
48)
0.54
9(0.4
88)
0.77
2(3.0
47)
0.09
5(0.0
12)
a 40.
752(0.0
09)
0.50
3(0.1
23)
0.18
3(1.2
54)
�0.
369(0.1
67)
0.26
6(0.5
85)
b 11
�0.
000(0.0
00)
�0.
479(0.1
40)
�0.
141(1.2
47)
�0.
000(0.0
02)
�0.
053(0.0
01)
b 21
0.58
2(0.0
28)
0.41
8(0.1
91)
0.42
4(0.0
49)
0.09
4(0.3
63)
�0.
371(0.4
86)
b 22
�0.
190(0.5
24)
�0.
000(0.0
00)
�0.
168(0.1
77)
�0.
175(0.2
37)
0.35
2(0.0
42)
b 31
�0.
887(0.1
70)
0.53
9(0.0
24)
�0.
582(2.1
69)
�0.
510(0.4
42)
0.28
4(0.3
20)
b 32
0.13
9(0.2
42)
0.23
3(0.0
64)
�0.
039(0.3
28)
0.10
1(0.1
93)
�0.
435(0.1
61)
b 33
�0.
417(0.9
40)
�0.
040(0.0
27)
�0.
261(0.1
72)
�0.
163(0.6
95)
�0.
346(0.0
82)
b 41
0.09
4(0.5
57)
�0.
641(0.4
79)
0.28
2(0.8
97)
0.20
2(0.5
03)
0.19
1(0.2
25)
b 42
�0.
438(0.2
08)
�0.
353(0.0
64)
0.24
6(0.5
31)
�0.
008(0.0
68)
0.03
0(0.1
06)
b 43
0.46
9(0.4
25)
�0.
309(0.1
50)
�0.
006(0.0
78)
0.06
9(0.3
44)
0.29
1(0.0
82)
b 44
0.19
4(2.4
70)
�0.
386(0.2
10)
�0.
193(0.7
13)
�0.
009(0.5
40)
�0.
183(0.1
41)
Obs
erva
tions
184,
888
96,3
4077,9
458,
874
1,72
9
Viol
atio
ns(%
)C
onve
xity
6997
9899
81M
onot
onic
ity40
3535
3524
Not
e:St
anda
rder
rors
inpa
rent
hese
s.1=
Deb
tsec
uriti
es,2
=Lo
ans
and
leas
es,3
=D
epos
its,4
=O
ther
debt
,5=
Equi
ty.
79
Tabl
e2.
6:N
orm
aliz
edQ
uadr
atic
Para
met
erEs
timat
es
Ass
ets
Less
than
$100
mill
ion
$1bi
llion
toM
ore
than
Para
met
erA
llba
nks
$100
mill
ion
to$1
billi
on$1
0bi
llion
$10
billi
on
b 12.
533(1.3
47)
0.60
0(0.0
96)
0.25
1(0.0
01)
0.91
8(0.0
03)
2.55
9(3.7
64)
b 26.
873(1.8
92)
2.12
7(0.2
32)
2.75
2(0.0
26)
3.56
7(0.1
05)
6.90
9(2.2
46)
b 38.
619(2.2
29)
2.67
3(0.1
58)
5.09
2(0.2
32)
4.89
0(0.2
87)
8.66
0(3.1
21)
b 4�
1.65
1(0.1
74)
�0.
033(0.0
86)
�0.
264(0.0
16)
�0.
266(0.0
41)
�1.
664(0.1
43)
b 5�
1.30
9(0.3
03)
�0.
275(0.0
37)
�0.
783(0.0
13)
�0.
723(0.0
14)
�1.
313(0.4
52)
k 11
1.46
0(0.2
13)
0.92
6(0.8
15)
�1.
288(0.0
66)
0.79
5(0.0
57)
1.50
3(1.3
16)
k 21
�0.
546(0.7
29)
0.12
5(0.0
63)
�0.
356(0.0
15)
0.10
5(0.3
50)
�0.
581(3.4
94)
k 22
3.08
3(0.0
42)
�1.
721(0.3
17)
3.29
9(0.2
51)
�2.
424(0.3
56)
3.09
3(0.0
27)
k 31
�3.
069(0.1
04)
0.34
1(0.3
47)
�0.
641(0.0
25)
0.76
7(0.1
14)
�3.
091(0.0
34)
k 32
0.55
5(1.9
65)
�0.
373(0.2
67)
�0.
131(0.8
05)
0.32
4(0.7
56)
�0.
576(6.9
06)
k 33
2.52
8(0.7
04)
0.27
2(0.8
71)
�0.
833(0.2
87)
1.52
6(0.3
87)
�2.
536(1.5
85)
k 41
�2.
292(0.7
78)
0.02
5(0.2
50)
0.58
9(0.3
10)
�1.
292(0.5
77)
2.31
7(0.9
78)
k 42
�0.
002(0.3
79)
0.00
0(0.0
00)
�0.
006(2.2
02)
0.00
0(0.0
16)
0.00
1(1.7
03)
k 43
0.00
1(0.2
76)
�0.
000(0.0
00)
0.00
3(1.2
79)
�0.
000(0.0
11)
�0.
001(1.2
36)
k 44
0.00
0(0.0
00)
0.00
0(0.0
00)
�0.
000(0.0
00)
0.00
0(0.0
00)
�0.
000(0.0
00)
Obs
erva
tions
184,
888
96,3
4077,9
458,
874
1,72
9
Viol
atio
ns(%
)C
onve
xity
00
00
0M
onot
onic
ity14
1214
1514
Not
e:St
anda
rder
rors
inpa
rent
hese
s.1=
Deb
tsec
uriti
es,2
=Lo
ans
and
leas
es,3
=D
epos
its,4
=O
ther
debt
,5=
Equi
ty.
80
Tabl
e2.
7:G
ener
aliz
edB
arne
ttPa
ram
eter
Estim
ates
Ass
ets
Less
than
$100
mill
ion
$1bi
llion
toM
ore
than
Para
met
erA
llba
nks
$100
mill
ion
to$1
billi
on$1
0bi
llion
$10
billi
on
a 11
1.71
9(0
.048
)-0
.980
(0.0
00)
1.10
6(0
.250
)1.
137
(0.8
12)
1.20
8(0
.445
)a 1
20.
523
(0.0
31)
-0.2
14(0
.004
)0.
314
(0.4
30)
0.11
1(4
.522
)-0
.000
(0.0
00)
a 13
0.09
0(0
.035
)-0
.206
(0.0
15)
-0.0
00(0
.000
)-0
.000
(0.0
00)
0.38
6(0
.063
)a 1
40.
419
(0.0
07)
-0.0
03(0
.003
)-0
.000
(0.0
01)
-0.2
14(0
.009
)-0
.456
(0.4
47)
a 15
0.45
1(0
.080
)0.
006
(0.0
05)
0.00
0(0
.000
)0.
000
(0.0
00)
-0.3
19(0
.327
)a 2
2-2
.526
(0.0
64)
1.37
7(0
.001
)1.
639
(0.1
59)
1.86
5(0
.424
)-1
.970
(0.9
44)
a 23
0.00
3(0
.002
)0.
175
(0.0
10)
-0.0
00(0
.000
)0.
000
(0.0
00)
-0.0
00(0
.000
)a 2
40.
173
(0.0
84)
0.00
9(0
.009
)0.
049
(0.0
67)
-0.0
00(0
.000
)-0
.000
(0.0
00)
a 25
-0.0
07(0
.002
)0.
028
(0.0
09)
-0.1
40(0
.081
)-0
.130
(0.1
48)
-0.1
65(0
.261
)a 3
3-3
.110
(0.1
88)
1.81
4(0
.001
)2.
231
(0.1
41)
2.54
8(0
.362
)2.
366
(0.3
95)
a 34
0.22
8(0
.108
)-0
.006
(0.0
06)
-0.0
00(0
.000
)0.
000
(0.0
00)
0.00
0(0
.000
)a 3
51.
237
(0.1
03)
-0.6
43(0
.000
)0.
836
(0.0
27)
0.96
0(0
.176
)0.
683
(0.0
66)
a 44
0.80
3(0
.193
)0.
198
(0.0
00)
0.29
7(0
.049
)0.
554
(0.0
18)
0.80
9(0
.671
)a 4
5-0
.046
(0.0
39)
-0.0
58(0
.002
)-0
.000
(0.0
00)
-0.0
00(0
.000
)0.
000
(0.0
00)
a 55
-0.0
68(0
.065
)0.
000
(0.0
00)
0.00
0(0
.000
)-0
.000
(0.0
00)
0.00
0(0
.000
)a 1
23-0
.000
(0.0
00)
0.00
0(0
.000
)-0
.000
(0.0
00)
0.00
0(0
.000
)0.
000
(0.0
00)
a 124
-0.0
00(0
.000
)-0
.001
(0.0
00)
-0.0
00(0
.000
)0.
000
(0.0
00)
0.00
0(0
.000
)a 1
250.
000
(0.0
00)
0.00
0(0
.000
)0.
000
(0.0
00)
0.00
0(0
.000
)-0
.000
(0.0
00)
a 134
-0.0
02(0
.001
)0.
000
(0.0
00)
-0.0
00(0
.000
)0.
000
(0.0
00)
-0.0
00(0
.000
)a 1
350.
000
(0.0
00)
-0.0
00(0
.000
)0.
000
(0.0
00)
0.00
0(0
.000
)0.
000
(0.0
00)
a 145
0.00
2(0
.001
)0.
000
(0.0
00)
0.00
0(0
.000
)-0
.000
(0.0
00)
0.00
0(0
.000
)a 1
560.
000
(0.0
00)
0.00
0(0
.000
)-0
.000
(0.0
00)
0.00
0(0
.000
)0.
000
(0.0
00)
a 155
-0.0
00(0
.000
)-0
.000
(0.0
00)
-0.0
00(0
.000
)-0
.000
(0.0
00)
0.00
0(0
.000
)a 2
130.
001
(0.0
01)
-0.0
00(0
.000
)0.
000
(0.0
00)
-0.0
00(0
.000
)0.
000
(0.0
00)
a 214
-0.0
00(0
.000
)0.
000
(0.0
00)
0.00
0(0
.000
)-0
.000
(0.0
00)
0.00
0(0
.000
)a 2
15-0
.000
(0.0
00)
-0.0
00(0
.000
)0.
000
(0.0
00)
0.00
0(0
.000
)0.
000
(0.0
00)
a 234
-0.0
00(0
.000
)-0
.000
(0.0
00)
0.00
0(0
.000
)0.
000
(0.0
00)
-0.0
00(0
.000
)a 2
350.
001
(0.0
00)
0.00
0(0
.000
)0.
001
(0.0
18)
0.00
0(0
.000
)-0
.000
(0.0
00)
81
Tabl
e7
(Con
t’d).
Gen
eral
ized
Bar
nett
Para
met
erEs
timat
es
Ass
ets
Less
than
$100
mill
ion
$1bi
llion
toM
ore
than
Para
met
erA
llba
nks
$100
mill
ion
to$1
billi
on$1
0bi
llion
$10
billi
on
a 245
0.00
3(0
.001
)-0
.000
(0.0
00)
-0.0
00(0
.000
)0.
000
(0.0
00)
-0.0
78(0
.038
)a 2
56-0
.004
(0.0
02)
-0.0
00(0
.000
)-0
.000
(0.0
00)
-0.0
00(0
.000
)-0
.000
(0.0
00)
a 255
0.00
3(0
.000
)0.
001
(0.0
01)
0.00
0(0
.000
)0.
000
(0.0
00)
0.00
0(0
.000
)a 3
120.
013
(0.0
01)
-0.0
00(0
.000
)0.
000
(0.0
00)
-0.0
00(0
.000
)-0
.000
(0.0
00)
a 314
-0.0
00(0
.000
)0.
000
(0.0
00)
0.00
0(0
.000
)0.
000
(0.0
00)
-0.0
00(0
.000
)a 3
15-0
.005
(0.0
01)
-0.0
00(0
.000
)0.
002
(0.0
04)
-0.0
09(0
.297
)-0
.000
(0.0
00)
a 324
-0.0
02(0
.001
)0.
000
(0.0
00)
0.00
0(0
.000
)0.
000
(0.0
00)
0.00
0(0
.000
)a 3
250.
001
(0.0
00)
-0.0
00(0
.000
)-0
.000
(0.0
00)
-0.0
00(0
.000
)-0
.000
(0.0
00)
a 345
0.00
2(0
.001
)0.
000
(0.0
00)
0.00
0(0
.000
)0.
000
(0.0
00)
-0.0
00(0
.000
)a 3
56-0
.007
(0.0
05)
0.00
0(0
.000
)-0
.000
(0.0
00)
0.00
0(0
.000
)-0
.000
(0.0
00)
a 355
-0.0
00(0
.000
)-0
.000
(0.0
00)
-0.0
00(0
.000
)0.
000
(0.0
00)
-0.0
00(0
.000
)a 4
12-0
.000
(0.0
00)
0.00
0(0
.000
)-0
.000
(0.0
00)
-0.0
00(0
.000
)-0
.000
(0.0
00)
a 413
0.00
0(0
.000
)0.
000
(0.0
00)
0.00
0(0
.000
)-0
.000
(0.0
00)
0.00
0(0
.000
)a 4
150.
000
(0.0
00)
-0.0
00(0
.000
)0.
000
(0.0
00)
0.00
0(0
.000
)-0
.000
(0.0
00)
a 423
-0.0
01(0
.000
)0.
000
(0.0
00)
-0.0
00(0
.000
)0.
000
(0.0
00)
-0.0
00(0
.000
)a 4
25-0
.001
(0.0
00)
-0.0
00(0
.000
)-0
.000
(0.0
00)
-0.0
00(0
.000
)0.
000
(0.0
00)
a 435
-0.0
00(0
.000
)0.
000
(0.0
00)
-0.0
00(0
.000
)-0
.000
(0.0
00)
-0.0
00(0
.000
)
Obs
erva
tions
184,
888
96,3
4077,9
458,
874
1,72
9
Viol
atio
ns(%
)C
onve
xity
00
00
0M
onot
onic
ity7
014
2614
Not
e:St
anda
rder
rors
inpa
rent
hese
s.1=
Deb
tsec
uriti
es,2
=Lo
ans
and
leas
es,3
=D
epos
its,4
=O
ther
debt
,5=
Equi
ty.
82
Tabl
e2.
8:El
astic
ities
ofTr
ansf
orm
atio
n,A
llB
anks
Elas
ticiti
esof
Tran
sfor
mat
ion
Fina
ncia
lgoo
di
Mod
els i
Ss i
Ls i
Ds i
Os i
E
Deb
tN
Q1.
898
(0.2
27)
secu
ritie
s(S
)G
B2.
469
(0.0
36)
Loan
sN
Q0.
003
(0.0
17)
0.02
5(0
.005
)an
dLe
ases
(L)
GB
-0.1
27(0
.016
)0.
019
(0.0
02)
Dep
osits
(D)
NQ
1.04
1(0
.083
)0.
070
(0.0
17)
0.75
6(0
.025
)G
B-0
.010
(0.0
09)
-0.0
00(0
.000
)3.
358
(0.0
10)
Oth
erde
bt(O
)N
Q-1
5.86
7(0
.505
)-0
.998
(0.1
05)
-11.
321
(0.9
76)
169.
785
(34.
796)
GB
-1.6
91(1
.273
)-0
.057
(0.0
25)
-0.2
76(0
.113
)9.
358
(11.
291)
Equi
ty(E
)N
Q1.
138
(0.2
14)
0.09
7(0
.032
)0.
882
(0.1
26)
-13.
166
(0.2
90)
1.04
1(0
.265
)G
B0.
700
(0.2
83)
0.00
0(0
.000
)2.
890
(0.0
06)
0.03
1(0
.038
)3.
022
(0.2
89)
Not
e:Sa
mpl
epe
riod,
annu
alda
ta19
92-2
013
(184,8
88ob
serv
atio
ns).
83
Tabl
e2.
9:El
astic
ities
ofTr
ansf
orm
atio
n,B
anks
With
Ass
ets
Less
Than
$100
Mill
ion
Elas
ticiti
esof
Tran
sfor
mat
ion
Fina
ncia
lgoo
di
Mod
els i
Ss i
Ls i
Ds i
Os i
E
Deb
tN
Q0.
480
(0.0
45)
secu
ritie
s(S
)G
B0.
375
(0.7
49)
Loan
sN
Q-0
.036
(0.0
07)
0.01
7(0
.000
)an
dLe
ases
(L)
GB
-0.0
52(0
.101
)0.
015
(0.0
07)
Dep
osits
(D)
NQ
-0.4
89(0
.024
)0.
018
(0.0
10)
0.52
6(0
.010
)G
B-0
.107
(0.3
11)
-0.0
24(0
.061
)2.
321
(0.0
54)
Oth
erde
bt(O
)N
Q3.
173
(0.4
92)
-0.5
90(0
.037
)-2
.756
(0.4
99)
29.4
99(3
.798
)G
B-0
.001
(0.0
60)
-0.0
02(0
.115
)-0
.002
(0.0
37)
9.99
5(7
342.
358)
Equi
ty(E
)N
Q-0
.226
(0.0
12)
0.06
6(0
.006
)0.
164
(0.0
13)
-2.6
76(0
.215
)0.
271
(0.0
28)
GB
0.00
0(0
.003
)0.
003
(0.0
25)
3.19
4(0
.526
)0.
835
(46.
275)
4.96
8(1
.377
)
Not
e:Sa
mpl
epe
riod,
annu
alda
ta19
92-2
013
(96,
340
obse
rvat
ions
).
84
Tabl
e2.
10:E
last
iciti
esof
Tran
sfor
mat
ion,
Ban
ksW
ithA
sset
sB
etw
een
$100
Mill
ion
and
$1B
illio
n
Elas
ticiti
esof
Tran
sfor
mat
ion
Fina
ncia
lgoo
di
Mod
els i
Ss i
Ls i
Ds i
Os i
E
Deb
tN
Q0.
452
(0.1
70)
secu
ritie
s(S
)G
B0.
509
(1.4
25)
Loan
sN
Q-0
.036
(0.0
22)
0.01
6(0
.008
)an
dLe
ases
(L)
GB
-0.0
88(0
.238
)0.
023
(0.0
48)
Dep
osits
(D)
NQ
-0.7
35(0
.210
)0.
130
(0.0
22)
1.59
9(0
.050
)G
B-0
.000
(0.0
00)
-0.0
00(0
.000
)2.
006
(0.3
14)
Oth
erde
bt(O
)N
Q-1
.582
(3.0
17)
-0.9
92(0
.881
)-3
.647
(5.3
35)
101.
493
(50.
103)
GB
-0.0
00(0
.001
)-0
.027
(0.1
02)
-0.0
00(0
.000
)1.
786
(9.8
08)
Equi
ty(E
)N
Q-0
.607
(0.1
24)
0.14
0(0
.018
)1.
502
(0.2
06)
-5.8
26(5
.334
)1.
473
(0.3
24)
GB
0.00
0(0
.000
)0.
034
(0.0
33)
2.35
2(0
.242
)0.
000
(0.0
00)
2.92
0(0
.025
)
Not
e:Sa
mpl
epe
riod,
annu
alda
ta19
92-2
013
(8,8
74ob
serv
atio
ns).
85
Tabl
e2.
11:E
last
iciti
esof
Tran
sfor
mat
ion,
Ban
ksW
ithA
sset
sB
etw
een
$1B
illio
nan
d$1
0B
illio
n
Elas
ticiti
esof
Tran
sfor
mat
ion
Fina
ncia
lgoo
di
Mod
els i
Ss i
Ls i
Ds i
Os i
E
Deb
tN
Q0.
116
(0.0
69)
secu
ritie
s(S
)G
B0.
153
(3.9
14)
Loan
sN
Q0.
006
(0.0
29)
0.03
1(0
.006
)an
dLe
ases
(L)
GB
-0.0
09(0
.735
)0.
007
(0.1
22)
Dep
osits
(D)
NQ
-0.1
19(0
.123
)0.
142
(0.0
27)
0.83
6(0
.036
)G
B0.
000
(0.0
00)
-0.0
00(0
.000
)1.
559
(0.3
45)
Oth
erde
bt(O
)N
Q-0
.385
(0.3
84)
-0.6
15(0
.098
)-2
.481
(0.4
44)
12.8
37(1
.974
)G
B-0
.737
(0.1
07)
-0.0
00(0
.013
)-0
.000
(0.0
00)
5.15
9(1
.124
)
Equi
ty(E
)N
Q-0
.062
(0.1
33)
0.17
6(0
.017
)0.
927
(0.0
68)
-3.2
68(0
.271
)1.
077
(0.1
23)
GB
0.00
0(0
.000
)0.
021
(0.0
52)
1.94
5(0
.241
)0.
000
(0.0
01)
2.50
8(0
.084
)
Not
e:Sa
mpl
epe
riod,
annu
alda
ta19
92-2
013
(8,8
74ob
serv
atio
ns).
86
Tabl
e2.
12:E
last
iciti
esof
Tran
sfor
mat
ion,
Ban
ksW
ithA
sset
sM
ore
Than
$10
Bill
ion
Elas
ticiti
esof
Tran
sfor
mat
ion
Fina
ncia
lgoo
di
Mod
els i
Ss i
Ls i
Ds i
Os i
E
Deb
tN
Q1.
898
(0.2
27)
secu
ritie
s(S
)G
B2.
497
(4.7
06)
Loan
sN
Q0.
003
(0.0
17)
0.02
5(0
.005
)an
dLe
ases
(L)
GB
0.00
0(0
.000
)0.
009
(0.0
18)
Dep
osits
(D)
NQ
1.04
1(0
.083
)0.
070
(0.0
17)
0.75
6(0
.025
)G
B-0
.231
(0.0
80)
0.00
0(0
.000
)0.
686
(0.1
20)
Oth
erde
bt(O
)N
Q-1
5.86
7(0
.505
)-0
.998
(0.1
05)
-11.
321
(0.9
76)
169.
785
(34.
796)
GB
-3.4
94(5
.084
)0.
000
(0.0
00)
-0.0
54(0
.121
)16
.892
(0.0
39)
Equi
ty(E
)N
Q1.
138
(0.2
14)
0.09
7(0
.032
)0.
882
(0.1
26)
-13.
166
(0.2
90)
1.04
1(0
.265
)G
B0.
948
(1.6
75)
0.04
9(0
.101
)1.
297
(0.2
62)
0.00
0(0
.000
)3.
899
(0.0
81)
Not
e:Sa
mpl
epe
riod,
annu
alda
ta19
92-2
013
(1,7
29ob
serv
atio
ns).
87
Tabl
e2.
13:P
rice
Elas
ticiti
es,A
llB
anks
Pric
eel
astic
ities
Fina
ncia
lgoo
di
Mod
elh i
Sh i
Lh i
Dh i
Oh i
E
Deb
tN
Q0.
377
(0.0
31)
0.12
2(0
.020
)0.
369
(0.0
15)
-0.5
93(0
.011
)-0
.274
(0.0
55)
secu
ritie
s(S
)G
B0.
281
(0.0
18)
-0.1
11(0
.017
)-0
.002
(0.0
01)
-0.0
52(0
.004
)-0
.117
(0.0
38)
Loan
sN
Q0.
010
(0.0
02)
0.03
4(0
.003
)0.
025
(0.0
02)
-0.0
40(0
.002
)-0
.029
(0.0
05)
and
Leas
es(L
)G
B-0
.014
(0.0
01)
0.01
6(0
.001
)-0
.000
(0.0
00)
-0.0
02(0
.002
)-0
.000
(0.0
00)
Dep
osits
(D)
NQ
0.17
8(0
.004
)0.
150
(0.0
09)
0.22
0(0
.004
)-0
.354
(0.0
15)
-0.1
94(0
.023
)G
B-0
.001
(0.0
01)
-0.0
00(0
.000
)0.
492
(0.0
25)
-0.0
09(0
.009
)-0
.482
(0.0
35)
Oth
erde
bt(O
)N
Q-0
.589
(0.0
25)
-0.4
93(0
.036
)-0
.726
(0.0
01)
1.16
8(0
.027
)0.
640
(0.0
89)
GB
-0.1
93(0
.119
)-0
.050
(0.0
20)
-0.0
40(0
.015
)0.
288
(0.0
79)
-0.0
05(0
.006
)
Equi
ty(E
)N
Q0.
109
(0.0
23)
0.14
2(0
.026
)0.
160
(0.0
24)
-0.2
57(0
.033
)-0
.154
(0.0
41)
GB
0.08
0(0
.036
)0.
000
(0.0
00)
0.42
3(0
.030
)0.
001
(0.0
02)
-0.5
04(0
.008
)
Not
e:Sa
mpl
epe
riod,
annu
alda
ta19
92-2
013
(184,8
88ob
serv
atio
ns).
88
Tabl
e2.
14:P
rice
Elas
ticiti
es,B
anks
With
Ass
ets
Less
Than
$100
Mill
ion
Pric
eel
astic
ities
Fina
ncia
lgoo
di
Mod
elh i
Sh i
Lh i
Dh i
Oh i
E
Deb
tN
Q0.
064
(0.0
06)
-0.0
31(0
.006
)-0
.081
(0.0
04)
0.01
4(0
.002
)0.
034
(0.0
02)
secu
ritie
s(S
)G
B0.
058
(0.1
23)
-0.0
41(0
.075
)-0
.016
(0.0
49)
-0.0
00(0
.000
)-0
.000
(0.0
00)
Loan
sN
Q-0
.005
(0.0
01)
0.01
4(0
.000
)0.
003
(0.0
02)
-0.0
03(0
.000
)-0
.010
(0.0
01)
and
Leas
es(L
)G
B-0
.008
(0.0
17)
0.01
2(0
.005
)-0
.004
(0.0
09)
-0.0
00(0
.000
)-0
.000
(0.0
03)
Dep
osits
(D)
NQ
-0.0
65(0
.003
)0.
015
(0.0
09)
0.08
7(0
.001
)-0
.012
(0.0
02)
-0.0
25(0
.002
)G
B-0
.017
(0.0
50)
-0.0
19(0
.048
)0.
356
(0.0
30)
-0.0
00(0
.000
)-0
.321
(0.0
28)
Oth
erde
bt(O
)N
Q0.
422
(0.0
66)
-0.5
01(0
.032
)-0
.455
(0.0
84)
0.12
7(0
.015
)0.
408
(0.0
34)
GB
-0.0
00(0
.004
)-0
.002
(0.0
28)
-0.0
00(0
.001
)0.
086
(0.2
39)
-0.0
84(0
.272
)
Equi
ty(E
)N
Q-0
.030
(0.0
02)
0.05
6(0
.005
)0.
027
(0.0
02)
-0.0
12(0
.001
)-0
.041
(0.0
04)
GB
0.00
0(0
.000
)0.
002
(0.0
22)
0.49
0(0
.001
)0.
007
(0.0
24)
-0.5
00(0
.000
)
Not
e:Sa
mpl
epe
riod,
annu
alda
ta19
92-2
013
(96,
340
obse
rvat
ions
).
89
Tabl
e2.
15:P
rice
Elas
ticiti
es,B
anks
With
Ass
ets
Bet
wee
n$1
00M
illio
nan
d$1
Bill
ion
Pric
eel
astic
ities
Fina
ncia
lgoo
di
Mod
elh i
Sh i
Lh i
Dh i
Oh i
E
Deb
tN
Q0.
052
(0.0
20)
-0.0
32(0
.020
)-0
.149
(0.0
43)
-0.0
08(0
.014
)0.
136
(0.0
29)
secu
ritie
s(S
)G
B0.
072
(0.1
87)
-0.0
72(0
.187
)-0
.000
(0.0
00)
-0.0
00(0
.000
)-0
.000
(0.0
00)
Loan
sN
Q-0
.004
(0.0
03)
0.01
4(0
.007
)0.
026
(0.0
04)
-0.0
05(0
.005
)-0
.032
(0.0
04)
and
Leas
es(L
)G
B-0
.012
(0.0
32)
0.01
8(0
.039
)-0
.000
(0.0
00)
-0.0
00(0
.001
)-0
.006
(0.0
06)
Dep
osits
(D)
NQ
-0.0
85(0
.024
)0.
118
(0.0
20)
0.32
3(0
.009
)-0
.018
(0.0
29)
-0.3
37(0
.043
)G
B-0
.000
(0.0
00)
-0.0
00(0
.000
)0.
403
(0.0
45)
-0.0
00(0
.000
)-0
.403
(0.0
45)
Oth
erde
bt(O
)N
Q-0
.183
(0.3
50)
-0.8
94(0
.795
)-0
.737
(1.0
79)
0.50
7(0
.313
)1.
308
(1.2
11)
GB
-0.0
00(0
.000
)-0
.022
(0.0
53)
-0.0
00(0
.000
)0.
022
(0.0
53)
-0.0
00(0
.000
)
Equi
ty(E
)N
Q-0
.070
(0.0
14)
0.12
7(0
.016
)0.
303
(0.0
41)
-0.0
29(0
.030
)-0
.331
(0.0
69)
GB
0.00
0(0
.000
)0.
028
(0.0
29)
0.47
2(0
.029
)0.
000
(0.0
00)
-0.5
00(0
.000
)
Not
e:Sa
mpl
epe
riod,
annu
alda
ta19
92-2
013
(77,
945
obse
rvat
ions
).
90
Tabl
e2.
16:P
rice
Elas
ticiti
es,B
anks
With
Ass
ets
Bet
wee
n$1
Bill
ion
and
$10
Bill
ion
Pric
eel
astic
ities
Fina
ncia
lgoo
di
Mod
elh i
Sh i
Lh i
Dh i
Oh i
E
Deb
tN
Q0.
017
(0.0
10)
0.00
5(0
.025
)-0
.029
(0.0
30)
-0.0
10(0
.009
)0.
017
(0.0
35)
secu
ritie
s(S
)G
B0.
023
(0.5
57)
-0.0
07(0
.563
)0.
000
(0.0
00)
-0.0
16(0
.006
)-0
.000
(0.0
00)
Loan
sN
Q0.
001
(0.0
04)
0.02
6(0
.005
)0.
035
(0.0
07)
-0.0
15(0
.003
)-0
.047
(0.0
05)
and
Leas
es(L
)G
B-0
.001
(0.1
06)
0.00
5(0
.095
)-0
.000
(0.0
00)
-0.0
00(0
.000
)-0
.004
(0.0
11)
Dep
osits
(D)
NQ
-0.0
18(0
.018
)0.
120
(0.0
23)
0.20
5(0
.009
)-0
.062
(0.0
13)
-0.2
47(0
.018
)G
B0.
000
(0.0
00)
-0.0
00(0
.000
)0.
388
(0.0
57)
-0.0
00(0
.000
)-0
.388
(0.0
57)
Oth
erde
bt(O
)N
Q-0
.057
(0.0
57)
-0.5
20(0
.082
)-0
.610
(0.1
08)
0.31
8(0
.060
)0.
869
(0.0
73)
GB
-0.1
09(0
.006
)-0
.000
(0.0
11)
-0.0
00(0
.000
)0.
109
(0.0
17)
-0.0
00(0
.000
)
Equi
ty(E
)N
Q-0
.009
(0.0
20)
0.14
9(0
.014
)0.
228
(0.0
17)
-0.0
81(0
.010
)-0
.286
(0.0
33)
GB
0.00
0(0
.000
)0.
016
(0.0
42)
0.48
4(0
.042
)0.
000
(0.0
00)
-0.5
00(0
.000
)
Not
e:Sa
mpl
epe
riod,
annu
alda
ta19
92-2
013
(8,8
74ob
serv
atio
ns).
91
Tabl
e2.
17:P
rice
Elas
ticiti
es,B
anks
With
Ass
ets
Mor
eTh
an$1
0B
illio
n
Pric
eel
astic
ities
Fina
ncia
lgoo
di
Mod
elh i
Sh i
Lh i
Dh i
Oh i
E
Deb
tN
Q0.
251
(0.0
30)
0.00
3(0
.015
)0.
309
(0.0
23)
-0.2
05(0
.008
)-0
.357
(0.0
59)
secu
ritie
s(S
)G
B0.
267
(0.3
54)
0.00
0(0
.000
)-0
.065
(0.0
01)
-0.0
80(0
.138
)-0
.122
(0.2
16)
Loan
sN
Q0.
000
(0.0
02)
0.02
2(0
.005
)0.
021
(0.0
05)
-0.0
13(0
.002
)-0
.031
(0.0
09)
and
Leas
es(L
)G
B0.
000
(0.0
00)
0.00
6(0
.014
)0.
000
(0.0
00)
0.00
0(0
.000
)-0
.006
(0.0
14)
Dep
osits
(D)
NQ
0.13
7(0
.011
)0.
061
(0.0
14)
0.22
4(0
.006
)-0
.146
(0.0
02)
-0.2
76(0
.033
)G
B-0
.025
(0.0
03)
0.00
0(0
.000
)0.
192
(0.0
35)
-0.0
01(0
.002
)-0
.166
(0.0
30)
Oth
erde
bt(O
)N
Q-2
.095
(0.0
70)
-0.8
71(0
.085
)-3
.356
(0.3
10)
2.19
3(0
.290
)4.
129
(0.0
06)
GB
-0.3
73(0
.201
)0.
000
(0.0
00)
-0.0
15(0
.037
)0.
388
(0.1
64)
0.00
0(0
.000
)
Equi
ty(E
)N
Q0.
150
(0.0
28)
0.08
5(0
.028
)0.
261
(0.0
36)
-0.1
70(0
.016
)-0
.327
(0.0
75)
GB
0.10
1(0
.129
)0.
035
(0.0
84)
0.36
4(0
.212
)0.
000
(0.0
00)
-0.5
00(0
.000
)
Not
e:Sa
mpl
epe
riod,
annu
alda
ta19
92-2
013
(1,7
29ob
serv
atio
ns).
92
Figu
re2.
1:D
ynam
ics
ofB
ank
Ass
ets
0 1 2 3 4 5 6 7 8 9
1992
19
94
1996
19
98
2000
20
02
2004
20
06
2008
20
10
2012
Figu
re 1
. Dyn
amic
s of B
ank
Ass
ets
Loan
s and
leas
es
Prem
ises
Oth
er a
sset
s
Deb
t sec
uriti
es
93
Figu
re2.
2:D
ynam
ics
ofB
ank
Liab
ilitie
s
0 2 4 6 8 10
12
1992
19
94
1996
19
98
2000
20
02
2004
20
06
2008
20
10
2012
Figu
re 2
. Dyn
amic
s of B
ank
Liab
ilitie
s
Oth
er d
ebt
Equi
ty
Dep
osits
94
Figu
re2.
3:A
vera
geR
efer
ence
Dis
coun
tRat
es
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
1992
199
3 19
94 1
995
1996
199
7 19
98 1
999
2000
200
1 20
02 2
003
2004
200
5 20
06 2
007
2008
200
9 20
10 2
011
2012
201
3
Figu
re 3
. Ave
rage
Ref
eren
ce D
isco
unt R
ates
Deb
t D
epos
its a
nd d
ebt
Dep
osits
, deb
t, an
d eq
uity
Fe
dera
l fun
ds ra
te
95
Figu
re2.
4:O
wn-
pric
eel
astic
ities
ford
ebts
ecur
ities
,hSS
Ͳ0.03
Ͳ0.025
Ͳ0.02
Ͳ0.015
Ͳ0.01
Ͳ0.005
0
0
0.1
0.2
0.3
0.4
0.5
0.6
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
2012
Figu
re 4
. Ow
n-pr
ice
elas
ticiti
es fo
r deb
t sec
uriti
es, Ș
SS
Gen
eralize
d�Ba
rnett
NQ
User�cost�o
f�deb
t�securities
96
Figu
re2.
5:O
wn-
pric
eel
astic
ities
forl
oans
and
leas
es,h
LL
Ͳ0.06
Ͳ0.05
Ͳ0.04
Ͳ0.03
Ͳ0.02
Ͳ0.01
0
0
0.00
5
0.01
0.01
5
0.02
0.02
5
0.03
0.03
5
0.04
0.04
5
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
2012
Figu
re 5
. Ow
n-pr
ice
elas
ticiti
es fo
r loa
ns a
nd le
ases
, ȘLL
Gen
eralize
d�Ba
rnett
NQ
User�cost�o
f�loans�and
�leases
97
Figu
re2.
6:O
wn-
pric
eel
astic
ities
ford
epos
its,h
DD
Ͳ0.01
Ͳ0.008
Ͳ0.006
Ͳ0.004
Ͳ0.002
00.00
2
0.00
4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
2012
Figu
re 6
. Ow
n-pr
ice
elas
ticiti
es fo
r dep
osits
, ȘD
D
Gen
eralize
d�Ba
rnett
NQ
User�cost�o
f�dep
osits
98
Figu
re2.
7:O
wn-
pric
eel
astic
ities
foro
ther
debt
,hO
O
Ͳ0.02
Ͳ0.015
Ͳ0.01
Ͳ0.005
00.00
5
0.01
0.01
5
0
0.2
0.4
0.6
0.81
1.2
1.4
1.6
1.8
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
2012
Figu
re 7
. Ow
n-pr
ice
elas
ticiti
es fo
r oth
er d
ebt, Ș O
O
Gen
eralize
d�Ba
rnett
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t
99
Figu
re2.
8:O
wn-
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eel
astic
ities
fore
quity
,hE
E
Ͳ0.3
Ͳ0.25
Ͳ0.2
Ͳ0.15
Ͳ0.1
Ͳ0.05
00.05
0.1
Ͳ0.6
Ͳ0.5
Ͳ0.4
Ͳ0.3
Ͳ0.2
Ͳ0.10
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
2012
Figu
re 8
. Ow
n-pr
ice
elas
ticiti
es fo
r equ
ity, Ș
EE
Gen
eralize
d�Ba
rnett
NQ
User�cost�o
f�equ
ity
100
Figu
re2.
9:Ef
fect
sof
a25
Bas
isPo
ints
Incr
ease
inth
eFe
dera
lFun
dsR
ate
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
1992
19
94
1996
19
98
2000
20
02
2004
20
06
2008
20
10
2012
Figu
re 9
. Effe
cts o
f a 2
5 B
asis
Poi
nts I
ncre
ase
in th
e Fe
dera
l Fun
ds R
ate
Oth
er d
ebt
Dep
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Lo
ans a
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ases
Eq
uity
D
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ecur
ities
101
Figu
re2.
10:E
ffec
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a25
Bas
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ints
Incr
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inth
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-0.2
-0.1
5
-0.1
-0.0
5 0
0.05
0.1
1992
19
94
1996
19
98
2000
20
02
2004
20
06
2008
20
10
2012
Figu
re 1
0. E
ffect
s of a
25
Bas
is P
oint
s Inc
reas
e in
the
Ret
urn
to
Inve
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ases
Eq
uity
D
ebt s
ecur
ities
102
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106
Chapter 3
Stochastic Volatility Demand Systems
Joint paper with Apostolos Serletis
3.1 Introduction
The measurement of consumer preferences and the estimation of demand systems has been one of
the most interesting and rapidly expanding areas of recent research. Following Diewert’s (1971)
influential paper, a large part of the empirical demand literature has taken the approach of using
a flexible functional form for the underlying aggregator function. Flexible functional forms, like
the locally flexible generalized Leontief of Diewert (1971), translog of Christensen et al. (1975),
almost ideal demand system (AIDS) of Deaton and Muellbauer (1980), and Minflex Laurent model
of Barnett (1983), and the globally flexible Fourier functional form of Gallant (1981) and ‘asymp-
totically ideal model’ (AIM) of Barnett and Jonas (1983) and Barnett and Yue (1988), have rev-
olutionized microeconometrics, by providing access to all neoclassical microeconomic theory in
econometric applications. See Barnett and Serletis (2008) for an up-to-date survey of the state-of-
the art in consumer demand analysis.
In recent years, economists and finance theorists have also been creating new models in which
stochastic variables are assumed to have a time-dependent variance (and are called ‘heteroscedas-
tic’, as opposed to ‘homoscedastic’). In fact, recent leading-edge research in financial econo-
metrics has applied the autoregressive conditional heteroscedasticity (ARCH) model, developed
by Engle (1982), to estimate time-varying variances in commodity prices. Other models include
Bollerslev’s (1986) generalized ARCH (GARCH) model and Nelson’s (1991) exponential GARCH
(EGARCH) model. The univariate volatility models have also been generalized to the multivari-
ate case. The multivariate models are similar to the univariate ones, except that they also specify
107
equations of how the conditional covariances and correlations move over time. See Bauwens et
al. (2006) and Silvennoinen and Terasvirta (2011) for surveys of this literature.
In this paper we merge the empirical demand systems literature with the recent financial econo-
metrics literature. Our primary interest lies in the estimation of stochastic volatility demand
systems. In particular, we relax the homoscedasticity assumption and instead assume that the
covariance matrix of the errors of demand systems is time-varying. Since most economic and
financial time series are nonlinear and time-varying, we expect to achieve superior modeling using
parametric nonlinear demand systems in which the unconditional variance is constant but the con-
ditional variance, like the conditional mean, is also a random variable depending on current and
past information.
The paper is organized as follows. Section 2 briefly reviews the neoclassical theory of con-
sumer choice whereas Section 3 provides a discussion of stochastic volatility demand systems.
Section 4 considers a model based on the Baba, Engle, Kraft, and Kroner (BEKK) representation
[see Engle and Kroner (1995)] for the conditional covariance matrix of the basic translog demand
system. Section 5 provides an empirical application of this model using monthly data on mone-
tary asset quantities and their user costs recently produced by Barnett et al. (2012) and maintained
within the Center of Financial Stability (CFS) program Advances in Monetary and Financial Mea-
surement (AMFM). The final section concludes.
3.2 Neoclassical Demand Theory
Consider n consumption goods that can be selected by a consuming household. The household’s
problem is
maxx
u(x) subject to p0x = y
where x is the n⇥1 vector of goods; p is the corresponding vector of prices; and y is the household’s
total expenditure on goods (often just called ‘nominal income’ in this literature). The solution of
108
the first-order conditions for utility maximization are the Marshallian ordinary demand functions,
x = x(p,y).
Demand systems are often expressed in budget share form, s = (s1, ...,sn)0, where si = pixi(p,y)/y
is the expenditure share of good i.
The maximum level of utility at given prices and income, h(p,y) = u [x(p,y)] , is the indirect
utility function. The direct utility function and the indirect utility function are equivalent represen-
tations of the underlying preference preordering. Using h(p,y), we can derive the demand system
by straightforward differentiation, without having to solve a system of simultaneous equations, as
would be the case with the direct utility function first order conditions. In particular, Diewert’s
(1974, p. 126) modified version of Roy’s identity,
si(v) =v j—h(v)v0—h(v)
, i = 1, ...,n (3.1)
can be used to derive the budget share equations, where v = [v1, ...,vn ] is a vector of expenditure
normalized prices, with the jth element being v j = p j/y, and —h(v) = ∂h(v)/∂v. See Barnett and
Serletis (2008) for more details.
Suppose that the indirect utility function h and, therefore, the share equation system, is defined
up to the set of parameters q . In order to estimate share equation systems using observations
of prices and shares (vt ,st)Tt=1, a stochastic version is specified. Also, since only exogenous
variables appear on the right-hand side of such systems, it seems reasonable to assume that at time
t the observed share in the ith equation deviates from the true share by an additive disturbance term
eit . Thus, the share equation system at time t is written in matrix form as
st = s(vt ,q)+ e t (3.2)
where e t = (e1t , ...,ent)0 and q is the parameter vector to be estimated. It has also been typically
assumed that
e t ⇠ N (0,W) (3.3)
109
where 0 is n-dimensional null vector and W is the n⇥n symmetric positive definite error covariance
matrix.
Finally, since the demand system (3.1) satisfies the adding-up property, i.e., the budget shares
sum to 1, the error covariance matrix W is singular. Barten (1969) shows that maximum likelihood
estimates can be obtained by arbitrarily dropping any equation in the system. McLaren (1990)
also establishes invariance by virtue of observational equivalence of the subsystems with different
deleted equations.
3.3 Stochastic Volatility Demand Systems
In this paper, we relax the homoscedasticity assumption in (3.3) and instead assume that
e t ⇠ N (0,W t) (3.4)
where W t is the time-varying covariance matrix of the errors.
As before, the error terms of the demand system sum to zero, (W t)Tt=1 are singular and we can
drop one equation to avoid singularity. The practical question is whether it matters which equation
is deleted. The following theorem establishes the result.
Theorem 1 Let the rank of the covariance matrices W be n� 1. Then any two subsystems ob-
tained from (3.2) by deleting different conditional mean equations, under assumption (3.4), are
observationally equivalent, in the sense that they have the same likelihood for any possible sam-
ple.
Proof. Since the rank of W t is n� 1, the covariance matrix for any subsystem of n� 1 equations
is not singular. Moreover, the error term of the deleted equation can be recovered from the error
terms of the other n� 1 equations using a linear injection (one-to-one mapping). Therefore, the
vector of n�1 error terms, with the nth error term being deleted, denoted by ut , can be transformed
to any other vector of n�1 error terms by eliminating the ith error term (i = 1, ...,n�1), denoted
110
by et , as follows
et = T ut (3.5)
where T is a unity (n�1)⇥ (n�1) transformation matrix with the ith row replaced by a vector of
�1. The transformation matrix T has the property that T T = I and, therefore, T = T�1. Hence,
the time-varying covariance matrices F t and Ht of the error vectors ut and et , respectively, are
linked by the following
F t = T�1Ht�
T 0��1= T HtT 0. (3.6)
Moreover, the Jacobian of the transformation is
J(T ) =�1. (3.7)
Suppose we have a random sample of normalized prices and shares (vt ,st)Tt=1. The Gaussian
log likelihood function of the demand system with the ith (i = 1, ...,n) equation deleted is1
L⇣
J�
�
�
(et)Tt=1
⌘
=�(n�1)T2
ln(2p)� 12
T
Ât=1
�
ln |Ht |+ e0tH�1t et
�
(3.8)
where J is the set of the parameters q of the mean equation (3.2) and covariance matrices (Ht)Tt=1
and the error vectors (e t)Tt=1 are determined by the subsystem (3.2) without the ith equation.
Using (3.5), (3.6), (3.7), and the fact that�
�
�
T�1Ht (T 0)�1�
�
�
= |Ht |, we can write (3.8) as
L⇣
J�
�
�
(et)Tt=1
⌘
=�(n�1)T2
ln(2p)� 12
T
Ât=1
ln�
�
�
T�1Ht�
T 0��1�
�
�
+u0t⇣
T�1Ht�
T 0��1⌘�1
ut
�
=�(n�1)T2
ln(2p)� 12
T
Ât=1
h
ln |F t |+u0t (F t)�1 ut
i
.
The last expression is exactly the log likelihood function based on observations of the error vector
u; and this proves the theorem. Q.E.D.
Theorem 1 is the counterpart of the invariance proposition for the homoscedastic case when
we impose no restrictions on the covariance matrices of the errors. The theorem posits that it1In this context, observational equivalence does not depend on the form of the probability distribution function.
However, transformation (3.5) does not preserve the form of the likelihood function in the general case as it does withthe Gaussian likelihood function.
111
does not matter which equation is eliminated from the demand system (3.2) in order to avoid
singularity. Moreover, under assumption (3.4), the covariance matrices of the different subsystems
are related by (3.6). However, if we introduce a particular representation for the covariance matrix,
generally, the observational equivalence does not hold, although it does hold for some particular
cases. In the next section we consider such a case, with the conditional mean equation given by the
basic translog demand system and the conditional variance equation being a GARCH(1,1) BEKK
specification.
3.4 A Specific Case
Consider the basic translog (BTL) reciprocal indirect utility function of Christensen et al. (1975)
lnh(v) = a0 +n
Âk=1
ak lnvk +12
n
Âk=1
n
Âj=1
g jk lnvk lnv j (3.9)
where G = [gi j] is an n⇥n symmetric matrix of parameters and a = (a0, ...,an) is a vector of other
parameters, for a total of�
n2 +3n+2�
/2 parameters. The BTL share equations, derived using
the logarithmic form of Roy’s identity, are
si =
ai +n
Âk=1
gik logvk
n
Âk=1
ak +n
Âk=1
n
Âj=1
g jk logvk
, i = 1, . . . ,n. (3.10)
Estimation of (3.10) requires some parameter normalization, as the share equations are homoge-
neous of degree zero in the a’s. Usually the normalization Âni=1ak = 1 is used.
As before, consider a stochastic version of the demand system (in matrix form)
st = s(vt ,q)+ e t (3.11)
where e t = (e1t , ...,ent)0 is an additive disturbance term and q is the parameter vector to be esti-
mated. Assume that the n-dimensional error vector is normally distributed with zero mean and
time-varying covariance matrix
e t |It�1 ⇠ N (0,W t) (3.12)
112
where W t is measurable with respect to information set It�1. To avoid singularity, delete (any) one
equation from (3.11) and consider the corresponding (n�1)⇥ (n�1) covariance matrix F t of the
error vector.
We assume the Baba, Engle, Kraft, and Kroner (BEKK) GARCH(p,q) representation for the
(n�1)⇥ (n�1) covariance matrix of the error vector with generality parameter K [see Engle and
Kroner (1995)]
F t =C0C+K
Âi=1
p
Âi=1
B0ikF t�iBik +
K
Âk=1
q
Âi=1
A0ikut�iu0t�iAik. (3.13)
The BEKK model has the attractive property of having the conditional covariance matrix, Ht ,
positive definite by construction. This model has [n(n+3)�2]/2 free parameters in the condi-
tional mean equations (3.10) and (n�1)n/2+K2 pqn2 free parameters in the conditional variance
equations (3.13), for a total of�
K2 pq+1�
n2 + n� 1 free parameters. The following theorem
claims the invariance of the maximum likelihood (ML) estimator with respect to the deleted equa-
tion for this model.
Theorem 2 Let the covariance matrices (W t)Tt=0 have rank n�1 and the initial covariance matrix
W 0 = L. Then any two subsystems of (3.11) consisting of n�1 equations with the corresponding
conditional variance equation (3.13) are observationally equivalent. Also, the ML estimates of the
parameters of one subsystem can be recovered from the ML estimates of the parameters of another
subsystem.
Proof. Without loss of generality, we provide the proof for the case of a GARCH(1,1) BEKK
with K = 1 representation for the conditional variance equation. Consider the subsystem of (3.11)
with the last equation deleted, i.e. the error vector is ut = (e1t ,e2t , ...,en�1,t). The Gaussian log
likelihood function based on the sample (vt ,st)Tt=1 can be written as
L⇣
Q|(ut)Tt=1
⌘
=�T (N �1)2
log(2p)� 12
T
Ât=1
�
log |F t |+u0tF�1t ut
�
(3.14)
where
F t =C0C+B0F t�1B+A0ut�1u0t�1A (3.15)
113
and Q = (a,vech(G ),vech(C),vec(B),vec(A)) is the vector of all parameters to be estimated and
(ut)Tt=1 is determined by the subsystem (3.11) without the nth equation.
As before, consider the subsystem of (3.11) without the ith equation and corresponding error
vector et = (e1t , ...,ei�1,t ,ei+1,t , ...,ent). It has been shown that the error vectors ut and et are
linked by the non-singular linear transformation (3.5). The Gaussian log likelihood function for
this subsystem is
L⇣
eQ�
�
�
(et)Tt=1
⌘
=�T (N �1)2
log(2p)� 12
T
Ât=1
⇣
log |Ht |+ e0t (Ht)�1 et
⌘
(3.16)
where
Ht = C0C+�
B�0Ht�1B+
�
A�0 et�i (et�i)
0 A (3.17)
and eQ = (ea,vech(eQ),vech(C),vec(B),vec(A)) is the vector of all the parameters.
Using (3.5), (3.7), and the fact that�
�
�
T�1Ht (T 0)�1�
�
�
= |Ht | we can write log likelihood function
(3.16)-(3.17) as
L⇣
eQ�
�
�
(et)Tt=1
⌘
=�T (N �1)2
log(2p)� 12
T
Ât=1
✓
log�
�
�
T�1Ht�
T 0��1�
�
�
+u0t⇣
T�1Ht�
T 0��1⌘�1
ut
◆
with
T�1Ht�
T 0��1= T�1C0C
�
T 0��1+T�1B0T
⇣
T�1Ht�1�
T 0��1⌘
T 0B�
T 0��1
+T�1A0Tet�ie0t�iT0A�
T 0��1.
Now introducing the following denotations
F t = T�1Ht�
T 0��1 (3.18)
C =Cholesky⇣
T�1C0C�
T 0��1⌘
(3.19)
B = T 0B�
T 0��1 (3.20)
A = T 0A�
T 0��1 (3.21)
where Cholesky(·) stands for the Cholesky decomposition, we get exactly the same expression for
the log likelihood as in (3.14)-(3.15).
114
Clearly, the ML estimators of Q and eQ are related as follows. The parameters of the conditional
mean equations are the same: a = ea and G = eG whereas the parameters of the conditional variance
equations are related through (3.19)-(3.21). Note that condition (3.18) is satisfied for the initial
covariance matrices which are predetermined by L. In addition, since both C and C are triangular
matrices in the BEKK representation, in order to preserve the triangular form, equation (3.19)
involves transformation using Cholesky decomposition.
Hence, the likelihood functions of different subsystems of (3.11) consisting of n�1 equations
with corresponding conditional variance equation (3.13) are the same up to the parameters trans-
formation which relates the maximum likelihood estimators for these subsystems. Q.E.D.
Theorem 2 states that the result of the maximum likelihood estimation of the system does not
depend on the choice of the n� 1 equations to be estimated from the n equations of the demand
system in the following sense. The estimators of the conditional mean equations are the same for
any set of n�1 equations; the estimators of the conditional variance equations for any set of n�1
equations can be obtained from the estimators of any other set of n�1 equations using the linear
transformation (3.19)-(3.21).
3.5 Empirical Application
Consider the model defined in the previous section with n = 3. Since (3.10) is a singular system
we delete one equation (say the third equation) and consider the following two conditional mean
equations
s1 =a1 + g11 logv1 + g12 logv2 + g13 logv3
n
Âk=1
ak +n
Âk=1
n
Âj=1
g jk logvk
+ e1; (3.22)
s2 =a2 + g12 logv1 + g22 logv2 + g23 logv3
n
Âk=1
ak +n
Âk=1
n
Âj=1
g jk logvk
+ e2. (3.23)
115
We assume a BEKK GARCH(1,1) with K = 1 representation for the covariance matrix of e1
and e2 in (3.22) and (3.23). In particular, the 2⇥2 covariance matrix of the errors can be written as
Ht =C0C+B0Ht�1B+A0et�1e0t�1A. (3.24)
Thus, the BTL demand system with a BEKK specification for the covariance matrix Ht , consists
of the conditional mean equations (3.22) and (3.23) and the following conditional variance and
covariance equations
h11,t = c211 +b2
11h11,t�1 +2b11b21h12,t�1 +b221h22,t�1
+a211e2
1,t�1 +2a11a21e1,t�1e2,t�1 +a221e2
2,t�1; (3.25)
h12,t = c11c12 +b11b12h11,t�1 +(b11b22 +b12b21)h12,t�1 +b21b22h22,t�1
+a11a12e21,t�1 +(a11a22 +a12a21)e1,t�1e2,t�1 +a21a22e2
2,t�1; (3.26)
h22,t = c212 + c2
22 +b212h11,t�1 +2b12b22h12,t�1 +b2
22h22,t�1
+a212e2
1,t�1 +2a12a22e1,t�1e2,t�1 +a222e2
2,t�1. (3.27)
This model has a total of 19 free parameters to be estimated.
Applied demand analysis uses two types of data, time series data and cross sectional data.
Time series data offer substantial variation in relative prices and less variation in income whereas
cross sectional data offer limited variation in relative prices and substantial variation in income
levels. In this application, we use the monthly time series data on monetary asset quantities and
their user costs recently produced by Barnett et al. (2012) and maintained within the Center of
Financial Stability (CFS) program Advances in Monetary and Financial Measurement (AMFM).
The sample period is from 1967:2 to 2011:12 (a total of 539 observations). For a detailed discussion
of the data and the methodology for the calculation of user costs, see Barnett et al. (2012) and
http://www.centerforfinancialstability.org.
116
In particular, we model the demand for three monetary assets: demand deposits (x1), small
time deposits at commercial banks (x2), and large time deposits (x3). As we require real per capita
asset quantities for the empirical work, we divided each quantity series by the CPI (all items) and
total population. The estimation is performed in Estima RATS. We first estimate equations (3.22)
and (3.23) under the homoscedasticity assumption in (3.3) and report the results in Table 1. To
verify the presence of ARCH effects in the residuals of (3.22) and (3.23), estimated under the
homoscedasticity assumption (3.3), we plot the estimated squared residuals e21 and e2
2 in Figures 1
and 2, respectively. Moreover, Lagrange multiplier tests for ARCH in each of e1 and e2 indicate
significant evidence of ARCH effects; the null hypotheses of no ARCH (of different orders) in
each of e1 and e2 are rejected with p-values less than 0.000001.
Next, we estimate the model under the heteroscedasticity assumption in (3.4), assuming the
BEKK specification (3.24) for the error covariance matrix, Ht . That is, we estimate the condi-
tional mean equations, (3.22) and (3.23), and the conditional variance equations, (3.25)-(3.27).
The estimation results are reported in Table 2. We also estimated the model using the subsystems
with the second and first equations deleted (see Table 3 and 4, respectively). Consistent with The-
orem 2 the parameters of the mean equations (a1,a2,a3,b11,b12,b13,b21,b22,b23,b31,b32,b33)
are exactly the same in all three estimations (see panels A in Tables 2-4). The parameters of the
variance equations are related by equations (3.19)-(3.21).
It is worth mentioning that the log likelihood function of the described model exhibits manifold
local maxima. To ensure reliability of the result it is necessary to implement the optimization with
different initial parameters (e.g. randomly assigned from the predetermined area) and/or different
eliminated equations. Furthermore, in most cases it is helpful to perform derivative-free search
(such as Simplex algorithm) as a preliminary step for the derivative-based optimization.
117
3.6 Conclusion
Uncertainty is a very important concept in economics and finance, if not the most important. Mo-
tivated by the fact that the current demand systems literature ignores the role of uncertainty, in this
paper we introduce recent advances in financial econometrics to model the covariance matrix of
the errors of flexible demand systems, thereby improving the flexibility of these systems to capture
certain important features of the data. We prove an important practical result of invariance of the
ML estimator with respect to the deleted equation for the BTL demand system with conditional
variance in the BEKK form. We also provide an empirical application based on the use of this
model.
Although we study the BEKK specification of the error covariance matrix of the basic translog,
our approach could be applied to any other known demand system (including those mentioned in
the Introduction). Moreover, other variance specifications could be used such as, for example,
the VECH model of Bollerslev et al. (1988), the constant conditional correlation (CCC) model of
Bollerslev (1990), as well as the dynamic conditional correlation (DCC) models of Engle (2002)
and Tse and Tsui (2002).
118
Figu
re3.
1:Sq
uare
dre
sidu
als
ofeq
uatio
n(2
2),e
2 1
19
70
19
75
19
80
19
85
19
90
19
95
20
00
20
05
20
10
0.0
0
0.0
1
0.0
2
0.0
3
0.0
4
0.0
5
0.0
6
0.0
7
0.0
8
0.0
9
Fig
ure
1:Squ
ared
resi
dual
sof
equat
ion
(22)
,e2 1
13
119
Figu
re3.
2:Sq
uare
dre
sidu
als
ofeq
uatio
n(2
3),e
2 2
19
70
19
75
19
80
19
85
19
90
19
95
20
00
20
05
20
10
0.0
00
0
0.0
02
5
0.0
05
0
0.0
07
5
0.0
10
0
0.0
12
5
0.0
15
0
Fig
ure
2:Squ
ared
resi
dual
sof
equat
ion
(23)
,e2 2
14
120
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