a rational expectations approach to macroeconometrics

This PDF is a selection from an out-of-print volume from the NationalBureau of Economic Research

Volume Title: A Rational Expectations Approach to Macroeconomics:Testing Policy Ineffectiveness and Efficient-Markets Models

Volume Author/Editor: Frederic S. Mishkin

Volume Publisher: University of Chicago Press

Volume ISBN: 0-226-53186-4

Volume URL: http://www.nber.org/books/mish83-1

Publication Date: 1983

Chapter Title: Monetary Policy and Interest Rates: An Efficient Markets-Rational Expectations Approach

Chapter Author: Frederic S. Mishkin

Chapter URL: http://www.nber.org/chapters/c10247

Chapter pages in book: (p. 76 - 109)

5 Monetary Policy andInterest Rates: AnEfficient Markets-RationalExpectations Approach

5.1 Introduction

The impact of a money stock increase on nominal interest rates is animportant issue. The most commonly held view-also a feature of moststructural macro models-has an increase in the money stock leading, atleast in the short and medium runs, to a decline in interest rates. In thesemacro models (see Brainard and Cooper 1976; Modigliani 1974), theinterest rate decline not only stimulates investment directly but also has afurther expansionary impact on investment and consumer expenditurethrough its effect on the valuation of capital. The decline in interest ratesis thus a critical element in the transmission mechanism of monetarypolicy. In addition, the view that increases in the money stock lead to animmediate decline in interest rates has important implications for theFederal Reserve System's conduct of monetary'policy when a decline ininterest rates is desired. This view is the basis for demands by governmentofficials that the Fed not keep the rate of money growth too low and soinduce an objectionable increase in interest rates.

Milton Friedman (1968, 1969) has criticized this view on the groundsthat it ignores the dynamic effects of a money stock increase. Friedmanconcedes that a so-called liquidity effect-where an excess supply ofmoney will create increased demand for securities, a rise in their price,and a resulting fall in interest rates-does work in the direction of adecline in interest rates when the money stock is increased. However,two other effects can counter this liquidity effect. The money stockincrease will, over time, have an expansionary effect on both real incomeand the price level. This "income and price level effect" will, through theusual arguments in the money demand function, tend to reverse thedecline in interest rates. More important for short-run effects on interestrates, increases in the money stock can also influence anticipations of

76

77 Monetary Policy and Interest Rates

inflation. Higher expected inflation as a result of money stock increaseswould, through a Fisher (1930) relation, increase nominal interest rates.This "price anticipations effect" can thus not only mitigate the decline ininterest rates stemming from the liquidity effect but could also overpowerit. That interest rates are highest in countries experiencing rapid"rates ofmonetary growth is casual evidence for this proposition.

Early work on the issue of money supply increases and interest rates,such as Cagan (1972), Gibson (1970), and Gibson and Kaufman (1968),tended to stress the "income and price level e~ffect" more than the "priceanticipations effect" because these researchers believed that adjustmentsof inflationary expectations proceeded slowly. Recent work on the theoryof rational expectations and market efficiency, starting with Muth (1961),indicates that inflationary expectations can adjust quite rapidly. Thus, the"price anticipations effect" should, and does~1 receive more weight in thischapter when the effect of money supply increases on interest rates isdiscussed.

Two lines of empirical work bear on the issue whether increases in themoney stock lead to a decline or to a rise in interest rates. "Keynesian"macroeconometric models impose a fair amount of structure in theirestimates of financial market and income-expenditure relationships. Inthese models, increases in the money stock do lead to a substantialdecline in interest rates in the short and medium run, as, for example, inModigliani (1974) and the simulation results in this chapter. This evi-dence is suspect, however, because these models ignore constraints thatshould be imposed if financial markets are efficient. Financial marketefficiency cannot be ignored because evidence supporting it is quitestrong (see Fama 1970). Furthermore, recent work (Mishkin 1978) indi-cates that a failure to impose financial markl~t efficiency on macroecon-ometric models can yield highly misleading results.

An alternative empirical approach to this issue is to estimate reducedform relationships where changes in interest rates are regressed on pastchanges in the money stock. Evidence from this approach (Gibson andKaufman 1968) does not support the view that increases in the moneystock result in a fall in interest rates. Unfortunately, this evidence suffersfrom a problem endemic to reduced form empirical work: it is difficult tointerpret the empirical results because tht:~ theoretical framework isobscure. Also, the absence of structure when changes in interest rates areregressed on changes in the money stock leads to a large number ofparameters being estimated, and this results in statistical tests with lowpower.

Neither approach discussed above distinguishes between the effectsfrom unanticipated versus anticipated monetary policy. Yet the theory ofefficient capital markets and rational expectations does make this distinc-tion, and this suggests an alternative approach to analyzing the rela-

78 Empirical Studies

tionship of money stock increases and interest rate movements. Thischapter develops efficient-markets (or, equivalently, rational expecta-tions) models for both long- and short-term interest rates and estimatesthem using postwar quarterly data. This approach has the advantage ofimposing a theoretical structure on the problem that allows both easierinterpretation of the empirical results and more powerful statistical testsof the proposition that increases in the money stock are correlated withdeclines in interest rates. Moreover, a Keynesian, liquidity preferenceview of interest rate determination can be embedded in the efficient-markets model and tested. Finally, as a side issue, attractive tests of bondmarket efficiency result from the approach used here.

5.2 The Model

The theory of efficient markets (or, equivalently, rational expecta-tions) implies that interest rates in a bond market should reflect allavailable information. More precisely, it implies that the market usesavailable information correctly in assessing the probability distribution ofall future interest rates and bond returns. Hence for long bond returns, Yt,and short-term interest rates, 't,

(1)

and

(2)

where

Yt == the one-period (from t - 1 to t) nominal return fromholding long-term bonds-it includes capital gainsplus interest payments,

't == the one-period (short-term) interest rate at time t,<Pt-1 == information available at time t - 1,

E (. . ., <Pt - 1) == the expectation conditional on <Pt - 1,Em( . .. '<Pt-1) == the expectation assessed by the market at t - 1.

In order to give this concept empirical content we must specify modelsof market equilibrium. For the case of long-term bonds, we assume, as inthe previous chapter, that the market equates expected one-period hold-ing returns across securities, allowing for risk (liquidity) premiums whichare constant over time. This model of market equilibrium implies that

(3) Em(Ytl<Pt-1) == 't-1 + dl,

where dl == a constant liquidity premium for long-term bonds.A more refined model of market equilibrium allowing the risk pre-

mium to vary over time is not used for long-term bonds because, asdiscussed in the previous chapter, it makes little difference to the empiri-


cal tests. Combining the model of market equilibrium above with marketefficiency yields the same condition as in C:hapter 4:

(4) E(Yt - 't-1 - dl l<f>t_1) == 0

and the same efficient-markets model

(5)

where an e superscript denotes expected values on all past information[i.e., a rational forecast is defined as X~ == E(Xtl<f>t-1)], and

X t == a variable (or vector of variables) relevant to the pricing of bonds,~l == a coefficient (or vector of coefficients),E~ == serially uncorrelated error process [because E( E~I<Pt-1) == 0].

In the analysis of short-term interest rates, we can no longer argue thatthe model of the market equilibrium has no effect on tests of the efficient-markets model. In this case, the model of the risk premium used heredoes contribute significantly to the explanation of the dependent vari-able. We assume, as in Fama (1976b), that the one-period-ahead forwardrate equals the one-period-ahead expected short rate, plus a risk pre-mium that now varies over time with the uncertainty in short-rate move-ments, that is,

(6)

and

(7)

where

t-iF; == forward rate for the one-period rate at time t implied by theyield curve at t - 1 ,

dJ == risk (liquidity) premium for t-1~"

ITt == measure of uncertainty in short rate movements.

Combining this model of market equilibrium with the rationality ormarket efficiency condition of (2) yields

(8)

and the corresponding efficient-markets model

(9)

where the s superscript is used to differentiate the ~ and E from theircounterparts in the long-term bond model.

The research question posed in the first section of this chapter suggeststhat money growth is an interesting piece of information relevant to thepricing of bonds and interest rates. Substituting for X t leads to thefollowing efficient-markets models:


(10) Yt= 't-1 + dl + ~~(MGt- MG~) + ef,(11) 't= t-1F;- ao- a10"t+ ~Sm(MGt- MG~) + e:,where MGt = the money growth rate at time t.

As is found in the foreign exchange market (see Mussa 1979), spot andforward rates move together, so that changes in short-term interest ratesare predominantly unanticipated. Because the long rate is closely linkedto the price of a long bond, over periods as short as a quarter thecorrelation of changes in long rates with unanticipated bond returns, Yt 't-1 - d, is very negative: -.96 in the sample period used in the followingempirical work. Changes in long interest rates will thus also be predomi-nantly unanticipated. We can see how the efficient-markets models abovediffer from earlier analysis: they stress that only unanticipated move-ments in money growth can have an effect on unanticipated movementsin interest rates. Since changes in interest rates are predominantly unan-ticipated, these efficient-markets models emphasize the effects of unan-ticipated money growth movements on changes in interest rates. Incontrast, the earlier work does not distinguish between the effects ofanticipated and unanticipated money growth.

If unanticipated increases in money growth are associated with adecline in long rates (as might be expected from "Keynesian" macro-econometric models), the coefficient on unanticipated money growthshould be significantly positive in the long bond equation above becauseYt - 't-1 and the change in long rates are negatively correlated: that is,~~>O. If unanticipated increases in money growth are associated with adecline in short rates, then the coefficient on unanticipated money growthin the short-rate equation should be significantly negative, that is, ~~<O.

An important caveat is in order. As noted in Chapter 2, the efficient-markets model does not guarantee that X t - X~ is exogenous so that theestimates of ~ are consIstent. Another way to make this point is toacknowledge that the efficient-markets model does not indicate whethera significant ~ coefficient implies causation from its unanticipated vari-able to bond prices and interest rates. As far as market efficiency isconcerned, causation could just as well run in the other direction, or itcould be nonexistent, as in the case where new information simulta-neously affects both interest rates and the right-hand-side variable. Thus,we must be careful in interpreting empirical results on the ~'s, not toascribe causation to the results without further identifying information.

The same caveat applies especially when we analyze the estimated ~m

coefficient. If the money supply process is seen as exogenous, the inter-pretation of the estimated ~m's is straightforward. The finding of asignificant positive ~~ and negative ~~ will then provide evidence sup-porting the "Keynesian" position that increased money growth leads, atleast in the short run, to declines in interest rates; and a failure to find this


result will cast doubt on this view. If the money supply process is notexogenous, however-the position taken by many critics of monetaristanalysis-then the estimated ~m coefficients may suffer from simul-taneous equation bias and give a misleading impression about how in-creases in the money supply affect interest rates. Because this chapterprovides no evidence on the exogeneity of the money supply process, the~m estimates must be viewed as providing information only on the cor-relations of unanticipated money growth and the movements in interestrates. Interpretation of these correlations is deferred to the concludingremarks at the end of the chapter.

The liquidity preference approach to the demand for money (seeGoldfeld 1973; and Laidler 1977) indicates that interest rates are relatednot only to money growth but also to movenlents in real income, the pricelevel, and inflation. Adding this information to the X vector of theefficient-markets models, noting that unanticipated inflation is equiva-lent to the unanticipated price level, leads to the following:

(12) Yt= 't-1 + dl + ~~(MGt-1VlG~) + ~~(YGt- YG~)

+ ~~(11't - 11'~) + e:(13)

where

't=t-1F;- ao- a1(Tt+ ~Sm(lMGt- MG~)

+ ~~(YGt- YG~) + ~~rr(11't- 11'~) + e:

YGt = growth rate of real income,11' = inflation rate,

~m, ~Y' ~7l" = coefficients.These equations are really efficient-markets analogs to the typical moneydemand relationship found in the literature. In addition, they captureelements of interest rate models of the Feldstein and Eckstein (1970)variety.

The magnitude and sign of the ~ coefficients in equations (10)-(13)depend on the time-series processes of the money supply, real income,and price level, even when the sign and magnitude of these coefficientsare assumed to reflect an underlying structural theory such as liquiditypreference. If the time-series processes of real income and the price levelare such that an unanticipated rise in these variables is not followed bymore than a compensating decline in these variables, then a liquiditypreference view implies that the coefficients of unanticipated incomegrowth and inflation should be negative in the long bond equation (12)-that is, ~~ < 0, ~~ < O-and positive in the short-rate equation (13)-thatis, J3~ > 0, J3~ > O. In this case, an unanticipated increase in incomegrowth should lead to higher interest rates:. currently and in the future.The negative effect of an unanticipated increase in inflation on bond


returns follows from the resulting reduction in real money balances,which also leads to rising interest rates. The unanticipated inflation effectshould be further strengthened if, as in the Cagan (1956) adaptive ex-pectations model, expected inflation rises when actual inflation is aboveits expected value. In this case, an unanticipated rise in inflation pro-motes a rise in nominal interest rates either through a Fisher (1930)relation or because expected inflation is a separate argument in themoney demand function, as in Friedman (1956).

Note also that the more persistent the time-series process of inflationand income growth-that is, the more an unanticipated increase in thesevariables leads to further increases-the more powerful the unantici-pated income and inflation effects on interest rates indicated by thetheoretical structure discussed above. Clearly, the importance of the"income and price level" and "price anticipations" effects also depend onthe time-series process of money growth. Thus the ~m coefficients alsowill not be invariant to changes in the money growth, time-series process.

We now turn to the actual estimation of the efficient-markets models ofequations (10)-(13), with the warning that some caution must be exer-cised when interpreting results from estimates of these equations becausethe direction of causation cannot be established in this framework.

5.3 Empirical Results

5.3.1 The Data

Postwar quarterly data is used in the empirical work below. The longbond models are estimated over the sample period 1954-1976. However,six-month Treasury Bills were not issued before 1959, and since thesix-month bill rate is needed to calculate the forward rate in the short ratemodels, these models are estimated over the 1959-1976 period. The datasources and definitions of variables used in these estimations are asfollows:

Yt = quarterly return from holding a long-term government bondfrom the beginning to the end of the quarter,

rt = the ninety-day Treasury Bill rate, the last trading day of thequarter in fractions at an annual rate in the short rate equa-tions, but rt -1 is at a quarterly rate in the long bond equations,

4[1 - (360 - 180 rSiXt -1)] ,(360 - 90 'i-1)

wherersixt = the six-month (180 days) bill rate at the end of quarter-in

fractions at an annual rate,


M1 Gt = growth rate of M1 (quarterly rate) = the first differencedseries of the log of the average level of M1 in the last month ofthe quarter,

M2G = growth rate of M2 (quarterly rate) = the first differencedseries of the log of the average level of M2 in the last month ofthe quarter,

lPGt = growth rate of industrial production (quarterly rate) = thefirst differenced series of the log of industrial production in thelast month of the quarter,

'iTt = the CPI inflation rate (quarterly rate) = the first differencedseries of the log of CPI in the last month of the quarter.

Unless otherwise noted, all these variables have been constructed fromseasonally adjusted data except for rt, t-IF;, and Yt, which do not requireseasonal adjustments. The bond return series was obtained from theCenter for Research in Security Prices (C:RSP) at the University ofChicago and is described in Fisher and Lorie (1977) and Mishkin (1978).The Treasury Bill data was supplied by the Federal Reserve Board. ThelPG, and 'iT, variables were constructed froln data in the Department ofCommerce's Business Statistics and Survey of Current Business. The M1and M2 data were obtained from the Board of Governors of the FederalReserve, Banking and Monetary Statistics, and the Federal Reserve Bulletin. All other variables used to specify the forecasting equations wereobtained from the NBER data bank.

As shown in Chapter 4, using averaged data in efficient-markets orrational expectations models can give misleading results. The data forbond returns and interest rates here are thus derived from security pricesat particular points in time. For the same reason, the derivation of theother variables here uses data as close to being end of quarter as possible.Industrial Production is thus made a proxy for real income in estimatingthe models rather than the more broadly based national income accountsmeasure. Similarly, the CPI has been used to calculate the inflationvariable rather than the GNP deflator. Endpoint data (or close to end-point) help unearth significant relationships between bond returns andunanticipated variables. Some experiments \vith quarterly averaged dataled to worse fits for efficient-markets models, fewer significant coef-ficients, and no appreciable differences as to the statistical significance ofthe ~m coefficients.

5.3.2 The Estimation ProcedureTo estimate the short and long rate models of equations (10)-(13),

measures of anticipated money growth, industrial production growth andinflation are needed. Here, these anticipations are assumed to be rationalforecasts obtained from linear forecasting equations. The model esti-mates are produced by estimating each short or long rate equation jointly


with the forecasting equations, and imposing the cross-equation restric-tions implied by rationality of expectations. See Chapter 2 for details ofthis procedure.

In Chapter 2 we saw that economic theory may not be a useful tool fordeciding on the specification of the forecasting equations. Thus atheore-tical statistical procedures are used here. If indeed economic theory is notparticularly useful in evaluating the forecasting equations, it is all themore important to check for the robustness of results to changes in thespecification of these equations. Therefore, two procedures for specify-ing the forecasting equations are used in the text, and results with severaladditional specifications are discussed in Appendix 5.2.

The simplest forecasting equations are univariate time-series models ofthe autoregressive type. Fourth-order autoregressions are usually suc-cessful in reducing residuals in quarterly data to white noise and are thusused here. Ordinary least-squares estimates for the MIG, M2G, lPG,and 11" equations for both sample periods used in estimation can be foundin Appendix 5.1. There is a fair amount of persistence in the time-seriesmodels for money growth and inflation, indicating that "income a~d pricelevel" and "price anticipation" effects of the sort that Friedman (1968,1969) discusses are potentially important. Although less persistence isevident in the time-series process of industrial production growth, it hasthe characteristic that a positive innovation does lead to a permanentlyhigher level of industrial production (although not in the rate of growth).Thus, as discussed in the preceding section, the unanticipated inflationand lPG coefficients may be expected to be negative in the long-ratebond equations and positive in the short-rate equations.

The univariate time-series models suffer from the problem of unstablecoefficients. Chow (1960) tests reported in Appendix 5.1, where thesample period has been split into equal lengths, reject in five out of eightequations the hypothesis that the coefficients of the univariate models areequal in the two subperiods. Multivariate forecasting equations thus havebeen specified by the procedure outlined in Chapter 2 which makes use ofGranger's (1969) concept of predictive content. Each of the four vari-ables-MIG, M2G, lPG, and 11"-was regressed on its own four laggedvalues as well as on four lagged values of each of the other three variablesand four lagged values of each of the following variables: the unemploy-ment rate; the ninety-day Treasury Bill rate; the balance of payments oncurrent account; the growth rate of real federal government expenditure,the high employment budget surplus; and the growth rate of federalgovernment, interest bearing debt, in the hands of the public. (Theseother variables were selected because a reading of the literature onFederal Reserve reaction functions-see Fair 1978 and the referencestherein-indicated that they might help explain money growth.) The fourlagged values of each variable were retained in the equation only if theywere jointly significant at the 5 percent level or higher.


The resulting multivariate time-series models for both sample periodscan be found in Appendix 5.1 along with F statistics of the joint signifi-cance test for whether the four lagged values of each variable should beincluded in the regression model. Not only do these models have a betterfit than the corresponding univariate models, but Chow tests reported inAppendix 5.1 now reject stability of the coefficients in only one out ofeight cases.

Before we turn to the empirical results, the measure of short-rateuncertainty, O"t, used here requires some discussion. Fama (1976b) calcu-lates O"t as the average of the absolute value of the changes in the spot rateduring the year before t and during the year following t. Because the risk(liquidity) premium must be set conditional on available information-inthis case that known at t-1-allowing O"t to be calculated from informa-tion not available at t-1 could be problematic. An alternative, thoughsimilar, measure of O"t is used in this study. l'he difference between theforward rate and the spot rate, that is, 't - t-lF;, was regressed on mea-sures of O"t, calculated as the average absolute change of the bill rate over anumber of previous quarters, where the number of quarters was varied.The best fit was obtained with O"t calculated from eight previous quartersof changes in the bill rate. The results are as follows:

(14) 1(- t-lF;= - .0001 - 1.0961 O"t+ Et

(.0017) (.2937)

R2 = .1659 standard error == .0068Durbin-Watson = 1.90

where8~ I't-i- 't-i-l 1

i= 1

8

As in Fama (1976b), increased uncertainty in short-rate movements doeslead to an increased risk premium, and this effect is statistically significantat the 1 percent level. In addition, the O"t measure used here outperformsthe Fama measure that is constructed from information unavailable att-1. The above O"t measure is used in the empirical tests that follow. Itsspecification is not a critical issue to the outcomes however: if we use aFama measure of O"t or exclude O"t from the model altogether, the resultsdo not alter appreciably.

5.3.3 The ResultsThere is no strong theoretical reason to estimate the long bond or

short-rate model with one monetary aggregat1e versus another. Unantici-pated growth rates of both Ml and M2 are therefore used and results withadditional monetary aggregates are explored in Appendix 5.2. The re-sulting estimates of the models appear in tables 5.1 and 5.2. Panel A of


Table 5.1 Nonlinear Estimates of the Long Bond ModelUsing Seasonally Adjusted Data

Coefficients of

Model M1 G - M1 Ge M2G-M2Ge /PG-/PGeConstantTerm

Panel A: Using Univariate Forecasting Models

1.1 .0482 -.0014(.5961) (.0032)

1.2 .0501 - .4242** -1.8482* -.0029(.5517) (.1260) (.8028) (.0032)

1.3 .9174 -.0013(.5459) (.0032)

1.4 .7174 -.4077** -1.7691 * -.0027(.5063) (.1243) (.7880) (.0031)

Panel B: Using Multivariate Forecasting Models

1.5 - .2621 -.0014(.7429) (.0032)

1.6 .4108 - .5039** -1.7529 -.0020(.7164) (.1568) (.9552) (.0032)

1.7 .9199 -.0014(.6738) (.0032)

1.8 1.0950 -.4987** -1.8206 -.0020(.6283) (.1492) (.9353) (.0032)

Note: Asymptotic standard errors of coefficients in parentheses.* = Significant at the 5 percent level.** = Significant at the 1 percent level.

these tables contains estimates which make use of the univariate forecast-ing equations, while panel B's estimates use the multivariate forecastingmodels of the form found in Appendix 5.1. The estimates of the ~

coefficients are not presented here because they are not particularlyinteresting.

An issue basic to these results is whether the efficient-markets (rationalexpectations) model used here is valid. Previous evidence on the effi-ciency of the bond market indicates that efficient-markets models of thetype used here are a reasonable characterization of bond market behav-ior. Table 5.3 contains likelihood ratio tests, described in Chapter 2, ofthe nonlinear constraints implied by both market efficiency (rationalexpectations) and the model of market equilibrium. The test statistics donot reject the constraints for any of the long bond models in table 5.1: themarginal significance level of the statistics are never lower than .05.These results then also provide additional evidence for the effi-cient-markets model of long bond behavior.


Table 5.2 Nonlinear Estimates of the Short Rate ModelUsing Seasonally Adjusted Data

Coefficients of

ConstantModel M1G-M1Ge M2G-M2Ge /PG-/PGe

TI -1Te Term IT


2.1 .2788* .0006 -1.2266**(.1088) (.0015) (.2714)

2.2 .2774** .0352 .6211** .0002 -1.1514**( .1075) (.0275) (.1989) (.0014) (.2618)

2.3 .1616 .0006 -1.2563**(.1085) (.0015) (.2851)

2.4 .1904 .0399 .6545** .0002 -1.1571 **(.1053) (.0278) (.2058) (.0015) (.2686)


2.5 .1677 .0006 -1.2761 **(.1283) (.0015) (.2863)

2.6 .2512 - .0455 .5199 .0004 -1.2109**(.1381) (.0493) (.3272) (.0016) (.3015)

2.7 .2562 .0001 -1.1807**(.1341) (.0016) (.2917)

2.8 .3039* -.0770 .6501 * -.0004 -1.0779**(.1409) (.0471) (.3314) (.0016) (.3069)

Note: See table 5.1.

The table 5.3 results for the short-rate nl0dels, however, reject thenonlinear constraints at the 5 percent level in six out of eight cases. Howshould we interpret these rejections? They can result from either thefailure of rationality (market efficiency) or of the model of marketequilibrium. Both models of market equilibrium used in the long bondand short-rate models are crude: neither risk premium is derived fromutility maximizing behavior. A theoretically more justifiable risk pre-mium would, for example, exploit the covariance of bill or bond returnswith returns on alternative assets. Yet, as the regression results in equa-tion (14) indicate, the model of market equilibrium is a significant ele-ment in explaining the movements of the dependent variable in theshort-rate equation. In this situation~ unlike that for the long-rate equa-tion where the model of market equilibrium appears to be unimportant inexplaining the dependent variable, its misspecification can lead to rejec-tions of the nonlinear constraints. Thus, rejections of the nonlinearconstraints occurring in the short-rate models, but not in the long bondmodels, can be attributed plausibly to misspecification in the model ofmarket equilibrium.


Table 5.3 Likelihood Ratio Tests of Nonlinear Constraints

Model

1.11.21.31.41.51.61.71.82.12.22.32.42.52.62.72.8

Likelihood Ratio Statistic

X2(4) = 6.45X2(12) = 14.00X2(4) = 3.20X2(12) = 12.43X2(12) = 18.10X2(24) = 33.81X2(8) = 10.67X2(24) = 32.60X2(4) = 12.76*X2(12) = 13.65X2( 4) = 12.46*X2(12) = 17.18X2(8) = 21.65* *X2(28) = 50.02**X2(8) = 25.69**X2(28) = 50.92**

Marginal Significance Level

.1680

.3007

.5249

.4118

.1127

.0881

.2211

.1128

.0125

.3235

.0143

.1430

.0056

.0067

.0012

.0051

Note: Marginal significance level is the probability of getting that value of the likelihoodratio statistic or higher under the null hypothesis.*Significant at the 5 percent level.**Significant at the 1 percent level.

A suitable transformation of the unconstrained system outlined inChapter 2 yields additional evidence on the potential misspecification ofthe model of market equilibrium. The unconstrained system where the 'Yare not equal in the forecasting and short-rate equations can be rewrittenas

(15)

't =t-lF;- ao- cral- Zt-l(l + (Xt - Zt_l'Y)~s+ Et ,

where the 'Y's are constrained to be equal in the two equations. There-fore, the nonlinear constraints tested in this paper are equivalent to (l = 0in the above system. It is now easy to see the following point: if the riskpremium is related to the variables in Z, yet they have been excludedfrom the model of the risk premium, then this may explain the rejectionsof the nonlinear constraints. To make this conjecture plausible, weshould expect that a model of the risk premium which is related to Zwould have reasonable characteristics. The Fama-type model, for exam-ple, does generate plausible values. The resulting risk premiums (atannual rates) have a mean of 57 basis points (1/100 of a percentage point)and a standard deviation of 30 basis points. They also move smoothly:their autocorrelations for lags of one through four are respectively .96,.91, .85, and .78. In the model which leads to the strongest rejection of


the nonlinear constraints, model 2.7, we could attribute this rejection tothe fact that a more appropriate specification of the risk premium is d: =

a o + al(Tt + Zt-l<X, where Zt-l contains the four lagged values of moneygrowth (M2G) and Treasury Bill rates (r). This latter specification leadsto values for the risk premium that are some'what more variable and lesssmooth than the equation (14) specification, but not appreciably so. Therisk premium from this expanded specification has a mean of 57 basispoints, a standard deviation of 46 basis points, and four lagged autocor-relations of .75, .56, .49, and .29.

Viewing the rejections with the benefit of the system (15) also has theadvantage of providing us with potentially interesting information on therisk premium. The results indicate that the premium could be related tomoney growth and interest rates as well as the variability measure (T.

However, they give no indication that the liquidity premium is in additionrelated to the other variables in table 5.A.2 of Appendix 5.1. The resultshere thus point out a direction for future research on the risk premium.Following Nelson (1972), I also conducted rnore direct experiments onthe relation of the risk premium to lagged interest rates and unemploy-ment with negative results. Experiments with lagged values of r - F alsodid not add explanatory power to the model of the risk premium.

If a misspecified model of the risk prernium is the source of therejections of the nonlinear constraints, the efficient markets-rationalexpectations model used here is fortunately still a valid framework foranalyzing the relationship of money growth and short rates. With rationalexpectations, the unanticipated X t - X~ variable will be uncorrelatedwith any past information, among which can be included the determi-nants of the risk premium which is set at {-I. Therefore, if somedeterminants of this risk premium have been excluded from the marketequilibrium model, they will be orthogonal to X t - X~. The exclusion ofthese variables, and the resulting rejection of the nonlinear constraints,will not lead to inconsistent estimates of the ~3 coefficients. Since it is notnecessary to derive a better model of the risk premium to achieve reliableestimates of the ~'s, this tricky research issue, which is beyond the scopeof this study, is left to future research.

The unanticipated MlG coefficients in table 5.1 do not lend support tothe view that an unanticipated increase in nnoney growth is correlatedwith a fall in long bond rates. In panel j\, although both of thesecoefficients have a positive sign, they are not significantly different fromzero at the 5 percent level: asymptotic t statistics are less than .1. Inaddition these coefficients are quite small. The ~ coefficients here denotethe percentage point change in the bond return from a 1 percent error inanticipations, and in our 1954-1976 sample period, a one percentagepoint rise in the quarterly bond return corresponds approximately to a 10


basis point (1/100 of a percentage point) fall in the long bond rate. Thus,the MIG coefficients in panel A indicate that a 1 percent surprise increasein M1 is associated with only a .5 basis point decline in the long bond rate.

The panel B estimates of the MIG coefficients indicate that the conclu-sion on the relationship of long rates and money growth is not altered byusing multivariate versus univariate forecasting models in estimation.Again, neither of the unanticipated MIG coefficients are significantlydifferent from zero at 5 percent, and they continue to be small, with thelargest of the coefficients indicating that a 1 percent surprise increase inM1 leads to only a 4.1 basis point decline in the long bond rate. Fur-thermore, one of the unanticipated MIG coefficients is now negative.

The coefficients on unanticipated M2 growth in table 5.1 are morepositive than the unanticipated MIG coefficients, they nevertheless donot lend strong support to the view that unanticipated money growthshould be negatively correlated with the change in long rates. They do notdiffer significantly from zero at the 5 percent level (although in 1.3 theunanticipated M2G coefficient is significantly different from zero at the10 percent level). Also note that the M2 results in panel A and in panel Bare so similar that it is again clear that the results on unanticipated moneygrowth are not particularly sensitive to specifications of the moneygrowth forecasting model.

How different are these findings from those that might be inferred from"Keynesian," structural macroeconometric models? Using a simulationtechnique discussed in Mishkin (1979) we can examine the response of amacromodel to a 1 percent surprise increase in MI. Equation 1.1 wasused to trace out the effect on M1 growth from a 1 percent innovation.The resulting M1 numbers were then used in a simulation experimentwith the MPS (MIT-Penn-SSRC 1977) Quarterly Econometric Model inorder to derive the response of this model to the 1 percent M1 innovationoccurring in the 1967: 1 quarter. The MPS model indicates that this 1percent Ml innovation would lead to an immediate decline of 18.1 basispoints in the long rate. Not only is this long-rate decline several timeslarger than the maximum 4.1 basis point decline implied by the empiricalevidence in table 5.1, but also it is significantly larger at the 5 percent levelfor three of the four estimates in table 5.1 (and is almost significantlylarger for the remaining estimate). Clearly, the coefficients on unantici-pated M1 growth are quite low relative to what might be expected from astructural macroeconometric model.

The unanticipated inflation and industrial production coefficients intable 5.1 conform to our priors. In both the M1 and M2 efficient-marketmodels, these coefficients are negative and are either significant or nearlysignificant at the 5 percent level. The results on the coefficients of unan-ticipated industrial production growth are especially strong, with both thepanel A and panel B estimates significantly different from zero at the 1


percent level. Although the unanticipated inflation coefficients are verysimilar in both panels, their asymptotic standard errors rise somewhatfrom panel A to panel B. They are thus not quite significant at the 5percent level in panel B, while they are significant at this level in panel A.

The similarity between the money growth as well as other coefficientestimates in panel A and panel B is encouraging, for it gives us confidencethat these results are robust to changes in the models describing expecta-tions. Further model estimates described in P\.ppendix 5.2 with additionalspecifications for the forecasting equations yield similar results. This is animportant finding. Poor specification of expectations formation appearsto be a major concern in this line of research because it leads to errors-in-variables bias in the coefficient estimates. The important question is,How severe would this bias be? Denoting the measured X t - X~ by an msuperscript and the true X t - X~ with a T superscript, we can write

(16) (Xt - x~)m= (Xt- X~)T+ Vt,

where Vt is the measurement error. If such variables as money growth,industrial production growth, and inflation are hard to forecast-whichseems likely-then the variance of the true .A~t - X~ forecast error will besubstantial. If the incremental predictive power of other informationbesides the past history of the forecasted variable is not large, then thevariance of the measurement error in expectations used here will be smallin relation to the variance of the true forecast error: that is, Var [eXt X~)T]»Var(Vt). If this occurs, the errors--in-variables bias would benegligible and should not be an important problem in this research.

The similarity of the model estimates despite substantial changes in thespecifications for the forecasting equations is found not only in thischapter, but also in the chapters preceding and following. This providesstrong support for the view that unanticipated increases in money growthare associated with interest rate declines. Moreover, the smaller standarderrors found for the coefficients estimated using univariate rather thanmultivariate forecasting equations provides some support for the positiontaken by Feige and Pearce (1976), that forecasts from univariate time-series models may be "economically rational" expectations. 1

The results for the short-rate model in Table 5.2 are even more damag-ing to the view that associates a decline in interest rates with an unantici-pated money growth increase. 2 All the coefficients on both unanticipatedMl and M2 growth are positive in table 5.5, and in three cases thecoefficients are statistically significant. They indicate that a 1 percent

1. Note that Sims (1977) has raised some questions about the statistical techniques usedby Feige and Pearce (1976), and this does cast some doubt on their evidence.

2. Urich and Wachtel (1981) obtain similar results using weekly data. Thus, reduction ofthe unit of observation in the analysis is likely to leave the findings here intact.


Table 5.4 Nonlinear Estimates of the Long Bond ModelUsing Seasonally Unadjusted Data

Coefficients of

ConstantModel M1G-M1Ge M2G-M2Ge IPG-IPGe

1T - 1Te Term


4.1 - .7339* -.0017(.3631) (.0031)

4.2 - .5879 -.2028* - 2.5145** -.0026(.3631) (.0857) (.6912) (.0032)

4.3 .0001 -.0014(.3610) (.0032)

4.4 .1426 -.2420** - 2.4438** -.0024(.3330) (.0838) (.7111) (.0032)


4.5 -1.2781 ** -.0014(.4504) (.0032)

4.6 - .8078 -.4105** -2.4472** -.0020(.4339) (.1371) (.8089) (.0032)

4.7 - .1404 -.0014(.4821) (.0032)

4.8 .1534 - .4741 ** -2.6226** -.0020(.4391) (.1396) (.8237) (.0033)


surprise increase in M1 or M2 is associated with a 16--30 basis pointunanticipated increase in the bill rate. The simulation experiment withthe MPS model that is described above indicates that a 1 percent M1surprise leads to an immediate decline of 88 basis points in the bill rate.This finding contrasts strongly with the finding here that even the leastpositive M1 coefficient is more than eight of its standard errors away fromthis figure.

The results on the unanticipated inflation coefficients are similar tothose in the long bond model. These coefficients are positive, as might beexpected, and are significantly different from zero in three out of fourcases. The results on the unanticipated industrial production growthcoefficients are not quite as strong as in the earlier table. They are neverstatistically significant, and in panel B they even have the wrong sign.

The efficient markets-rational expectations model does not specifywhether the X - xe variables should be described by seasonally adjustedor seasonally unadjusted data. This empirical issue cannot be settledeasily on theoretical grounds because it is not clear whether marketparticipants concentrate on seasonally adjusted versus unadjusted in-


Table 5.5 Nonlinear Estimates of the Short Rate ModelUsing Seasonally Unadjusted Data

Coefficients of

ConstantModel M1G-M1Ge M2G-M2Ge /PG-/PGe 'Tt' - 'Tt'e Term IT


5.1 .3029** .0003 -1.1255**(.0652) (.0014) (.2530)

5.2 .2458** .0274 .4687** .0001 -1.1267**(.0671) (.0171) (.1716) (.0014) (.2464)

5.3 .1926* .0003 -1.1468**(.0644) (.0015) (.2765)

5.4 .1967** .0440* .5459** .0001 -1.1260**(.0624) (.0176) (.1746) (.0014) (.2526)


5.5 .3431 ** .0007 -1.2403**(.0831) (.0015) (.2639)

5.6 .2484** .0386 .5079 .0004 -1.2037**(.0956) (.0376) (.2623) (.0016) (.2861)

5.7 .3285** -.0007 - .9891 **(.0918) (.0015) (.2841)

5.8 .2011* .0400 .5788* - .0003 -1.0791 **(.0986) (.0374) (.2674) (.0016) ( .3021)


formation. For this reason, the table 5.1 and table 5.2 models have alsobeen estimated with seasonally unadjusted data for the X's. The resultingestimates and test statistics appear in tables 5.4, 5.5, and 5.6 and wereobtained by the same procedures as the previous estimates with season-ally adjusted data.

A comparison of tables 5.4-5.6 with tables 5.1-5.3 indicates that thechoice of adjusted or unadjusted data is not a critical factor in thisresearch. The likelihood ratio tests of the nonlinear constraints yieldsimilar conclusions. In addition the coefficient estimates are similar,although their standard errors tend to be sITlaller in the seasonally unad-justed results. In the short-rate models, all the industrial productiongrowth coefficients now have the "correct" positive sign.

The seasonally unadjusted data are even less favorable to the view thatincreased money growth is associated with a decline in interest rates.Now all the M1 coefficients in the long bond :model are negative, implyinga positive correlation of movements in money growth and long interestrates, and two of these coefficients are significantly different from zero atthe 5 percent level. In addition, the M2 coefficients in the long-rate model


Table 5.6 Likelihood Ratio Tests of Nonlinear Constraints

Model

4.14.24.34.44.54.64.74.85.15.25.35.45.55.65.75.8


Likelihood Ratio Statistic

X2(4) = 5.08X2(12) = 20.83X2(4) = 3.27X2(12) = 19.53X2(12) = 12.10X2(24) = 28.48X2(8) = 7.23X2(24) = 27.11X2(4) = 9.14X2(12) = 14.81X2(4) = 12.02*X2(12) = 15.01X2(8) = 19.30*X2(28) = 49.71**X2(8) = 24.90**X2(28) = 49.96**

Marginal Significance Level

.2792

.0529

.5137

.0765

.4377

.2403

.5120

.2994

.0578

.2521

.0172

.2407

.0133

.0070

.0016

.0065

are less positive. For the short-rate models, all the money growth coef-ficients in table 5.4 are positive and are now statistically significant at the 5percent level, with six out of eight significant at the 1 percent level. Theseasonally unadjusted data, then, lend support to the contrary view thatunanticipated movements in money growth and interest rates are posi-tively correlated.

5.4 Concluding Remarks

A wide range of empirical tests of the relationship between moneygrowth and interest rates have been conducted in this chapter and inAppendix 5.2. A guiding principle of this research has been to use manydifferent empirical tests of the model in order to provide information onthe robustness of the results. In pursuit of this goal, model estimationshave been varied along the following dimensions: (1) the choice of themonetary aggregate, (2) the choice of the relevant variables to include inthe X vector, (3) the use of seasonally adjusted versus seasonally unad-justed data, (4) the specification of the forecasting models used to de-scribe expectations formation, and (5) the sample period. The largenumber of estimates provide information on the sensitivity and reliabilityof the results reported here.

The results point to the following conclusions. There is no empiricalsupport for the view that an unanticipated increase in the money stock isassociated with a decline in interest rates. However, there are two aspects


of the research methodology used here which raise questions about thegeneral validity of this conclusion.

As we have seen, the ~ coefficients in the efficient markets-rationalexpectations models are not invariant to changes in the time-series pro-cesses of the money growth, income growth, and inflation variables. Thusthe conclusions from the estimates in this chapter provide information onthe relationship between money growth and interest rates only for thepostwar sample period. However, realize that many structural macro-econometric models displaying a negative relationship between moneygrowth and interest rates have been estinlated using sample periodswhich overlap those used here. The results reported in this chapter arecertainly of interest in evaluating these models.

A further difficulty with the present research methodology is thatmisspecification of the forecasting equations describing expectationsformation can invalidate the results on the relationship between moneygrowth and interest rates. Specifically, misspecification of expectationsformation can lead to inconsistent and biased ~ coefficients. However,the robustness of results to different specifications of the time-seriesmodels describing expectations provides evidence that the misspecifica-tion problem is probably not very severe.

How should we interpret the conclusion reached above? If we arewilling to accept exogeneity of the money supply process in the postwarperiod and the efficient-markets models as true reduced forms, theinterpretation is clear-cut. The evidence here would then cast doubt onthe commonly held view that an unanticipated increase in the moneystock will lead to a decline in interest rates. N·ot only does this suggest thatthe Federal Reserve cannot lower interest rates by increasing the rate ofmoney growth, but it also requires some rnlodification of the monetarytransmission mechanism embodied in structural macroeconometric mod-els. It is plausible that an unanticipated increase in money growth will notinduce a decline in interest rates because it leads to an immediate upwardrevision in expected inflation. Thus, there is still a potential effect on realinterest rates from unanticipated money growth, and the evidence in noway denies that there are potent effects of money supply increases onaggregate demand.

As was mentioned in Section 5.2, if unanticipated money growth is notexogeneous, then the ~m coefficient estimates are inconsistent and canlead to misleading inference. Particularly disturbing in this regard is thecase where the Federal Reserve smooths interest rates so that an unantic-ipated increase is interest rates causes the Ft~deral Reserve to react by anunanticipated increase in money growth. 'The resulting correlation ofMGt - MG~ with the Et error terms (negativt~ with E: and positive with E:1.tends to bias the results toward a positive association of money growthand interest rates. Thus, we cannot rule out the view in structural mac-


roeconometric models that an exogenous increase in money growth leadsto a decline in interest rates, despite the empirical results of this chapter.

Note, however, the nature of money growth endogeneity required forthis reservation to hold. If money growth is endogenous in the sense thatthe Federal Reserve modifies money growth within a quarter only inresponse to past public information available at the start of the quarter,MGt - MG~ will not be correlated with e: or e:. Hence the existence ofGranger (1969) "causality" running from interest rates to money growthdoes not imply that the estimates of ~m will be inconsistent. "Causality"tests of the Sims (1972) variety, therefore, cannot shed light on theconsistency of the ~m estimates. If we are not to reject the common viewthat increases in money growth lead to interest rate declines, research of afairly subtle sort is needed to demonstrate that unanticipated moneygrowth is negatively correlated with e: and positively correlated with e:.Hence, this issue cannot be resolved without further research.

App

endi

x5.

1:E

stim

ates

ofth

eF

orec

asti

ngE

quat

ions

Tab

leS.

A.1

Uni

vari

ate

Tim

e-Se

ries

Mod

els

Mod

el:

A1.

1A

1.2

A1.

3A

I.4

AI.

5A

I.6

A1.

7A

1.8

Dep

ende

ntV

aria

ble:

MIG

MIG

M2

GM

2G

IPG

IPG

7T7T

Sam

ple

Peri

od:

1954

---19

7619

59-1

976

1954

---19

7619

59-1

976

1954

---19

7619

59-1

976

1954

---19

7619

59-1

976

-- C

onst

antt

erm

.003

8.0

053

.004

9.0

076

.009

3.0

092

.001

2.0

014

(.00

13)

(.00

19)

(.001

7)(.0

024)

(.028

3)(.0

035)

(.006

6)(.0

008)

M1

G(-1

).3

777

.315

8(.1

078)

(.12

34)

M1

G(-

2)

.220

5.2

166

(.115

2)(.

1289

)M

1G

( -3

).0

550

.030

6(.1

151)

(.127

5)M

1G

(-4

)-.

0341

-.03

71(.

1076

)(.1

236)

M2

G(-

1)

.659

8.6

113

(.108

0)(.

1220

)M

2G

(-2

)-.

0542

-.04

12(.

1280

)(.1

421)

M2

G(-3

).1

736

.161

9(.1

286)

(.14

08)

M2

G(-

4)

-.07

58-

.152

5(.

1094

)(.

1230

)

Tab

leS.

A.1

(con

tinue

d)

Mod

el:

AL

IA

1.2

A1.

3A

1.4

A1.

5A

1.6

A1.

7A

1.8

Dep

ende

ntV

aria

ble:

MIG

MIG

M2

GM

2G

IPG

IPG

1T1T

Sam

ple

Peri

od:

1954

-197

619

59-1

976

1954

-197

619

59-1

976

1954

-197

619

59-1

976

1954

-197

619

59-1

976

IPG

(-1

).4

254

.351

4(.

1003

)(.1

187)

IPG

(-2

)-.

2346

-.21

00(.

1091

)(.

1250

)IP

G(-3

).1

507

.144

9(.1

091)

(.123

8)IP

G(-4

)-.

2516

-.20

25(.

1003

)(.1

131)

1T(-

1)

.400

8.3

991

(.104

4)(.1

209)

1T(-

2).5

520

.616

2(.1

112)

(.126

8)1T

(-3)

.106

3-.

0179

(.111

9)(.

1275

)1T

(-4)

-.1

837

-.1

613

(.10

33)

(.111

2)

R2

.295

2.2

023

.456

3.3

766

.214

8.1

496

.730

5.7

555

SE.0

062

.006

6.0

066

.007

0.0

237

.023

9.0

039

.003

8D

-W1.

971.

961.

961.

942.

001.

982.

032.

01

Not

e:St

anda

rder

roro

fthe

coef

ficie

ntsi

npa

rent

hese

s.D

efin

ition

sofv

aria

bles

:M1G

=qu

arte

rly

rate

ofgr

owth

ofM

1,M

2G

=qu

arte

rly

rate

ofgr

owth

ofM

2,IP

G=

quar

terl

yra

teof

grow

thof

indu

stri

alpr

oduc

tion,

1T=

quar

terl

yra

teof

grow

thof

CPI

,U

N=

unem

ploy

men

trat

e,r

=ni

nety

-day

bill

rate

,B

OP

=ba

lanc

eof

paym

ents

oncu

rren

tacc

ount

,G

=qu

arte

rly

rate

ofgr

owth

ofre

alfe

dera

lgov

ernm

ente

xpen

ditu

res,

GD

EB

T=

quar

terl

yra

teof

grow

thof

gove

rnm

entd

ebt,

SUR

P=

high

empl

oym

ents

urpl

us.

Tabl

e5.

A.2

Mul

tivar

iate

Tim

e-Se

ries

Mod

els

Mod

el:

A2.

1A

2.2

A2.

3A

2.4

A2.

5A

2.6

A2.

7A

2.8

Dep

ende

ntV

aria

ble:

MIG

MIG

M2

GM

2G

IPG

IPG

'IT'IT

Sam

ple

Perio

d:19

54-1

976

1959

-197

619

54-1

976

1959

-197

619

54-1

976

1959

-197

619

54-1

976

1959

-197

6

Con

stan

tter

m-.

0004

.001

5.0

026

.004

4-.

0148

.001

7.0

053

.003

3(.

0017

)(.0

021)

(.00

16)

(.002

7)(.0

125)

(.011

1)(.0

019)

(.002

1)M

1G

(-1

).0

906

-.1

031

.690

4.1

835

(.165

5)(.

1875

)(.

3938

)(.0

635)

M1

G(-2

).5

233

.533

6.6

883

.037

6(.

1680

)(.

1906

)(.4

089)

(.064

9)M

1G

(-3

)-

.276

5-

.318

41.

0207

-.0

663

(.178

5)(.

1990

)(.4

200)

(.065

7)M

1G

(-4

)-

.175

7.0

052

.274

5.1

600

(.179

1)(.1

739)

(.433

7)(.0

727)

M2

G(-

1)

.359

0.5

998

.505

0.5

211

.713

2(.

1618

)(.

1666

)(.1

119)

(.127

1)(.3

814)

M2

G(-

2)

-.3

972

.561

2.0

985

.095

5.9

460

(.175

4)(.1

935)

(.12

66)

(.14

26)

(.43

00)

M2

G(-

3)

.351

0.4

112

.151

7.1

179

.721

7(.1

757)

(.207

7)(.1

167)

(.132

5)(.

4122

)M

2G

(-4

).1

396

.024

0-.

0806

-.08

38-

.204

9(.1

645)

(.184

3)(.

1036

)(.

1180

)(.

4182

)

Tab

le5.

A.2

(con

tinue

d)

Mod

el:

A2.

1A

2.2

A2.

3A

2.4

A2.

5A

2.6

A2.

7A

2.8

Dep

ende

ntV

aria

ble:

MIG

MIG

M2

GM

2G

IPG

IPG

1T1T

Sam

ple

Perio

d:19

54-1

976

1959

-197

619

54-1

976

1959

-197

619

54-1

976

1959

-197

619

54-1

976

1959

-197

6

IPG

(-1

)-.

0947

-.27

78(.1

622)

(.112

9)IP

G(-2

)-.

6171

-.3

626

(.172

7)(.

1089

)IP

G(-3

)-.

0902

-.04

57(.1

709)

(.108

9)IP

G(-4

)-.

2807

-.1

869

(.122

4)(.1

010)

1T(-

1)

-.5

539

-2.2

77.2

245

.120

9(.6

501)

(.793

)(.1

079)

(.12

73)

1T(-

2)-.

5186

-1.7

53.5

527

.770

5(.6

229)

(.701

)(.1

053)

(.117

1)1T

(-3)

-.8

774

.398

.283

3.3

911

(.653

3)(.8

46)

(.109

0)(.

1339

)1T

(-4)

-.7

464

1.25

2-.

0636

-.23

28(.6

533)

(.591

)(.1

116)

.(.11

76)

UN

(-1

)-.

0904

-.00

36-.

0052

(.009

6)(.0

009)

(.001

4)U

N(-2

)-.

0018

.001

2-.

0016

(.012

0)(.0

016)

(.002

1)U

N(-3

).0

173

.001

8.0

034

(.011

4)(.0

015)

(.002

1)U

N(-4

)-.

0001

-.00

04-.

0004

(.009

1)(.0

095)

(.001

3)

r(-1

)-.

2647

-.49

66-

.521

4.9

813

(.106

7)(.

1000

)(.

1221

)(.4

501)

r(-

2).2

086

.485

3.4

827

.525

3(.1

305)

(.137

3)(.

1631

)(.4

819)

r(-3

).1

250

.033

2.0

551

.330

7(.1

396)

(.14

72)

(.166

9)(.

4793

)r(

-4)

-.02

77.0

519

.033

3-1

.830

(.112

1)(.1

132)

(.127

9)(.4

37)

BO

P(-1

).0

058

.000

1(.0

035)

(.000

6)B

OP

(-2

).0

050

.000

3(.0

047)

(.000

8)B

OP

(-3

).0

016

-.0

005

(.004

6)(.0

008)

BO

P(-4

)-.

0094

-.00

10(.0

037)

(.000

6)

R2

.542

7.4

024

.623

2.5

472

.515

9.7

108

.794

9.8

789

SE.0

053

.005

9.0

056

.006

1.0

200

.016

0.0

035

.002

9D~W

1.97

2.01

1.96

1.95

2.03

2.00

1.96

2.21

Not

e:Se

eta

ble

5.A

.1.

Tabl

e5.

A.3

FSt

atis

tics

for

Sign

ifica

ntEx

plan

ator

yPo

wer

inFo

reca

stin

gEq

uatio

nsof

Four

Lags

ofEa

chV

aria

ble

Mod

el

Var

iabl

eA

2.1

A2.

2A

2.3

A2.

4A

2.5

A2.

6A

2.7

A2.

8

MIG

3.10

*2.

442.

322.

176.

53**

.09

1.66

4.14

M2

G3.

98**

5.28

**15

.37*

*10

.42*

*.0

511

.09*

*1.

121.

13IP

G.9

81.

261.

701.

653.

71**

4.70

**1.

081.

551T

2.04

1.81

1.19

.40

7.61

**5.

05**

79.3

2**

48.3

3**

UN

.77

1.42

.51

.25

3.00

*2.

326.

51**

5.88

**r

2.85

*2.

469.

19**

5.93

**1.

955.

62**

.69

.94

BO

P1.

41.7

91.

04.7

01.

833.

54*

2.32

2.87

*G

1.54

.47

.10

.22

.89

.46

2.25

1.59

GD

EB

T.6

3.4

3.9

0.4

4.3

61.

84.5

71.

33SU

RP

.57

.62

1.09

1.28

.54

.70

1.57

2.01

Not

e:Th

eF

stat

istic

ste

stth

enu

llhy

poth

esis

that

the

coef

ficie

nts

onth

efo

urla

gsof

each

ofth

ese

vari

able

seq

uals

zero

.The

Fst

atis

tics

are

dist

ribu

ted

asym

ptot

ical

lyas

F(4

,x)

,w

here

xru

nsfr

om47

to83

.The

criti

calv

alue

sof

Fat

the

5pe

rcen

tlev

elar

e2.

5-2.

6an

dat

the

10pe

rcen

tlev

elar

e3.

6-3.

7.*S

igni

fican

tatt

he5

perc

entl

evel

.**

Sign

ifica

ntat

the

1pe

rcen

tlev

el.


Table S.A.4 Chow Tests for the Models of Tables S.A.1 and S.A.2

Model

ALIAl.2A1.3Al.4Al.5A1.6A1.7A1.8A2.1A2.2A2.3A2.4A2.5A2.6A2.8

F Statistic

F(5,82) = 3.50**F(5,62) = 2.73*F(5,82) = 2.49F(5,62) = 1.24F(5,82) = 2.81 *F(5,62) = 3.80**F(5,82) = 1.59F(5,62) = 1.83F(13,66) = 1.81F(9,54) = 2.94**F(9,74) = 1.60F(9,54) = 1.80F(17,58) = 1.39F(21,30) = 1.76F(17,38) = 1.24

MarginalSignificanceLevel

.0064

.0271

.0377

.3014

.0216

.0046

.1721

.1200

.0597

.0065

.1310

.0895

.1753

.0765

.2824

Note: Tests that the coefficients are equal in the two halves of the sample period.*Significant at the 5 percent level.**Significant at the 1 percent level.

Appendix 5.2: Additional Experiments Using the Two-Step Procedure

Because the two-step procedure used by Barro (1977) yields consistentparameter estimates and is far easier to execute than the joint nonlinearprocedure used in the text, it is used in tables 5.A.5 and 5.A.6 to provideadditional estimates of the long bond and short-rate models.

The two-step procedure here does not correct for heteroscedasticitywithin each long bond and short-rate equation even though Goldfeld-Quandt (1965) tests frequently reveal that it exists. This simplifies estima-tion and does not affect the consistency of the parameter estimates.However, this two-step procedure may yield incorrect standard errorsand test statistics. Thus although tables 5.A.5 and 5.A.6 provide usefulinformation, some caution about statistical inference is warranted.

The first four models of panels A and B in both tables reestimate themodels in the text by the two-step procedure. As we might expect, theparameter estimates are similar to those generated by the nonlinearprocedures of the text and yield similar conclusions. This gives us someconfidence that the two-step procedure can be used to gain furtherinformation on the robustness of this chapter's empirical results. Usingthe two-step procedure, long bond and short-rate models were alsoestimated with alternative specifications of the forecasting equations.

Tab

leS.

A.S

Est

imat

esof

the

Lon

gB

ond

Mod

el:

Usi

ngSe

ason

ally

Adj

uste

dD

ata

and

the

Tw

o-St

epP

roce

dure

(Sam

ple

Per

iod

1954

-197

6)

Coe

ffic

ient

sof

Mod

elM

IGt-

M1

Gt

M2G

t-M

2Gre

MB

Gt-

MB

Gre

UR

Gt-

UR

Gre

UB

Gt

-U

BGre

IPG

t-IP

Gre

'7Tt-7T~

Con

stan

tTe

rm

Pane

lA:

Usi

ngR

esid

uals

from

Uni

vari

ate

Fore

cast

ing

Mod

els

A5.

1.3

507

-.00

08(.5

753)

(.003

4)A

5.2

.365

8-.

4447

-2.1

36-

.000

8(.5

460)

(.141

2)(.8

580)

(.003

3)A

5.3

1.11

3-.

0008

(.530

)(.0

034)

A5.

4.8

762

-.4

265

-1.9

87-.

0008

(.500

7)(.1

385)

(.845

5)(.0

033)

A5.

5-.

3502

-.0

008

(.538

8)(.0

034)

A5.

6.1

974

-.44

09-2

.295

-.00

08(.5

195)

(.14

18)

(.879

3)(.0

033)

A5

.7.1

955

-.00

08(.1

936)

(.003

4)A

5.8

.221

4-.

3952

-2.4

87-

.000

8(.1

909)

(.143

7)(.8

788)

(.003

3)A

5.9

.370

3-

.000

8(.4

937)

(.003

4)A

5.tO

.419

6-.

4064

-2.4

10-.

0008

(.487

9)(.1

436)

(.879

6)(.0

033)

A5.

11.2

788

(.714

9)A

5.12

.626

2(.6

958)

A5.

13

A5.

14

A5.

15

A5.

16

A5.

17

A5.

18

A5.

19

A5.

20

Not

e:Se

eta

ble

5.A

.1.

1.10

5(.6

42)

1.19

7(.6

08)

Pane

lB:

Usi

ngR

esid

uals

from

Mul

tivar

iate

Fore

cast

ing

Mod

els

-.00

08(.0

034)

-.5

626

-1.9

97-

.000

8(.1

863)

(.994

)(.0

033)

-.00

08(.0

034)

-.5

547

-1.9

73-.

0008

(.179

5)(.9

77)

(.003

3).3

019

-.0

008

(.656

3)(.0

034)

.130

8-.

5339

-2.0

77-.

0008

(.622

9)(.1

854)

(1.0

08)

(.003

3).1

587

-.00

08(.2

204)

(.003

4).1

558

-.5

063

-2.1

95-

.000

8(.

2164

)(.1

855)

(1.0

16)

(.003

3).3

936

-.00

08f

"'Q

'7"'

\(.0

034)

\•..

JU

/..J

)

.639

2-

.521

9-

2.31

1-.

0008

(.565

7)(.

1818

)(1

.018

)(.0

033)

Tabl

e5.

A.6

Estim

ates

ofth

eSh

ort-

Rat

eM

odel

:U

sing

Seas

onal

lyA

djus

ted

Dat

aan

dth

eTw

o-St

epPr

oced

ure

(Sam

ple

Peri

od19

59-1

976)

Coe

ffic

ient

sof

Con

stan

tM

odel

M1G

t-M

1G,e

M2G

t-M

2G,e

MB

Gt-

MB

G,e

UR

Gt-

UR

G,e

UB

Gt-

UB

G,e

IPG

t-IP

G,e

1Tt-1T~

Ter

m(J

Pane

lA:

Usi

ngR

esid

uals

from

Uni

vari

ate

Fore

cast

ing

Mod

els

A6.

1.2

777

.000

2-1

.164

(.122

3)(.0

017)

(.287

)A

6.2

.293

0.0

538

.653

4.0

001

-1.1

02(.1

131)

(.031

5)(.1

968)

(.001

6)(.2

66)

A6.

3.0

766

.000

0-1

.123

(.119

7)(.0

018)

(.298

)A

6.4

.163

2.0

615

.672

3.0

002

-1.0

79(.1

133)

(.032

4)(.2

081)

(.001

6)(.2

76)

A6.

5.2

291

-.00

03-1

.078

(.128

7)(.0

017)

(.290

)A

6.6

.139

0.0

601

.509

4-.

0008

-1.0

03(.1

252)

(.032

7)(.2

096)

(.001

6)(.2

77)

A6.

7.0

095

-.00

02-1

.093

(.045

0)(.0

017)

(.296

)A

6.8

.011

1.0

646

.552

7-.

0008

-1.0

05(.0

444)

(.034

2)(.2

102)

(.001

6)(.2

80)

A6.

9.0

752

-.0

002

-1.0

97(.1

157)

(.001

6)(.2

95)

A6.

10.0

934

.070

3.5

310

-.0

008

-1.0

03(.1

139)

(.034

1)(.2

091)

(.001

6)(.2

78)

Pane

lB:

Usi

ngR

esid

uals

from

Mul

tivar

iate

Fore

cast

ing

Mod

els

A6.

11.1

990

.000

1-1

.130

(.144

2)(.0

015)

(.293

)A

6.12

.206

0.0

323

.365

0.0

001

-1.1

39(.1

451)

(.060

9)(.

3147

)(.0

018)

(.296

)A

6.13

.089

4.0

001

-1.0

92(.1

397)

(.001

7)(.2

95)

A6.

14.1

143

.034

0.3

781

-.0

001

-1.1

01(.

1429

)(.0

619)

(.327

2)(.0

018)

(.298

)A

6.15

.205

4-.

0003

-1.0

56(.1

617)

(.001

7)(.2

94)

A6.

16.1

616

.011

1.6

450

.000

3-1

.126

(.15

69)

(.042

7)(.2

398)

(.001

7)(.2

89)

A6.

17.0

168

-.00

01-1

.091

(.049

4)(.0

017)

(.296

)A

6.18

-.00

06.0

131

.669

4.0

002

-1.1

58(.0

549)

(.043

5)(.2

443)

(.001

7)(.2

90)

A6.

19.1

493

-.00

04-1

.046

(,135

7)(.0

017)

(.296

)A

6.20

.090

1.0

161

.637

7-.

0000

-1.1

21(.1

325)

(.043

5)(.2

443)

(.001

7)(.2

93)

Not

e:Se

eta

ble

5.A

.1.


Models were estimated with residuals from the eighth-order autoregres-sive forecasting equations, as well as from multivariate models whichincluded the four lagged values of a variable even if it was significant onlyat the 10 percent level (rather than the 5 percent level as in the text). Theresults were quite close to those reported here and the results againappear robust to changes in the specification of the forecasting equation.

Because the Federal Reserve might have changed its reaction functionin the 1970s by paying more attention to monetary aggregates than it didpreviously, it is possible that the results here might substantially change ifthe 1970s are excluded from the sample period. Two-step estimates of thelong bond and short-rate models over the sample period ending in the1969:4 quarter fail to support this conjecture. The unanticipated IPG and11' coefficients remain similar to those in tables 5.A.5 and 5.A.6: for thelong bond model, the IPG coefficients range from - .36 to - .46 and the 11'

coefficients from -1.69 to -2.17; and for the short-rate model, the IPGcoefficients range from - .04 to .03 and the 11' coefficients from .37 to .57.Similar conclusions about the relationship of money growth and interestrates result also from estimates using the shorter-sample periods. For thelong bond model, the unanticipated MIG coefficients are now negative,ranging from - .26 to - .54 and the M2G coefficients range from .24 to.79. For the short-rate model, the money growth coefficients remainpositive, with the MIG coefficients ranging from .11 to .20 and the M2Gcoefficients from .03 to .12.

The most obvious choice for the monetary aggregate that is exoge-nously determined by the Federal Reserve are not M1 and M2. Asbecomes clear from such debates as those between Anderson and Jordan(1969) and De Leeuw and Kalchenbrenner (1969), other aggregates maybe a more sensible control variable for the Fed. If these aggregates aremore likely than M1 or M2 to be exogenous, their use in the models hereshould give a clearer picture of the effect of monetary policy on interestrates. For this reason, tables 5.A.5 and 5.A.6 also contain two-stepestimates of the models using the following additional variables:

MBG = growth rate of the monetary base (quarterly rate),URG = growth rate of unborrowed reserves (quarterly rate),UBG = growth rate of the unborrowed base (quarterly rate).

These variables are constructed analogously to MIG and M2G from thesame data source, and the specifications for the forecasting equationswere obtained with the same procedures used for MIG and M2G.

In some applications the monetary base has been chosen as the Fed'scontrol variable (e.g., see Anderson and Jordan 1968), while in monetarysectors of the large structural macroeconometric models such as the MPS(see Modigliani 1974) unborrowed reserves are often the exogenouscontrol variable. On the other hand, the unborrowed base is the mone-


tary aggregate corresponding most closely \\lith open market operations.All three of the monetary aggregates are thus worthy candidates to beincluded in the long bond and short-rate mlodels.

The results from using alternative monetary aggregates do not alter theconclusions or the relationship of monetary policy and interest rates. Inthe long bond models, the coefficients for the alternative aggregates aresomewhat less positive than those for Ml or M2. They provide even lesssupport for the view that an increase in monetary aggregates is associatedwith a fall in long interest rates. The coefficients in the short-rate modelsare almost always positive, and this is consistent with the results for Mland M2, that a surprise increase in the monetary aggregate is associatedwith a rise in short rates.

a rational expectations approach to macroeconometrics

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