fuhrer and moore 1995

American Economic Association

Monetary Policy Trade-offs and the Correlation between Nominal Interest Rates and RealOutputAuthor(s): Jeffrey C. Fuhrer and George R. MooreSource: The American Economic Review, Vol. 85, No. 1 (Mar., 1995), pp. 219-239Published by: American Economic AssociationStable URL: http://www.jstor.org/stable/2118005Accessed: 15/06/2010 10:47

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Monetary Policy Trade-offs and the Correlation between Nominal Interest Rates and Real Output

By JEFFREY C. FUHRER AND GEORGE R. MOORE*

We present a structural model of the U.S. economy that combines our price-contracting specification with a term-structure relationship, an aggregate demand curve, and a monetary-policy reaction function. The model matches important features of postwar data well and provides a structural explanation of the correlation between real output and the short-term nominal rate of interest. We perform a battery of monetary-policy experiments which show that, as viewed through the lens of this model, monetary policy has struck a good balance recently among competing monetary-policy objectives. (JEL E52, E43)

We develop a simple rational-expect- ations structural model of the U.S. macroeconomy. The model comprises an overlapping-contracts specification of prices, detailed in Fuhrer and Moore (1995); an aggregate demand or "IS" equation that links the output gap to its own lags and the ex ante long-term real interest rate; and a monetary-policy reaction function in which the short-term nominal interest rate responds to deviations of inflation and the output gap from their targets.

We find that our simple model matches the dynamic properties of the primary variables of interest quite well. In addition, the model allows a structural interpretation of the well-documented but puzzling negative correlation between real output and the short-term nominal interest -rate. We argue that this reduced-form correlation between output and the short-term nominal rate arises from the structural interactions among

monetary policy, aggregate supply, and aggregate demand. In particular, the systematic way in which monetary policy has leaned against the wind, in combination with slug- gish inflation adjustment and a structural IS curve that relates output to the rationally expected long-term real rate of interest, has made the sample path of the long-term real rate look like the short-term nominal rate.

Finally, we subject the structural model to a battery of monetary-policy experiments. We explore the trade-offs among competing monetary-policy objectives that arise under policies that differ substantially from recent experience.

The next section explores one explanation for the output-gap/nominal-rate correlation, namely that the ex ante long-term real rate implied by a stationary vector autoregression behaves like a short-term nominal rate in our sample. Section II presents full-information maximum-likelihood estimates of our structural model. The contracting model differs from the models of Edmund S. Phelps (1978) and John B. Taylor (1980) in that agents negotiate nominal contracts with a concern for relative real wages, rather than relative nominal wages. In Fuhrer and Moore (1995), we show that this variation of the Phelps-Taylor model captures the time-series properties of the inflation process better than the original nominal-contracting model. In Section III we use the structural model to demonstrate

*Federal Reserve Bank of Boston, 600 Atlantic Ave., Boston, MA 02106. Our opinions are not necessarily shared by the Federal Reserve Bank of Boston, the Board of Governors of the Federal Reserve System, or other members of their staffs. We thank Tim Cogley, Bennett McCallum, participants at the Stanford/ Federal Reserve Bank of San Francisco conference on Macroeconomic Stabilization Policy, and two anony- mous referees for their comments. We thank Meeta Anand and Alicia Halligan for excellent research assis- tance.

219

220 THE AMERICAN ECONOMIC REVIEW MARCH 1995

that the systematic component of monetary policy since the mid-1960's can account for the reduced-form correlation of real output and the nominal bill rate. We then perform counterfactual monetary-policy experiments with the model, assessing the effect of alternative policies on competing policy objectives. Our conclusions are presented in Sec- tion IV.

I. The Real-Output/Short-Nominal-Rate Correlation

A substantial body of research, typified by Ben S. Bernanke and Alan S. Blinder (1992), has demonstrated a strong correlation between measures of the short-term nominal rate of interest and real output. This correlation is somewhat puzzling, since most textbook theoretical models link long-term real rates with output, and it is generally pre- sumed that there is at best an imperfect link between short nominal rates and long real rates. In this section, we estimate an unconstrained vector autoregression in order to characterize the dynamic interactions among short rates, inflation, and real output. We then compute the ex ante long real rate that is consistent with the VAR. We find that the long real rate and the short nominal rate behave remarkably alike in our sample, and we use the dynamic correlations implied by the VAR to provide a reduced-form explanation of why this has been so.

A. The Data

The variables in our analysis are quarterly series for the inflation rate, the Treasury bill rate, and the deviation of log per capita output from a deterministic trend; series definitions are listed in Table 1.1 Tests for

TABLE 1-QUARTERLY DATA, 1965:1-1992:4

Variable Definition

7rt inflation in the implicit deflator for nonfarm business output

rt 3-month Treasury bill rate Yt log of per capita nonfarm business output it deviation of yt from trend

the order of integration of inflation and the bill rate yield mixed results, but they are broadly consistent with the work of others (see e.g., Charles R. Nelson and Charles I. Plosser, 1982; John Y. Campbell and Robert J. Shiller, 1987; James H. Stock and Mark W. Watson, 1988).2 The log of per capita output appears to be trend-stationary over the sample period. From here on, references to "output" should be understood to mean "the deviation from trend of the log of per capita nonfarm business output."3 See Fuhrer and Moore (1995) for more details of the time-series properties of the data.

B. Reduced-Form Data Descriptions

We explore the dynamic interactions among the bill rate, inflation, and the output gap in the context of an unconstrained vector autoregression. We initially estimate the vector autoregression with six lags of each variable. Then we reduce the lag length until the last lag remains statistically significant and the residuals appear to be uncor- related. The final specification includes four lags each of inflation and the output gap and thre-e lags of the bill rate. Table 2

'We have replicated our calculations using gross domestic product and its implicit deflator in place of nonfarm business output and its deflator, and the results of the two exercises are similar. We have also experimented with a variety of detrending methods, and although the conclusions that we draw here are not sensitive to detrending method, we find that the simple linear trend yields the sharpest results.

2Campbell and Shiller (1987) find some ambiguity in the results for short-term interest rates. Andrew K. Rose (1988) finds evidence of a stationary inflation rate.

3The debate over the deterministic/stochastic trend in output has generated an extensive literature (see e.g., Pierre Perron, 1989; Lawrence J. Christiano and Martin Eichenbaum, 1990; Campbell and Perron, 1991). Ultimately, determination of the nature of the trend is probably not possible (see the "near- observational-equivalence" papers of Stephen R. Blough [1992] and Jon Faust [1993]).

VOL. 85 NO. 1 FUHRER AND MOORE: MONETARY POLICY TRADE-OFFS 221

TABLE 2-ROOTS OF THE VECTOR AUTOREGRESSION

Modulus Period

0.95 69.3 0.89 20.9 0.66 2.8 0.63 3.4 0.54 4.7 0.25 2.0

displays the nonzero roots of the vector autoregression (VAR). The dominant roots of the VAR are a complex pair with a modulus of 0.95. While we cannot reject the hypothesis of unit roots in the data, the unrestricted vector autoregression is actu- ally quite stable, converging to its long-run equilibrium at a rate of about 20 percent per year.

Figure 1 graphs the vector autocorrelation function implied by this vector autoregression.4 The diagonal elements show the univariate autocorrelation functions of the three variables in the system, and the off- diagonal elements show the lagged cross correlations. Inflation and the bill rate are quite persistent, with positive autocorrela- tions out to lags of about four years, while output is somewhat less persistent. Much of the conventional macroeconomic wisdom about the dynamic interaction of inflation, interest rates, and output can be found in the off-diagonal elements of the vector autocorrelation function. In the second and third elements of the first row, for example, a high level of the bill rate is followed by a low level of inflation some 12 quarters later, while a high level of output is followed by a high level of inflation about six quarters later. In the second element of the third row, a high level of the bill rate is strongly correlated with a low level of output about six quarters later. This strong correlation

between output and the short nominal rate is consistent with the evidence presented in Bernanke and Blinder (1992). We consider the vector autocorrelation function in Fig- ure 1 to be (i) verification of the strong short-nominal-rate/real-output correlation, and (ii) a compact, comprehensive graphical representation of the dynamics which our structural model must match.

Given uncertainty about the order of integration of the data, we explore the robustness of the output/nominal-rate correlation to the assumption of nonstationarity. Inter- estingly, the estimated cointegrating vector in a Johansen vector autoregression (see S0ren Johansen and Katerina Juselius, 1990) reinforces the evidence obtained from the vector autocorrelation function.5 The coefficient on the inflation rate is not signifi- cantly different from zero, and the deviation of output from trend moves almost one-for- one with the short-term nominal rate. Thus the correlation between output and the nominal rate, regardless of the assumed order of integration, is one of the strongest in this data set. Still, the cointegration analysis leaves much to be desired. After all, most of the existing evidence suggests that the de- terministically detrended output gap is stationary, while the bill rate may not be; yet the cointegration analysis links the output gap and the bill rate in the most significant cointegrating vector in the multivariate system. We find it implausible that the link between short rates and the real output gap is really a long-run cointegrating relationship. Because we find the results of the cointegration analysis somewhat hard to interpret, and because we find the implications of the stationary vector autoregression intuitively appealing, we focus for the rest

4The computations underlying Figure 1 use the residual covariance matrix and the companion form of the vector autoregression to derive, first, the uncondi- tional covariance matrix of the variables in the system, then the autocovariance function, and finally the vector autocorrelation function. Wald tests for the significance of lagged variables across equations and within each equation are used in the selection of lag length.

5Using the critical values from table A.3 in Johansen and Juselius (1990), the maximum eigenvalue and trace statistics for the Johansen VAR indicate that we can reject the hypothesis that the vector autoregression contains three unit roots in favor of two unit roots (and thus one cointegrating vector) at the 1-percent significance level. But we can reject two unit roots (and two cointegrating vectors) in favor of one, and one unit root in favor of zero unit roots, only at the 20-percent significance level.


0 ~~~~0 0 c

0)

c(Uci. (U 0.~~~~~~~~~~~~~~~~~~~~~

fE ca~~~~~~0c

0

0~~~~~~~~~~~~~~~~~~~~~~

ci)c C)

(D (D 0D

0)3 ci 1)C

0~

U') 0LO0 - 1 0 1 o cic

VOL. 85 NO. 1 FUHRERANDMOORE: MONETARYPOLICYTRADE-OFFS 223

of the paper on stationary representations of the fundamental series in the model. In the next subsection, we explore the properties of a true ex ante long-term real interest rate implied by the stationary VAR.

C. The Expected Long-Term Real Rate

The concept of duration, introduced by Frederick R. Macaulay (1938), is useful in computing the sample path of the long-term real rate implied by the vector autoregression. Duration unifies the representation of holding-period yields on discount bonds and coupon bonds. The duration of a discount bond is simply its maturity. The duration of a coupon bond is a weighted average of the time until the payments on the bond are received; the weighting function is the ratio of the present value of the payment stream to the value of the bond.

Let Rt be the yield to maturity on a coupon bond selling at par, and let M be the maturity of the bond at the end of quarter t. Then the duration of the bond is given by

1 - eRtM (1) Dt=

Rt

and the holding-period yield on the bond from quarter t to quarter t + 1 is (2) Rt-Dt(Rt+l-Rt) For the sake of a convenient linear approximation, we set duration to a constant in our calculations.6

Let pt be the real yield to maturity on a hypothetical real coupon bond. The in- tertemporal arbitrage condition that equal- izes the expected real holding-period yields on the long-term bond and Treasury bills is then7

(3) ptD_tpt nplrt-t,rt _ _ r_

Solving equation (3) for Pt in terms of Pt+i and rt - Et(t+ ), then recursively substituting the result into itself, the long-term real rate is an exponentially weighted moving average of the forecast path of the real rate of return on Treasury bills. The longer the duration of the bond, the greater is the weight on the future:

(4) Pt= :-1+ D 5(+ D) Et(rt+i - 7t+i,)

We compute the realization of Pt by first combining the estimated vector autoregression with equation (3), an identity defining the expected long-term real rate, and solving for the backward-looking vector auto- regressive reduced form of the four- dimensional system. Then we simulate the reduced form, exogenizing the estimated vector autoregression residuals, thus computing the sample path of Pt that is consistent with the VAR and the sample observa- tions on the bill rate, inflation, and the output gap.8 In effect, the calculation uses the vector autoregression to generate an infinite-horizon forecast of the bill rate and the inflation rate at each point in the sample, and then it discounts the implied short-term real-rate forecasts according to equation (4).

D. The Long Real Rate and the Short Nominal Rate

A duration (D) assumption is required in order to compute the ex ante long real rate. Over the sample period, the average duration of Moody's BAA corporate bond rate is 40 quarters. We use the duration on this benchmark bond to construct the VAR-consistent long real rate, setting D = 40 in the empirical work that follows. The vector autoregression implies a 40-quarter duration long-term real rate that looks like the short-

6 A constant-duration approximation is conventional

in the term-structure literature. See Shiller et al. (1983) for an example.

The operator E,(*) denotes mathematical expectation conditioned on all model variables dated t and earlier.

8Details of the computation are reported in the appendix of Fuhrer and Moore (1995). The method uses a slightly different approach but should yield estimates identical to those derived from the method in, for example, Campbell and Shiller (1987, 1991).


0.04 Long real rate --.--.-.Bill rate

0.16

0.035 0.14

0.03 0.12

0.025 0.1

0.08 0.02

0.06

0.015

0.04

0 .0 1 .. . . .. . . .. .. .. . .

65Q1 68Q1 71Q1 74Q1 77Q1 8OQ1 83Q1 86Q1 89Q1 92Q 1 Year

FIGURE 2. COMPARISON OF LONG REAL RATE AND BILL RATE

term nominal rate. Figure 2 plots p and the bill rate together.9 Adjusted for scale differences, the two series track one another remarkably well.10

Why does the 40-quarter-duration real rate implied by the vector autoregression look like the short-term nominal rate? We

9The conformance of long-real-rate and short nominal-rate behavior holds for all long-duration real rates, including real rates of duration 20, 30, and 35 quarters.

10Because we use no long-term nominal rate data, we verify the plausibility of the computed ex ante real rate by comparing it with other estimates of the long real rate. We construct three other real-rate proxies. The first subtracts the ex post 40-quarter inflation rate from the BAA corporate bond rate. The second subtracts the expected 40-quarter-duration inflation rate implied by the VAR from the corporate bond rate. The third subtracts the Hoey survey's ten-year inflation expectation from the corporate bond rate. The correlation of each of these series with the computed long real rate falls between 0.73 and 0.93; the rate based on the Hoey survey has the lowest correlation. Plots confirm the similar qualitative behavior of all the real-rate series. As noted in Section II, the volatility of the model's ex ante real rate is considerably smaller than the volatility of these alternatives.

gain insight into this question by splitting the real rate, pt, into its long-term nominal rate and long-term inflation components, R* and 7>*:

(5) R* - D[Et(R* 1) -R* =rt

(6) 7r*- D[ Et(7r* 1) - 7*] =Et(7,+1).

The (unobserved) rationally expected long real rate and its components in equations (5) and (6) may be expressed as restricted linear combinations of the lags of the variables in the VAR. Summing the lag coefficients in the reduced-form equations for R*, -ITw, and p,11

(7) R* = +0.075r +0.1357T +0.051y

7* = -0.146r +0.124-r- +0.068y

p = + 0.217r + 0.0117- 0.017y.

The long-term nominal rate depends upon

1"The solutions to all of the linear models in this paper may be expressed as restricted vector autore- gressions. See Appendix A in Fuhrer and Moore (1995) for details of the solution methodology.


inflation with about the same coefficient as the long-term inflation rate does. When rr* is subtracted from R* to derive p, the inflation coefficients essentially cancel. But since the long-term nominal rate and the long- term inflation rate depend upon the short nominal rate with opposite signs, these coefficients reinforce each other when rr* is subtracted from R* to derive p.

The upper-left two-by-two block of the vector autocorrelation function in Figure 1 provides some intuition for this pattern in the reduced-form coefficients. Interest rates and inflation are tightly coupled in the VAR. Inflation and the bill rate are positively correlated with their own future values, consistent with the positive sum of reduced-form coefficients linking current r with expected future bill rates, R*, and linking current rr with expected future inflation, rr*. But the bill rate is negatively correlated with future values of inflation (beyond five quarters), while inflation is positively correlated with future values of the bill rate. In the sample, high inflation rates are followed by high bill rates; and high bill rates are followed by low inflation rates. Thus in the reduced form, the expected long nominal rate, R*, depends positively on current inflation, while the expected long-term inflation rate, rr*, depends negatively on the current short nominal rate. We suspect that both of these dynamic correlations are heavily influenced by the monetary policy regime. They prove to be crucial in our structural analysis of the effects of alternative monetary policies.12

II. The Structural Model

The simple structural model that we use to investigate policy trade-offs and policy- dependence of long-rate behavior includes three stochastic equations: an IS curve that relates output to the long-term real interest rate, a monetary-policy reaction function that moves the short-term nominal interest rate in response to inflation and output, and a price-contracting specification in which agents set nominal wage contracts with a concern for relative real wages.13

A. Model Specification

IS Curve.-The real economy is represented in the structural model with a simple IS curve that relates the output gap to its own lagged values and to one lag of the long-term real interest rate, Pt-1:

(8) Yt =ao+ajt-1+aJt-2 ppt- + yt.

The long-term real rate is the yield to maturity on a hypothetical long-term real bond. In the estimate of the realization of Pt computed above, the expected holding-period yield on the long-term real bond is set equal to the expected real return on Treasury bills forecast by the unconstrained vector autoregression for the inflation rate, the bill rate, and output. In the structural model dis- cussed below, the realization of Pt is set equal to the expected real return on Trea- sury bills forecast by the restricted structural model. Thus the series of Pt realiza- tions changes as the estimated structural parameters change.

Reaction Function.-Monetary policy is represented with a policy reaction function

12Note that under the assumption of nonstationary data, we cannot interpret the real-output/nominal-rate correlation as a reduced-form correlation between an ex ante long real rate and real output. The nominal rate and the inflation rate both have unit roots under this assumption, and both behave almost like pure random walks. Thus the long real rate for any assumed duration behaves just like the short real rate: if Et(r,+i - r+iE+1)=rt-E + 1 Vi, then all discounted sums of expected real short rates will be identical and equal to the current real short rate. Yet the p value for the restriction that the bill rate and inflation enter with equal and opposite signs in the cointegrating vector described in Subsection I-B is 2.2x 0-4. This is yet another reason why we find it difficult to interpret the data under the assumption of nonstationarity.

13In Fuhrer and Moore (1995) we present a more extensive discussion of the model, and it compares the relative contracting specification with the contracting specification of Phelps (1978) and Taylor (1980).

226 THE AMERICAN ECONOMIC REVIEW MA,RCH 1995

that relates the quarterly change in the bill rate to lagged changes in the bill rate, lagged levels of the inflation rate, and the contem- poraneous level of the output gap.

The beginning of the sample period, 1965, is dictated by our use of the short-term nominal interest rate as the fundamental instrument of monetary policy. The federal funds rate, the overnight rate on interbank loans, was less than the Federal Reserve discount rate prior to the mid-1960's. Since that time, the funds rate has generally traded above the discount rate, and there has been a direct link between Federal Re- serve open-market transactions and movements in the funds rate.

While the details of reserve accounting and the tactics of monetary policy have changed several times since the mid-1960's, it has always been the case that required reserves have been essentially predeter- mined over the course of a reserve mainte- nance period. By draining nonborrowed reserves from the banking system, the Federal Reserve forces banks to borrow at the discount window. When the federal funds rate is trading above the discount rate, the demand for discount-window borrowing is negatively related to the spread between the funds rate and the discount rate. As discount-window borrowing increases, Fed- eral Reserve district banks apply increasing administrative pressure to the borrowers, and banks in need of reserves are willing to pay an increasing premium in the interbank federal funds market to avoid these "frowns." When the trading desk drains reserves, it forces the funds rate up relative to the discount rate.14

The 3-month Treasury bill rate proxies for the overnight federal funds rate in our quarterly model. Monetary policy is represented with a policy reaction function that relates the quarterly change in the bill rate to lagged changes in the bill rate, lagged

levels of the inflation rate, and the contem- poraneous level of output,

2

(9) A r= a0 + Ea,rArti i=l

3

+ E t _j+ a,y,t + Ert' j=1

In addition to the interest-rate-smoothing terms, Art-j, policy leans against the wind in the sense that both Ea-, and ay are positive. Policy raises the funds rate when either inflation or output is above its target.

There has no doubt been some change in the behavior of monetary policy over this sample period. However, the changes appear to be empirically unimportant for the simple model explored here. In the estimation section below, we present evidence bearing on this point.

Note that using the first difference of the bill rate in the reaction function does not imply a unit root in the bill rate process. The nominal rate, inflation, and the output gap are all stationary in this structural model. To see why, consider the steady state. The contracting specification detailed below insures that the output gap y is zero in the steady state. Thus the IS curve, equation (8), implies a constant steady-state value for p of ao /aP. The definition of the long real rate, equation (3), then links the nominal rate and inflation in the steady state according to

rt - 7t+1 = aO lap

which implies that the rates of change of r and v in the steady state are equal. Denote the constant steady-state rate of change of r and iT by ,u. The reaction function, equation (9), with 9=0 and Art= Art = u in the steady state, implies that the steady-state value of inflation at lag 3, 7Tt, must be

Ss qTt 3= k

where k is a constant that depends on p,

14See Peter Keir (1981), Marvin Goodfriend (1983), D. H. Resler et al. (1985), and Ann-Marie Meulendyke (1990) for more extensive discussions of this mecha- nism.


ao, the a, 's, and the ar's, but not on time. Thus the inflation rate is stationary in the steady state, and the steady-state rate of change of inflation, ,u, must be zero. But then the definition of p in equation (3) implies that the nominal rate must also be stationary. Note that if the nominal rate does not respond to the level of inflation (acz = 0 Vi), the steady-state level of inflation is not independent of time. Monetary policy determines the order of integration of inflation in this model.

Contracting Equations. -Agents negotiate nominal contracts that remain in effect for four quarters. The contracts that we have in mind in this specification are nominal-wage contracts. However, with a fixed markup from wages to prices, as in Taylor (1980), there is little to distinguish wages and prices. Thus, in what follows, we refer to contract prices, and we use price data in the estimation of the model.

The aggregate log price index in quarter t, Pt, is a weighted average of the log contract prices, xt-j, that were negotiated in the current and the previous three quarters and are still in effect. The weights, fi, are the proportions of the outstanding contracts that were negotiated in quarters t - i,

3

(10) Pt= Efixt-i i=O

where fi > 0 and Efi = 1. We characterize the distribution of contract prices with a downward-sloping linear function of contract length,

(11) fi = 0.25 + (1.5 - i) s

with 0 < s < 1/6 and i = 0,..., 3. This distribution depends on a single slope parameter, s, and it is invertible. When s = 0, it is the rectangular distribution of Taylor (1980), and when s = 1/6, it is the triangular distribution."5

Let vt be the index of real contract prices that were negotiated on the contracts cur- rently in effect,

3

(12) Vt = fi(xt-i-Pt-i) i=O

Agents set nominal contract prices so that the current real contract price equals the average real contract price index expected to prevail over the life of the contract, adjusted for excess-demand conditions:

(13) xt -pt

3

= E fiEt( vt+i + yYt+) + Ept. i=O

Substituting equation (12) into equation (13) yields the "relative" version of Taylor's (1980) contracting equation,16

3

(14) xt-pt = i(xt-i-Pt-i) i=l

3

+ E iEt(xt+i-pt+i) i=l

3

+ Y* E fiEt( Yt+i) + sext. i=O

In their contracting decisions, agents care about the relative real contract price in effect during the life of their contracts. Thus they compare the current real contract price with an average of the real contract prices that were negotiated in the recent past and those that are expected to be negotiated in the near future; the weights in the average measure the extent to which the past and

15We find that more heavily parameterized contract price distributions are not supported by the data.

16Compare equation (14) with equation 1 on page 4 of Taylor (1980). The coefficients in equation (14) are pi = Ejjfif+j/(1-Ej1f7) and y* = y/(l -

Ejf 2)


future contracts overlap the current one. When output is expected to be high, the current real contract price is high relative to the real contract prices on overlapping contracts.17 In contrast, the Taylor (1980) specification assumes that agents care about relative nominal contract wages (and prices) in effect during the life of their contracts.

The main virtue of this contracting specification is that it imparts the degree of persistence to inflation that we find in the data (as summarized in the vector autocorrelation function of Fig. 1, for example); the conventional contracting specification of Taylor (1980) cannot. While the Taylor specification can be shown to imply that prices depend symmetrically on past and expected future prices, thus imparting significant inertia to the price level, it implies that inflation is extremely flexible. In contrast, the relative contracting specification of equation (14) can be shown to imply that inflation depends symmetrically on past and expected future inflation, thus imparting significant inertia to both inflation and the price level. In addition, the relative contracting model, because it implies a link between inflation and excess demand, is better able to replicate the correlation between excess demand and inflation; the standard model links the price level and excess demand and is thus less able to cap- ture this important feature of the data. For an extensive discussion of the properties of the new contracting specification, especially in comparison to the standard Phelps/ Taylor specification of overlapping contracts, see Fuhrer and Moore (1995).

B. FIML Estimation

The equations listed in Table 3 comprise the structural model. We estimate all the model's parameters by full-information

TABLE 3-REAL CONTRACrING MODEL

Equation name Equation Status

IS curve (8) stochastic Reaction function (9) stochastic Contract price (13) stochastic Long-term real rate (3) identity Price index (10) identity Real contract price (12) identity

TABLE 4-FIML PARAMETER ESTIMATES

Standard Equation Parameter Estimate error t statistic

(8) ao 0.008 0.005 1.4 a, 1.340 0.094 14.2 a2 -0.372 0.090 -4.1 ap 0.360 0.216 1.7

(9) ao -0.003 0.001 -2.6 arI 0.126 0.082 1.5 ar2 -0.397 0.081 -4.9 a7r, 0.060 0.043 1.4 aV2 0.130 0.037 3.5 aV3 -0.116 0.040 -2.9 ay 0.102 0.028 3.7

(10),(12) s 0.081 0.013 6.2 (13) y 0.008 0.005 1.7

Sample: 1965:1-1992:4

Q(12) statistic (p value): IS curve: 28.6 (0.005) Reaction function: 19.3 (0.082) Contracting equation: 26.1 (0.010)

maximum likelihood. The sample period is 1965: 1 through 1992:4. In the quarterly data, we use the 3-month Treasury bill rate to proxy for the overnight federal funds rate.

Consistent, inefficient estimates of the model parameters are used as starting values for the estimation. Parameter estimates, standard errors, t statistics and error sum- mary statistics are reported in Table 4.18

17In interpreting the empirical estimates presented here, it is important to remember that y measures the impact of the output gap on the log real contract price, not on inflation or on the price index. The inflation rate is related to the real contract price via a complex lag/lead polynomial.

18Note the negative sign for a, is embedded in equation (8). The parameter covariance matrix is computed as the inverse of the Hessian of the log-likelihood function. We use Matlab's sequential quadratic programming algorithm with numerical derivatives to perform the optimization, and the Hessian is computed using the BFGS update formula. For more on the BFGS update, see Phillip E. Gill et al. (1981).

VOL. 85 NO. 1 FUHRER AND MOORE: MONETARYPOLICY TRADE-OFFS 229

Some serial correlation remains in the residual of the IS equation, largely due to a significant autocorrelation at lag 8. The residual in the contract-price equation has a significant first-order moving-average component with a coefficient of -0.33. The estimated slope of the contract-price distribution lies at the midpoint of its feasible range, and it is estimated with considerable precision. The other parameters are of the expected sign and magnitude, with the exception of the high real-rate elasticity in the IS curve.

The explanation for the magnitude of this elasticity derives from the construction of the ex ante long real rate under the assumption of stationary nominal rates and inflation. Note from Figure 2 that the volatility of the ex ante real rate is quite small; the rate varies over the sample between 1 percent and 3.5 percent. The stationarity of the nominal rate and of inflation implies that at long horizons, nominal rates and inflation are forecast to be at their means. As a result, the long real rate that is constructed assuming a long duration will not exhibit much volatility (compared with, say, estimated ex post long-term real rates). This observation has been made in the substantial "variance bounds" literature. While the correlation of long real rates with the output gap is essentially unaffected by the scale of long-rate volatility, the scale of the parameter in the IS equation is inversely propor- tional to volatility. Thus the low-volatility ex ante real rate in this model requires a large IS coefficient. Ultimately, the plausibility of this and the other model coefficients is verified by the matchup between the vector autocorrelation function implied by the model and the vector autocorrelation function of the unconstrained VAR. This comparison is made below.

The monetary-policy reaction function determines the order of integration of the inflation rate in the structural model. Only if policy responds to the level of the inflation rate will inflation be stationary (see the discussion of the steady state of the model above). This feature is common to many models of price behavior, including the standard Phillips curve. Given that the sam-

ple mean of the output gap is zero by construction, the estimated mean inflation rate is the constant in the reaction function di- vided by the sum of the coefficients on lagged inflation. While the coefficient on the first lag of inflation is estimated impre- cisely, the constant and the sum of the lag coefficients are estimated quite precisely. As a result, the equilibrium inflation rate is estimated precisely at 4.5 percent per year with an asymptotic standard error of 0.9 percent per year.19 This estimate suggests that policy has targeted the level of inflation over the sample period and induced stationarity in the inflation rate. In light of this evidence, and because we wish to work with a stationary model, we maintain emphasis on the level of inflation in our policy experiments.

Separate limited-information estimates of the reaction function for the pre- and post- 1979 subsamples show some differences in estimated coefficients. The emphasis on inflation increases in the second half of the sample, while the emphasis on interest-rate smoothing decreases somewhat. However, these changes are not large enough to show up as significant serial correlation in the estimated reaction-function errors in Table 4. Furthermore, the improvement in fit from allowing different reaction functions in the two subsamples is not statistically significant. Substituting the subsample reaction functions for the single-sample reaction function improves the likelihood by 3.5; the p value for the likelihood-ratio test of the seven restrictions imposed by a single reaction function is 0.35. While a full explo- ration of the subsample stability of all of the parameters in the model lies outside the scope of this paper, we take the evidence presented here as supporting the hypothesis that the parameters of the contracting specification and the IS curve are stable enough across policy regimes to be used in the

19The sum of the coefficients on lagged inflation is 0.07 with an asymptotic standard error of 0.02.


broad-brush policy comparisons we perform below.20

As final evidence of the goodness-of-fit of the model, we compute the vector autocorrelation function implied by the structural model. The vector autocorrelation function for inflation, the bill rate, and output implied by the structural model is shown in Figure 3. Comparing Figures 1 and 3, the structural model fares well. With half the number of free parameters, the structural model accurately reproduces the dynamic correlations among inflation, the bill rate, and output implied by the unconstrained vector autoregression. We find this qualitative result to be compelling evidence that the structural model is a suitable framework for the policy experiments we pursue in the next section.

III. Policy Experiments

A. A Structural Interpretation of the Nominal-Rate/Output Correlation

One should not be surprised to find that reduced-form relationships, such as the dynamic correlations between inflation and the bill rate examined in Section I, vary with changes in the systematic behavior of economic agents.21 We use the structural interpretation of the vector autoregression de- veloped in the previous section to explore the effect of alternative monetary-policy parameters on the cross correlations between inflation and interest rates, and thus on the behavior of long-term real interest rates.

In our policy experiments we use a sim- plified version of the estimated reaction function,

When the coefficients of the IS and the contracting equations are set approximately to their estimated values, and a. = ay = 0.1, the vector autocorrelation function of the structural model closely mimics the autocorrelation functions of both the unconstrained vector autoregression and the estimated structural model.

While all of the model's autocovariances and cross covariances respond to changes in the policy parameters, the cross covariances between inflation and the bill rate are especially sensitive. Figure 4 shows the cross correlations (the left panels) and covariances (the right panels) between inflation and the bill rate when the policy parameters are set at their baseline values (a., = ay = 0.1) and at a more aggressive policy setting (a,, = ay = 1).

In the vector autoregression and in the baseline structural model, the combination of the negative correlation between the bill rate and future inflation and a relatively weak positive covariance between current inflation and future bill rates makes the long-term real rate look like the short-term nominal rate. The sign of the correlation between the bill rate and future inflation and the magnitude of the covariance between inflation rate and the future bill rate depend upon the vigor with which monetary policy responds to inflation and output. When a. = ay =1 (the dashed curves in Fig. 4), the correlation between the bill rate and future inflation (the top left panel) turns positive, and the positive covariance between inflation and future bill rates (the bottom right panel) strengthens.

To confirm the link between the dynamic covariances and the qualitative behavior of the long-term real rate, we compute ordinary least-squares (OLS) regressions of the implied long rate on inflation, the bill rate, and output for a variety of policy settings. The simulations of the structural model are organized like the simulation of the reduced-form model that yields the long- term real rate in Figure 2. Table 5 displays the regression coefficients, comparable to the last lines in equation (7).

Note that the OLS coefficients are an approximation to the sum of the solution

20 A complete investigation of the parameter stability

of this model appears in Fuhrer (1994); that paper concludes that, with the possible exception of some instability in the lags of output in the IS curve, the parameters in this model are quite stable across monetary policy regimes.

tThis is, after all, the thrust of the arguments made by Robert E. Lucas, Jr. (1976).

VOL.-85 NO. 1 FUHRER AND MOORE: MONETARY POLICY TRA4DE-OFFS 231

a-

v ~~~~~~~~~~~0 0) 0)

o~~~~~~~~~~~~~~~oc

C - 0~~~~~~~~~~~~~~~C

_____ LE) C) 0

0 ciC

d~~~~~~~~~~~~~~

V

4)~~~~~~~~~~~~~~~~~~~~~~4

4) 4~~~~~~~~~~~~~~~~)

- ~~~~~~~~~~~~~~-a- L0 Z

~~~~~0 ~~~~~~~~~~~~0 -o~~~~~~~~~~~~0C

0) ci)~~~~~~~~~~~~~v 0)

C 4) - C~~~~~~~~~~~~~~~~~~~~4

C -~~~~~~~~~~~~~~~~~~~I 0 I

C)~~~~~~~~~~~~~~~~~


Autocorrelation Autocovariance Inflation, lagged bill rate xInflation, agged bill rate

os C I~~~~~~~~~~~~~~~~~~~~~~( SI' eX3

0.4 - 1.5 2

0.2

0.1 0 .. .. . .. .

(0.5; 0 10 20 30 40 0 10 20 30 40

Bill rate, lagged inflation Bill rate, lagged inflation 1 (x 1e-3)

4 0.8

0.6 .-

0.4 2.

0.2 1

(0W ............. .. -----

0 10.21

(0.4)

(0O6) (2) 0 10 20 30 40 0 10 20 30 40

Lag, Quarters Lag, Quarters

FIGURE 4. AUTOCORRELATION AND AUTOCOVARIANCE FUNCTIONS WHEN POLICY PARAMETERS ARE SET AT BASELINE VALUES (SOLID CURVES) AND FOR THE MORE AGGRESSIVE POLICY (DASHED CURVES)

TABLE 5-ORDINARY LEAST-SQUARES (OLS) REGRESSIONS OF P ON r, XT, AND Y

Policy parameters OLS coefficient on:

a, ay rt It

0.01 0.01 1.74 - 0.06 - 0.05 0.05 0.05 0.40 -0.05 0.07 0.05 0.10 0.31 -0.07 0.12 0.10 0.05 0.29 -0.01 0.08 0.10 0.10 0.25 -0.03 0.12 0.35 0.35 0.13 0.04 0.21 1.00 1.00 0.09 0.12 0.32

Note: Each regression includes a constant.

coefficients as presented in equation (7). One cannot compare directly the solution coefficients from the VAR in Section I with the solution lag coefficients in the reduced form of the structural model, because the structural contracting model is not nested within the vector autoregression. The fundamental price series in the vector autore-

gression is the inflation rate, while the fundamental price series in the structural model is the log of the price index. The real rate that is consistent with the structural model is thus a linear combination of lags of the bill rate, the output gap, the price index, and the contract price; the contract price implicitly contains infinite lags of the price index [invert equation (10) to see this]. Thus we can obtain only approximate projections of the real rate on lags of the bill rate, inflation, and the output gap.

For relatively modest policy responses to inflation and output, the long real rate is most closely associated with the bill rate. As policy targets inflation and output more vigorously, the regression weight on the short nominal rate decreases, while the weight on the inflation rate increases. At a", = 1.0, the weights are about equal: the long real rate looks approximately like the sum of the nominal rate and inflation. None of these policies generates equal and opposite


alphapi alphay= 0.1 0.1

Bill rate 0.08 ...........

- ..

. Inflation rate

0.06

0.04

0.02

0.

(0.02)

(0.04)

(0.06) (20) (10) 0 10 20 30 40 00

Quarters

alphapi = alphay = 1.0 0.2

.---- Bill rate

- Inflation rate 0.15

0.1

a_ ...... . . .___... . _..'

0L 0.05

\~~~~~~~~~~~~~.. .... ., ._ ................ .............

(0.05)

(20) (10) 0 10 20 30 40 50 Quarters

FIGURE 5. DISINFLATION DYNAMICS

coefficients on the bill rate and inflation. Thus there is no evidence that, under any of the policies considered, output would be correlated with the short real rate.

Why does the policy in the last row of Table 5 reverse the sign of the correlation between the bill rate and future inflation and fundamentally change the behavior of the long-term real rate? The disinflation simulations displayed in Figure 5 provide a clue. The deterministic simulations begin with inflation and the nominal rate at their steady-state values. At time 0, the target level of inflation is permanently decreased to 0, and the simulation unfolds from that point until inflation and the nominal rate settle at their new lower equilibrium levels.

When policy responds about as it has over the last 25 years, as shown in the top panel, the short nominal rate rises slightly in response to above-target inflation. Infla-

tion falls rapidly, raising the real rate of interest and depressing real activity. Be- cause this policy places a small weight on output deviations, policy only gradually lowers the nominal rate to reverse the downward pressure on economic activity. As a result, the output gap remains negative after inflation reaches its new target level, and inflation overshoots its equilibrium level, dropping 2 percentage points below its new equilibrium. One can read the dynamic covariances between the short rate and inflation directly from Figure 5. Above-equilibrium nominal rates at the beginning of the simulation are followed by below-equilibrium inflation rates later in the simulation. Above-equilibrium inflation rates at the beginning of the simulation are followed by above-equilibrium bill rates. Thus the dynamic covariances between the short nominal rate and inflation that we see in both the unconstrained and the structural models arise from the relatively moderate policy responses associated with this monetary policy rule.

The disinflation proceeds quite differently under the more vigorous policy response displayed in the bottom panel of Figure 5. Monetary policy raises short rates quickly and vigorously, depressing real activity, and then it lowers them just as quickly, exerting only enough downward pressure on real activity to lower inflation monotonically to its new target level. Inflation does not overshoot its long-run equilibrium. One important distinction between this simulation and the previous one is that, in this simulation, policy exhibits a sufficient distaste for output deviations to avoid the undamped swings in output that caused inflation to overshoot in the first simulation. Once again, one can read the covariances between the nominal rate and inflation directly from the simulation: above-equilibrium short rates at the beginning of the simulation are followed by above-equilibrium inflation rates later in the simulation. Above-equilibrium inflation rates at the start of the simulation are followed by short rates that are well above equilibrium. The stronger response to both output and inflation deviations in this simulation produces markedly different dynamic


covariances between inflation and nominal rates, thus yielding markedly different behavior of the long real rate.22

B. Monetary Policy Trade-offs

One might ask why, if the Fed could have avoided inflation overshooting with a more vigorous but balanced policy response, it appears not to have done so. Here, we provide a more complete accounting of the costs and benefits of pursuing this more aggressive policy, as well as other policy options. We associate costs and benefits with four system characteristics of the model. The first is the sacrifice ratio, defined as the cumulative percentage-point deviation of output from potential, discounted at 3 percent per year, for each percentage-point reduction in the rate of inflation. The re- maining three characteristics are the uncon- ditional variances of inflation, the bill rate, and the output gap. Table 6 provides a direct comparison of these system characteristics when monetary policy responses are varied over a two-order-of-magnitude (log- spaced) grid centered around the baseline response arr = ay = 0.1.

ar =0.01, a t = 0.01: Policy is responsible for stabilizing inflation and output fluctua- tions in this model. When both of the baseline policy parameters are reduced by a factor of 10, the variances of inflation and output increase by more than an order of magnitude, and the variance of the bill rate falls by 20 percent. This timid policy rule produces an enormous sacrifice ratio of 12.2.

To demonstrate why the sacrifice ratios and variances skyrocket under this policy response, in Figure 6 we display a disinflation simulation of the type shown in Figure 5. Because the model incorporates no credibility effects, even this minute bill-rate response to inflation eventually yields a suc- cessful disinflation. Thus inflation, shown in the top panel, falls at the beginning of the simulation. However, inflation wildly overshoots its target, and policy moves the bill rate tentatively in response to enormous deviations of both output and inflation around their target values. Movements in the short real rate (shown in the bottom panel), which drive the movements in the long real rate, output, and inflation in this simulation, are dominated by the swings in inflation. In contrast, in the active policy of Figure 5, it is the movements of the bill rate, in response to deviations of output and inflation from target, that dominate the movements in real rates and thus in output. In the weak targeting simulation of Figure 6, policy has essentially abandoned its re- sponsibility for stabilizing output and inflation, and hence their variances soar. In a mechanical sense, this policy eventually achieves its objective of zero inflation. How- ever, the key assumption that policy is fully credible is quite dubious under this policy rule. To be considered credible, we believe that policymakers must be seen to be fight- ing inflation, raising interest rates in response to deviations of inflation from its target. This policy cannot be considered credible by that metric.

ca = 1.0, a = 1.0: Compared with the baseline simuiation, the primary effect of this very strong targeting of inflation and output is to drive up the variance of the bill rate by an order of magnitude. The sacrifice ratio and the variance of inflation improve a bit.

a! = 1.0, ay =0.01: While it controls the inflation rate very closely, this combination of strong inflation-targeting and weak output-targeting drives up the variance of the bill rate by an order of magnitude and

22Inflation overshooting is not alleviated by vigorous inflation-targeting alone, but by the combination of relatively strong output-targeting and inflation-targeting. A simulation under a vigorous inflation-targeting policy coupled with a weak output-targeting policy yields inflation overshooting, a negative correlation between the short rate and future inflation rate, and an implied long real rate is closely correlated with the nominal rate. For further discussion of this point in the context of the MIT-Penn-SSRC (MPS) quarterly model, see Flint Brayton and Peter A. Tinsley (1992 pp. 27-29).

VOL. 85 NO. 1 FUHRER AND MOORE: MONETARY POLICY TRADE-OFFS 235

TABLE 6-SYSTEM CHARACTERISTICS

Policy Variance

acer ay Sacrifice ratio Inflation Bill rate Output

0.10 0.10 1.91 0.0028 0.0015 0.0013 0.01 0.01 12.24 0.1285 0.0013 0.0620 1.00 1.00 1.25 0.0013 0.0162 0.0012 1.00 0.01 1.91 0.0013 0.0171 0.0035 0.01 1.00 0.32 0.0100 0.0132 0.0006

alphapi - alphay = 0.01 0.6

iBill rate

-Inflation rate 0.4

02

0.

(0.2)

(0.4)

(20) (10) 0 10 20 30 40 50 Quarters

alphapi alphay 0.01 0.6

- ho (1 real rate

Long real rate

0.4

02

0.

(0.2)

(0.4) . (20) (10) 0 10 20 30 40 50

Quarters

FIGURE 6. DISINFLATION UNDER WEAK INFLATION- AND OUTPUT-TARGETING

triples the variance of the output gap compared to the baseline case.

la = 0.01, aoy = 1.0: This combination of weak inflation-targeting and strong output- targeting shares the credibility problems of the policy response with weak inflation- targeting and weak output-targeting. As one might expect, output hardly deviates from

trend, and the sacrifice ratio is extremely low in this experiment.

Figures 7-10 plot the four system characteristics as surfaces above the (a7,, ay) plane. To gain clear views of the surfaces in the upper panels, we rotate the plots about the vertical axis by varying amounts. The view- point can be determined by inspecting the policy coordinates at the corners of the sur-


14- 0.016-

12- 0.016-

0.014

0.012 -

8- 0.01

6- 0.00 - :

0.006- 4-

0.004-

0.002

.01 803 1

0 0 .32~~~~~~~.3

FIGURE 7. SACRIFICE RATIO FIGURE 9. BILL-RATE VARIANCE

0.14-

0.12-

0.06-~~~~~~~~~~~~~~~~~~~~~00

0.1~~~~~~~~~~~~~~~~~~~~.3

0.06-

0.06~~~~~~~~~~~~~~~~~~~~~~~~~~~~.1*

0.02 . 0.01

01.03

a I

FIGURE 8. INFLATION VARIANCE FIGURE 10. Oumpur-GAP VARiANCE

faces. In each figure, the baseline policy setting is indicated by (0.1,0.1). Note that, in each case, the highest point on the surface should be thought of as folding back away from the reader; the low plateaus of each surface are "sticking out of the page" toward the reader. Darker shading corre- sponds to lower values of the system charac- teristic; lighter shading denotes higher values.

If our estimated characterization of recent monetary policy is approximately cor- rect, ar = ay = 0.1, then policy has been striking a good balance among competing policy objectives. Given policymakers' concern for the sacrifice ratio and the variance of inflation, interest rates, and output,

movement from the center toward any cor- ner of the policy parameter grid entails a substantial deterioration in at least one measure of system performance without substantial improvement in any of the other measures.

More aggressive targeting of inflation and output yields little or no improvement in the sacrifice ratio, inflation variance, or output variance, but it produces higher bill-rate variance. Starting in the center of the plots, increases in both a,, and acy correspond to movements along the level contours of the sacrifice-ratio, inflation-variance, and output-variance functions and to uphill movement on the bill-rate variance function. Weaker targeting of inflation and output


yields a small improvement in the bill-rate variance at the expense of a substantial deterioration in the sacrifice ratio and the variance of inflation and output.

Movements toward the other two corners of the policy grid yield predictable results. If policy emphasizes inflation at the expense of output, then output variance rises; if policy emphasizes output at the expense of inflation, then inflation variance rises. Each of these policy changes is associated with movement across a flat region of the sacrifice-ratio function, and each is accom- panied by a substantial increase in the variance of the bill rate.

IV. Conclusion

Our simple structural model of the macroeconomy fits the data well, accurately replicating the dynamic correlations among inflation, the bill rate, and the output gap. Thus we feel comfortable in using the model to interpret the well-documented correlation between short nominal rates and output, and to explore the trade-offs among competing monetary-policy objectives that face policymakers.

In the textbook IS-LM model, the IS curve links real output with the short-term nominal rate. In quarterly data since 1965, real output is strongly negatively correlated with the short-term nominal rate. Theoretically, however, we prefer an IS curve in which real output depends upon a long-term real rate. When we construct expected long-term real rates from an unconstrained vector autoregression for inflation, the 3-month Treasury bill rate, and real output, we find that the vector autoregression implies a long-term real rate that looks like the bill rate, and consequently the long real rate explains output quite well.

Upon examining the dynamic covariances for the VAR, we find that the behavior of the implied long rate derives from the negative correlation in our sample between the current bill rate and future inflation rates, coupled with a relatively weak positive covariance between current inflation rates and future bill rates.

We use a simple structural model to ex- amine the robustness of these covariances to variations in monetary policy. We find that, when monetary policy responds considerably more vigorously to both output and inflation than it appears to have done in the sample, the sign of the reduced-form correlation between bill rates and future inflation rates is reversed, the positive covariance between inflation and future bill rates is strengthened, and the implied long real rate no longer mimics the short nominal rate. One qualitative distinction between the baseline monetary policy and the more vigorous policy is the presence of inflation overshooting in the baseline policy. Under the baseline policy, when inflation exceeds its target value, policy raises the bill rate, depressing real activity, and inflation approaches and then overshoots its target value. This description of the dynamic path of short rates and inflation articulates the negative correlation between the current bill rate and future inflation rates: a high bill rate today is associated with below-target inflation rates in the future. Under the more vigorous policy, when inflation exceeds its target value, policy raises the short rate, depressing real activity, but then it lowers the short rate as inflation approaches its target, eliminating overshooting. This changes the sign of the dynamic correlation: high short rates today are associated with above-target inflation rates in the future.

One might view the absence of inflation overshooting under the more vigorous monetary policy as preferable to the inflation overshooting that appears to have charac- terized actual policy behavior. However, we show that the modest decrease in inflation variability associated with the more vigorous policy is purchased at the cost of signifi- cantly higher variability in both output and the bill rate. More generally, we find in conjunction with the estimated IS curve, the relative-contracting model implies significant trade-offs among monetary-policy goals -the output sacrifice ratio and the variances of inflation, interest rates, and output. Given our parameter estimates and our characterization of monetary policy in


recent years, we conclude that policymakers have been striking a good balance among competing monetary-policy objectives.

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