Economics Letters 62 (1999) 63–67

The inflation and output variability tradeoff: evidence from a Garchmodel

*Jim LeeDepartment of Economics and Finance, Fort Hays State University, Hays, KS 67601, USA

Received 6 April 1998; accepted 18 September 1998

Abstract

This paper empirically investigates the possibility of a tradeoff between the variability of inflation and output gap over theperiod 1960–1997 in light of a bivariate GARCH model. There is strong evidence of temporal instability in the GARCHprocess, and the variability tradeoff is more apparent for the post-October 1979 subperiod. The slope of the tradeoff,nevertheless, appears to be considerably flat. 1999 Elsevier Science S.A. All rights reserved.

Keywords: Conditional variance; GARCH; Tradeoff

JEL classification: E3; E5; C5

1. Introduction

A major hallmark for evaluating monetary policy is the Phillips-curve type tradeoff betweeninflation and output or unemployment. Recently, instead of the conventional thinking of a tradeoff inlevels, Taylor (1993), (1994) argued that the policy tradeoff is better described in terms of arelationship between variability in output and variability in inflation. While it is less intuitive to viewmonetary policy objectives in terms of variances, targeting a certain level of inflation or outputnaturally affects these variables’ relative variability. For instance, in an event of an aggregate demandshock, the Federal Reserve’s efforts to keep inflation stable would result in relatively larger outputfluctuations.

This concept of variability tradeoffs has been shown by Fuhrer (1997) to be an essential guide forconducting monetary policy. Lacking in the existing literature, however, is direct evidence thatsupports its presence in retrospect. One explanation for this deficiency is that realizations of volatilitybehavior are directly unobservable. In this paper, we overcome this problem by modeling stochasticvolatility based on conditional variances estimated within a generalized autoregressive conditionalheteroscedasticity (GARCH) framework. Fitting a bivariate GARCH model to U.S. historical dataover the period 1960–1997, we attempt to provide insight into the plausibility of an output-inflationvariability tradeoff.

*Corresponding author. Tel.: 11 785 6285868; fax: 11 785 6285398; e-mail: jlee@fhsu.edu

0165-1765/99/$ – see front matter 1999 Elsevier Science S.A. All rights reserved.PI I : S0165-1765( 98 )00212-2

64 J. Lee / Economics Letters 62 (1999) 63 –67

2. Empirical model and results

The tradeoff relationship is explored in light of a bivariate GARCH model involving the joint1processes governing inflation and output. The full observation period runs from 1960Q1 through

1997Q4. In line with Fuhrer (1997); Taylor (1994), the output variable is measured by the percentdeviation of real GDP from potential GDP. The inflation variable is measured by the annualpercentage change of the core Consumer Price Index (CPI), which measures consumer inflationexcluding changes in food and energy prices. The core CPI instead of the index for all consumer items(or the GDP deflator) is used in order to alleviate effects emerging from major aggregate supplyshocks, such as the OPEC oil price increases in the 1970s, which have been considered to bepredominant factors shifting the output-inflation level tradeoff. In addition, as asserted by Cecchetti(1997); Motley (1997), developments in core CPI inflation might better characterize the underlying

2trend in inflation to which monetary policymakers pay particular attention.The estimation model is specified as a GARCH(1,1) process, which has been widely used in the

literature since Bollerslev and Wooldridge (1992). This parsimonious specification is supported byEngle (1983) Lagrange multiplier test results, which reveal strong evidence of ARCH effects of a highorder. To illustrate the estimation model, let y ; [y y ]9 be a 231 vector containing the inflationt 1t 2t

and output variables such that the innovation vector e ; [e e ]9 contains their correspondingt 1t 2t

de-meaned values based on the regression:

y 5 m 1 e , e | N(0, H ) (1)t t t t

where m is a 231 vector of constants, and H is a 232 time-varying conditional variance-covariancet

matrix measured at time t. The stochastic behavior of H can be expressed as:t

9H 5 g 9g 1 a9e e a 1 b9H b, (2)t t21 t21 t21

where g is a 232 lower triangular matrix with three intercept parameters, a and b are 232 squarematrices of parameters. For estimation, we use the BEKK parameterization, as described in Engle andKroner (1995), to vectorize H . For the bivariate GARCH(1,1) model under consideration, there aret

effectively 11 parameters. Consistent estimates of the parameters are obtained using the quasi-maximum likelihood procedure suggested by Bollerslev and Wooldridge (1992).

Estimation results for the GARCH(1,1) model are displayed in Table 1. Panel A (second column)displays results based on full-sample data. The parameters in g represent the mean levels of theconditional variances of the inflation (g ) and output variables (g ), along with their mean11 22

covariance level (g ). The estimates indicate that, over the full-sample period, the mean conditional21

variance is relatively smaller for inflation than for the output gap.The parameters in a reveal the extents to which the conditional variances of inflation and output

are correlated with past squared errors (i.e., deviations from their means). Of particular interest are the

1The inflation data are quarterly averages of the monthly core CPI data. All data are obtained from the database of theFederal Reserve Bank of St. Louis, except those for potential GDP, which are from the Board of Governors of the FederalReserve System. Alternative measures of potential GDP levels, including trend estimates obtained from the Hodrick andPrescott (1997) filter, yield no appreciably different results than those reported below.2Despite that the core CPI is used in preference (on conceptual grounds) to the all-items CPI or the GDP deflator, our mainfindings, particularly estimates for the slope of the volatility tradeoff, are robust to alternative measures.

J. Lee / Economics Letters 62 (1999) 63 –67 65

Table 10Estimation results for the GARCH(1,1) model

A B C1960Q1–1997Q4 1960Q1–1979Q3 1979Q4–1111997Q4

Regression: H 5g 9g 1a9e e a 1b9H b, where e ;[e e ]9t t21 t2 t21 t 1t 2t

Intercept matrix (g )g 0.38* (3.51) 1.19* (5.64) 0.24*** (1.62)11

g 0.02 (0.09) 0.12 (0.21) 0.11 (0.33)21

g 0.63* (6.01) 0.37 (0.97) 0.45* (3.51)22

Volatility transmissionsa 0.72* (6.93) 0.84* (3.42) 0.68* (4.21)11

a 20.14* (3.50) 20.13* (3.11) 20.12** (2.24)12

a 0.03 (0.83) 0.01 (0.12) 20.07 (1.30)21

a 0.89* (7.01) 0.97* (3.65) 0.96* (3.89)22

Volatility tradeoffsb 0.04 (0.62) 20.17*** (1.87) 20.10** (2.18)12

b 0.01 (0.11) 20.12 (0.53) 20.15** (1.96)21

Wald tests for cross-effectsOutput Gap→InflationH : a 5b 50 12.78* 12.51* 5.14**0 12 12

Inflation→Output GapH : a 5b 50 1.68 0.28 6.72**0 21 21

Volatility persistenceb 0.68* (8.01) 0.50* (2.81) 0.67* (10.53)11

b 0.29** (2.57) 0.16*** (1.67) 0.13*** (1.68)22

Likelihood function 2301.56 2168.49 2115.59Likelihood ratio 34.96*

Notes: e is the de-meaned value of the inflation series, and e is the de-meaned value of the output gap series. The1t 2t

parameters g , a , and b are the ij (i, j51,2) elements of g, a, and b, respectively. Robust t-statistics (in absolute values)ij ij ij2are reported in parentheses next to parameter estimates. Wald statistics for volatility cross-effects have a x (2) distribution.

The likelihood ratio is for testing parameter constancy in the GARCH model across the 1960Q1–1979Q3 and 1979Q4–1997Q4 subperiods.*, **, and *** denote statistical significance at the 1%, 5%, and 10% levels, respectively.

off-diagonal elements, which depict how the past squared error of one variable affects the conditionalvariance of another variable. The estimate for a suggests a negative cross-effect running from the12

lagged output error to the inflation variance. In contrast, the estimate for a , which depicts a21

cross-effect in the opposite direction, is statistically indifferent from zero.The parameters in b depict the extents to which the current levels of conditional variances are

correlated with past conditional variances. More specifically, the diagonal elements (i.e., b and b )11 22

reflect the levels of persistence in the conditional variances. The estimates reveal that inflationvolatility is relatively more persistent than output volatility. The off-diagonal elements in b (i.e., b12

and b ), on the other hand, depict the extent to which the conditional variance of one variable is21

correlated with the lagged conditional variance of another variable. Clearly, there is little evidencesupporting a correlation between inflation and output volatility.

In light of the off-diagonal parameter estimates in a and b, we formally test for volatilitytransmissions between inflation and output. More specifically, we perform joint tests under the null

66 J. Lee / Economics Letters 62 (1999) 63 –67

hypothesis that a 5b 50 for i±j. Based on the Wald test statistics shown in Table 1 (Panel A), theij ij

null hypothesis of no cross-effects can be rejected for a transmission running from the output gap toinflation, but not vice versa.

Fuhrer (1997), however, argues that changes in monetary policy responses to inflation and outputaffect the nature of possible variability tradeoffs. Under this perspective, he divides a similarobservation period into subsamples corresponding to changes in the monetary policy rule. Thebreakpoint corresponds to a change in the Federal Reserve’s operating procedure from targeting aninterest rate instrument earlier to targeting nonborrowed reserves in October 1979. The timing alsocoincides with the onset of the Fed’s explicit commitment to controlling inflation. Along this line, wefurther assess the robustness of the preceding findings by re-estimating the GARCH model based on

3two subsamples: 1960Q1–1979Q3 and 1979Q4–1997Q4.Subsample estimation results are reported in panels B and C of Table 1. There are several

interesting observations. First, there is a remarkable difference in the g parameter estimates,11

revealing a much higher level of conditional volatility of inflation for the first than the secondsubsample. Other parameter estimates, on the other hand, are largely similar, except those for theoff-diagonal parameters in b. More specifically, the latter show stronger evidence of a tradeoff in bothdirections during the more recent than earlier subperiod. This finding is reinforced by the Waldstatistics, as shown below the parameter estimates.

In light of the casual evidence of temporal instability in parameter estimates, we perform alikelihood ratio test for structural change in the entire bivariate GARCH process. Based on the valuesof likelihood functions, the likelihood ratio statistic is 39.48, which strongly rejects parameterconstancy across the two subsamples. Importantly, this finding implies that the estimation results forthe observation period as a whole are misleading. This is especially crucial for the inference on thevolatility tradeoff between inflation and output. Based on subsample estimations, the off-diagonalparameter estimates in b are overall less than 0.2 in absolute value, which implies a considerably flattradeoff frontier.

3. Conclusion

In this paper, we have empirically explored the possibility of a variability tradeoff between inflationand output in light of a bivariate GARCH model estimated over the period 1960–1997. Despite weakevidence from full-sample data, a tradeoff between the conditional variances of output and inflation isapparent within subsamples, particularly the one corresponding to the post-1979 monetary policyregime. We, therefore, interpret the empirical results as support for Taylor (1994) view that the Fed’sefforts in controlling inflation may risk meeting the other objective of stable output growth. Inretrospect, however, the slope of such a tradeoff appears to be considerably flat.

3We have also considered the possibility of excluding a period between 1979Q4 and 1982Q3, in which the Fed operated aprocedure targeting nonborrowed reserves. The qualitative results are nevertheless similar to those reported for the entire1979Q4–1997Q4 subsample.

J. Lee / Economics Letters 62 (1999) 63 –67 67

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