The Inflation-Output Variability Relationship in the G3: A Bivariate GARCH (BEKK) Approach

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1 Risk Letters, 005, 1 (), 17- Te Inflation-Output Variabilit Relationsip in te G3: A Bivariate GARCH (BEKK) Approa Menelaos Karanasos a, and Jinki Kim b a Brunel Universit, UK b Gangwon Development Resear Institute, Korea Abstrat Tis paper emplos bivariate GARCH models of inflation and output growt to investigate te relationsip between nominal and real unertaint in te G3. Our estimated models are used to generate te onditional varianes of inflation and output growt as proxies of inflation and output variabilit and test for bidiretional effets. Our evidene support a number of important onlusions. For te entire sample period , in all tree ountries, tere is no ausal relation between nominal and real unertaint. For te USA over te period , wi followed te anges in Fed operating proedures, te inflation volatilit as a signifiant impat on output volatilit. Tis finding is in agreement wit Logue-Sweene's (1981) potesis. In sarp ontrast, in Japan during te eigties and nineties te effet of output variabilit on inflation variabilit is signifiant as predited b Devereux (1989). Finall, in te sixties and seventies no effet in eiter diretion is present in an of te tree ountries. Kewords: Bivariate GARCH; Inflation variabilit; Real unertaint 1. INTRODUCTION One of te most fierel debated issues in maroeonomis is te nature of te relationsip between te levels of inflation and output or unemploment. Logue and Sweene (1981) and Devereux (1989) analzed te long-term relationsip between inflation and output in a different wa. In partiular, Logue and Sweene (1981) point out tat nominal unertaint as a positive impat on real unertaint wereas aording to Devereux (1989) iger output variabilit leads to iger inflation variabilit. Logue and Sweene (1981), using rosssetional tests and data from 4 ountries tat are members of te OECD, find tat te variabilit in real growt is strongl and positivel related to te variabilit in inflation. Te extent to wi tere is a relationsip between nominal and real unertaint is an issue tat annot be resolved on merel teoretial grounds. Tis paper builds on earlier eonometri work on estimating su relations. Te studies of Lee (1999), Arestis et al. (00), Fountas et al. (00), and Karanasos and Kim (005) are te onl attempts to investigate te inflation-output variabilit relationsip using measures of onditional volatilities. However, tere is a lear need for furter empirial investigation. In tis paper, te above issue is analzed empiriall for te G3 ountries wit te use of a bivariate GARCH model tat inludes inflation and output growt. Among oters, Grier and Perr (000), and Grier et al. (004) use bivariate GARCH models to simultaneousl estimate te onditional means, varianes and ovariane of inflation and output growt. Our estimated model is used to generate te onditional varianes of inflation and output growt as proxies of inflation and output growt variabilit. Tis model allows us to examine te ausal relationsip between nominal and real unertaint. Tis approa provides a simple wa to illustrate te existene or absene of a variane relationsip. Moreover, we examine weter te transition from te ig inflation of te sixties and seventies to an era of low inflation during te 1980s and 1990s affets te inflation-output variabilit relationsip b dividing te wole sample period into two subperiods. * Corresponding autor. menelaos.karanasos@brunel.a.uk We would like to tank M. Karanassou for er elpful omments and suggestions. Tis resear was onduted wile bot autors were at te Universit of York. ISSN Global EoFinane Limited. All rigts reserved. 17

2 Karanasos and Kim 18 Our evidene supports a number of important onlusions. First, for te entire sample period tere is no ausal relation between nominal and real unertaint. In oter words, no effet in eiter diretion is present in an of te tree ountries. Seond, for te USA, in te eigties and nineties tere is evidene of ausalit running onl from real unertaint to nominal unertaint. Tis finding provides support for te Devereux potesis. In sarp ontrast, for Japan, in te post-1979 period tere is evidene tat inreased inflation variabilit inreases real variabilit, onfirming te teoretial preditions made b Logue and Sweene. Finall, in all tree ountries in te sixties and seventies tere is no evidene of a ausal relation between te variabilit of inflation and output. Te laout of te paper is as follows. Setion desribes te teoretial model used for estimation. Setion 3 presents te empirial analsis and results. Conluding remarks are in Setion 4.. THE MODEL We use bivariate VAR models to estimate te onditional means of te rates of inflation and output growt. Let π t and t denote te inflation rate and real output growt respetivel, and define te residual vetor t as t = ( π t, t ). Regarding t, we assume tat it is onditionall normal wit mean vetor 0 and varianeovariane matrix H t, were ve ( Ht ) = ( π t, π, t, t ). Tat is, ( t Ωt 1) N(0, Ht ), were Ω is te information set up to time. In our empirial work, we estimate several bivariate VAR speifiations for inflation and output growt. Our oie between te various models was based on te use of Granger ausalit tests (Wald tests). Following Engle and Kroner (1995), tese Granger ausalit tests were performed on te assumption tat te onditional ovariane matrix follows te BEKK representation 1. Tat is, Ht is parameterized as wit H CC + A A + BH, (.1) t = t 1 t 1 t 1B C = π π π β βπ, A =, B =. π β π β Beause of te presene of a paired transposed matrix fator for ea of tese tree matries non-negative definiteness of te onditional matrix is assured. Also, in te above BEKK model, { t } is ovariane stationar if and onl if all te eigenvalues of A A + B B (were stands for Kroneker produt) are less tan one in modulus (see Engle and Kroner, 1995). We estimate our bivariate sstem using te Berndt et al. (1974) numerial optimization algoritm (BHHH) to obtain te maximum likeliood estimates of te parameters..1. Commonalit in volatilit movements Te notion of `persistene' of a sok to volatilit witin te GARCH lass of models is onsiderabl more ompliated tan te orresponding onept of persistene in te mean for linear models. One definition of persistene would be to sa tat soks fail to persist wen }( i = π,, π ) is stationar and ergodi. Te persistene of soks an also be defined in terms of foreast moments; i.e., to sa tat soks to it fail to persist if and onl if for ever s, E s ( it ) onverges, as t, to a finite limit independent of time s information. In tis stud we will adopt te latter one. Note tat te two onditional (o)varianes in equation (BEKK) an be expressed as { it π, t, t = + + β = + π + β π π, π π, π, + β + β β π π, ππ, π β π, π π,, π π + β + β,, π,,,, π,, + + (.a) (.b) 1 In te presene of onditional eterosedastiit, Vilasuso (001) suggests tat ausalit tests be arried out in te ontext of an empirial speifiation tat models bot te onditional means and onditional varianes.

3 Karanasos and Kim 19 π, t = π + β β + π π π, + ( β ππ, β + β π + ( β ) π π, π + β ) π β π,,. π, π, (.) Te ross-equation restritions implied b (.) make it diffiult to link te persistene in a partiular omponent of te onditional variane-ovariane matrix to partiular parameters. As a measure of persistene we use te largest eigenvalue of A A + B B. Moreover, from te expressions in (.) it is easil seen tat te off-diagonal elements of te matrix A ( B ) depit ow te past squared error (onditional variane) of one variable affets te onditional variane of anoter variable. In oter words, π, π, β π and β π an be viewed as providing information on te orrelation between real and nominal unertaint. 3. EMPIRICAL ANALYSIS 3.1. Data Te data set omprises montl Produer Prie Index (PPI) and Industrial Prodution Index (IPI) series for te USA, Japan and German. In our empirial analsis we use te PPI and te IPI as proxies for te prie level and output respetivel. Te index for te USA and Japan overs te period of Februar 1957 to August 000 and onsists of 53 observations for ea series. For German te sample is Februar 1958 to Jul 000. Inflation is omputed as [100 ( log ( PPI t ) log ( PPI )) ] were PPI t and PPI are montl Produer Prie Indies at time t and respetivel. Real output growt is measured b te annualized montl differene in te log of te IPI [100 ( log ( IPI t ) log ( IPI )) ]. 3.. Results Table 1 reports parameter estimates for te tree BEKK GARCH(1,1) models. Wit all ountries, te potesis of unorrelated standardized and squared standardized residuals is well supported 3. Te bivariate AR(1)-GARCH(1,1) models seem to fit te means and varianes of bot inflation and output growt well. Based on te t-statistis te null potesis of no ross effets is aepted. In oter words, in all tree ases te statistial insignifiane of te estimates of π, β π, π and β π sows te lak of an assoiation between te variabilit of inflation and output growt. Clearl, tere is no support for an relationsip between real and nominal unertaint. Table reports tree alternative measures of te persistene in te onditional varianes of inflation and output growt. For te BEKK model, te largest eigenvalue of A A + B B is reported in te seond olumn of Table. Te estimated eigenvalue for German is markedl lower tan te orresponding values for Japan and te USA. Tese two ountries generated ver similar persistene parameters (0.97 and 0.96 respetivel). A simple wa to ompare te persistene in te two onditional varianes in te BEKK model is to regress ˆ ĥ it ( i = π, ) on a onstant and it 1 (see olumn 3, Table ). In te USA it is lear tat inflation volatilit is more persistent tan output volatilit. However, for Japan and German real unertaint is more persistent tan nominal unertaint. Furtermore, te four deades under investigation are araterized b persistentl ig inflation, as was te ase from late 1960s troug te earl 1980s, followed b te relativel sok-free 1990s were bot inflation and real growt were more stable tan te were in te 1980s. Terefore, we tougt it neessar to partition te total sample period into two subperiods. In partiular, te full sample, wi runs from 1957:0 troug 000:08, is broken into two subsamples, orresponding to assumed sifts in te monetar poli regime. Te first subperiod goes from te beginning of te sample to te end of Te seond subperiod starts in 1980 and ontinues till to te end of te sample. In te USA te subsamples for te 1980 breakpoint are defined a priori as orresponding to te periods before and after te nonborrowed reserves operating proedure. Table 3 reports parameter estimates for te BEKK parameterizations of te tree bivariate GARCH(1,1) models. Table 3a reports te results for te pre-1980 period. In all tree ountries all te off-diagonal estimates in A and B are statistiall insignifiant. Tat is, in te sixties and seventies tere is no ausal relation between nominal and real unertaint. Te BEKK estimates of te inflation and output unertaint are based upon a bivariate VAR(1) model. On te basis of te Wald tests and te requirement of wite residuals we deide to inlude twelve lags in te VAR. We do not report te estimated results for te mean equation for spae onsiderations. 3 Te results from te Ljung-Box tests for serial orrelation in te standardized residuals, teir squares and teir ross produts are not presented to preserve spae.

4 Karanasos and Kim 0 Table 1. Parameter Estimates for te BEKK GARCH(1,1) Models (Entire Sample) C (6.) C 0.91 (0.65) C (1.37) (11.97) 0.04 (0.51) (0.68) 0.74 (13.13) 0.85 (36.85) (0.88) (0.1) (5.43) USA JAPAN GERMANY (10.1) (0.9).454 (1.7) (11.1) (0.) (1.03) 0.00 (4.73) (5.1) (0.8) (0.8) (5.13) Tis table reports parameter estimates for te BEKK(1,1) models. Te numbers in parenteses are t-statistis (4.94) (0.7) 6.16 (3.74) (6.41) (0.46) (0.45) (4.85) (6.84) 0.45 (0.58) (0.39) (4.0) Table. Persistene for te BEKK GARCH(1,1) Models Eigenvalue Slope oeffiients USA a b b JAPAN GERMANY a Te largest eigenvalue of A A + B B is reported. b Te estimated slope oeffiient from te regression of it (i=,) on a onstant and i,t-1 is reported. Table 3a. Parameter Estimates for te BEKK GARCH(1,1) Models (Subsample: ) USA JAPAN GERMANY (8.6) (0.19) 0.08 (0.60) (7.5) (16.08) (0.94) (0.83) (.16) (4.94) (0.95) (1.10) (1.41) (4.8) -0.4 (1.08) (0.78) (.57) 0.45 (4.9) 0.33 (0.50) (0.03) (3.38) (5.69) 0.34 (0.73) (0.77) (1.70) Te numbers in parenteses are t-statistis.

5 Karanasos and Kim 1 Table 3b. Parameter Estimates for te BEKK GARCH(1,1) Models (Subsample: ) USA JAPAN GERMANY 0.43 (4.4) 0.35 (.50) (0.81) (.07) 0.85 (1.78) (1.6) (0.45) (3.6) (1.90) (0.8) (.55) (.34) (8.49) (1.00) (.48) (7.00) (0.0) (0.08) (0.37) (.06) (3.35) -.17 (0.79) (0.76) (0.36) Te numbers in parenteses are t-statistis. Te results for te post-1979 period are reported in Table 3b. Te piture is different to tat of te pre-1980 period. In te USA te estimate of, wi depits te extent to wi te onditional variane of output β π growt is orrelated wit te lagged onditional variane of inflation, is statistiall signifiant. In Japan te estimate of β, wi depits a ross-effet in te opposite diretion, is igl statistiall signifiant. Te π former finding is in agreement wit Logue-Sweene's teor wereas te latter supports Devereux's potesis. In sarp ontrast, for German tere is no ausal relation between te two volatilities. 4. CONCLUSIONS In tis paper we ave emploed bivariate GARCH models to generate te onditional varianes of montl inflation and output growt for te G3. We ten used tese varianes as proxies of nominal and real unertaint to examine te bidiretional relationsip between te two variables. Te following observations, among oter tings, are noted about te inflation-output variabilit relationsip. First, in te entire sample period, tere is no ausal relation between te two volatilities. Seond, for te USA, during te eigties and nineties tere is evidene of a unidiretional feedbak between te variabilit of inflation and of output wit te line of ausation running from te former to te latter. Tis finding of a positive effet of nominal unertaint on real unertaint is in agreement wit Logue-Sweene's (1981) potesis. Tird, for Japan, during te period te variabilit of output as a positive impat on te variabilit of inflation as predited b Devereux (1989). Finall, in te sixties and seventies, no effet in eiter diretion is present for all tree ountries. REFERENCES Arestis, P.G., M. Caporale and A. Cipollini (00) Does inflation targeting affet te trade-off between output-gap and inflation variabilit? Manester Sool, 70, Berndt, E., B. Hall, R. Hall and J. Hausman (1974) Estimation and inferene in nonlinear strutural models, Annals of Eonomi and Soial Measurement, 3, Devereux, M. (1989) A positive teor of inflation and inflation variane, Eonomi Inquir, 7, Engle, R. and K.F. Kroner (1995) Multivariate simultaneous generalized ARCH, Eonometri Teor, 11, Fountas, S., M. Karanasos and J. Kim (00) Inflation and output growt unertaint and teir relationsip wit inflation and output growt, Eonomis Letters, 75, Grier, K.B. and R. Grier (003) On te real effets of inflation and inflation unertaint in Mexio, Unpublised Manusript, Universit of Oklaoma.

6 Karanasos and Kim Grier, K.B. and M. Perr (000) Te effets of real and nominal unertaint on inflation and output growt: some GARCH- M evidene, Journal of Applied Eonometris, 15, Grier, K.B, Ó.T. Henr, N. Olekalns and K. Sields (004) Te asmmetri effets of unertaint on inflation and output growt, Journal of Applied Eonometris, 19, Karanasos, M. and J. Kim (005) On te existene or absene of a variane relationsip: a stud of maroeonomi unertaint, WSEAS Transations on Computers, 4, Lee, J. (1999) Te inflation and output variabilit tradeoff: evidene from a GARCH model, Eonomis Letters, 6, Logue, D. and R. Sweene (1981) Inflation and real growt: some empirial results, Journal of Mone, Credit and Banking, 13, Vilasuso, J. (001) Causalit tests and onditional eterosedastiit: Monte Carlo evidene, Journal of Eonometris, 101, 5-35.