Department of Economics East Carolina University Greenville, NC Phone: Fax:

Transcription

1 March 3, 999 Time Series Evidence on Wheher Adjusmen o Long-Run Equilibrium is Asymmeric Philip Rohman Eas Carolina Universiy Absrac The Enders and Granger (998) uni-roo es agains saionary alernaives wih asymmeric adjusmen is applied o he exended Nelson and Plosser daase. The es rejecs he uni-roo null roughly as frequenly as does he ADF es. JEL Classificaion C3, E3 Correspondence: Philip Rohman Deparmen of Economics Eas Carolina Universiy Greenville, NC 7858 Phone: Fax: rohmanp@mail.ecu.edu

2 . Inroducion Over he pas fifeen years a subsanial body of evidence documening asymmeric adjusmen in he dynamic behavior of key macroeconomic and financial ime series has developed; see, e.g., Poer (995) and Ramsey and Rohman (996) and he references herein. In ligh of his evidence, Enders and Granger (998) (EG) emphasize ha sandard uni-roo ess are run agains saionary alernaives wih symmeric adjusmen mechanisms. This leaves open he possibiliy ha he well-known failure of Dickey-Fuller-ype uni-roo ess o rejec he null hypohesis of inegraion for many economic ime series may be due o hese ess being run wih a misspecified alernaive hypohesis. To his effec EG develop generalizaions of he Dickey- Fuller es o allow for asymmeric adjusmen under he saionary alernaive hypohesis. The purpose of his paper is o apply he EG ess o he well-known Nelson and Plosser (98) long-run daase exended by Schoman and van Dijk (99). Using heir ime reversibiliy es, Ramsey and Rohman (996) found srong evidence of Aseepness@ and Adeepness@ asymmery (as defined by Sichel (993)) in hese ime series. The resuls of his paper, hen, can shed ligh on wheher he ime irreversibiliy in hese daa deeced by Ramsey and Rohman may be due o asymmeric daa generaing processes of he ype considered under he alernaive hypohesis of he ess employed. The paper proceeds as follows. The EG ess are described in Secion. Secion 3 presens he resuls of applying hese ess o he exended Nelson and Plosser daase and Secion 4 concludes. I hank Dick van Dijk for his very helpful commens.

3 3. The Asymmeric Adjusmen Uni-Roo Tess The following represenaion of an auoregressive process of order (AR()) for a ime series { y }, =,...,N, is he basis for he Dickey-Fuller es: y = ρ y + ε, () where ε is a whie-noise innovaion and ρ = α -, where α is he AR() coefficien. Under he alernaive hypohesis of saionariy, - < ρ < 0. The long-run equilibrium poin or aracor under his alernaive hypohesis is given by y = 0, around which he process adjuss symmerically. EG consider hreshold generalizaions of he linear AR() process given by equaion (); see Tong (990) for exensive discussion of hreshold ime series models. The following firs-order hreshold auoregressive (TAR) model also has an aracor a y = 0 : where I is he Heaviside indicaor funcion y = I ρ y +( - I ) ρ y + ε, (), if y- I = 0, if y < 0 0. (3) Model () is a special case of () and (3), i.e., when = ρ () reduces o () and here is ρ symmeric adjusmen owards he aracor. EG also analyze wo modificaions of he TAR process given by (). The firs allows a non-zero consan aracor as follows: where y = I ρ [ y - a0 ] +( - I ) ρ [ y - a0 ] + ε, (4)

4 4 I =, if 0, if y y < a0 a0. In he uni-roo case, ρ = 0 and (4) is a pure random walk. If < ( ρ = ) < 0, hen = ρ - ρ (4) reduces o a saionary linear AR() process wih a non-zero consan erm. The second modificaion allows for a linear deerminisic rend aracor: where y = a+ I ρ [ y - a0 - a( - )] +( - I ) ρ [ y - a0 - a( - )] + ε, (5) I =, if 0, if y y < a a a( ) a( ). Noe ha his differs from equaion (5) in Enders and Granger (998), in ha heir model does no include a as a consan on he righ-hand side; his consan should be included since, under he uni-roo null { y } is a random walk wih drif. The sequence { y } is rend saionary if, in (5), < ( ρ, ) < 0 and ( a0,a ) 0. Higher-order processes can be considered in - ρ equaions (), (4), and (5) by augmening hem wih lagged changes in { y }. The indicaor funcion in (3) depends on he level of y -. EG also examine he case in which he swiching beween regimes is a funcion of he lagged change in y -, as in: I =, if y- 0, if y < 0 0. (6)

5 They call models given by () and (6) momenum hreshold auoregressive (M-TAR) models, in he sense ha he regime swiching is a funcion of he sign of he lagged Amomenum@ of { y }. The aracor in () combined wih (6) is 0. An imporan exension of (6) is: 5, if ( y- - a0 - a( )) 0 I = (7) 0, if ( y- - a0 - a( )) < 0. Replacing he indicaor funcions in (4) and (5) wih (6) and (7), respecively, yields M-TAR models wih a non-zero consan aracor and a deerminisic linear aracor; in he laer case he swiching is made condiional on wheher he lagged change in y is greaer han a. As is he case for he TAR models, he M-TAR models can be augmened by adding lagged y erms. Through Mone Carlo simulaions EG provide criical values for he F-saisic for he null hypohesis ha = ρ in (), (4), and (5). For he TAR models, EG call his F-saisic ϕ ; for ρ he M-TAR models, using he indicaor funcions (6) and (7), hey call he F-saisic for his null hypohesis ϕ. To se { I } according o he aracors of eiher (4) or (5), he parameer a0 or he parameers a0 and, a respecively, need o be esimaed. The criical values for he ϕ and ϕ are sensiive o wheher hese parameers are esimaed. To use he aracors of (4) and (5), as his paper does, he daa are firs regressed on he deerminisic componens and he residuals are sored. Then he following regression is esimaed: yˆ = I ρ yˆ +( - I ) ρ yˆ + ˆ ε, (8)

6 where yˆ } is eiher he demeaned or derended series, seing he indicaor funcion I for { eiher TAR-ype or M-TAR-ype swiching and obaining he F-saisic for he null hypohesis ρ = ρ = 0. The observed value of he F-saisic can be compared agains he appropriae 6 criical values abulaed by EG. Under he alernaive hypohesis, he esimaors for ρ and ρ converge o mulivariae normal disribuions, so ha a es of symmeric adjusmen agains asymmeric adjusmen can be carried ou wih he F-saisic for he null hypohesis = ρ ρ using he sandard criical values. EG recommend checking he residual sequence { εˆ } using sandard diagnosics o deermine if i appears o be a whie-noise process. If he residual analysis suggess significan deparures from whie-noise, hen equaion (8) should be re-esimaed by adding lagged changes in yˆ } as addiional regressors. The appropriae number of lags o include can be deermined by such residual analysis and/or use of model selecion crieria such as he Akaike informaion crierion (AIC) or he Schwarz informaion crierion (SIC). { 3. The Tes Resuls To ease comparison wih sandard uni-roo esing, Table repors he resuls of applying he augmened Dickey-Fuller (ADF) es o he exended Nelson and Plosser daase. The number of lagged y } used in calculaing he ADF τ τ and τ µ es saisics was se equal o he number { of such lags used in compuing he φ es saisics in Table. In all bu wo cases, he unemploymen rae and he bond yields series, he logarihms of he daa were used and deerminisic linear rends were included in he ADF regressions. The raw levels daa were used for he unemploymen rae and bond yields series and for each of hese he only deerminisic

7 7 regressor used for he ADF es was a consan erm. The ADF es residuals appear o be whienoise for all series. For nine ou of he foureen series, he uni-roo null hypohesis is no rejeced a he 0% significance level. The ADF es rejecs he uni-roo null a he 0% significance level for he real GNP, indusrial producion, and employmen series. For he real per capia GNP series he uni-roo null is rejeced a he 5% significance level wih he ADF es, and for he unemploymen rae daa he es rejecs he null a he % significance level. The Cheung and Lai (995) response surface regressions were used o esimae criical values for he ADF ess. The TAR ϕ uni-roo es resuls appear in Table. The number of lagged changes in y o include was deermined by boh sandard diagnosic checking of he regression residuals and he SIC. The p-values for he Ljung-Box Q saisics are all relaively high, suggesing ha he residual series are whie-noise processes. To a large exen he findings repored in Table mirror raher closely he ADF uni-roo es resuls, wih one excepion which provides marginally sronger evidence agains he uni-roo null. Tha is, wih he ϕ es he I() null hypohesis is rejeced for he real GNP series a he 5% significance level. While he uni-roo null hypohesis is rejeced a convenional significance levels for five ou of he foureen series in he exended Nelson and Plosser daase using he ϕ es, here is lile evidence ha adjusmen o he aracors of hese series is asymmeric in he TAR sense, i.e., he p-values for he F es of he symmery null hypohesis are all relaively high. A 4%, he p- value for his F-es is lowes for he real per capia GNP series. The M-TAR ϕ uni-roo es resuls appear in Table 3. Wih he excepion of wo series, he number of lagged y } series o include as regressors maches up wih he number of such lags {

8 used for he TAR ϕ ess repored in Table. Likewise, in all cases he Ljung Box Q(4) saisics indicae ha he residual series are serially uncorrelaed. The uni-roo ϕ es resuls differ somewha from hose obained wih he ϕ es in ha for he real GNP, real per capia GNP, and he indusrial producion series, he he ϕ es is higher han for he ϕ es; he ADF es p-values for he real per capia GNP and indusrial producion series are also lower han he corresponding p-values for he M-TAR es. Using he ϕ es leads o rejecion of he uni-roo null for only four of he series in he exended Nelson and Plosser daase, wih he evidence agains I() behavior for he unemploymen rae series once again being very srong. In all of hese cases, however, he symmeric adjusmen null hypohesis is no rejeced a arguably low significance levels Conclusions For mos series he ADF and Enders and Granger (998) ϕ and ϕ es resuls concur when applied o he exended Nelson and Plosser daase. More specifically, he resuls obained wih he ADF and φ ess srongly mirror one anoher, and for no series is here evidence in favor of a TAR-ype asymmeric adjusmen mechanism oward long-run equilibrium. The ϕ es resuls provide slighly weaker evidence agains he uni-roo null hypohesis. Finally, i is useful o compare he resuls of his paper wih hose repored by Ramsey and Rohman (996). These auhors find very srong evidence in favor of ime asymmeries in he daa generaing processes for mos of he series in his daase. The ϕ and ϕ es resuls of his paper sugges, hen, ha he Ramsey and Rohman (996) findings may no be due o TAR and M-TAR ype dynamics owards long-run equilibrium.

9 9 Table ADF Tess for Exended Nelson and Plosser Daa Series Sample Period ADF Tes Saisic Real GNP (T) c Nominal GNP (T) Per Capia Real GNP (T) b Indusrial Producion (T) c Employmen (T) c Unemploymen Rae (µ) a GNP Price Deflaor (T) CPI (T) Nominal Wage (T) Real Wage (T) Money Supply (T) Velociy (T) Bond Yields (µ) S&P500 Index (T) Noes: Logarihms of all series were used, excep for he unemploymen rae and he bond yields daa. Column 3 repors he Augmened Dickey-Fuller τ τ or τ µ es saisics. The number of lagged differenced series included in he ADF regressions is he same as repored in column of Table. A(µ)@ indicaes ha he only deerminisic erm included in he ADF regression was a consan, while A(T)@ indicaes ha a deerminisic linear rend was included. a indicaes significance a he % level, b indicaes significance a he 5% level, and c indicaes significance a he 0% level, using he Cheung and Lai (995) criical values.

10 0 Table TAR φ Tes Resuls for he Exended Nelson and Plosser Daase Series Lags ˆρ ˆρ SIC φ µ /T H0 : = ρ ρ Q(4) Real GNP -0.6 (-.83) -.4 (-.38) (T) b Nominal GNP (-.3) (-.60) (T) Per Capia Real GNP -0.8 (-.98) -0.4 (-.36) (T) b Indusrial Producion (-.33) -0.4 (-.9) (T) c Employmen -0.6 (-.44) -0.5 (-.47) (T) c Unemploymen Rae (-.97) -0.8 (-.8) (µ) a GNP Price Deflaor (-.7) (-.08) (T) CPI (-.0) -0.0 (-.6) (T) Nominal Wage (-.7) (-.68) (T) Real Wage (-.) -0.0 (-0.7) (T) Money (-.63) (-.47) (T) (coninued on nex page)

11 Table, Coninued Series Lags ρˆ ρˆ SIC ϕ µ / T H 0 : = ρ ρ Q(4) Velociy (-0.73) (-.63) (T) Bond Yields (-0.36) -0.4 (-0.46) (µ) S&P500 Index (-.75) (-.78) (T) Noes: Logarihms of all series were used, excep for he unemploymen rae and he bond yields daa. Column repors he number of lagged differences included in he TAR regression. Columns 3 and 4 repor he poin esimaes of ρ and ρ wih - saisics in parenheses below for he null hypohesis ha he respecive parameer equals 0. The SIC, repored in column 5, is calculaed as T@log(SSR/T) + n@log(t), where T is he number of useable observaions and n is he number of regressors. Column 6 repors he sample values of ϕ µ and ϕ T, where A(µ)@ indicaes ha he TAR model was esimaed on he deviaions from he sample mean and A(T)@ indicaes ha he TAR model was esimaed on he linear derended series. Column 7 presens he p- value for he F-es of he null hypohesis ha ρ = ρ. The las column gives he he Ljung-Box Q saisic for he null hypohesis ha he firs four residual auocorrelaions are joinly equal o 0. a indicaes significance a he % level, b indicaes significance a he 5% level, and c indicaes significance a he 0% level. Criical values for he ϕ µ and ϕ T saisics are given in Table, Panels C and E, of Enders and Granger (998).

12 Table 3 M-TAR ϕ Tes Resuls for he Exended Nelson and Plosser Daase Series Lags ρˆ ρˆ SIC ϕ µ / T H 0 : = ρ ρ Q(4) Real GNP -0.8 (-.58) -0.7 (-.8) (T) c Nominal GNP (-.77) (-0.96) (T) Per Capia Real GNP -0.9 (-.57) -0.8 (-.4) (T) c Indusrial Producion (-.60) -0.5 (-.5) (T) Employmen -0.3 (-.) -0.8 (-.6) (T) c Unemploymen Rae 3-0. (-.88) -0.8 (-.68) (µ) a GNP Price Deflaor (-.39) -0.0 (-0.83) (T) CPI (0.8) (-.3) (T) Nominal Wage (-.65) (-.8) (T) Real Wage (-0.57) (-0.96) (T) Money (-.6) (-.35) (T) (coninued on nex page)

13 3 Table 3, Coninued Series Lags ρˆ ρˆ SIC φ µ H 0 : ρ = ρ /T Q(4) Velociy (-0.73) (-.63) (T) Bond Yields 0.0 (0.7) (-0.90) (µ) S&P500 Index (-.44) -0.0 (-.97) (T) See Noes for Table, excep ha he uni-roo es saisics are now given by hese saisics are given in Table, Panels D and F, of Enders and Granger (998). ϕ µ and ϕ T. Accordingly, criical values for

14 4 References Cheung, Y.-W and K. S. Lai, 995, Lag order and criical values of he augmened Dickey-Fuller Journal of Business and Economic Saisics 3, Enders, W. and C. W. J. Granger, 998, Uni-roo ess and asymmeric adjusmen wih an example using he erm srucure of ineres raes, Journal of Business and Economic Saisics 6, Nelson, C. R. and C. I. Plosser, 98, Trends and random walks in macroeconomic ime series, Journal of Moneary Economics 0, Poer, S., 995, A nonlinear approach o U.S. GNP, Journal of Applied Economerics 0, Ramsey, J. B. and P. Rohman, 996, Time irreversibiliy and business cycle asymmery, Journal of Money, Credi, and Banking 8, -. Schoman, P. C. and H. K. van Dijk, 99, On Bayesian roues o uni roos, Journal of Applied Economerics 6, Sichel, D. E., 993, Business cycle asymmery: a deeper look, Economic Inquiry 3, Tong, H., 990, Non-linear ime series: a dynamical approach (Oxford Universiy Press, Oxford).