A Quasi-Bayesian Analysis of Structural Breaks: China s Output and Productivity Series

Transcription

1 Inernaional Journal of Business and Economics, 2004, Vol. 3, No. 1, A Quasi-Bayesian Analysis of Srucural Breaks: China s Oupu and Produciviy Series Xiao-Ming Li * Deparmen of Commerce, Massey Universiy Albany, New Zealand Absrac A quasi-bayesian model selecion approach is employed o deec he number and daes of srucural changes in China s GDP and labour produciviy daa. I is shown ha he predicive likelihood informaion crierion is valid only among models wih well-behaved residuals. Key words: srucural change; predicive likelihood; GDP; labour produciviy; China JEL classificaion: C22; E30 1. Inroducion The lieraure on srucural change in macroeconomic ime series has been growing rapidly over he pas decade. From a mehodological poin of view, such sudies may be classified ino wo caegories: one ha adops a hypohesis esing approach, and he oher ha akes a model selecion approach (see Maddala and Kim, 1998, p. 406). Regarding hypohesis esing, here are wo broad avenues o view. Tess of he uni roo null hypohesis wih or wihou srucural change agains he rend-saionariy alernaive ha allows for one or more breaks in he rend funcion fall ino he firs. Conribuions in his area include Perron (1989), Inwood and Sengos (1991), Zivo and Andrews (1992), Raj and Sloje (1994), Perron (1997), Nunes e al. (1997), Lumsdaine and Papell (1997), Kanas (1998), Ben-David e al. (1999), and Newbold e al. (2001). The second line involves esing for alernaive hypoheses agains each oher wih respec o he number of breakpoins in he series, disregarding wheher or no a uni roo is presen. Several esing echniques have recenly been developed by Vogelsang (1997), Bai and Perron (1998), and Bai (1999) and applied in Ben-David and Papell (1998) and Ben-David and Papell (2000). The problem of choosing beween difference-saionary models and rend-saionary models ha accoun for srucural change and beween models wih Received April 17, 2003, revised February 23, 2004, acceped March 8, * Correspondence o: Deparmen of Commerce, Massey Universiy Albany, Privae Bag , Norh Shore Mail Cenre, Auckland, New Zealand. x.n.li@massey.ac.nz. The auhor hanks an anonymous Associae Edior and he Managing Edior for very useful commens and suggesions ha helped improve he qualiy and presenaion of his paper.

2 58 Inernaional Journal of Business and Economics differen numbers of breaks can also be ackled by a model selecion approach. In fac, i should properly be so, as Maddala and Kim (1998) argue ha ulimaely deerminaion of he number of breaks is bes viewed as a model selecion problem. However, he srucural-change lieraure wih model selecion does no seem o be as vas as wih hypohesis esing. Kim and Maddala (1991) propose a Bayesian approach which uses he BIC crierion for selecing he bes model among he models wih differen numbers of breaks. A recen paper by Wang and Zivo (2000) also employs similar mehods o selec he mos appropriae model from he daa. While he above wo sudies may be caegorised as following a Bayesian approach, Kashiwagi (1991) suggess wha Maddala and Kim (1996a) erm a quasi-bayesian approach ha uses he predicive likelihood in analysing he same problem of deecing srucural changes. Bianchi (1995) exends his quasi-bayesian approach o he case of auoregressive errors, and Bianchi and Kan (1996) apply he exended mehod in a sudy of he rend behaviour of he US GNP series. For differences beween he Bayesian and quasi-bayesian procedures, see Maddala and Kim (1996b). More work needs o be done o compare he above wo independenly developed procedures for selecing he bes model among alernaives wih differen numbers of breaks (Maddala and Kim, 1996b). This paper is par of he empirical work for ha purpose. I repors he resuls of applying he quasi-bayesian approach o deecing he srucural breaks in China s GDP and labour produciviy daa, which may be used laer for comparisons wih he resuls of applying he Bayesian approach. Secion 2 briefly describes he approach employed, Secion 3 presens empirical resuls, and Secion 4 concludes. 2. Predicive Likelihood Consider a ime series model wih muliple breaks as follows: y = a+ b+ a DU + bdt + e. (1) n n i= 1 i i i= 1 i i In his model, y is a ime series under invesigaion, represens ime rend, n is he number of breakpoins, e denoes he error erm, and a, b, a i, and b i are model parameers o be esimaed. The break dummy variables, DU i and DT i, ake he following values: DU { 1 if i = TB 0 oherwise i i 1 if i DTi = TB + TB { 0 oherwise wih he ranges of n break daes being: 1 < TB 1 < < TB i < < TB n < T. Here T denoes he effecive sample size and TB 1,, TB n are n break daes. We as-

3 Xiao-Ming Li 59 sume ha he errors e follow an AR(p) process wih he order p o be deermined. The breakpoins (including heir number and iming) are o be searched for wih he aid of he predicive likelihood informaion crierion as briefly described below. If p = 0, he errors are serially independen and so he predicive likelihood crierion suggesed by Kashiwagi (1991) is relevan, and is value, PL, is compued as: T T 2 ( 1) ln(2 ) T k + PL = πσ, (2) 2 2 T k 2 where σ 2 is he variance of e and k is defined as k 2(n+1) m 1. Here m 1 is he number of he periods ha have only one observaion. If p > 0, he errors e become an auoregressive process of order p: e = e + e + + e +, e ~ N(0, σ Ω p ), ε ~ N(0, σ ), (3) µ 1 1 µ µ p p ε where µ 1,, µ p are parameers and ε represens a normally, independenly and idenically disribued (NIID) error erm. In his case, a heurisic crierion proposed by Bianchi and Kan (1996) may be employed. I is expressed as: T T 2 ( 1) ln(2 ) ln ( ) T k + PL = πσ ε ΩP, (4) 2 2 T k 2 wih k 2(n+1) m 1 + p. The idea behind adding p o he expression for k is o penalise for he number of auoregressive erms in he errors. Equaion (1) is esimaed wih he ordinary leas square (OLS) mehod when p is assumed o be 0 and wih he maximum likelihood mehod when p is assumed o be greaer han 0. In he former case, use (2) o calculae PL. In he laer case, simply deducing from he esimaed value of he log-likelihood funcion he bias correcion erm T(k+1)/(T k 2) will give PL. 2 ε 2 ε 3. Empirical Resuls We apply he above-oulined mehods o China s oupu (GDP) and labour produciviy (PRO) annual daa (in log form) obained from he China Saisical Yearbooks for 1998 and 2000 and from hp:// _15395.hm: The 2000 Saisical Communiqué of Naional Economy and Social Developmen, February 28, 2001 (in Chinese). The sample spans from 1952 o 2000, giving us a oal of 49 observaions. Tables 1 o 4 display he models wih he number of breaks associaed wih he highes PL values under four differen assumpions abou he error erms. They also repor diagnosic saisics abou he behaviour of he error erms. In paricular, boh he Ljung-Box Q and he Breusch-Godfrey (B-G) saisics are used o check for he presence of serial correlaion. Since he former is valid only for a pure auoregressive process while he laer is also valid in he case where such variables as rends and dummies appear as regressors, we mainly rely on he laer for inference.

4 60 Inernaional Journal of Business and Economics If only he predicive likelihood crierion is used, one would cerainly pick ou he models in Table 1 and conclude ha GDP has 10 srucural breaks and labour produciviy has 9. However, alhough he Q saisics sugges ha he assumpion of independen errors canno be rejeced a he 5% level, he B-G saisics indicae a decisive rejecion a a lower han 1% level for he wo series. In addiion, he produciviy daa suffer he problem of heeroscedasiciy. Thus, he models ha assume e o be an NIID error are no valid, and he wo numbers of breakpoins suggesed by he highes PL values may well be overesimaed. Table 1. The Seleced Model When he Errors Are Assumed o Be AR(0) Series n PL Q(10) B-G(10) Normaliy ARCH(3) GDP [0.090] [0.001] [0.890] [0.121] Produciviy [0.051] [0.003] [0.497] [0.035] Noes: For GDP, he 10 break daes are 1958, 1960, 1962, 1967, 1969, 1970, 1976, 1984, 1989, and For produciviy he 9 break daes are 1961, 1963, 1966, 1969, 1971, 1976, 1984, 1990, and Numbers in parenheses are he lag lenghs, and numbers in brackes are he p-values. Table 2. The Seleced Model When he Errors Are Assumed o Be AR(1) Series n PL Q(10) B-G(10) Normaliy ARCH(3) GDP [0.000] [0.001] [0.780] [0.154] Produciviy [0.008] [0.000] [0.851] [0.476] Noes: For GDP, he 7 break daes are 1958, 1960, 1962, 1966, 1969, 1971, and For produciviy he 11 break daes are 1954, 1960, 1962, 1966, 1969, 1971, 1976, 1981, 1984, 1990, and Numbers in parenheses are he lag lenghs, and numbers in brackes are he p-values. Table 3. The Seleced Model When he Errors Are Assumed o Be AR(2) Series n PL Q(10) B-G(10) Normaliy ARCH(3) GDP [0.356] [0.640] [0.187] [0.130] Produciviy [0.218] [0.884] [0.729] [0.432] Noes: For GDP, he 5 break daes are 1957, 1961, 1966, 1975, and For produciviy he 4 break daes are 1956, 1961, 1976, and Numbers in parenheses are he lag lenghs, and numbers in brackes are he p-values. Table 4. The Seleced Model When he Errors Are Assumed o Be AR(3) Series n PL Q(10) B-G(10) Normaliy ARCH(3) GDP [0.001] [0.002] [0.931] [0.234] Produciviy [0.032] [0.031] [0.929] [0.282] Noes: For GDP, he 7 break daes are 1956, 1958, 1960, 1968, 1971, 1974, and For produciviy he 4 break daes are 1957, 1961, 1976, and Numbers in parenheses are he lag lenghs, and numbers in brackes are he p-values. The models wih AR(1) and AR(3) specificaions for e do no qualify for a clean bill of healh eiher. Tables 2 and 4 show ha boh he Q and B-G saisics signal serial correlaion in ε, alhough normaliy and homoscedasiciy are presen. The only valid models seem o be hose given in Table 3: incorporaing he AR(2)

5 Xiao-Ming Li 61 assumpion for e now makes ε a NIID process, as suggesed by all he repored diagnosic es saisics. Accordingly, and are he only valid highes PL values for he GDP and produciviy daa respecively, and he associaed numbers of breaks are 5 for he former and 4 for he laer. Table 5 presens more deails abou he PL values and break daes for he models wih AR(2) error erms, and he wo columns wih bold numbers represen he wo bes models among hem for he wo series respecively. Table 5. The PL Values and Break Daes When he Residuals Are Assumed o Be AR(2) GDP n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 PL TB TB TB TB TB TB Produciviy n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 PL TB TB TB TB TB TB I is ineresing o compare our resuls wih hose in Bianchi and Kan (1996). They show ha he predicive likelihood approach is robus o differen assumpions abou he error erms e, as he number of srucural breaks urns ou o be he same (n = 2), and he parameer esimaes for differen error specificaions are no very differen. However, his conclusion does no apply in our sudy here, since we find ha he number and he daes of breaks are sensiive o differen error processes assumed. To furher verify wheher his is indeed he case, we conduced a small Mone Carlo sudy. More specifically, we generaed a random variable y using equaion (1) bu assuming 2 breaks (i.e., DU 1, DU 2, DT 1 and DT 2 ) wih heir daes and he values of all coefficiens being given. The error erm e was generaed following an AR(1) process wih he auoregressive coefficien predeermined. When he (correc) AR(1) specificaion is used for he error srucure, he compued highes PL value equals which corresponds o he wo rue breakpoins. However, if he (incorrec) AR(0) process was assumed for he error erm, we obain a higher PL value of associaed wih 4 breaks insead of 2; ha is, we end up wih an over-inflaed number of breakpoins. This suggess ha he validiy of he PL model selecion approach is condiional on he correc specificaion of he error srucure, a leas in his applicaion. Merely acceping he highes PL value among all models wih differen srucural changes and varying error processes could herefore resul in overesimaing or underesimaing he number of breakpoins and possibly in misidenifying heir locaions. The bes models we have seleced, i.e., he models ha have he highes PL values and well-behaved residuals ε, yield he following maximum likelihood esimaes:

6 62 Inernaional Journal of Business and Economics ln GDP = DU DU DU (85.411) (10.705) ( 3.212) ( 7.629) (2.256) DU DU DT DT (4.294) ( 2.776) ( 3.615) ( 0.807) DT DT DT + e (2.403) ( 5.019) (6.555) e = 1.034e e + ε 1 2 (17.053) ( ) (5) and ln PRO = DU DU DU (98.06) (4.428) (3.919) ( 9.480) ( 4.653) DU DU 0.031DT 0.023DT ( 5.685) ( 2.852) (2.773) (7.796) 0.021DT (5.125) 90 + e e = 0.908e e + ε. 1 2 (10.966) ( 9.834) (6) Based on equaions (5) and (6), Figures 1 and 2 depic he long-run growh pahs of China s GDP and labour produciviy. The growh raes of GDP during he six break periods can be calculaed using he poin esimaes of he slopes given in (5). They are: 15.6% ( ), 6.0% ( ), 6.0% ( ; noe ha he coefficien of DT 61 in (5) is insignifican), 9.0% ( ), 2.7% ( ), and 11.3% ( ). By he same calculaion mehod and using he relevan informaion provided in equaion (6), we obain he growh raes of labour produciviy for he five break periods as follows: 8.6% ( ), 1.1% ( ), 4.2% ( ), 6.5% ( ), and 8.6% ( ). Figure 1. The Long-Run Growh Pah of Logarihmic Oupu 8 GDP Trend

7 Xiao-Ming Li 63 Figure 2. The Long-Run Growh Pah of Logarihmic Labour Produciviy Produciviy Trend These changes in long-run growh raes can all be relaed o paricular hisorical evens, bu here are hree noably surprising resuls o poin ou here. Firs, he Culural Revoluion ( ) did no have a negaive impac on labour produciviy growh bu had a posiive growh effec on GDP, conrary o wha people normally believe. Second, i is since he end of he Culural Revoluion in 1976, and no since he inauguraion of economic reforms in 1979, ha boh GDP and labour produciviy have experienced growh akeoff. Tha is, economic reforms did no impac he long-run growh pahs of hese wo series, again a seemingly couner-inuiive resul. Third, 1990 is he firs year of implemening a drasic program of macroeconomic auseriy and is also he immediaely subsequen year of he 1989 poliical crisis (he June 4h even ). However, how hese unfavourable shocks should cause a rise in he long-run growh rae of labour produciviy is a mysery. Clearly, he currenly adoped approach is unable o solve hese puzzles. 4. Conclusions In his paper, a quasi-bayesian model selecion approach developed by Kashwagi (1991) and exended by Bianchi and Kan (1996) is adoped o deec he number and iming of srucural changes in China s GDP and labour produciviy daa. We have shown ha one canno mechanically or blindly use he highes value of he predicive likelihood informaion crierion o pick ou he bes model among models wih and wihou well-behaved residuals: he crierion is valid only when he whie-noise assumpion abou he error erms is saisfied. In our case, only wihin he class of models wih AR(2) error erms can we compare he PL values and use validly he highes one o deermine he number and daes of breakpoins.

8 64 Inernaional Journal of Business and Economics The empirical resuls sugges 5 breakpoins for GDP in 1957, 1961, 1966, 1975, and 1977 and 4 breakpoins for labour produciviy in 1956, 1961, 1976, and These breakpoins divide he sample ino differen growh periods. While mos of he esimaed growh resuls are expeced, some changed/unchanged long-run growh raes do no seem o agree wih common sense. Accordingly, furher research o adop oher model selecion approaches such as he Bayesian one is needed o confirm hese new findings or check heir validiy. References Bai, J. and P. Perron, (1998), Esimaing and Tesing Linear Models wih Muliple Srucural Changes, Economerica, 66, Bai, J., (1999), Likelihood Raio Tess for Muliple Srucural Changes, Journal of Economerics, 91, Ben-David, D. and D. H. Papell, (1998), Slowdowns and Meldowns: Poswar Growh Evidence from 74 Counries, Review of Economics and Saisics, 80, Ben-David, D. and D. H. Papell, (2000), Some Evidence on he Coninuiy of he Growh Process among he G7 Counries, Economic Inquiry, 38, Ben-David, D., R. L. Lumsdaine, and D. H. Papell, (1999), Uni Roos, Pos-war Slowdowns and Long-run Growh: Evidence from Two Srucural Breaks, NBER Working Papers, No Bianchi, M., (1995), Time Series Modelling in he Presence of Srucural Change, Ph.D. Disseraion, Universie Caholique de Louvain. Bianchi, M. and K. Kan, (1996), Segmened Trend Modelling of he US GNP Series, Applied Economics, 28, Inwood, K. and T. Sengos, (1991), Disconinuiies in Canadian Economics Growh, , Exploraions in Economic Hisory, 28, Kanas, A., (1998), Tesing for a Uni Roo in ERM Exchange Raes in he Presence of Srucural Breaks: Evidence from he Boosrap, Applied Economics Leers, 5, Kashwagi, N., (1991), Bayesian Deecion of Srucural Changes, Annals of he Insiue of Saisical Mahemaics, 43, Kim, I. M. and G. S. Maddala, (1991), Muliple Srucural Breaks and Uni Roos in Exchange Raes, Paper presened a he Economeric Sociey Meeing a New Orleans, Dec Lumsdaine, R. L. and D. H. Papell, (1997), Muliple Trend Breaks and he Uni-Roo Hypohesis, Review of Economics and Saisics, 79, Maddala, G. S. and I. M. Kim (1996a), Bayesian Deecion of Srucural Changes, in Bayesian Analysis in Saisics and Economerics, D. A. Berry, K. M. Chaloner, J. K. Geweke, G. S. Maddala, and I. M. Kim eds., New York: Wiley, Maddala, G. S. and I. M. Kim, (1996b), Srucural Change and Uni Roos, Journal of Saisical Planning and Inference, 49,

9 Xiao-Ming Li 65 Maddala, G. S. and I. M. Kim, (1998), Uni Roos, Coinegraion, and Srucural Change, Cambridge, Cambridge Universiy Press. Newbold, P., S. Leybourne, and M. E. Wohar, (2001), Trend-Saionariy, Difference-Saionariy, or Neiher: Furher Diagnosic Tess wih an Applicaion o U.S. Real GNP, , Journal of Economics and Business, 53, Nunes, L., C. P. Newbold, and C. M. Kuan, (1997), Tesing for Uni Roos wih Breaks: Evidence on he Grea Crash and he Uni Roo Hypohesis Reconsidered, Oxford Bullein of Economics and Saisics, 59, Perron, P., (1989), The Grea Crash, he Oil Price Shock, and he Uni Roo Hypohesis, Economerica, 57, Perron, P., (1997), Furher Evidence on Breaking Trend Funcions in Macroeconomic Variables, Journal of Economerics, 80, Raj, B. and D. J. Sloje, (1994), The Trend Behaviour of Alernaive Income Inequaliy Measures in he Unied Saes from and he Srucural Break, Journal of Business & Economic Saisics, 12, Vogelsang, T. J., (1997), Wald-Type Tess for Deecing Breaks in he Trend Funcion of a Dynamic Time Series, Economeric Theory, 13, Wang, J. and E. Zivo, (2000), A Bayesian Time Series Model of Muliple Srucural Changes in Level, Trend, and Variance, Journal of Business and Economic Saisics, 18, Zivo, E. and D. W. K. Andrews, (1992), Furher Evidence on he Grea Crash, he Oil-Price Shock, and he Uni-Roo Hypohesis, Journal of Business and Economic Saisics, 10,