Time Series Test of Nonlinear Convergence and Transitional Dynamics. Terence Tai-Leung Chong

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1 Time Series Tes of Nonlinear Convergence and Transiional Dynamics Terence Tai-Leung Chong Deparmen of Economics, The Chinese Universiy of Hong Kong Melvin J. Hinich Signal and Informaion Sciences Laboraory Applied Research Laboraories Universiy of Texas a Ausin Venus Khim-Sen Liew Labuan School of Inernaional Business and Finance, Universii Malaysia Sabah Kian-Ping Lim Labuan School of Inernaional Business and Finance, Universii Malaysia Sabah Firs Draf: 24 May 2006 This Version: 21 Augus 2007 Absrac: A growing number of recen empirical sudies reveal ha oupu differenials are nonlinear. This paper revisis he income convergence hypohesis by using he nonlinear uni roo es of Kapeanios e al. (2003). The es is applied o OECD counries. Ou of he 12 income gaps in which nonlineariy has been deeced, wo cases of long-run converging and four cases of caching up are found. Our resuls are complemenary o hose of Greasley and Oxley (1997) and Benzen (2005). JEL Classificaions: C32, F43, O40 Key Words: OECD, long-run convergence, caching up, nonlinear uni roo es 1. Inroducion The income convergence hypohesis posulaes ha he growh raes of poor and rich counries will converge in he long run. Bernard and Durlauf (1995) use a ime series approach o es he 1

2 income convergence hypohesis. However, Oxley and Greasley (1995) argue ha he rejecion of convergence by he ime series es should no be necessarily aken as evidence of divergence, because counries may sill be in he ransiional process of convergence. They refine he concep of convergence ino long-run converging and caching up. Long-run converging refers o he aainmen of long-run seady-sae equilibrium in he oupu differenial beween wo conrasing counries. From definiion 2 in Bernard and Durlauf (1996), counry i and j converge if lim E(y i,+k -y j,+k I )=0 as k goes o infiniy, where y i is he log real GDP per capia in counry i, I is he informaion se available a period. In conras, caching up refers o he siuaion in which he narrowing of oupu gap beween he wo counries is observed over ime bu he convergence process has ye o be compleed. Daa (2003) and Benzen (2005) re-examine he convergence debae by relaxing he assumpion of srucural sabiliy (Chong, 2001). Daa (2003) argues ha income dispariies among counries are mos likely aribuable o caching up raher han divergence. He poins ou ha nonlineariy may affec he power of he ime series based es, which is under he linear and ime-invarian assumpions. To explore he nonlineariy issue, his paper ess if incomes are converging in a nonlinear manner by using he es of Kapeanios e al. (2003). 2. Nonlinear Tes of Income Convergence Le Y OECD and US Y be he real per capial gross domesic producs of he individual OECD counry and he Unied Saes respecively. Consider he model z = µ + z 1 α + + n k = 1 δ k z k + ε, (1) where z = logy OECD logy US, µ is he mean of z and ε refers o he error erm. Comparing o he es of Bernard and Durlauf (1995), he above specificaion of he income convergence es has he advanage of disinguishing he long-run converging from caching up. The es of caching up and long-run converging requires he oupu differenial o be saionary. Empirically, he absence of uni roo ( < 0), implies eiher caching up in he presence of a deerminisic rend (α 0), or long-run converging if he deerminisic rend is absen (α = 0). If he oupu differenial conains a uni roo ( = 0), hen he per capial oupu of he wo counries are said o diverge over ime. Equaion (1) may no be able o deec convergence if z is nonlinear. 2

3 Kapeanios e al. (2003) exend he augmened Dickey-Fuller uni roo es o incorporae nonlineariy as characerized by he Smooh Transiion Auoregressive (STAR) process: p x = ρ j x + δx + υ, (2) j= 1 j 3 1 where x = z ˆ α ˆ β, is he de-meaned and de-rended series wih ˆα and ˆβ being he leas squares esimaors obained from regressing z on a consan and a rend erms. The null hypohesis of H 0 : δ = 0 (nonsaionary) agains he alernaive H 1 : δ < 0 (saionary) can be esed. Alhough his es is useful in he sudy of nonlinear income convergence, i canno ell he significance of he deerminisic rend, so i is no direcly applicable o our conex here. Subsequenly, here is no way o disinguish beween long-run converging and caching up, even if nonlinear saionariy is found. In he ligh of his, his paper modifies he Kapeanios e al. (2003) uni roo es and Oxley and Greasley (1995) ime series es of income convergence. We incorporae an addiive inercep ( µ ) and rend componen [ G (rend) ] ino Equaion (2) o yield: p 3 µ ρj j δ 1 j = 1 y = + y + y + φ G (rend) + ξ, (3) where y is he original series under sudy, which is differen from he de-meaned and derended series x. G (rend) is he rend componen of specific funcional form. Two commonly used rend variable are he linear rend and he square of he rend (hereafer, referred as linear and nonlinear rend). ξ is he error erm. The saisical inerpreaion of Equaion (3) is analogous o ha of Oxley and Greasley (1995). The absence of nonlinear uni roo ( δ < 0) implies eiher nonlinear caching up, given he 3

4 presence of deerminisic rend (φ 0), or nonlinear long-run converging if deerminisic rend is absen (φ = 0). However, if he oupu differenial conains a nonlinear uni roo (δ = 0), he per capial oupu of he wo conrasing counries are said o diverge over ime. As in he case of Kapeanios e al. (2003), he saisical significance of δ and φ can be esed using saisics. Since he asympoic disribuion of he saisic in his case is also unknown, he corresponding criical values are simulaed from 5000 replicaions of various sample sizes. The resuling criical values are given in Tables 1a and 1b. Tables 1a and 1b Here 3. Resuls and Conclusions The real GDP per capia of 15 OECD counries relaive o ha of he US are examined. The primary daa se, wih he sample period ranging from 1950 o 2000, is obained from he Penn World Tables 1 (PWT). We firs check he lineariy of he resuling series of income gaps by he lineariy es of Luukkonen e al. (1988). The following model is esimaed: p 2 3 z= θ0 + ( θ1 kz k+ θ2kz kz d+ θ3kz kz d) + θ4z d+ ε, (4) k = 1 where he mainain hypohesis of lineariy ( θ2k = θ3k = θ4 =0 for all k ) is esed agains he alernaive hypohesis of nonlineariy using he F-ype es saisic. The opimal auoregressive lag lengh (k) and he opimal delay lag lengh (d), which are deermined empirically based on sample daa, are chosen from k {1,...,4} and d {1,...,4} such ha he F-es saisic is opimized. The marginal significance value (msv) of he implied F-es saisic is hen boosrapped. The resuls of lineariy es are repored in Table 2. I is obvious from Table 2 ha he null hypohesis of linear income gap canno be rejeced for Denmark, Germany and Ialy. However, for he res of he OECD counries, here is evidence of nonlinear income gaps a convenional significance levels. Nonlinear income gaps are found in 12 ou of 15 OECD counries. Table 2 Here 1 Available a hp:// 4

5 The modified KSS nonlinear uni roo es is applied o he income gaps (wih respec o he Unied Saes) of hese 12 OECD counries. Table 3 shows he resuls of he KSS es wih consan and linear rend. Table 3 Here The esimaors of he parameers of ineres in Equaion (3), δ and φ, ogeher wih he corresponding -saisics are repored. Noe ha he 10, 5 and 1% simulaed criical values for 50 observaions are -3.06, and respecively (Table 1a). Uni roo is found in 8 OECD income gaps (Belgium, Canada, Finland, France, Japan, Norway, Sweden and Swizerland), which provides evidence agains income convergence beween hese counries wih respec o he Unied Saes. On he oher hand, no uni roo is found in he income gaps of Ausralia, Ausria, he Neherlands and he Unied Kingdom, implying he rejecion of income divergence. Upon he rejecion of income divergence, we can furher examine wheher hese four counries are in he process of caching up or have aained long-run converging wih respec o he Unied Saes. The 10, 5 and 1% simulaed criical values of he lef (righ) ail are 2.63 (2.62), (3.02) and (3.76) respecively. I is observed ha he rend erm is insignifican in he case of Ausralia, Ausria and he Neherlands, which provides evidence supporing long-run converging. We also perform he KSS es wih a consan and a nonlinear rend for comparison and he resuls are repored in Table 4. Table 4 Here Noe ha he 10, 5 and 1% simulaed criical values for 50 observaions are -3.10, and respecively (Table 1b). As for φ, he corresponding criical values of he lef (righ) ail are (2.65), (2.99) and (3.75) respecively. From he saisics of he esimaed δ, Belgium, Canada, Finland, France, Japan and Norway exhibi income divergence wih respec o Unied Saes. Meanwhile, Ausria and he Neherlands have aained he sae of long-run converging wih he U.S., whereas Ausralia, Sweden, Swizerland and he Unied Kingdom are in he process of caching up. Our resuls are complemenary o hose of Greasley and Oxley (1997) and Benzen (2005). References 5

6 Benzen, J., Tesing for caching-up periods in ime-series convergence. Economics Leers 88, Bernard, A. B. and S. N. Durlauf, Convergence in inernaional oupus. Journal of Applied Economerics 10, Bernard, A. B. and S. N. Durlauf, Inerpreing ess of he convergence hypohesis. Journal of Economerics 71, Chong, T. T. L., Srucural change in AR(1) models. Economeric Theory 17, Chong, T. T. L., Esimaing he differencing parameer via he parial auocorrelaion funcion. Journal of Economerics 97, Daa, A., Time series es of convergence and ransiional dynamics. Economics Leers 81, Greasley, D. and L. Oxley, Time-series based ess of he convergence hypohesis: Some posiive resuls. Economics Leers 56, Kapeanios, G., Shin, Y. and A. Snell, Tesing for a uni roo in he nonlinear STAR framework. Journal of Economerics 112, Luukkonen, R., Saikkonen, P. and T. Teräsvira, Tesing lineariy agains Smooh Transiion Auoregressive Models. Biomerika 75, Oxley, L. and D. Greasley, A ime series perspecive on convergence: Ausralia, UK and USA since Economic Record 71,

7 Table 1a: The Simulaed Criical Values of he -saisics for δ in Equaion (3): Sample Size Specificaion of Trend Linear ( rend ) 2 Nonlinear ( rend ) 10% 5% 1% 10% 5% 1% Table 1b: The Simulaed Criical Values of he -saisics for φ in Equaion (3): Sample Size Simulaed Criical Values Lef-ail Righ-ail 10% 5% 1% 10% 5% 1% Panel A: Specificaion of Trend: Linear ( rend ) Panel B: Specificaion of Trend: Nonlinear ( rend )

8 Table 2. Lineariy Tes Series k d F-saisic Boosrap msv Ausralia [0.0003] I Ausria [0.0576] X Belgium [0.0039] I Canada [0.0883] X Denmark [0.1917] Finland [0.0029] I France [0.0703] X Germany [0.2719] Ialy [0.2224] Japan [0.0029] I Neherlands [0.0050] I Norway [0.0278] V Sweden [0.0010] I Swizerland [0.0141] V Unied Kingdom [0.0067] I Noes: Superscrips I,V and X denoe significan a 1, 5 and 10% respecively. 8

9 Table 3. KSS Tes wih Consan and Linear Trend Series lag δ φ Esimaor saisic Esimaor saisic Ausralia I Ausria V Belgium Canada Finland France Japan Neherlands I Norway Sweden I Swizerland V Unied Kingdom I I Table 4. KSS Tes wih Consan and Nonlinear Trend Series lag δ φ Esimaor saisic Esimaor ( 10 3 ) saisic Ausralia I X Ausria V Belgium Canada Finland France Japan Neherlands I Norway Sweden V I Swizerland X V Unied Kingdom V V 9