Time Series Test of Nonlinear Convergence and ... - CiteSeerX

Time Series Test of Nonlinear Convergence and Transitional Dynamics

Terence Tai-Leung Chong Department of Economics, The Chinese University of Hong Kong Melvin J. Hinich Signal and Information Sciences Laboratory Applied Research Laboratories University of Texas at Austin Venus Khim-Sen Liew Labuan School of International Business and Finance, Universiti Malaysia Sabah Kian-Ping Lim Labuan School of International Business and Finance, Universiti Malaysia Sabah First Draft: 24 May 2006 This Version: 21 August 2007

Abstract: A growing number of recent empirical studies reveal that output differentials are nonlinear. This paper revisits the income convergence hypothesis by using the nonlinear unit root test of Kapetanios et al. (2003). The test is applied to OECD countries. Out of the 12 income gaps in which nonlinearity has been detected, two cases of long-run converging and four cases of catching up are found. Our results are complementary to those of Greasley and Oxley (1997) and Bentzen (2005).

JEL Classifications: C32, F43, O40 Key Words: OECD, long-run convergence, catching up, nonlinear unit root test

1. Introduction The income convergence hypothesis postulates that the growth rates of poor and rich countries will converge in the long run. Bernard and Durlauf (1995) use a time series approach to test the

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income convergence hypothesis. However, Oxley and Greasley (1995) argue that the rejection of convergence by the time series test should not be necessarily taken as evidence of divergence, because countries may still be in the transitional process of convergence. They refine the concept of convergence into long-run converging and catching up. Long-run converging refers to the attainment of long-run steady-state equilibrium in the output differential between two contrasting countries. From definition 2 in Bernard and Durlauf (1996), country i and j converge if lim E(yi,t+k-yj,t+k|It)=0 as k goes to infinity, where yi is the log real GDP per capita in country i, It is the information set available at period t. In contrast, catching up refers to the situation in which the narrowing of output gap between the two countries is observed over time but the convergence process has yet to be completed. Datta (2003) and Bentzen (2005) re-examine the convergence debate by relaxing the assumption of structural stability (Chong, 2001). Datta (2003) argues that income disparities among countries are most likely attributable to catching up rather than divergence. He points out that nonlinearity may affect the power of the time series based test, which is under the linear and time-invariant assumptions. To explore the nonlinearity issue, this paper tests if incomes are converging in a nonlinear manner by using the test of Kapetanios et al. (2003). 2. Nonlinear Test of Income Convergence Let YOECD and YUS be the real per capital gross domestic products of the individual OECD t

t

country and the United States respectively. Consider the model n

∆zt =

µ + ∂z t −1 + αt + ∑ δ k ∆z t − k + ε t ,

(1)

k =1

where zt = log YOECD − log YUS , µ is the mean of z t and ε t refers to the error term. Comparing to t

t

the test of Bernard and Durlauf (1995), the above specification of the income convergence test has the advantage of distinguishing the long-run converging from catching up. The test of catching up and long-run converging requires the output differential to be stationary. Empirically, the absence of unit root ( ∂ < 0), implies either catching up in the presence of a deterministic trend ( α ≠ 0), or long-run converging if the deterministic trend is absent ( α = 0). If the output differential contains a unit root ( ∂ = 0), then the per capital output of the two countries are said to diverge over time. Equation (1) may not be able to detect convergence if z t is nonlinear.

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Kapetanios et al. (2003) extend the augmented Dickey-Fuller unit root test to incorporate nonlinearity as characterized by the Smooth Transition Autoregressive (STAR) process: p

∆xt = ∑ ρ j ∆xt − j + δxt3−1 + υ t ,

(2)

j =1

where xt = zt − αˆ − βˆ t ,

is the de-meaned and de-trended series with αˆ and βˆ being the least squares estimators obtained from regressing zt on a constant and a trend terms. The null hypothesis of H 0 : δ = 0 (non-

stationary) against the alternative H 1 : δ < 0 (stationary) can be tested. Although this test is useful in the study of nonlinear income convergence, it cannot tell the significance of the deterministic trend, so it is not directly applicable to our context here. Subsequently, there is no way to distinguish between long-run converging and catching up, even if nonlinear stationarity is found. In the light of this, this paper modifies the Kapetanios et al. (2003) unit root test and Oxley and Greasley (1995) time series test of income convergence. We incorporate an additive intercept ( µ ) and trend component [ G (trend ) ] into Equation (2) to yield: p

∆yt = µ + ∑ ρ j ∆yt − j + δ yt3−1 + j =1

φ G (trend ) + ξ t ,

(3)

where y t is the original series under study, which is different from the de-meaned and de-

trended series xt . G (trend ) is the trend component of specific functional form. Two commonly used trend variable are the linear trend and the square of the trend (hereafter, referred as linear and nonlinear trend). ξ t is the error term. The statistical interpretation of Equation (3) is analogous to that of Oxley and Greasley (1995). The absence of nonlinear unit root ( δ < 0) implies either nonlinear catching up, given the

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presence of deterministic trend ( φ ≠ 0), or nonlinear long-run converging if deterministic trend is absent ( φ = 0). However, if the output differential contains a nonlinear unit root ( δ = 0), the per capital output of the two contrasting countries are said to diverge over time. As in the case of Kapetanios et al. (2003), the statistical significance of δ and φ can be tested using t statistics. Since the asymptotic distribution of the t statistic in this case is also unknown, the corresponding critical values are simulated from 5000 replications of various sample sizes. The resulting critical values are given in Tables 1a and 1b. Tables 1a and 1b Here 3. Results and Conclusions

The real GDP per capita of 15 OECD countries relative to that of the US are examined. The primary data set, with the sample period ranging from 1950 to 2000, is obtained from the Penn World Tables1 (PWT). We first check the linearity of the resulting series of income gaps by the linearity test of Luukkonen et al. (1988). The following model is estimated:  p  zt = θ 0 +  ∑ (θ1k zt − k + θ 2 k zt − k zt − d + θ 3 k zt − k zt2− d )  + θ 4 zt3− d + ε t ,  k =1 

(4)

where the maintain hypothesis of linearity ( θ 2 k = θ 3k = θ 4 =0 for all k ) is tested against the alternative hypothesis of nonlinearity using the F-type test statistic. The optimal autoregressive lag length (k) and the optimal delay lag length (d), which are determined empirically based on sample data, are chosen from k ∈ {1,..., 4} and d ∈ {1,..., 4} such that the F-test statistic is optimized. The marginal significance value (msv) of the implied F-test statistic is then bootstrapped. The results of linearity test are reported in Table 2. It is obvious from Table 2 that the null hypothesis of linear income gap cannot be rejected for Denmark, Germany and Italy. However, for the rest of the OECD countries, there is evidence of nonlinear income gaps at conventional significance levels. Nonlinear income gaps are found in 12 out of 15 OECD countries. Table 2 Here

1

Available at http://www.bized.ac.uk/dataserv/penndata/penn.htm.

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The modified KSS nonlinear unit root test is applied to the income gaps (with respect to the United States) of these 12 OECD countries. Table 3 shows the results of the KSS test with constant and linear trend. Table 3 Here

The estimators of the parameters of interest in Equation (3), δ and φ , together with the corresponding t-statistics are reported. Note that the 10, 5 and 1% simulated critical t values for 50 observations are -3.06, -3.38 and -4.05 respectively (Table 1a). Unit root is found in 8 OECD income gaps (Belgium, Canada, Finland, France, Japan, Norway, Sweden and Switzerland), which provides evidence against income convergence between these countries with respect to the United States. On the other hand, no unit root is found in the income gaps of Australia, Austria, the Netherlands and the United Kingdom, implying the rejection of income divergence. Upon the rejection of income divergence, we can further examine whether these four countries are in the process of catching up or have attained long-run converging with respect to the United States. The 10, 5 and 1% simulated t critical values of the left (right) tail are 2.63 (2.62) , -3.07 (3.02) and -3.78 (3.76) respectively. It is observed that the trend term is insignificant in the case of Australia, Austria and the Netherlands, which provides evidence supporting long-run converging. We also perform the KSS test with a constant and a nonlinear trend for comparison and the results are reported in Table 4. Table 4 Here

Note that the 10, 5 and 1% simulated critical t values for 50 observations are -3.10, -3.44 and 4.07 respectively (Table 1b). As for φ , the corresponding critical values of the left (right) tail are -2.66 (2.65), -3.02 (2.99) and -3.86 (3.75) respectively. From the t statistics of the estimated δ , Belgium, Canada, Finland, France, Japan and Norway exhibit income divergence with respect to United States. Meanwhile, Austria and the Netherlands have attained the state of long-run converging with the U.S., whereas Australia, Sweden, Switzerland and the United Kingdom are in the process of catching up. Our results are complementary to those of Greasley and Oxley (1997) and Bentzen (2005). References

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Bentzen, J., 2005. Testing for catching-up periods in time-series convergence. Economics Letters 88, 323-328. Bernard, A. B. and S. N. Durlauf, 1995. Convergence in international outputs. Journal of Applied Econometrics 10, 97–108. Bernard, A. B. and S. N. Durlauf, 1996. Interpreting tests of the convergence hypothesis. Journal of Econometrics 71, 161–173. Chong, T. T. L., 2001. Structural change in AR(1) models. Econometric Theory 17, 87-155. Chong, T. T. L., 2000. Estimating the differencing parameter via the partial autocorrelation function. Journal of Econometrics 97, 365-381. Datta, A., 2003. Time series test of convergence and transitional dynamics. Economics Letters 81, 233-240.

Greasley, D. and L. Oxley, 1997. Time-series based tests of the convergence hypothesis: Some positive results. Economics Letters 56, 143-147. Kapetanios, G., Shin, Y. and A. Snell, 2003. Testing for a unit root in the nonlinear STAR framework. Journal of Econometrics 112, 359–379. Luukkonen, R., Saikkonen, P. and T. Teräsvirta, 1988. Testing linearity against Smooth Transition Autoregressive Models. Biometrika 75, 491–499. Oxley, L. and D. Greasley, 1995. A time series perspective on convergence: Australia, UK and USA

since

1870.

Economic

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Record

71,

259–270.

Table 1a: The Simulated Critical Values of the t-statistics for δ in Equation (3):

Sample Size

25 50 100 200 400 800

Specification of Trend Linear ( trend ) 10% 5% 1% -3.10 -3.42 -4.33 -3.06 -3.38 -4.05 -3.05 -3.35 -3.96 -3.03 -3.31 -3.90 -3.00 -3.29 -3.89 -2.99 -3.29 -3.88

Nonlinear ( trend 2 ) 10% 5% -3.13 -3.50 -3.10 -3.44 -3.07 -3.40 -3.06 -3.39 -3.04 -3.35 -3.04 -3.35

1% -4.31 -4.07 -4.02 -3.96 -3.94 -3.94

Table 1b: The Simulated Critical Values of the t-statistics for φ in Equation (3):

Sample Size

25 50 100 200 400 800 25 50 100 200 400 800

Simulated Critical Values Left-tail Right-tail 10% 5% 1% 10% Panel A: Specification of Trend: Linear ( trend ) -2.66 -3.09 -4.10 2.67 -2.63 -3.07 -3.78 2.62 -2.57 -2.94 -3.68 2.59 -2.56 -2.91 -3.65 2.57 -2.54 -2.90 -3.60 2.54 -2.51 -2.89 -3.54 2.53 Panel B: Specification of Trend: Nonlinear ( trend 2 ) -2.69 -3.12 -4.14 2.69 -2.66 -3.02 -3.86 2.65 -2.65 -3.98 -3.74 2.63 -3.63 -2.96 -3.63 2.60 -2.62 -2.95 -3.62 2.60 -2.62 -2.94 -3.62 2.60

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5%

1%

3.10 3.02 2.93 2.91 2.87 2.90

4.12 3.76 3.65 3.63 3.58 3.56

3.12 2.99 2.97 2.96 2.94 2.94

4.16 3.81 3.70 3.66 3.63 3.62

Table 2. Linearity Test

Series

k

d

F-statistic

Bootstrap msv

Australia

1

4

10.1151

[0.0003]I

Austria

1

3

3.0525

[0.0576]X

Belgium

1

1

6.3405

[0.0039]I

Canada

2

4

2.3312

[0.0883]X

Denmark

1

1

1.7166

[0.1917]

Finland

1

1

4.1310

[0.0029]I

France

2

1

2.5320

[0.0703]X

Germany

4

1

1.3331

[0.2719]

Italy

1

3

1.5571

[0.2224]

Japan

1

4

6.7077

[0.0029]I

Netherlands

2

2

3.9866

[0.0050]I

Norway

2

4

3.3597

[0.0278]V

Sweden

1

1

8.1277

[0.0010]I

Switzerland

2

2

4.7174

[0.0141]V

United Kingdom

1

1

5.6389

[0.0067]I

Notes: Superscripts I,V and X denote significant at 1, 5 and 10% respectively.

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Table 3. KSS Test with Constant and Linear Trend

Series Australia Austria Belgium Canada Finland France Japan Netherlands Norway Sweden Switzerland United Kingdom

lag

δ

1 3 3 1 3 3 3 3 1 3 3 3

Estimator -2.9640 -0.1329 -0.1787 -1.4445 -0.2621 -0.1504 -0.0107 -0.9767 -0.5318 -0.9132 -3.3806 -1.2478

φ t statistic -4.1020I -3.4391V -1.3084 -1.7594 -2.7844 -1.1670 -0.0794 -4.2263I -2.2411 -2.8819 -2.8182 -4.2524I

Estimator -0.0007 -0.0001 -0.0002 -0.0003 0.0005 -0.0013 -0.0017 -0.0002 0.0010 -0.0013 -0.0015 -0.0012

t statistic -2.3915 -0.1774 -0.4399 -1.6423 0.7874 -1.7631 -1.8197 -0.7735 1.6689 -3.8379I -3.5410V -4.1191I

Table 4. KSS Test with Constant and Nonlinear Trend

Series Australia Austria Belgium Canada Finland France Japan Netherlands Norway Sweden Switzerland United Kingdom

lag

δ

1 3 3 1 3 9 9 3 1 3 3 3

Estimator -3.3568 -0.1308 -0.1766 -1.8253 -0.2256 -0.4650 -0.0516 -0.9993 -0.3015 -1.2110 -4.0956 -1.0648

φ t statistic -4.4737I -3.9963V -1.6516 -2.0877 -2.9112 -1.9385 -0.8412 -4.5127I -1.6570 -3.5960V -3.1699X -3.4541V

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Estimator ( ×10−3 ) -0.0164 -0.2962 -0.5024 -0.0007 -0.0376 -0.3721 -0.0195 -0.3948 -0.7277 -0.0247 -0.0280 -0.0170

t statistic -2.8727X -0.4028 -0.7190 -1.9476 0.4229 -1.9172 -1.8146 -0.8827 0.8305 -3.9633I -3.4426V -3.1517V