Authors
^{1} Allameh Tabataba'i University,
^{2} Department of Finance, Feng Chia University
^{3} Department of Finance, National Su Yatsen University
^{4} Faculty of Economics, University of Mazandaran
Abstract
Keywords
Introduction
The income convergence hypothesis is one of the controversial predictions of the neoclassical growth theory. The hypothesis predicts that countries with similar initial conditions will move toward a perpetual common balanced growth path, and the difference of per capita income among countries will disappear. But this prediction has rejected by endogenous growth theory is also due to inconclusive empirical results therein. Its empirical validity remains to be controversial. In terms of the empirical analyses to the convergence hypothesis, researchers have examined convergence with several wayssuch as absolute convergence, conditional convergence, and catchingup (stochastic convergence) hypothesis and they examine employing various methodologies like crosssectional approach, distribution approach, and time series approach.
The crosssectional and the time series approaches are used to test the absolute and conditional notions of the convergence hypothesis. In the crosssectional approach, the growth rate of per capita income is based on initial per capita income, and a negative (partial) relation or reverse correlation between two variables is interpreted as evidence of the absolute (conditional) convergence.
In the time series framework, the convergence hypothesis is examined by employing the unit roots or stationarity tests. Hence, its empirical validity depends on the unit root or stationarity tests (Lee et al., 2013). In this approach, the deterministic terms (intercept and/or linear trend) do not allow to be included in the unit root or stationarity test with investigating the absolute convergence hypothesis. The unit root or stationarity tests are used to test the conditional convergence hypothesis instead. In addition, the unit root and/or stationarity tests contain intercept and linear trend while testing the catchingup hypothesis. As noted by Cunado & Gracia (2006: 156), the catchingup hypothesis states “it might be appropriate in a context in which convergence is an ongoing process”. Sigma convergence is based on the crosssectional distribution of per capita income among countries. Researchers examine sigma convergence using dispersion indexes as such standard deviation and Gini coefficients and also using distribution dynamics approach. If standard deviation of per capita income across countries decreases over time, it represents that there exists the sigma convergence.
In this paper we attempt to reexamine the catchingup (stochastic convergence) hypothesis among the selected 16 Organization for Economic Cooperation and Development (OECD) countries using the time series approach of convergence hypothesis with annual data over one century. As noted by Cunado & Gracia (2006), examining the catchingup hypothesis needs two steps. The first step (necessary condition) involves testing the unit root hypothesis in the relative per capita GDP, and the second step (sufficient condition) is related to estimate trend function for relative per capita GDP series. In other words, in the second step, the relative per capita real GDP dynamics is modeled as a trend function.
The main target and motivation of our study are that we try to develop the second step of testing catchingup process by introducing a trend function that composes both the sharp breaks and smooth shifts using annual data over one century in length. The reason for us to develop the second step of testing is that according to the demonstration of previous studies to the GDP or per capita GDP behaviors, they are well characterized by sharp breaks (e.g. BenDavid & Papell, 1998; CarrioniSilvestre et al., 2005) and by smooth shifts (e.g. Su & Chang, 2011; Chang et al., 2012). But to the best of our knowledge, none of them incorporates the sharp breaks and smooth shifts together in one investigated model, which is ascribed to that all mentioned above papers examine based on unit root or stationarity tests, but there is no unit root or stationarity test taking both types of structural breaks consideration in a model yet.
In this paper, followed by CarrioniSilvestre et al. (2005, CBL hereafter) and Becker et al. (2006, BEL hereafter) stationarity tests we show that the relative per capita real GDP series are well characterized with sharp breaks (using CBL, 2005) and with smooth shifts (using BEL, 2006). Whereas our stationarity tests identify that the relative per capita real GDP series experience both types of structural breaks, and we thereby, as the second step, specify a trend function to capture the effects of both types of breaks – that is, sharp breaks and smooth shifts. The sharp breaks are modeled by dummy variables and used to identify the break locations with the procedure of Bai & Perron (1998).^{[1]} The smooth shifts are modeled using Fourier terms as proposed by Becker et al. (2004). We also employ this methodology to estimate the trend function, by which the coefficients of intercept and the slope of the trend function can be timevarying for any subperiod (for example between two break dates), and we can be better indicating convergence or divergence in subperiods.
Several previous studies, such as Greasley & Oxley (1997), Li & Papell (1999), Freeman & Yerger (2001), Strazicich et al. (2004), Datta (2003), Dawson & Sen (2007), Christopoulos & LeonLedesma (2008), Chong et al. (2008), and Costantini & Sen (2012) have tested the convergence hypothesis for the OECD countries using the time series framework and found that the convergence hypothesis over two decades 1990s and 2000s is successive advances in the econometric treatment of unit root tests. For example, Greasley & Oxley (1997) tested the bivariate convergence between eight OECD countries using the ADF and Perron’s (1989) unit root tests. Li & Papell (1999) used the Perron (1997) unit root test for 16 OECD countries. In order to relax the structural breaks and capture dynamic behavior, Datta (2003) used the Kalman filtering to test the convergence hypothesis aiming at 15 OECD countries. Strazicich et al. (2004) used the Lee & Strazicich (2003) LM unit root test with two structural breaks. Chong et al. (2008) tested the convergence hypothesis toward the USA for 15 OECD countries using Kapetanios et al. (2003) nonlinear unit root test and found that 12 out of 15 OECD income gaps present nonlinear dynamics. Christopoulos & LeonLedesma (2008) developed a simple neoclassical growth theory and showed that some determinants of convergence rate may vary with time. They thereby tested the convergence hypothesis for 14 OECD countries using stationarity covariates and found strong evidence for the convergence in 12 countries. Freeman & Yerger (2001), Fleissig & Strauss (2001), and Cheung & Pascual (2004) tested the convergence hypothesis using panel unit root tests.
The remainder of paper is organized as follows: section 2 presents our methodology. In section 3 we present our data and empirical results. Conclusions are presented in the final section.
2. Methodology
2.1 Time Series Framework of Catchingup Hypothesis
The time series approach of the convergence hypotheses is introduced by Carlino & Mills (1993) and it is extended by Bernard & Durlauf (1995), Evans & Karras (1996), and Li & Papell (1999). By this approach, country i will be converged toward the country j (as a leader or a benchmark country) if and only if:
(1)
Where y is the per capita real GDP in log, is relative per capita real GDP, and is the information set at time t. The indices i and j denote country i and country j, respectively. We can define three versions of the convergence hypothesis using equation (1). If then it shows absolute convergence. In order to test this definition, researchers use unit root or stationarity test without any intercept and linear trend. If and the series () performs level stationarity, then it can be named as conditional convergence or deterministic convergence. If and the series () performs trend stationarity, then it can be named as stochastic convergence or catchingup process.
Testing the catchingup process proceeds with the following two steps. The first step or the necessary condition relates to testing existing unit root/stationarity to the relative per capita real GDP series. The second step or the sufficient step involves the estimation of trend functions for relative per capita real GDP series that the unit root hypothesis is rejected for it (Ranjbar et al., 2013).
2.2 Necessary Condition: Stationarity Tests
In order to test the first step or the necessary condition of catchingup hypothesis, this paper uses the univariate and panel data versions of the CBL (2005) stationarity test and also the BEL (2006) Fourier stationarity test, in which the former allows for sharp breaks and the later allows for an unknown form and a number of smooth drifts.
2.2.1CarrioniSilvestre et al. (2005) Stationarity Test
The CBL (2005) stationarity test is adopted in this study due to several advantages. First, the reversal of the null and alternative hypotheses is the most appealing for the CBL test, because most of the panel unit root tests are equipped with the null hypothesis, in which the rejection of the unit root null implies only parts (but not all) of countries are stationary. By contrast, the null hypothesis of the CBL stationarity test is based on the stationarity throughout all countries. Therefore, if the null hypothesis of the CBL test is rejected, then we say that all of the series in the panel are nonstationary. Second, the CBL method enables us to consider multiple structural breaks positioned at different unknown dates in addition to a different number of breaks for each individual. Allowing the existence of structural breaks can potentially strengthen our results more correctly in respect of specifying the model. Third and finally, we can allow for more general forms of crosssectional correlation than previous studies through the conventional crosssectional demeaning of the data, which assumes that a common factor affects all units with the same intensity. CarrioniSilvestre and GermanSoto (2009) also indicate that the lack of consideration of the crosssectional dependence might bias the analysis to conclude in favor of the stationarity of the panel data even in the case that it is nonstationary. It is important to note that the panel stationarity test controls nonparametrically for serial correlation in the error through the estimation of the longrun variance via kernels. In our study, we employ the bootstrap distribution, tailored to the error structure of panel data, in order to accommodate general forms of crosssectional dependence.
CBL (2005) extended the approach of Hadri (2000) by further allowing for multiple structural breaks through incorporating dummy variables into the deterministic specification of a model.[2] In this case, the data generation process under the null of stationarity is based on following model:
In equation (2), is relative per capita real GDP of country i in year t, as well as , t, and m are intercept, linear trend, and the optimal number of breaks, respectively. The other regressors, and are l^{th} break in intercept and slope of linear trend in year t respectively and are defined as the following:
The univariate test statistic () is computed as Kwiatkowski et al. (1992, KPSS hereafter) test with multiple breaks:[3]
where is the partial sum of the estimated OLS residuals from equation (2), denotes a heteroscedasticity and autocorrelation consistent estimate of the longrun variance of . We estimate the consistent longrun variance using the new boundary condition rule proposed by Sul et al. (2005). For this end, we estimate an AR(p) autoregressive process for each unit by the OLS method:
where are the estimated OLS residuals from equation (2). Then the estimator of longrun variance () is constructed using Sul et al. (2005) boundary condition rule as follows:
where is the autoregressive coefficient estimates from equation (6), and the optimum lag length (p) in equation (6) is determined using the BIC information criterion. is the location of the breaks related to the entire time period (T). The test statistic is dependent on the , which is important in identifying the location and the number of breaks correctly. For this end, the CBL recommend for Bai & Perron (1998) procedure, which is based upon the global minimization of the sum of squared residuals (SSR) expressed as follows:
where is m^{t}^{h} break date. The optimal number of breaks is selected by CBL criterion of Liu et al. (1997). CBL calculated the test statistic for the null of a stationary panel with multiple breaks as follows:
where and are the mean and standard deviation of. We computed the empirical distribution of using Bootstrap techniques following as Maddala & Wu (1999). In step one, we run a regression with equation (2), imposing the null hypothesis of stationarity and then save the resulting residuals () and fitted. In step two, we generate bootstrap residuals following the sampling strategy suggested by Maddala & Wu (1999), with replacement samples of t+100 values (and then discard the first 100 values) from the residual matrix. In step three, we calculate the bootstrap samples of observations as . In step four, we construct the pseudo individual and panel statistics based on equations (5) and (9), respectively. And in step five, we repeat steps 14 for 20,000 times to derive the empirical distribution of under the null hypothesis of regimewise stationary.
2.2.2Becker et al. (2006) Stationarity Test
BEL (2006) developed the standard KPSS stationarity test with a Fourier function that allows the deterministic term in regression to be a timedependent function. Hence the test does not need to prespecified number and form of structural breaks. It can control for unknown number and form of structural breaks using a selected frequency component of a Fourier function. Hence this test is suitable for various series with various types of smooth structural breaks with unknown number and form. Following the BEL (2006), we consider the following data generating process (DGP):
where are stationarity errors and are independent and identically distributed (i.i.d) with variance . Under the null hypothesis that , the process described by equation (10) is stationarity. The rational for selecting is based on the fact that a Fourier expression is capable of approximating functions which are not graded, to any desired degree of accuracy, where k represents the frequency selected for the approximation, and measures the amplitude and displacement of the frequency component.[4] A desire feature of equation (10) is that the standard linear specification is regarded as a special case while setting . It also follows that at least one frequency component must be present if there is a structural break. Here, if it is possible to reject the null hypothesis, the series must have a nonlinear component.[5] Becker et al. (2004) use this property of equation (10) to develop a test which is more powerful to detect breaks of an unknown form than the standard Bai & Perron (1998) test. As the DGP in equation (10) nests used to generate the common KPSS (1992) test, the BEL’s stationarity test with a Fourier function needs only a slight modification of the KPSS statistic. First, one needs to obtain the residuals from the following equation:
Equation (11) tests the null of trend stationarity. The test statistic is given by:
where and are the OLS residuals from regression (11) and the long run variance. In this paper we follow CarrioniSilvestre & Sansó (2006) and use the Sul et al. (2005) method to compute the long run variance. BEL (2006) suggests that the frequencies in equation(11) should be obtained via the minimization of the sum of squared residuals. However, their Monte Carlo experiments suggest that no more than one or two frequencies should be used because of the loss of power associated with a larger number of frequencies.^{[6]}
As seen in equation (11), the conventional KPSS test is one variety of BEL (2006), in which trigonometric component has been ignored. In order to test the presence of nonlinear terms, BEL offered an Ftest expressed as follows:
where denotes the SSR from equation (11), T is time period, q is the number of regressors, and denotes the SSR from the regression without the nonlinear terms. As noted by BEL (2006), the presence of the nuisance parameters causes the distribution of not to be nonstandard. Therefore, in this paper we calculate the critical values for any series. To this end, we first generate 20,000 random series under the null of linearity. Then by using optimum frequency to any actual series, we calculate the Fstatistic to any of 20,000 pseudo series and finally, we obtain the critical values from the sorted vector of pseudo Fstatistic.
2.3 Sufficient Condition: Estimation of Trend Function
As noted by Tomljanovich & Vogelsang (2002), and Cunado & Gracia (2006), for the catchingup hypothesis, the trend stationarity is a necessary condition. In order to test the sufficient condition, we introduce a new methodology for estimating the trend function. Suppose series is trend stationarity and we can specify its trend function as following:
where m is the optimal number of breaks. The other regressors, and are l^{th} break points in intercept and slope of linear trend in year t respectively and are defined as the following:
Variables DU and DT are incorporated into the model for capturing the sharp drifts. Following Gallant (1981) in respect of obtaining a global approximation from the smooth shifts, we use the Fourier approximation and incorporate terms and into the model. n and k indicate the number of frequencies, which are contained in the approximation and equal to and particular frequency, respectively.
The estimation of equation (14) involves with three questions, the choice of m, the choice of n, and the choice of k. As noted by Becker et al. (2004) it is reasonable to restrict n=1 because if is rejected for one frequency, then the null hypothesis of time invariance should be also rejected. Also Enders & Lee (2012) noted that imposing the restriction n=1 is useful to save the degrees of freedom and prevent from overfitting problem. Hence we respecify the equation (14) as follows:
In order to estimate the equation (17), we propose a twostep procedure. In the first step, we determine the optimum break points, m, and optimum frequency, k. For this propose, the allowed maximum k is set to be 5, and we select an integer frequency until allowing smooth shifts temporarily.[7] Then for any K=k, we estimate the equation (17) using the procedure proposed in Bai & Perron (1998) and save the sum of squared residuals (SSR). We select frequency k^{*} as an optimum frequency minimizing the SSR, and we further reestimate the equation (17) with K= k^{*} and select the obtained number and location of break points as optimum number and location of break points. In the second step, we test the absence of the nonlinear component by the equation (17). To this end, following the Becker et al. (2004; 2006), we use the usual Ftest statistic as follows:
and denote the SSR from equation (17) with and without nonlinear component, respectively, and q is the number of regressors. As noted by Becker et al. (2006), due to the presence of nuisance parameter, the Ftest has not standard distribution and we then calculate its critical values employing Monte Carlo simulation.
For testing the catchingup (stochastic convergence) hypothesis, we follow the CarrioniSilvestre and GermanSoto (2009) procedure. We can say that there exists evidence of catchingup process or stochastic convergence when the coefficients of the parameters of each regime are significant at least at the 10% level of significance, and and average slope of trend function () have opposite sign, i.e., when , average slope of trend function is positive; or when , average slope of trend function is negative. If both and average slope of trend function of each regime have the same sign, we conclude that the divergence has occurred. If both parameters ( and ) are insignificant, it suggests that catchingup process has occurred.
3. Data and Empirical Results
3.1 Data and Variable
We collect annual per capita real GDP in 1990 GearyKhamis PPPadjusted dollars for 16 selected OECD countries including; Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Italy, Japan, Netherlands, Norway, Sweden, Switzerland, and the United Kingdom, and the U.S., over the period 18702010. The source of the data is from the New Maddison Project Database. To test the catchingup (stochastic convergence) process, we calculate the ratio of real GDP per capita of each OECD country to real GDP per capita of OECD group exhibited as , where is per capita real GDP for the ith country, and is the yearly average value in the sample span; i and t indicate country i in year t, respectively. The advantage of using relative per capita real GDP is that we can utilize this approach to discuss the income convergence effect in OECD country.
3.2 Empirical Results
As mentioned in the previous section, to test the necessary condition for income convergence, we need to first run the univariate and panel CBL (2005) stationarity test. The results are presented in Table 1. Hence it is possible that relative per capita real GDP time series in our panel are dependent – that is, there is possibility that our panel data suffer from crosscountry dependence, and as we know that the presence of crosssectional dependence might bias our analyses and result in favor of the stationarity of panel data (Lee, 2013; Lee et al., 2013). Hence, before we test the panel stationarity hypothesis using CBL test, we use the Pesaran (2004) crosssection dependence (PCD, hereafter) test so that we can test the crosssectional dependence. Pesaran (2004) developed a simple test for error cross sectional dependence, which has correct size, sufficient power, and is applied to both stationarity and nonstationarity panels. Pesaran’s cross section dependence test proceeds with following three steps. First, the residuals are obtained from the ADF regression for any member of panel. Second, the average of pairwise correlation coefficients of residuals is calculated as follows:
where are residuals from standard ADF regression. The PCD test statistics is computed as follows:
The PCD statistic tests the null hypothesis of cross independence, and they are distributed in standard normality.[8] The PCD statistic test results are shown in the first row of panel A of Table 1. As seen that Pesaran (2004) statistic takes the value of 6.158 with a pvalue 0.000, and it means that we are able to reject the null hypothesis of crosssectional independence at the 1% significance level; whereas the panel statistic of CBL (2005) stationarity test requires the individual statistics to be crosssectional independence. Hence in order to overcome this shortcoming, we compute the empirical distribution of panel statistic of CBL test using the bootstrap techniques as suggested by Maddala & Wu (1999). The critical values for panel statistic are computed by 20,000 replications. As we can see that both versions of panel statistics (homogenous and heterogeneous long run variances) are less than the critical values at the 10% significance level. The findings show that the stationarity of all countries of panel data is not rejected.
Table 1: CarrioniSilvestre et al. (2005) Stationarity Test Results.
Panel A: Pesaran (2004) and Panel Carrioni Silvestre et al. (2005) Test 

Pesaran (2004) cross sectional dependence test 

Test 


Pvalue 


6.158 


0.000 

Carrion i Silvestre et al (2005) stationarity test 
Bartlett 

Critical values 


90 
95 
97.5 
99 

Homogenous long run variance 
0.861 

4.125 
4.841 
5.669 
6.474 

Heterogeneous long run variance 
1.641 

4.184 
4.769 
5.655 
6.327 

Panel B: Univariate Carrioni Silvestre et al. (2005) Stationarity Test with Breaks in Intercept & Trend 

Country 
Bartlett 


90% 


95% 

97.5% 

99% 
Australia 
0.020 


0.025 


0.028 

0.031 

0.034 
Austria 
0.011 


0.048 


0.060 

0.072 

0.088 
Belgium 
0.035 


0.106 


0.134 

0.165 

0.200 
Canada 
0.028 


0.032 


0.037 

0.042 

0.049 
Denmark 
0.026 


0.091 


0.112 

0.136 

0.164 
Finland 
0.044 


0.064 


0.077 

0.090 

0.107 
France 
0.047 


0.073 


0.089 

0.106 

0.130 
Germany 
0.018 


0.049 


0.060 

0.072 

0.089 
Italy 
0.038 


0.102 


0.132 

0.164 

0.207 
Japan 
0.020 


0.048 


0.061 

0.074 

0.090 
Netherlands 
0.029 


0.073 


0.091 

0.110 

0.134 
Norway 
0.008 


0.046 


0.057 

0.068 

0.083 
Sweden 
0.010 


0.034 


0.038 

0.043 

0.049 
Switzerland 
0.037 


0.046 


0.054 

0.061 

0.070 
United Kingdom 
0.015 


0.056 


0.071 

0.086 

0.106 
United States 
0.021 


0.054 


0.068 

0.082 

0.100 
Notes: Critical values for univariate version computed using Monte Carlo simulation and critical values for panel version computed using Bootstrap techniques. Maximum number of breaks fixed at 5.
The results of univariate version of CBL (2005) stationarity test are presented in panel B of Table 1. The critical values for univariate version are computed using Monte Carlo simulation and 20,000 replications. As we can see that the null hypothesis of stationarity is not rejected at the 10% significance level for any country.[9]
The results for BEL (2006) stationarity test are provided in Table 2. In order to run the BEL test, we set maximum frequencies at 5, and we use the Sul et al. (2005) method to choose the kernel and the estimation of long run variance. The significant Fstatistic showed in the third column indicates that both sine and cosine terms should be included in the estimated model for all countries. The numbers in the second column show the optimum frequency for each country. The results show that the optimum frequency of K=5 are fitted for France and Germany, K=4 for Netherlands, K=2 for Australia, Norway, and United Kingdom, and K=1 for the other countries. From comparing the BEL test statistics (Bartlett) with their critical values, we know that the null hypothesis of stationarity is not rejected for each country. In addition, both CBL and BEL stationarity tests do not reject the necessary condition in any country, and we therefore further test the sufficient condition for all the OECD countries.
Table 2: Becker et al.(2006) Stationarity Test Results.
Country 
Optimum frequency 
F statistic 
Truncation lag 
Bartlett 
Critical Values 

90% 
95% 
97.50% 
99% 

Australia 
2 
38.255 
3 
0.034 
0.102 
0.129 
0.154 
0.187 
Austria 
1 
86.1484 
1 
0.031 
0.046 
0.051 
0.053 
0.059 
Belgium 
1 
83.310 
2 
0.029 
0.114 
0.141 
0.184 
0.223 
Canada 
1 
50.133 
3 
0.029 
0.109 
0.139 
0.152 
0.165 
Denmark 
1 
108.734 
2 
0.022 
0.092 
0.108 
0.123 
0.155 
Finland 
1 
40.148 
2 
0.041 
0.092 
0.111 
0.136 
0.161 
France 
5 
10.630 
5 
0.083 
0.114 
0.137 
0.144 
0.15 
Germany 
5 
10.175 
2 
0.038 
0.121 
0.148 
0.161 
0.19 
Italy 
1 
132.992 
3 
0.032 
0.129 
0.169 
0.180 
0.203 
Japan 
1 
35.617 
1 
0.018 
0.049 
0.056 
0.064 
0.071 
Netherlands 
4 
15.695 
2 
0.092 
0.098 
0.116 
0.131 
0.158 
Norway 
2 
66.393 
1 
0.038 
0.051 
0.054 
0.058 
0.066 
Sweden 
1 
291.370 
2 
0.028 
0.104 
0.123 
0.136 
0.149 
Switzerland 
1 
243.067 
5 
0.055 
0.090 
0.102 
0.107 
0.124 
United Kingdom 
2 
29.443 
2 
0.057 
0.096 
0.13 
0.188 
0.211 
USA 
1 
39.639 
2 
0.025 
0.094 
0.117 
0.149 
0.175 
Notes: The finite sample critical values for flexible Fourier KPSS test (Bartlett) statistic were calculated with 20000 replications.
In order to run the second step or test the sufficient condition for the catchingup hypothesis, we estimate the equation (17) for 16 countries and report the results in Table 3. To this end, we set a maximum break point at 8 and a maximum frequency at 5. The results of a gridsearch for finding the best frequency, presented in the second column of panel A in Table 3, indicate that the case K=1 (frequency) is fitted for Belgium and Italy, K=2 for Finland, France, Norway, and Sweden, K=3 for Japan and Netherland, K=4 for Denmark and the United Kingdom, and K=5 for Australia, Austria, Canada, Germany, Switzerland, and the U.S.
Table 3: Estimation Results for Trend Function in Equation (17), 18702010.
Panel A: The Results for Optimum Frequency, F Statistic and its Critical Values 

Country 
Optimum frequency 
F stat 
90% 
95% 
97.50% 

99% 
Australia 
5 
20.863** 
2.125 
2.457 
2.727 

4.022 
Austria 
5 
6.278** 
2.297 
3.478 
4.369 

4.691 
Belgium 
1 
97.278** 
2.365 
3.596 
3.872 

4.212 
Canada 
5 
5.412** 
2.495 
2.883 
3.164 

4.599 
Denmark 
4 
3.817** 
2.124 
2.662 
3.635 

5.606 
Finland 
2 
28.986** 
2.923 
3.577 
4.246 

6.956 
France 
2 
0.168 
2.310 
2.670 
3.381 

3.434 
Germany 
5 
8.124** 
2.763 
2.923 
3.447 

5.041 
Italy 
1 
139.049** 
2.244 
2.617 
3.201 

4.183 
Japan 
3 
19.984** 
2.642 
3.434 
4.372 

4.762 
Netherlands 
3 
3.255** 
2.080 
2.898 
3.179 

3.735 
Norway 
2 
108.428** 
2.235 
3.082 
3.574 

5.249 
Sweden 
2 
5.755** 
2.727 
3.336 
4.085 

5.243 
Switzerland 
5 
16.130** 
2.335 
3.091 
3.334 

4.452 
United Kingdom 
4 
22.721** 
2.464 
2.858 
3.312 

4.265 
United States 
5 
4.461** 
2.402 
3.120 
3.909 

5.094 
Panel B: The Results for Sharp Drift Dates in Equation (17) 

Country 
Break dates 

First 
Second 
Third 
Fourth 
Fifth 
Sixth 

Australia 
1891 [1889,1892] 
1927 [1924,1928] 
1941 [1939,1942] 
1966 [1964,1967] 
1990 [1987,1991] 

Austria 
1913 [1911,1914] 
1930 [1929,1931] 
1944 [1942,1945] 
1958 [1956,1959] 


Belgium 
1900 [1899,1901] 
1919 [1917,1920] 
1944 [1942,1945] 
1977 [1975,1978] 
1993 [1991,1994] 

Canada 
1900 [1898,1906] 
1918 [1916,1921] 
1932 [1930,1933] 
1946 [1944,1947] 
1961 [1959,1963] 

Denmark 
1929 [1928,1930] 
1943 [1941,1948] 
1982 [1980,1989] 



Finland 
1894 [1890,1895] 
1916 [1914,1917] 
1944 [1942,1945] 
1991 [1989,1992] 


France 
1916 [1915,1917] 
1931 [1928,1932] 
1945 [1944,1946] 
1977 [1975,1978] 


Germany 
1913 [1910,1914] 
1931 [1928,1932] 
1945 [1943,1946] 
1959 [1957,1960] 


Italy 
1886 [1884,1887] 
1942 [1940,1943] 
1995 [1994,1996] 



Japan 
1889 [1887,1890] 
1916 [1915,1917] 
1930 [1928,1931] 
1944 [1942,1945] 
1973 [1971,1974] 
1990 [1988,1991] 
Netherlands 
1906 [1903,1907] 
1931 [1930,1932] 
1945 [1944,1946] 
1981 [1979,1982] 


Norway 
1891 [1890,1892] 
1913 [1912,1914] 
1929 [1927,1931] 
1944 [1942,1946] 
1987 [1985,1988] 

Sweden 
1890 [1888,1891] 
1913 [1912,1914] 
1929 [1926,1930] 
1944 [1942,1945] 
1960 [1958,1962] 
1991 [1988,1992] 
Switzerland 
1884 [1881,1885] 
1930 [1929,1931] 
1944 [1942,1946] 
1995 [1993,1996] 


United Kingdom 
1914 [1913,1915] 
1928 [1926,1929] 
1946 [1945,1947] 
1971 [1969,1972] 


United States 
1931 [1930,1932] 
1945 [1943,1946] 
1976 [1974,1977] 



Panel C: Catchingup Phase after any Breaks 

Country 
Before the first break 
1th 
2th 
3th 
4th 
5th 
6th 
Australia 
C 
C 
D 
C 
C 
C 

Austria 
C 
C 
C 
C 
C 


Belgium 
C 
D 
C 
C 
D 
C 

Canada 
C 
D 
C 
C 
C 
C 

Denmark 
C 
C 
C 
C 



Finland 
D 
D 
C 
D 
C 


France 
C 
C 
C 
C 
C 


Germany 
C 
D 
C 
C 
D 


Italy 
C 
C 
C 
D 



Japan 
D 
C 
D 
C 
C 
C 
C 
Netherlands 
C 
D 
C 
C 
C 


Norway 
C 
D 
D 
D 
C 
C 

Sweden 
D 
C 
D 
C 
C 
C 
C 
Switzerland 
D 
C 
C 
C 
D 


United Kingdom 
D 
C 
C 
C 
D 


United States 
C 
C 
D 
C 
C 


Notes: Critical values for F statistic were calculated with 20000 replications. Maximum breaks were fixed at 8 and maximum frequencies were fixed at 5. C and D denote the catchingup and divergence process after any break. The figures in the bracket in panel B are 95% confidence interval.
Next, the results for Fstatistics and its critical values (computed from 20,000 replications) are presented in the columns 37 of panel A of Table 3. The results indicate that both sine and cosine terms should be included in the estimated model for all countries except for France.
Third, we show the estimated break point locations and corresponding 95% confidence intervals which provide information about the degree of uncertainty in the estimation of the break dates in panel B of Table 3. The 95% confidence intervals appear to be very tight for all the break dates in all countries, which make us very confident that the break locations are properly estimated. The dispersion of break point dates is shown in panel B of Table 3, and it shows that all countries experience at least three sharp breaks. Japan and Sweden experience 6 sharp breaks, Australia, Belgium, Canada, and Norway experience 5 breaks, Austria, Finland, France, Germany, Netherland, Switzerland, the United Kingdom experience 4 breaks, and Denmark, Italy, and the U.S. experience 3 breaks in their catchingup processes to the real GDP per capita in OECD countries.
From 69 estimated break points, 10 out of them (i.e., Austria [1913], Belgium [1919], Canada [1918], Finland [1916], France [1916], Germany [1913], Japan [1916], Norway [1913], Sweden [1913], and United Kingdom [1914]) occurred in 1920s that are coincided with years World War I (WWI). 16 out of 69 break points (i.e., Australia [1941], Austria [1944], Belgium [1944], Canada [1946], Denmark [1943], Finland [1944], France[1945], Germany [1945], Italy [1942], Japan [1944], Netherlands [1945], Norway [1944], Sweden [1944], Switzerland [1944], United Kingdom [1946], and USA[1945]) occurred in 1940s that are coincided with years World War II (WWII). 13 out of 69 break points (i.e., Australia [1927], Austria [1930], Canada [1932], Denmark [1929], France [1931], Germany [1931], Japan [1930], Netherlands [1931], Norway [1929], Sweden [1929], Switzerland [1930], United Kingdom [1928], and USA [1931]) occurred over the period 192930 that are coincided with year Great depression. 12 out of 69 break points (i.e., Australia [1990], Belgium [1977], Denmark [1982], Finland [1991], France [1977], Japan [1973 and 1990], Netherlands [1981], Norway [1987], Sweden [1991], United Kingdom [1971], and USA [1976]) are coincided with oil shocks in the early and end stages of 1970s and the early stage of 1990s.
Also our results in panel C of Table 3 show that from 69 estimated break points that occurred over the period 18702010, 52 cases (75%) result in catchingup, and the others result in divergence. In panel C, terms C and D denote the catchingup and divergence process after any break, respectively. Figure 1 displays the time paths of the relative per capita real GDP (blue line) and the estimated flexible trend function (red line) for each country. As we know that the actual nature of break(s) is generally unknown, and there is no specific guide as to where and how many breaks to use in testing for a unit root or stationarity. Using an incorrect specification for the form and number of breaks can be as problematic as ignoring the breaks altogether. A further examination of the figures, we can clearly observe both forms of breaks; i.e., sharp breaks and smooth shifts in the trend of the data. According to the graphs, it seems that the estimated break points using both the dummy variables and the Fourier approximations are reasonable, and these results further support our hypothesis that trend function can experience both types of breaks (i.e., sharp and smooth breaks).
Figure 1: Log Relative per capita Real GDP (Blue Line) and Estimated Trend Function with Sharp Breaks and Smooth Shifts (Red Line) for 16 Selected OECD Countries, 18702010.

4. Conclusion
In this paper we attempt to retest the catchingup hypothesis among the 16 OECD countries using the time series approach of stochastic convergence hypothesis with annual data over one century. To reach this aim, we propose a model which specifies a trend function, incorporating both types of structural breaks (i.e., sharp breaks and smooth shifts) using dummy variable and Fourier function, respectively. In order to detect the sharp breaks, we use the multiple break models proposed in Bai & Perron (1998), and we apply the Fourier function proposed in Becker et al. (2004) to capture the smooth shifts. Evidences demonstrate that the null hypothesis of stationarity is not rejected by CBL (2005), and BEL (2006) tests for each countries we focus. The tests also show that all relative per capita real GDP series have experienced shapes breaks and smooth shifts. Therefore, to investigate the sufficient condition of catchingup hypothesis, we specify a new trend function that incorporates with both types of structural breaks. The results show that most of sharp breaks are coincided with WWI, WWII, great depression, and oil shocks, and most divergence process occur over WWI and WWII.
[1]. Whereas the procedure of Bai & Perron (1998) needs all regressors in the model be stationary, hence we estimate our trend function for series that the null of stationary is not rejected for them.
[2]. The null hypothesis of CBL (2005) implies regimewise stationarity for all countries, versus the alternative of nonstationarity for some countries.
[3]. Hadri (2000) proposed an LM panel data stationarity test without breaks. However, CBL (2005) extended the analysis to account for the presence of multiple breaks in a panel framework.
[4]. As see in equation (10), the conventional KPSS test is a one variety of BEL (2006) when trigonometric component is ignored. As noted by BEL (2006, p: 391) “the usual KPSStype stationary tests will diverge when nonlinear trends are ignored. This leads to overrejections of the true stationary null hypothesis in favor of the false unitroot hypothesis.”
[5]. In order to test for presence of nonlinear terms, BEL offered a F(k) test. As noted byBEL, the presence of the nuisance parameter causes that the distribution of F(k) does not have be nonstandard. Hence, we calculate the critical values for any series herein. To this end, we first generate 20,000 random series using the Gauss (version 10.0.0) RNDN procedure under the null of linearity. Then using optimum frequency to any actual series, we calculate the Fstatistic to any of 20,000 pseudo series. In final step we obtain the critical values from the sorted vector of pseudo Fstatistic.
[6]. In order to determine the optimum frequency, we follow BEL (2006) and first determine the maximum frequency equal to 5 and then calculate the sum of squared residuals (SSR hereafter) for any frequency. The optimum frequency is that minimize the SSR.
[7]. See Christopoulos & LeonLedesma (2011) for more details.
[8]. Pesaran (2004) indicates that the PCDtest has exactly mean zero for fixed T and N and is robust to heterogeneous dynamic models including multiple breaks in slope coefficients and/or error variances.
[9]. In order to save the space, we do not report the estimated break dates.