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Studies in Nonlinear Dynamics & Econometrics

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Volume 18, Issue 2


Are income differences within the OECD diminishing? Evidence from Fourier unit root tests

Alan King / Carlyn Ramlogan-Dobson
Published Online: 2013-08-27 | DOI: https://doi.org/10.1515/snde-2013-0008


We investigate the income convergence hypothesis for 24 OECD countries using Fourier-type unit root tests that can account for structural breaks of unknown number, timing and functional form in a series’ data generating process. Our results indicate that all 24 countries have followed a nonlinear underlying growth path (relative to the US) over the post-war era. These growth paths indicate that as many as half of the countries sampled are systematically catching-up with the US. Only a handful show evidence of being in relative decline, but the fact that this group includes several G7 economies is of concern.

This article offers supplementary material which is provided at the end of the article.

Keywords: Fourier function; income convergence; nonlinear mean-reversion; nonlinear trend


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About the article

Corresponding author: Alan King, Department of Economics, University of Otago, New Zealand, e-mail:

Published Online: 2013-08-27

Published in Print: 2014-04-01

It is worth noting that, although some studies use their findings to choose between the neoclassical and the endogenous growth models, developments in the theoretical literature mean that “it is now possible, generally speaking, to explain both convergence and non-convergence behavior by appropriately chosen models of growth theory of both these varieties” (Islam 2003, p. 311). Therefore, as Durlauf, Johnson, and Temple (2005, p. 586) also note, caution must be exercised when making such inferences regardless of the testing approach employed.

A distinction can be drawn between conditional (beta) convergence, which reflects the presence of the control variables in equations (1) and (2), and unconditional (beta) convergence, when these variables have been excluded from the regression equation.

Time-series convergence is equivalent to unconditional beta convergence.

However, E&L do show that, even when the true trend takes the form of linear segments separated by sharp breaks, the power of their unit root test is comparable to that of a standard discrete-break test.

It should be noted that, although this representation of the deterministic components for x is very flexible, it does have some restrictions of its own. In particular, the sinusoidal oscillations about the linear trend component of equation (8) are regular.

Chong et al. (2008), C&L and King and Ramlogan-Dobson (2011) have all investigated the convergence hypothesis for the OECD with unit-root tests incorporating nonlinear mean-reversion. However, C&L is the sole study, to our knowledge, to use a Fourier-type test with this feature.

In their nonlinear mean-reversion test C&L assume the DGP for x follows a logistic, rather than an exponential, STAR process in order to allow for the possibility that the mean-reversion process following especially large negative shocks (e.g., the Great Depression or World War II) may differ from that following a positive shock.

The importance of estimating each test statistic’s distribution in this way can be illustrated by the 10% critical value for the FLM test (for k^=1.0). The value E&L report for T=100 is −3.82. Reducing the sample size to T=59 raises the critical value to −4.31. The test statistic’s distribution is relatively insensitive to the presence of a large (+0.9) AR term in the first difference of the data series (10% CV=−4.42). However, it is sensitive to the presence of a quite small (+0.3) MA term (10% CV = −4.80).

The 10% (5%) [1%] critical values obtained in relation to equation (11) are 6.550 (8.194) [12.062] for T=59. The 10% (5%) [1%] critical values obtained in relation to equation (12) are 6.003 (7.460) [10.820] for T=59.

As C&L note, a standard F-test of the null that αk = βk = 0 may be used to assess the presence of breaks or nonlinearity in the trend, providing the null of a unit root can be rejected.

C&L state that only the sign of βk matters for determining the sign of the trend function’s slope for low and moderate values of k^ (given that γ=0) on the grounds that sin(z) tends to zero as z tends to zero. However, this overlooks the fact that, as z tends to zero, cos(z) tends to a constant (and so becomes part of the intercept term). Hence, as k^ tends to zero, the value of αk needed to offset the effect of a given value of βk on the sign of the trend’s slope becomes relatively small.

It will be noted that the trend functions for some countries do not fit the initial observations of yd very well. This largely reflects the fact that, in order to be consistent with the unit root tests (from which the trend’s value of k is obtained), the trend functions are estimated over the interval [m+1, T], where m is the number of augmentation lags, rather than the full sample period [1, T]. Closer fitting trend functions could be obtained by the use of a multiple-frequency Fourier function. This option is not pursued, however, as a single frequency has proved sufficient to identify a deterministic trend for all countries and increasing the number of frequencies risks over-fitting the data and treating short-run cyclical effects as structural breaks.

Rapid catching-up is defined as sustained (over at least 15 years) growth in yd’s trend of at least 0.5% per annum. All countries in this group comfortably exceed this standard.

Denmark’s alternative trend (for k=0.9) has a significant γ coefficient, but as both trends largely coincide from the late 1960s onwards, the estimate of γ for k=1.5 in effect captures the slope of both trends during the latter part of the sample period.

Germany’s estimate of γ implies a rate of relative decline of only 0.1% per annum. However (and bearing in mind the point made in footnote 12), the first half of its plotted trend appears to capture the winding-down of the country’s post-war reconstruction process and this has been followed by a sustained relative decline similar rate to that observed in the three other countries.

Citation Information: Studies in Nonlinear Dynamics & Econometrics, Volume 18, Issue 2, Pages 185–199, ISSN (Online) 1558-3708, ISSN (Print) 1081-1826, DOI: https://doi.org/10.1515/snde-2013-0008.

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