# Convergence across the United States: evidence from panel ESTAR unit root test.

Abstract Many empirical studies try to test whether there is income convergence across metropolitan areas in the continental United States. Drennan et al. (Journal of Economic Geography 4(5), 2004) claim that income among metropolitan economies is diverging for the period 1969-2001, after applying univariate unit root tests to the time series data. This paper brings new information to this area of study by using the nonlinear panel unit root test of the Exponential Smooth Auto-Regressive Augmented Dickey-Fuller (ESTAR-ADF) unit root test on the time series data for the period 1929-2005. Our results find evidence of stationarity for time series and thereby support beta and sigma convergence among states in a nonlinear setup. However, when the non-linear test encompasses cross section dependence as advocated by Cerrato et al. (2008), the evidence is attenuated.Keywords United States income convergence * Nonlinear panel unit root test * ESTAR * Cross section dependence

JEL C10 * F01 * J10 * O10 * O40

Introduction

The analysis of convergence is based on Solow's (1956) model; this neo-classical growth model predicts that a poor economy tends to grow faster than a rich one. Based on Solow's model, there are vast amounts of studies devoted to economic growth and convergence (see Baumol 1986; Barro and Sala-i-Martin. 1991; Mankiw et al. 1992; De Long 1988; Jones 1997; Pritchett 1997 among others). The assumption of diminishing returns is crucial for the convergence hypothesis to hold. This is because economic agents will allocate resources (e.g. labor and capital) across different locations so as to maximize their wealth. As a result, differences in returns to labor or capital among different regions will diminish over time. However, we argue in our paper that only when all economies are able to access the same technology will convergence eventually occur in the long run.

One channel for technology spillover across borders is the inter-regional trade of manufactured goods and production specialization. Fan (2004) incorporated the role of product quality and international trade to explain the East Asian Miracle and the empirical finding of conditional convergence. He assumed that quality means superior goods and the demand for it increases with income. Following the implications of his model, it suggests a conflict in the preferences for the ideal quality of consumption between rich and poor regions. A poor region may choose an inferior autarkic production technology so that a greater quantity of low quality goods can be produced given the resources' availability. By making such a decision, the poor region forgoes the opportunity of joining the global markets and catches up with its neighbors by division of labor, production specialization, and technology spillover.

Nevertheless, the poor economies will grow eventually when their human capital accumulates over time. When human capital of the poor region approaches the average levels of other regions, the chance of participating in global industrial specialization may occur. We denote a threshold level, c, of human capital accumulation at which the economy will experience a jump in its per capita human capital and income. Therefore, in our paper, we model the growth dynamic of metropolitan areas in such a way that the economy may only experience a high economic growth rate when it reaches the threshold level of human capital accumulation and starts to engage in trade with other regions. We use the ESTAR model to estimate the growth dynamic across states so as to capture the likelihood that the growth rate of different regions will converge provided that they reach the threshold level of human capital accumulation.

The rest of this paper is organized as follows. The second section provides a brief literature review on beta and sigma convergence. The third section describes the empirical methodologies that we employed. The fourth section evaluates the empirical findings and the fifth section concludes.

Income Convergence

Beta Convergence

The conditions of free factor mobility and free trade are essential, and they contribute to the acceleration of the convergence process through the equalization of the prices of goods and factors of production. In this context, the tendency for income disparities to decline over time is explained by the hypothesis that factor costs are lower and profit opportunities are higher in poor regions as compared to rich regions. Therefore, low-income regions will tend to grow faster and catch up with the leading regions. In the long run, factor prices, income differences, and growth rates will be equalized across regions.

The most common measures of convergence are beta ([beta]) and sigma ([sigma]) convergence in their conditional and unconditional versions. Beta convergence identifies a negative relationship between the growth of per capita incomes and the initial level of income across regions over a give time period. Some empirical studies find evidence in support of unconditional beta convergence across states (Barro and Sala-i-Martin 1991). However, we believe that unconditional convergence may not be expected when heterogeneous factors' endowment across countries is obvious. Therefore, we expect conditional convergence instead of its unconditional version as shown by Barro and Sala-i-Martin (1995).

In order to control for differences in the steady-state growth path, Barro and Sala-i-Martin (1991, 1992), and Mankiw et al. (1992) include explanatory variables that change across countries, like population growth, rate of capital depreciation, and technological progress, in their studies. The universal consensus is that, while there is no evidence of unconditional convergence among countries with very different initial endowments, evidence in support of conditional convergence is found for groups of countries with homogenous endowment. Barro and Sala-i-Martin (1991) find evidence in support of unconditional beta convergence as well as conditional beta convergence for states by introducing regional and sectoral dummy variables to capture the origin of the heterogeneous characteristics across states. In the same notion, Mankiw et al. (1992) also find evidence of unconditional as well as conditional beta convergence for different countries by introducing saving, population growth, and human capital accumulation variables.

Sigma Convergence

Another measure of convergence is sigma convergence; its magnitude is measured by the standard deviation of per capita income across states over time (Quah 1993). A continuous decline in annual standard deviations of income across states over time implies sigma convergence. Moreover, the use of cross-sectional regression for testing beta convergence may commit Galton's fallacy of regression to the mean and it implies biased estimates and invalid test statistics. In response to this fallacy, Friedman (1992) and Quah (1993) argue that sigma convergence is the only valid measure of convergence. Barro and Sala-i-Martin (1991) test for sigma convergence using state per capita income data from 1880 to 1988. Their results support sigma convergence for all decades except the 1920s and the 1980s by using standard deviation of the log of per capita income as the series of interest in cross-section regression. Another reason for using sigma convergence is that the validity of beta convergence implies convergence of its variance to the steady-state level, but it does not imply that its variance is diminishing over time. In this paper, we test both unconditional beta and sigma convergence so as to provide robust results and we believe that the legal system, language, currency, financial markets, and culture are likely to be homogeneous across states.

Starting from the 1960s, there are vast amounts of studies concerning income convergence in which the hypothesis is examined for states (Borts 1960; Borts and Stein 1964) and for regions (Perloff 1963). In most cases, there is evidence in support of income convergence. In contrast, numerous studies find evidence against convergence hypothesis across states (e.g., Browne 1989; Barro and Sala-i-Martin 1991; Blanchard and Katz 1992; Carlino 1992; Mallick 1993; Crihfield and Panggabean 1995; Glaeser et al. 1995; Drennan et al. 1996; Vohra 1996; and Drennan and Lobo 1999).

Our research differs from other research on regional income convergence and growth in the United States by several major attributes. Firstly, the sample period is extended to the most recent years so that the trend of U.S. regional convergence is better understood. Secondly, we allow the convergence process to follow nonlinear dynamics across states because the convergence process occurs through the equalization of prices of goods and factors of production, and this law of one price follows nonlinear dynamics as supported by recent empirical evidence. Thirdly, we use a relative growth differential as a complement to the standard deviation across states so that the sigma converging member can be identified.

Methodology

Beta Convergence

The data set used in this study consists of annual panel data of per capita personal income (PCPI) for the states for the period 1929 to 2005, which are collected from the Bureau of Economic Analysis's Regional Economic Information Systems (REIS). (1) The sample excludes Alaska and Hawaii because the data are not fully available. Table 4 of Appendix 1 provides the states included in our sample. In order to obtain a robust result, unconditional beta and sigma hypotheses are estimated by both the panel data unit root test and nonlinear unit root test.

Table 4 States included New England Mid-East Great Lakes Plains Southeast Maine District of Indiana Kansas Arkansas Columbia New New Jersey Ohio Missouri Georgia Hampshire Rhode Island New York Wisconsin Nebraska Kentucky Vermont Pennsylvania North Dakota Louisiana South Dakota Mississippi North Carolina South Carolina Tennessee Virginia West Virginia New England Southwest Rocky Mountain Far West Maine New Mexico Idaho California New Hampshire Texas Utah Nevada Rhode Island Wyoming Oregon Vermont Washington U.S. Bureau of Economic Analysis

Traditionally, the most commonly used regressions in growth studies are cross sectional (see Baumol 1986 for beta convergence). The basic idea is to estimate the coefficients of the following equation and evaluate the null hypothesis of divergence (that is, [beta] = 0) against the alternative hypothesis convergence, when [beta] [member of](-1, 0): the pooled data regression is represented in Eq. 1.

[[[y.sub.[i, T]] - [y.sub.[i, 0]]]/T] = [alpha] + [beta][y.sub.[i, 0]] + [u.sub.[i, 0]] i = 1, ..., N (1)

where [alpha] is a constant (which captures the regions' steady state), [beta] captures the rate or speed of convergence, and u is a disturbance term. Note that we only consider the growth rate of output in the whole period of analysis (between t = 0 and T = 1). One modification of Eq. 1 is the panel data regression as represented in Eq. 2.

[[[y.sub.[i, T]] - [y.sub.[i, t - 1]]]/T] = [alpha] + [beta][y.sub.[i, t - 1]] + [u.sub.[i, t]] (2)

where, in this case, T denotes the number of periods or years between t and t - 1. One of the advantages of this technique is that it lets us take advantage not only of the cross-sectional dimension, but also of the time dimension, thus providing greater degrees of freedom. However, a valid criticism of regressions between the per capita GDP growth rate and initial per capita GDP is that the test does not have a standard distribution under the null hypothesis ([beta] = 0), so making a comparison using the traditional statistics and related critical values can lead to an erroneous conclusion. The term of income convergence may be defined as the convergence of long-run output differences as the forecasting horizon increases. It implies that per capita GDP in any pair of provinces tends to converge to the same level in the long run. In statistical terms, the convergence to the provincial per capita income means that the income gap between any two provinces must be mean-reverting or stationary. Therefore, one possibility is to examine whether each regional income series independently presents a unit root (Dickey and Fuller 1976). However, it is well known that such a procedure suffers from serious power problems (see, for example, the Fisher-ADF and Fisher-Phillips-Perron tests proposed by Maddala and Wu 1999; Choi 2001; Levin et al. 2002; Breitung and Das 2004).

Linear Panel Unit Root Test

The notion of income convergence can be formalized by applying the concept of cointegration. If [y.sub.[i, t]] and [y.sub.[j, t]], the capita GDP for provinces i and j respectively at time t, no stationary, [y.sub.[i, t]] and [y.sub.[j, t]] are covering the same income level in the long run once they are cointegrated. A common approach in the bivariate set-up is to examine whether the income differential between cities i and j is stationary. In the following, the empirical framework was started by carrying out the ADF test for stationarity for pair-wise relative income differential series [y.sub.[ij, t]] = ln([y.sub.[j, t]]/[y.sub.[i, t]])], where [y.sub.[ij, t]] is the income differential between the benchmark city i and city j at time t. Notice that California was selected as the benchmark city. We also use other cities and the arithmetic mean as a benchmark and the results are robust. The standard ADF regression takes the form:

[y.sub.[ij, t]] = [[alpha].sub.j] + [[theta].sub.j][y.sub.[ij, t-1]] + [K.summation over (k = 1)][[delta].sub.[j, k]][DELTA][y.sub.[ij, t-k]] + [[mu].sub.[ij, t]] t = 1, ..., T; j = 1, ..., N (3)

Rearranging Eq. 3 becomes;

[DELTA][y.sub.[ij, t]] = [[alpha].sub.j] + [[beta].sub.j][DELTA][y.sub.[ij, t - 1]] + [K.summation over (k = 1)] [[delta].sub.[j, k]][DELTA][y.sub.[ij, t-k]] + [[mu].sub.[ij, t]] t = 1, ..., T; j = 1, ..., N (4)

where [DELTA] is the first difference operator, k is the number of augmenting terms and {[u.sub.[ij, t]} (j = 1, 2, .., N) are white noise series independently distributed across N = 50 provinces, i.e. [u.sub.[ij, t]] ~ id (0, [[sigma].sub.[ij, t].sup.2]). The number of augmenting terms is determined by using the Akaiki information criteria (AIC). We need to include a constant term, [alpha], for each city in order to account for province-specific fixed effects such as initial endowment, employees' educational attainment, and the preferential policy implemented by the central government for different states. The purpose of including the constant term is to differentiate between the concept of conditional convergence ([alpha] [not equal to] 0) and unconditional convergence ([alpha] = 0). There are numerous linear types of panel unit root tests, as referring to second-generation unit root test are developed, for example, Levin and Lin (1992); Im et al. (1997); Maddala and Wu (1999).

Nonlinear Panel Unit Root Test with Cross Section Dependence

We believe that the growth dynamics across states follows nonlinear patterns. Firstly, we anticipate that the economy may only experience a high growth rate when it reaches the threshold level of human capital accumulation and starts to engage in trade with other regions. Secondly, the equalization of prices of goods and factors of production follows a nonlinear dynamics, as shown by many researchers (e.g. Michael et al. 1997; Taylor et al. 2001; Sarno et al. 2004). Therefore, we use the ESTAR model to specify the growth dynamic across states so as to capture the likelihood that the growth of different regions will converge only if the region reaches a threshold level of growth rate. Cerrato et al. (2008) developed a new nonlinear panel ADF test under cross sectional dependence, which is based on the following Exponential Smooth Transition Autoregressive (ESTAR) specification applying to the de-meaned series of interest: In its general form, we have:

[[~.y].sub.it] = [[xi].sub.i][[~.y].sub.[i, t-1]] + [[xi]*.sub.i][[~.y].sub.[i, t-1]]Z([[theta].sub.i]; [[~.y].sub.[i, t-d]]) + [[mu].sub.it], t = 1, ..., T, i = 1, ..., N, (5)

where

Z([[theta].sub.i]; [[~.y].sub.[i, t - d]]) = 1 - exp [-[[theta].sub.i][([[~.y].sub.[i, t-d]] - c).sup.2]] (6)

where [[theta].sub.i] is a positive coefficient an c is the equilibrium value of income difference between region i and mean difference across provinces due to heterogeneous human capital accumulation between region i and the mean value. The initial value, [~.[y.sub.i0]], is given, and the error term, [[micro].sub.it], has the one-factor structure:

[[mu].sub.it] = [[gamma].sub.i][f.sub.i] + [[epsilon].sub.it], [([[epsilon].sub.it]).sub.t]~ i.i.d. (0. [[sigma].sub.i.sup.2]) (7)

in which [f.sub.t] is the unobserved common factor, and [[epsilon].sub.it] is the individual-specific (idiosyn-cratic) error. Following the existing literature, the delay parameter d is set to be equal to one so that Eq. 6 may be rewritten in first difference form in general as:

[DELTA][[~.y].sub.[i, t]] = [[alpha].sub.i] + [[xi].sub.i][[~.y].sub.[i, t - 1]] + [[h - 1].summation over (h = 1)][[delta].sub.ijh][DELTA][[~.y].sub.[ij, t-h]] + ([[~.[alpha]]*.sub.i] + [[xi]*.sub.i][[~.y].sub.[i, t-1]] + [[h - 1].summation over (h = 1)][[delta]*.sub.ih][DELTA][[~.y].sub.[i, t-h]]) *Z([[theta].sub.i]; [[~.y].sub.[i, t-d]]) + [[gamma].sub.i][[Florin].sub.i] + [[epsilon].sub.it] (8)

Notice that when [~.[y.sub.i,[t - d]] = c, Z(*) = 0 and Eq. 8 is equivalent to a standard linear ADF model of Eq. 4. However, when the magnitude of income divergence between [~.[y.sub.i,[t - d]] c becomes large and hence Z(*) [approximately equal to] 1 will generate a new linear ADF model with parameter [[beta].sub.i] = [[xi].sub.i] + [[xi]*.sub.i]. In contrast when income divergence is negligible, [[xi].*.sub.i] affects the flow of income differential in this case. However, when income divergence becomes more serious, [[xi]*.sub.i] plays a more important role in governing the adjustment process. We should take note that the [[xi].sub.i] + [[xi]*.sub.i] < 0 is the necessary condition for "Global Stability" to hold. Once the condition of [[xi].sub.i] + [[xi]*.sub.i] < 0 is fulfilled it is legitimate to have [[xi].sub.i] [greater than or equal to] 0, if this is occurred the implication is that income divergence follows a nonstationary growth path (e.g. random-walk or explosive innovation within the "band of inaction" of c), and eventually it converge back to its equilibrium once the magnitude of income divergence is outside the band. If we assumed [y.sub.it] follows a unit root processes in the middle regime, then [[xi].sub.i] = 0, and Eq. 10 can be rewritten as:

[DELTA][[~.y].sub.[i, t]] = [[xi]*.sub.i][[~.y].sub.[i, t-1]][1 - exp(-[[theta].sub.i][[~.y].sub.[i, t-1].sup.2]] + [[gamma].sub.i][[Florin].sub.t] + [[epsilon].sub.[i, t]] (9)

The null hypothesis of nonstationarity is [H.sub.0]: [[theta].sub.i] = 0[for all]i, against the alternative of

[H.sub.1]: [[theta].sub.i]> 0 for i = 1, 2, ..., [N.sub.1] and [[theta].sub.i] = 0 for i = [N.sub.1] + 1, ..., N

Because [[xi]*.sub.i] in Eq. 9 is not identified under the null, it is not feasible to directly test the null hypothesis. Thus, Cerrato et al. (2008) reparameterize Eq. 9 by using a first-order Taylor series approximation and obtain the auxiliary regression

[DELTA][[~.y].sub.[i, t]] = [a.sub.i] + [delta][[~.y].sub.[i, t - 1].sup.3] + [[gamma].sub.i][[Florin].sub.t] + [[epsilon].sub.[i, t]] (10)

For a more general case where the errors are serially correlated, regression (10) is extended to

[DELTA][[~.y].sub.[i, t]] = [a.sub.i] + [delta][[~.y].sub.[i, t-1].sup.3] + [[h - 1].summation over (h = 1)][[upsilon].sub.ih][DELTA][y.sub.[i, t-h]] + [[gamma].sub.i][[Florin].sub.t] + [[epsilon].sub.[i, t]] (11)

Cerrato et al. further prove that the common factor [[Florin].sub.t] can be approximated by;

[[Florin].sub.t][approximately equal to][1/[[approximately equal to].[gamma]]][DELTA][[[approximately equal to].y].sub.t] - [[bar.b]/[bar.[gamma]]][[[approximately equal to].y].sub.[t, 1].sup.3], (12)

where [[[approximately equal to].y].sub.t] is the mean of [[~.y].sub.t] and [bar.b] = [1/N] [N.summation over (i=1)][b.sub.i] Therefore, it follows that Eq. 12 can be written as the following nonlinear cross sectionally augmented DF(NCADF) regression:

[DELTA][[~.y].sub.[i, t]] = [a.sub.i] + [b.sub.i][[~.y].sub.[i, t-1].sup.3] + [c.sub.i][DELTA][[[approximately equal to].y].sub.t] + [d.sub.i][[[approximately equal to].y].sub.[t-1].sup.3] + [[epsilon].sub.[i, t]] (13)

Given the framework above, the authors develop a unit root test in heterogeneous panel model based on Eq. 13. Extending the idea in Kapetanios et al. (2003), the authors suggest using model based on Eq. 13 and t-statistics on [b.sub.i], that is denoted by:

[t.sub.[iNL]](N, T) = [[[^.b].sub.i]/[s.e. ([[^.b].sub.i])]] (14)

This is a nonlinear cross sectionally augmented version of IPS test based (NCIPS). Consequently, the authors calculate critical values of both individual and panel NCADF tests for varying cross section and time dimensions.

Sigma Convergence

Barro and Sala-i-Martin (1991) test for sigma convergence using state per capita income data from 1880 to 1988 and find evidence in support of sigma convergence for all decades except the 1920s and the 1980s. However, Drennan et al. (2004) find evidence against sigma convergence using data for all metropolitan areas in the continental United States for the period 1969-2001. The series of interest is the standard deviation of the natural logarithm of metropolitan PCPI at time t, denoted as [y.sub.t]. We apply the unit root test to this series and, once evidence is found in support of the unit root hypothesis for the standard deviations of the natural logarithm of PCPI, it will provide evidence against the convergence hypothesis. However, the use of standard deviation is not satisfactory due to the limited observations and bias of outliers. Therefore, we use the relative income differential as a complement to the standard deviation across states' income level so that the sigma converging member can be identified.

Empirical Results

Beta Convergence

In the case of regional output, the period of analysis is from 1923 to 2005. The first piece of evidence in favor of beta convergence is presented in Fig. 1, which shows the negative relationship between the average growth rate (from 1929 to 2005) and the initial level of regional GDP per capita (in 1929). The estimated coefficient in Eq. 1 is negative, as expected, and statistically significant. The convergence rate is 1.15% (see Table 1). To gain an idea of the speed (in years) with which this convergence should take place, we calculated the half-life for closing the gap between the GDP per capita of the relatively poorer regions and the relatively richer ones. In this case, the convergence rate implies that half the gap should be closed in 60 years (see Table 1).

[FIGURE 1 OMITTED]

Table 1 Traditional test for [beta] convergence in income: cross section data Coefficient [beta] -0.01146 (0.000783) [0.000] R2 0.8169 No. of obs. 50 Years to close half tile gap 60 (a) Standard errors are in parentheses; p-values are in brackets (b) Hal-life is calculated as In( 112) divided by the respective beta coefficient

Table 2 shows that when nonlinearity is incorporated into the testing procedure, the nonlinear tests support beta convergence more often than using a linear ADF test. The findings confirm our hypothesis that the growth dynamics across states follow nonlinear patterns and a region may only experience a high economic growth rate when it reaches the threshold level of human capital accumulation and starts to engage in trade with other regions.

Table 2 Tests for [beta] convergence in income: univariate unit root tests Linear Test Nonlinear Test Panel Nonlinear (ADF) (ESTAR) ESTAR(a) Percentage of 16/50 or 32% 34/50 or 68% Convergence Test-Statistics -2.13 (b) (a) Table 12: Critical values of Panel NCADF Test-Cerrato et al. (2008) (b) indicates the test is at the 10% significant level

Sigma Convergence

Figure 2 plots the standard deviation of the natural logarithm of states' PCPI from 1929 to 2005. Over 77 years, if the theory of sigma convergence is correct, one would expect to sec a persistent downward trend in the variable. Again, Table 3 shows that, when nonlinearity is incorporated into the testing procedure, the nonlinear tests support sigma convergence.

[FIGURE 2 OMITTED]

Table 3 Results of augmented Dickey-fuller unit root test: standard deviation on natural log personal income Linear Test Nonlinear Test Test statistics 1.7 -4.33 (a) (a) denotes 1% sig. level

However, the use of a single standard deviation is not satisfactory because there are limited observations available and the bias of outliers exists. Therefore, we use the relative income differential as a complement to the standard deviation across states' income level so that the sigma converging member can be identified.

Conclusion

In this paper, we examine the empirical validity of both beta and sigma convergence across states using states' per capita personal income during the period 1929 to 2005. Using both linear and panel nonlinear unit root tests, we found strong evidence in support of beta and sigma convergence across states on average, as suggested by Barro and Sala-i-Martin (1991, 1995). Our results suggest future research on the question of why the income is not converging for some states even in the long run. There could be some possible explanations for the above puzzle. Firstly, as argued by Drennan et al. (2004), transportation technology may be one important factor affecting the convergence process. After the mid-1970s, transportation technology did not improve significantly. In contrast, the transaction costs of exchanging services and manufactured goods across states were reduced dramatically during 1940 to 1970. For example, railroads, trucks, refrigeration cars, the interstate highway system, and jet air transportation served to raise the mobility of labor, capital, and commodities, and hence to equalize returns and prices across states. However, we believe that this convergence process follows nonlinear dynamics as supported by various empirical studies in the field of the Law of One Price (LOP) across states.

Secondly, the conclusion of unconditional convergence could be derived from the neoclassical growth model only if certain assumptions hold in the nature of input factors and agents' maximizing behavior (Mankiw et al. 1992). The model assumes diminishing marginal returns to labor and capital, which in turn implies income convergence across states because labor and capital are moving around seeking the highest returns. However, Acemoglu (2002) argued that skill-biased technical change might favor rich regions and lead to the violation of diminishing marginal returns to labor at the initial stage of technical progress. Therefore, we would expect a sudden jump in the growth rate when the income level of a region exceeds a particular threshold level. Again, this suggests that we should model the growth dynamic in a nonlinear setup.

Lastly, product quality and intra-regional trade may play a role in explaining conditional convergence. There may be a conflict between the preferences for the ideal quality of consumption between rich and poor metropolitan areas across states. A poor region may choose an inferior autarkic production technology so that a greater quantity of low quality goods can be produced with the given resources' availability. By making such a decision, the poor region forgoes the opportunity of joining the global markets and boosts its growth by division of labor, production specialization, and technology spillover. Again, we would expect a sudden jump in the growth rate when the income level of a region exceeds a particular threshold level.

Appendix

References

Acemoglu, D. (2002). Technical change, inequality, and the labor market. Journal of Economic Literature, 40, 7-72.

Barro, R. J., & Sala-i-Martin, X. (1991). Convergence across states and regions. Brookings Papers on Economic Activity, 1, 107-182.

Barro, R. J., & Sala-i-Martin, X. (1992). Convergence. Journal of Political Economy, 100(2), 223-251.

Barro, R. J., & Sala-i-Martin, X. (1995). Economic growth. Boston: McGraw Hill.

Baumol, W. (1986). Productivity growth, convergence, and welfare: what the long-run data show. American Economic Review, 76, 116-131.

Blanchard, O. J., & Katz, L. F. (1992). Regional evolutions. Brookings Papers on Economic Activity, 1, 1-75.

Borts, G. H. (1960). The equalization of returns and regional economic growth. American Economic Review, 50, 319-334.

Borts, G. H., & Stein, J. L. (1964). Economic growth in a free market. New York: Columbia University Press.

Breitung, J., & Das, S. (2004). Panel unit root tests under cross sectional dependence. Mimeo, University of Bonn.

Browne, L. E. (1989). Shifting regional fortunes: The wheel turns. New England Economic Review, Federal Reserve Bank of Boston, May/June, 27-40.

Carlino, G. A. (1992). Are regional per capita earning diverging? Business Review, Federal Reserve Bank of Philadelphia, March/April, 3-12.

Cerrato, M., de Peretti, C, & Sarantis, N. (2008). A nonlinear panel unit root test under cross section dependence. Discussion Paper 2008-08, Department of Economics, University of Glasgow.

Choi, I. (2001). Unit root tests for panel data. Journal of International Money and Finance, 20(2), 249-272.

Crihfield, C. J., & Panggabean, M. (1995). Growth and convergence in US cities. Journal of Urban Economics, 38, 138-165.

De Long, J. (1988). Productivity growth, convergence, and welfare: comment. American Economic Review, 78, 1138-1154.

Dickey, D. A., & Fuller, W. A. (1976). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association, 74, 427-431.

Drennan, M. P., & Lobo, J. (1999). A simple test for convergence of metropolitan income in the United States. Journal of Urban Economics, 46, 350-359.

Drennan, M. P., Lobo, J., Strumsky, D. (2004). Unit root tests of sigma income convergence across metropolitan areas of the U.S. Journal of Economic Geography, 4(5).

Drennan, M. P., Tobier, E., & Lewis, J. (1996). The interruption of income convergence and income growth in large cities in the 1980s. Urban Studies, 33, 63-82.

Fan, C. S. (2004). Quality, trade, and growth. Journal of Economic Behavior and Organization, 55(2), 271-291.

Friedman, M. (1992). Do old fallacies ever die? Journal of Economic Literature, 29, 2129-2132.

Glaeser, E. L., Scheinkman, J. A., Shleifer, A. (1995). Economic growth in a cross-section of cities. Journal of Monetary Economics, 36, 117-143.

Im, K. S., Pesaran, H. M., Shin, Y. (1997). Testing for unit roots in heterogeneous panels. Working paper, Department of Applied Economics, University of Cambridge.

Jones, C. (1997). Convergence revisited. Journal of Economic Growth, 2, 131-153.

Kapetanios, G., Shin, Y., & Snell, A. (2003). Testing for a unit root in the Nonlinear STAR Framework. Journal of Econometrics, 112, 359-379.

Levin, A., & Lin, C. (1992). Unit root tests in panel data: Asymptotic and finite sample properties. Discussion Paper No. 92-93, Department of Economics, University of California, San Diego

Levin, A., Lin, C.-F., & Chu, C.-S. J. (2002). Unit root tests in panel data: asymptotic and finite-sample properties. Journal of Econometrics, 108, 1-25.

Maddala, G. S., & Wu, S. (1999). A comparative study of unit root tests with panel data and a new simple test. Oxford Bulletin of Economics and Statistics, 61, 631-652.

Mallick, R. (1993). Convergence of state per capita incomes: an examination of its sources. Growth and Change, 325-334.

Mankiw, N. G., Romer, D., & Weil, D. N. (1992). A contribution to the empirics of economic growth. Quarterly Journal of Economics, 107, 407-437.

Michael, P., Nobay, A. R., & Peel, D. A. (1997). Transaction costs and nonlinear adjustment in real exchange rates: an empirical investigation. Journal of Political Economy. 105(4), 862-879.

Perloff, H. S. (1963). How a region grows. New York: Committee for Economic Development.

Pritchett, L. (1997). Divergence, big time. Journal of Economic Perspectives, 11(3), 3-17.

Quah, D. (1993). Galton's fallacy and tests of the convergence hypothesis. Scandinavian Journal of Economics, 93, 427 443.

Sarno, L., Taylor, M. P., & Chowdhury, I. (2004). Nonlinear dynamics in deviations from the law of one price: a broad-based empirical study. Journal of International Money and Finance, 23(1), 1-25.

Solow, R. (1956). A contribution to the theory of economic growth. Quarterly Journal of Economics, 70 (1), 65-94.

Taylor, M. P., Peel, D, A., & Sarno, L. (2001). Nonlinear mean-reversion in real exchange rates: toward a solution to the purchasing power parity puzzles. International Economics Review, 42(4), 1015-1042.

Vohra, R. (1996). How fast do we grow? Growth and Change, 47-54.

(1) http://www.bea.doc.gov/bea/regional/reis

Published online: 23 October 2009

[c] International Atlantic Economic Society 2009

DOI 10.1007/s11294-009-9241-8

C.-K. M. Lau ([??])

ITC, Hong Kong Polytechnic University, Hung Hom, Hong Kong

e-mail: 06901279r@polyu.edu.hk

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Title Annotation: | Exponential Smooth Auto-Regressive |
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Author: | Lau, Chi-Keung Marco |

Publication: | International Advances in Economic Research |

Geographic Code: | 1USA |

Date: | Feb 1, 2010 |

Words: | 5361 |

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