Real convergence in Taiwan: a fractionally integrated approach

Journal of Asian Economics 15 (2004) 529–547

Real convergence in Taiwan: a fractionally integrated approach J. Cunado, L.A. Gil-Alana*, F. Pe´rez de Gracia Facultad de Ciencias Economicas, Universidad de Navarra, Edificio Biblioteca, Entrada Este, E-31080 Pamplona, Spain Received 4 March 2003; received in revised form 2 February 2004; accepted 15 March 2004

Abstract In the context of the extraordinary economic growth of Taiwan, this article examines its real convergence process in terms of real per capita GDP with respect to the US and Japan. For this purpose, we examine the order of integration of the real per capita GDP of the Taiwanese economy together with its differences with respect to these two economies. We use both parametric and semiparametric techniques, and the results support the hypothesis of conditional real convergence of this economy, especially with respect to Japan for the period 1903–1999. Splitting the sample around World War II, the results substantially differ from one sub-sample to the other, suggesting that a structural break may occur at that time. Including a dummy variable to take into account the break, the results show that convergence is satisfied with respect to Japan if the underlying disturbances are white noise and with respect to both (Japan and the US) with autocorrelated disturbances. # 2004 Elsevier Inc. All rights reserved. JEL classification: C32; O41 Keywords: Real convergence; Fractional integration; Long memory

1. Introduction In a recent paper, Cunado, Gil-Alana and Pe´rez de Gracia (2004) analyze the real convergence hypothesis for 11 emerging Latin American and Asian countries. For the Asian countries (India, Indonesia, Taiwan, and South Korea), they only find evidence of convergence with respect to Japan for the case of Taiwan. This economy has experienced * Corresponding author. Tel.: þ34 948 425 625; fax: þ34 948 425 626. E-mail address: [email protected] (L.A. Gil-Alana).

1049-0078/$ – see front matter # 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.asieco.2004.03.005

530

J. Cunado et al. / Journal of Asian Economics 15 (2004) 529–547

an extraordinarily rapid and sustained growth of output per capita (see, Young, 1995). From 1966 to 1990, the GDP per capita growth rate averaged 8.5%, investment rate grew from a constant price investment to GDP ratio of 10% in the early 1950s to 20% in 1990, and human capital accumulation has also been rapidly raised (the proportion of the working population with a secondary education or more increased from 25.8% of the working population in 1966 to 67.6% in 1990). Furthermore, there have been large intersectoral reallocations of labor from the agricultural sector towards the services and industry sectors, and a rapid increase in its manufacturing exports. However, the scenario of this economy was different in the period before World War II. The economy was almost totally based on agriculture, and the growth rates were lower than those of the OECD countries, which explains the difference in the convergence process of the Taiwanese economy in the preand post-war period. In this paper, we concentrate on the real (deflated by prices) convergence process of the Taiwanese economy with respect to both the US and Japan for a long period of time, which covers both its pre- and post-war growth episodes (1903– 1999). The great differences in per capita output and growth rates among different economies and, in particular, the extraordinary post-war growth of some countries such as the four Asian Tigers (Singapore, South Korea, Taiwan, and Hong Kong) help to explain the number of empirical studies on real convergence including the East-Asian economies (see, e.g., Easterly, 1995; Lim & McAleer, 2000; Young, 1992, 1995). From a theoretical viewpoint, the interest on convergence studies may be explained in terms of a prediction test of the neoclassical growth model (Solow, 1956) as opposed to the ‘‘new’’ endogenous growth models (Lucas, 1988; Romer, 1986). While the neoclassical model predicts (under some assumptions) that per capita output will converge to each country’s steady-state (conditional convergence) or to a common steady-state (unconditional convergence), regardless of its initial per capita output level, in endogenous growth models there is no tendency for income levels to converge, since divergence can be generated by relaxing some of the neoclassical assumptions (e.g., incorporating nonconvexities in the production function). Empirical tests of the convergence hypothesis provide several definitions of convergence and thus, different methodologies to test it. In a cross-section approach, a negative (partial) correlation between growth rates and initial income is interpreted as evidence of unconditional (conditional) beta-convergence. In this context, one of the most generally accepted results is that while there is no evidence of unconditional convergence among a broad sample of countries, the conditional convergence hypothesis holds when examining a more homogenous groups of countries (or regions) or when conditioning on additional explanatory variables (see, e.g., Barro, 1991; Barro & Sala-i-Martin, 1991, 1992, 1995; Baumol, 1986; De Long, 1988; Dowrick & Nguyen, 1989; Grier & Tullock, 1989; Mankiw, Romer, & Weil, 1992). In a time series approach, stochastic convergence implies that output differences among economies cannot contain unit roots. Empirical tests on this hypothesis have been carried out by Campbell and Mankiw (1989), Cogley (1990), Bernard (1991), Carlino and Mills (1993), Bernard and Durlauf (1995) and, in general, they do not find evidence of convergence. In this paper, we define real convergence as mean reversion in the differences in (real) per capita output among countries and we test this hypothesis using both parametric and

J. Cunado et al. / Journal of Asian Economics 15 (2004) 529–547

531

semiparametric techniques based on fractional integration.1 The outline of the paper is as follows. Section 2 briefly describes the connection between fractional integration and real convergence. Section 3 covers the empirical analysis while Section 4 offers some conclusions.

2. Long-memory processes and convergence For the purpose of the present paper, we define an I(0) process {ut, t ¼ 0, 1, . . .} as a covariance stationary process with spectral density function that is positive and finite at the zero frequency. In this context, we say that a given raw time series {xt, t ¼ 0, 1, . . .} is I(d) if: ð1  LÞd xt ¼ ut ;

t ¼ 1; 2; :::;

xt ¼ 0;

t0

(1) 2

where ut is I(0) and where L means the lag operator (Lxt ¼ xt1 ). Note that the polynomial above can be expressed in terms of its Binomial expansion, such that for all real d, 1   X dðd  1Þ 2 d d L  ð1Þj Lj ¼ 1  dL þ ð1  LÞ ¼ j 2 j¼0 implying that: ð1  LÞd xt ¼ xt  dxt1 þ

dðd  1Þ xt2  : 2

The macroeconomic literature has stressed the cases of d ¼ 0 and 1, however, d can be any real number. Clearly, if d ¼ 0 in (1), xt ¼ ut, and a ‘weakly autocorrelated’ xt is allowed for. However, if d > 0, xt is said to be a long-memory process, also called ‘strongly autocorrelated,’ so-named because of the strong association between observations widely separated in time, and as d increases beyond 0.5 and through 1, xt can be viewed as becoming ‘‘more nonstationary,’’ in the sense, for example, that the variance of partial sums increases in magnitude. These processes were initially introduced by Granger (1980, 1981), Granger and Joyeux (1980) and Hosking (1981) (though earlier work by Adenstedt, 1974, and Taqqu, 1975, shows an awareness of its representation). They were theoretically justified in terms of aggregation of ARMA processes with randomly varying coefficients by Robinson (1978) and Granger (1980).3 Empirical applications based on fractional models like (1) are among others Diebold and Rudebusch (1989), Baillie and Bollerslev (1994), Gil-Alana and Robinson (1997), and Gil-Alana (2000), and other recent papers based on these methods are Smith, Sowell, and Zin (1997) and Cavaliere (2001). 1 By ‘‘mean reversion’’ we mean that the time series reverts to the mean. In other words, any shock affecting the level of the series will disappear in the long run. 2 In Eq. (1), xt ¼ 0, t  0 is a standard assumption to be made in the context of I(d) statistical models (see, e.g., Gil-Alana & Robinson, 1997). 3 Similarly, Cioczek-Georges and Mandelbrot (1995), Taqqu, Willinger, and Sherman (1997), Chambers (1998), and Lippi and Zaffaroni (1999) also use aggregation to motivate long memory processes, while Parke (1999) uses a closely related discrete time error duration model.

532

J. Cunado et al. / Journal of Asian Economics 15 (2004) 529–547

To determine the appropriate degree of integration in a given raw time series is important from both economic and statistical viewpoints. If d ¼ 0, the series is covariance stationary and possesses ‘short memory,’ with the autocorrelations decaying fairly rapidly. If d belongs to the interval (0, 0.5), xt is still covariance stationary, however, the autocorrelations take much longer time to disappear than in the previous case. If d 2 [0.5, 1), the series is no longer covariance stationary, but it is still mean reverting, with the effect of the shocks dying away in the long run. Finally, if d  1, xt is nonstationary and non-mean reverting. Thus, the fractional differencing parameter d plays a crucial role in describing the persistence in the time series behavior: higher the d, higher will be the level of association between the observations. There exist many approaches of estimating and testing the fractional differencing parameter d (see, e.g., Dahlhaus, 1989; Geweke & Porter-Hudak, 1983; Sowell, 1992; Tanaka, 1999). In this article we use both parametric and semiparametric methods. First, we present a parametric testing procedure due to Robinson (1994a) that permits us to test I(d) statistical models in raw time series. Then, a semiparametric procedure will also be described. 2.1. A parametric testing procedure Robinson (1994a) proposed a Lagrange Multiplier (LM) test of the null hypothesis: H 0 : d ¼ d0 :

(2)

in a model given by: y t ¼ b 0 zt þ xt ;

t ¼ 1; 2; . . . ;

(3)

and (1), for any real value d0, where yt is the time series we observe; b ¼ (b1, . . ., bk)0 is a (kx1) vector of unknown parameters; and zt is a (kx1) vector of deterministic regressors that may include, for example, an intercept (e.g., zt B 1), or an intercept and a linear time trend (in case of zt ¼ (1, t)0 ). Specifically, the test statistic is given by: ^r ¼

T 1=2 ^ 1=2 ^ A a; ^2 s

(4)

^ are given in ^2 , and A where T is the sample size and the technical definitions of a^, s Appendix A. Based on the null hypothesis H0 (2), Robinson (1994a) established that under certain regularity conditions:4 ^r !d Nð0; 1Þ as T ! 1;

(5)

and also the Pitman efficiency theory of the tests against local departures from the null. Thus, we are in a classical large sample-testing situation: an approximate one-sided 100a% level test of H0 (2) against the alternative: Ha: d > d0 (d < d0) will be given by the rule: ‘‘Reject H0 if ^r > za (^r < za ),’’ where the probability that a standard normal variate exceeds za is a. This version of the tests of Robinson (1994a) was used in empirical 4

These conditions are very mild, regarding technical assumptions to be satisfied by the model in (1) and (3).

J. Cunado et al. / Journal of Asian Economics 15 (2004) 529–547

533

applications in Gil-Alana and Robinson (1997) and Gil-Alana (2000), and other versions of his tests, based on seasonal (quarterly and monthly) and cyclical data, can be, respectively, found in Gil-Alana and Robinson (2001) and Gil-Alana (1999, 2001a). Another remarkable feature of this procedure is that its standard null limit distribution holds independently of the way of modeling the I(0) disturbances and whether or not we include deterministic trends in the model, unlike what happens with other procedures for testing unit and/or fractional roots (e.g., Dickey & Fuller, 1979; Kwiatkowski, Phillips, Schmidt, & Shin, 1992; Phillips & Perron, 1988), where the limit distribution of the tests changes with features of the regressors (see, e.g., Schmidt & Phillips, 1992).5 A problem with the parametric procedures is that the model must be correctly specified. Otherwise, the estimates are liable to be inconsistent. In fact, misspecification of the short run components of the process may invalidate the estimation of the long run parameter d. This is the main reason for using also in this article a semiparametric procedure that we are now to describe. 2.2. A semiparametric estimation procedure There exist several methods for estimating the fractional differencing parameter in a semiparametric way. Examples are the log-periodogram regression estimate (LPE), initially proposed by Geweke and Porter-Hudak (1983) and modified later by Ku¨ nsch (1986) and Robinson (1995b), the average periodogram estimate (APE; Robinson, 1994b) and the quasi maximum likelihood estimate (QMLE, Robinson, 1995a). In this article we use the QMLE. It is basically a ‘Whittle estimate’ in the frequency domain, considering a band of frequencies that degenerates to zero. The estimate is implicitly defined by: ! m 1X ^ d ¼ arg mind logCðdÞ  2d log lj ; m j¼1 CðdÞ ¼

m 1X Iðlj Þl2d j ; m j¼1

lj ¼

2pj ; T

m ! 0; T

(6)

where I(lj) is the periodogram of the raw time series, and d 2 (0.5, 0.5).6 Under finiteness of the fourth moment and other mild conditions, Robinson (1995a) proved that: pffiffiffiffi mðd^  d0 Þ !d Nð0; 14Þ as T ! 1; where d0 is the true value of d and with the only additional requirement that m ! 1 slower than T. Robinson (1995a) shows that m must be smaller than T/2 in order to avoid aliasing effects. A multivariate extension of this procedure can be found in Lobato (1999). The other methods also based on semiparametric models (like the APE and the LPE) have been 5 Moreover, the tests of Robinson (1994a) do not impose Gaussianity, with a moment condition only of order 2 required. 6 Velasco (1999a, 1999b) has recently showed that the fractionally differencing parameter can also be consistently semiparametrically estimated in nonstationary contexts by means of tapering.

534

J. Cunado et al. / Journal of Asian Economics 15 (2004) 529–547

successfully applied to economic time series (see, e.g., Gil-Alana, 2002). However, we have decided to use in this article the QMLE firstly because of its computational simplicity. Note that using the QMLE, we do not need to employ any additional user-chosen numbers in the estimation (as is the case with the LPE and the APE). Also, we do not have to assume Gaussianity in order to obtain an asymptotic normal distribution, the QMLE being more efficient than the LPE. In addition, several Monte Carlo experiments carried out, for example, by Gil-Alana (2004) showed that, in finite samples, the QMLE has better statistical properties compared with the other procedures. In the following section, we use these procedures to examine the convergence hypothesis in Taiwan with respect to the US and Japan by looking at their orders of integration. Thus, if the order of integration of the original Taiwanese series is higher than that of its differences, the latter series is ‘less nonstationary’ than the original one, implying that is less explosive if d > 1 (and thus reducing its degree of divergence) or mean reverting if d < 1 (and thus showing real convergence). Other authors have also examined the convergence hypothesis in several countries by means of methodologies based on fractional integration. (e.g., Michelacci & Zaffaroni, 2000; Silverberg & Verspagen, 1999), though the results are mixed. Michelacci and Zaffaroni (2000) could not reject the hypothesis that the differenced series across all the 14 OECD countries examined, were nonstationary and mean reverting (0.5 < d < 1). Therefore, according to these authors, the convergence hypothesis cannot be rejected, and thus, convergence takes place, although at a very slow hyperbolic rate. However, Silverberg and Verspagen (1999) obtain a different result: they analyze the GDP per capita series relative to the US for the same OECD countries as in Michelacci and Zaffaroni (2000), and find no evidence of convergence except for Australia, Austria, the UK, and Japan, although their overall conclusions strongly depend on the specific model used. For the emerging countries, these authors find evidence of convergence with respect to the US for Argentina, Brazil, Colombia, Peru, Venezuela, Indonesia, and Taiwan.7

3. Data and test results The data used in this section are annual log real GDP per capita in 1990 Geary–Khamis PPP-adjusted dollars for US, Japan, and Taiwan for the period 1903–1999. The data for the period 1903–1994 have been obtained from Maddison (1995) and these series have been updated using the Groningen Growth and Development Center (GGDC) Database 2002. We start by presenting a graphical analysis of the behavior of the real GDP per capita (in logs) in Taiwan and its differences with respect to the US and Japan (see Fig. 1). It is worth noting the following aspects: first, both the economic growth and its convergence process to the leading economies experienced an important change after World War II (WWII). In the pre-war period, there is not a decrease in the differences in per capita output between 7 Michelacci and Zaffaroni use the Geweke and Porter-Hudak (GPH, 1983) method as modified by Robinson (1995b), which is known to be highly biased in small samples. On the other hand, Silverberg and Verspagen (1999) employ Beran’s (1994) nonparametric FGN estimator and the results are here very sensitive to the specific choice of the model.

J. Cunado et al. / Journal of Asian Economics 15 (2004) 529–547

535

Fig. 1. Plots of the log real GDP per capita in Taiwan and the differences with respect to the US and Japan.

the US and Taiwan nor between Japan and Taiwan. In contrast, in the post-war period, there is a decrease in the differences with respect to the US for the whole subsample, while only for the period 1967–1999 with respect to Japan, due to the highest growth rates of this economy from the end of the WWII to 1966. This graphical analysis suggests that we should analyze the convergence process of this economy during both the whole sample and for the pre- and post-war periods. The inspection of the per capita GDP differences between the economies of Taiwan and that of the two leading economies, Japan and the US, suggests that during the pre-war period, there is not a clear tendency of catching-up, while this tendency changes in the postwar period. In the former period and from the early days of the Japanese colonialism of Taiwan (since 1895), Japan focused upon building-up the agricultural sector and the development of agriculture took top priority. In such a period (1911–1940), Taiwan’s average growth rate was 1.18%, while that of Japan and the US were, respectively, 2.64 and 1.15%. In the post-war period (1951–1992), Taiwan grew at a 6.03% rate while the growth rates of Japan and the US were 5.57 and 1.93%. However, this economy does not show a higher growth rate during the whole post-war period since, unlike Japan and the US, the post-war recovery in this economy occurred later, in the mid-1960s, which explains that the real per capita GDP differences between the Japanese economy and Taiwan show an increasing trend until the 1970s, while they experience an important decrease since that period. While there could be many factors contributing to the differences between the pre- and post-war performance, according to Hsiao and Hsiao (2003), the most important factors are the different pace of world technological progress and innovation, the type of technology between the two periods, and the increasing openness of this economy. In the pre-war period, Taiwan traded almost exclusively with Japan, while after the war, Japan continued to be its major trading partner until the mid-1960s and after this period, the United States became another major trading partner. These authors argue that the trade and economic relationships of Taiwan with these two economies is the major driving force of its ‘‘miracle growth.’’ We initially perform the tests of Robinson (1994a) to the real GDP per capita in Taiwan as well as to its differences with respect to the US and Japan. Denoting each of the time

536

J. Cunado et al. / Journal of Asian Economics 15 (2004) 529–547

series yt, we employ throughout the model given by (1) and (3), with zt ¼ (1, t)0 , t  1, zt ¼ (0, 0)0 otherwise. Thus, under the null hypothesis H0 (2): y t ¼ b 0 þ b1 t þ x t ; d0

ð1  LÞ xt ¼ ut ;

t ¼ 1; 2; . . . t ¼ 1; 2; . . .

(7) (8)

and we treat separately the cases b0 ¼ b1 ¼ 0 a priori; b0 unknown and b1 ¼ 0 a priori; and b0 and b1 unknown, i.e., we consider, respectively, the cases of no regressors in the undifferenced regression (7), an intercept, and an intercept and a linear time trend. In other words, and for the differences in GDP per capita levels, we distinguish between long-run unconditional and conditional convergence (the cases b0 ¼ b1 ¼ 0 and b0 unknown and b1 ¼ 0) and convergence as a catch-up (b0 and b1 unknown). Note that in the case of unconditional convergence, shocks to income of one country relative to another must be temporary. That is, there is a tendency for per capita income differences to return to zero, and shocks to relative per capita income are temporary. Conditional convergence implies that following a shock, there is a tendency for relative income to converge or return to a country-specific differential (which according to growth models will depend on differences in saving rates, population growth rates, etc.). Finally, convergence as a catch-up suggests that per capita income differences are expected to decrease at a rate given by the coefficient of the linear trend (b1). Following a shock, there is a tendency to return to the same decreasing behavior of relative income. We start with the assumption that ut in (8) is white noise. Thus, when d ¼ 1, for example, the differences (1  L)yt behave, for t > 1, like a random walk when b1 ¼ 0, and a random walk with drift when b1 6¼ 0. However, we report test statistics not merely for the null d0 ¼ 1 in (2) but for d0 ¼ 0, 0.25, 2, thus including also a test for stationarity (d0 ¼ 0.5); for I(2) processes (d0 ¼ 2), as well as other fractionally integrated possibilities. Furthermore, we also present 95% confidence intervals of those values of d0 where H0 (3) cannot be rejected, using a shorter grid of 0.01 values. The test statistic reported across Table 1 (and also in Tables 2–6), is the one-sided one corresponding to ^r in (4), so that significantly positive values of this are consistent with orders of integration higher than d0, whereas significantly negative ones are consistent with alternatives of form: d < d0. A notable feature observed in Table 1 (panel i), in which ut is taken to be white noise and b0 ¼ b1 ¼ 0 a priori (when the form of ^r significantly simplifies) is that the unit root hypothesis is rejected for the per capita GDP in Taiwan in favor of higher orders of integration. In fact, the only non-rejectable value takes place at d ¼ 1.25. However, if we look at the differences with respect to the US, this hypothesis is rejected while the unit root is not. The results for the differences with respect to Japan are even more conclusive and the unit root is now rejected in favor of smaller degrees of integration, the values of d where H0 (2) cannot be rejected ranging between 0.71 and 0.99. Looking at the confidence intervals, we observe that the intervals are lower for the differenced series (with respect to both the US and Japan), although only for the Japanese case the unit root is excluded, implying that mean reversion occurs in this case, and giving us evidence of real convergence between Taiwan and Japan. Including an intercept or an intercept and a linear time trend, the results are similar in both cases, with the non-rejection values occurring at the same (d0/series) combination in both cases, and the unit root null is

J. Cunado et al. / Journal of Asian Economics 15 (2004) 529–547

537

Table 1 Values of Robinson’s (1994a) statistic, testing H0 (2) in a model given by (1) and (3) with white noise disturbances Series

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Confidence interval

19.18 14.96 11.73

16.87 10.22 5.65

10.13 4.69 0.97

2.78 0.59 1.70

0.48 1.65 2.90

1.83 2.91 3.51

2.61 3.69 3.90

3.16 4.20 4.19

[1.07–1.45] [0.93–1.24] [0.71–0.99]

(ii) With an yt US-yt Japan-yt

intercept 19.59 18.18 19.36 17.75 20.17 15.23

16.00 13.90 7.46

10.71 7.73 1.24

2.95 2.75 2.77

0.49 0.14 3.00

1.84 1.27 3.58

2.59 2.21 3.94

3.11 2.89 4.21

[1.07–1.45] [1.09–1.58] [0.73–0.98]

(iii) With an yt US-yt Japan-yt

intercept 23.03 21.31 17.94

and a linear time trend 21.28 16.79 9.43 18.73 13.59 7.37 14.21 7.32 1.24

2.92 2.75 1.77

0.37 0.17 3.00

1.83 1.27 3.59

2.57 2.21 3.94

3.08 2.89 4.21

[1.08–1.45] [1.09–1.58] [0.73–0.98]

(i) No regressors yt 19.59 US-yt 19.36 Japan-yt 20.17

In bold, the non-rejection values of the null hypothesis at the 5% significance level are shown.

rejected for the case of the differenced series with respect to Japan, suggesting again the existence of convergence between these two Asian economies. The significance of the above results, however, might be due in large part to the unaccounted for I(0) autocorrelation in ut. Thus, we also fitted AR models to ut. The results are not reported, though is important to stress that we observed a lack of monotonicity in ^r with respect to d0 in practically all cases. This lack of monotonicity could be explained in terms of model misspecification as is argued, for example, in Gil-Alana and Robinson Table 2 Values of Robinson’s (1994a) statistics, testing H0 (2) in a model given by (1) and (3) with Bloomfield (1) disturbances Series

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Confidence interval

8.81 6.16 3.81

6.60 3.80 1.14

2.47 1.19 0.84

0.96 0.74 2.42

2.78 1.83 3.41

3.40 2.54 3.84

3.78 2.98 4.09

3.99 3.31 4.27

[0.80–1.07] [0.70–1.20] [0.44–0.85]

(ii) With an intercept 9.43 8.29 yt US-yt 8.95 7.25 Japan-yt 8.85 6.01

6.31 4.43 2.58

2.38 1.04 0.25

1.17 1.36 2.23

2.83 2.51 3.44

3.56 3.09 3.94

3.86 3.49 4.23

4.05 3.71 4.37

[0.79–1.05] [0.71–1.05] [0.59–0.90]

(iii) With an intercept and a linear time trend 10.72 8.64 5.56 1.73 yt US-yt 10.15 7.37 3.98 0.79 Japan-yt 7.08 5.29 2.48 0.25

1.11 1.37 2.23

2.76 2.47 3.44

3.55 3.19 3.95

3.83 3.49 4.21

4.01 3.71 4.31

[0.76–1.06] [0.68–1.06] [0.58–0.90]

(i) No regressors yt 9.43 8.95 US-yt Japan-yt 8.85

In bold, the non-rejection values of the null hypothesis at the 5% significance level are shown.

538

J. Cunado et al. / Journal of Asian Economics 15 (2004) 529–547

Table 3 Values of Robinson’s (1994) statistic, testing H0 (2) in a model given by (1) and (3) with white noise disturbances, based on the data after World War II Series

0.00

0.25

0.50

12.19 9.86 12.80

11.29 6.32 10.10

(ii) With an yt US-yt Japan-yt

intercept 12.40 11.02 12.59 11.38 14.91 14.60

9.21 9.74 13.56

(iii) With an yt US-yt Japan-yt

intercept 10.86 11.44 14.93

(i) No regressors yt 12.40 US-yt 12.59 Japan-yt 14.91

0.75

1.00

1.25

1.50

1.75

2.00

Confidence interval

7.63 2.35 5.95

0.96 0.54 1.19

1.88 2.11 1.75

2.83 2.94 2.99

3.36 3.41 3.52

3.68 3.71 3.83

[0.97–1.21] [0.81–1.15] [0.98–1.23]

7.99 7.46 11.32

2.38 1.13 6.92

0.57 1.08 2.05

1.37 1.82 0.76

1.95 2.34 2.04

2.39 2.75 2.71

[1.04–1.61] [0.98–1.42] [1.28–1.65]

and a linear time trend 9.60 8.37 5.75 9.55 7.52 4.30 14.09 13.10 11.13

2.38 1.19 6.94

0.06 0.62 2.14

1.14 1.51 0.80

1.72 1.90 1.95

2.13 2.28 2.47

[1.07–1.70] [0.96–1.56] [1.29–1.65]

In bold, the non-rejection values of the null hypothesis at the 5% significance level are shown.

(1997): frequently misspecification inflates both numerator and denominator of ^r to varying degrees, and thus affects ^r in a complicated way. However, it may also be due to the fact that the AR coefficients are Yule–Walker estimates and thus, though they are smaller than one in absolute value, they can be arbitrarily close to 1. A problem then may occur in that they may be capturing the order of integration of the series by means, for example, of a coefficient of 0.99 in case of using AR(1) disturbances. In order to solve this problem, we decided to use other less conventional forms of I(0) processes. One that seems especially relevant and convenient in the context of Table 4 Values of Robinson’s (1994a) statistics, testing H0 (2) in a model given by (1) and (3) with Bloomfield (1) disturbances, based on the data after World War II Series

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Conficence interval

(i) No regressors yt 5.47 5.61 US-yt Japan-yt 8.05

5.34 4.00 6.29

5.17 2.55 4.99

3.95 0.94 3.57

1.34 0.39 1.75

0.80 1.45 0.11

1.82 2.12 1.44

2.42 2.55 2.22

2.87 2.91 2.68

[0.98–1.45] [0.65–1.35] [1.01–1.54]

(ii) With an yt US-yt Japan-yt

intercept 5.47 4.62 5.61 5.09 8.05 7.82

3.98 4.80 6.25

2.19 4.58 4.96

0.13 0.07 2.93

2.38 2.26 0.62

2.90 2.71 1.26

3.19 2.99 2.16

3.37 3.12 2.62

[0.94–1.12] [0.91–1.13] [1.14–1.58]

(iii) With an yt US-yt Japan-yt

intercept 2.90 3.23 7.04

trend 0.49 0.77 4.61

0.68 0.63 2.84

1.87 1.71 0.60

2.50 2.12 1.21

2.68 2.81 1.99

2.72 3.66 2.28

[0.37–1.20] [0.56–1.24] [1.14–1.59]

and a linear time 2.01 1.17 2.25 1.65 5.76 5.14

In bold, the non-rejection values of the null hypothesis at the 5% significance level are shown.

J. Cunado et al. / Journal of Asian Economics 15 (2004) 529–547

539

Table 5 Values of Robinson’s (1994a) statistic, testing H0 (2) in a model given by (1) and (3) with white noise disturbances and dummy variables for the break Dummy variable

0.00

0.25

0.50

0.75

17.87 13.45 12.70

15.98 8.72 7.93

11.23 3.87 3.33

Analysis based on US-yt Model A 15.73 12.66 Model B 19.72 15.23 Model C 19.77 15.24

7.22 7.32 7.30

Analysis based on Japan-yt Model A 19.91 18.43 Model B 19.56 15.78 Model C 19.43 15.46

14.92 10.03 9.65

Analysis based on yt Model A 19.03 Model B 16.63 Model C 16.13

1.00

1.25

1.50

1.75

2.00

Confidence interval

3.19 0.64 0.43

0.50 1.08 1.03

1.93 2.05 1.89

2.63 2.67 2.58

3.07 3.13 3.09

[1.09–1.43] [0.90–1.40] [0.89–1.40]

1.21 0.98 0.99

1.79 1.83 1.85

2.64 3.00 2.98

2.90 3.62 3.38

3.05 4.00 3.46

3.21 4.16 3.54

[0.73–0.98] [0.72–0.97] [0.72–0.96]

9.25 4.85 4.62

3.70 1.61 1.62

0.11 0.25 0.10

1.69 1.31 0.90

2.66 1.97 2.10

3.26 2.66 2.99

[1.08–1.40] [0.99–1.62] [1.00–1.64]

In bold, the non-rejection values of the null hypothesis at the 5% significance level are shown. Model A corresponds to the ‘‘crash model’’ in Perron (1989), model B refers to the ‘‘changing growth model,’’ and model C is a combination of both.

the present tests is that proposed by Bloomfield (1973), in which the spectral density function is given by: ! m X s2 exp 2 tr cosðlrÞ : (9) f ðl; tÞ ¼ 2p r¼1 Table 6 Values of Robinson’s (1994a) statistic, testing H0 (2) in a model given by (1) and (3) with Bloomfield (1) disturbances and dummy variables for the break Dummy variable

0.00

0.25

0.50

8.55 3.56 3.43

7.20 2.41 2.21

Analysis based on US-yt Model A 4.45 3.20 Model B 8.30 6.01 Model C 8.38 6.01 Analysis based on Japan-yt Model A 9.78 8.18 Model B 8.44 5.16 Model C 8.01 4.98

Analysis based on yt Model A 9.35 Model B 5.73 Model C 5.756

0.75

1.00

1.25

1.50

1.75

2.00

Confidence interval

3.66 1.86 1.82

0.49 1.16 1.04

2.54 3.03 3.45

3.36 3.57 3.59

3.86 3.93 4.03

4.10 4.06 4.33

[0.88–1.06] [0.90–1.07] [0.92–1.10]

2.16 2.60 2.59

0.34 0.36 0.48

2.28 2.41 2.71

3.62 3.30 3.42

4.09 3.43 3.88

4.34 4.15 4.19

4.49 4.34 4.35

[0.66–0.90] [0.70–0.89] [0.68–0.92]

5.47 2.13 1.82

2.76 0.44 0.72

0.42 1.99 2.07

1.28 2.75 2.81

2.35 3.09 3.15

3.06 3.52 3.15

3.48 3.82 3.72

[0.96–1.29] [0.68–0.90] [0.69–0.88]

In bold, the non-rejection values of the null hypothesis at the 5% significance level are shown.

540

J. Cunado et al. / Journal of Asian Economics 15 (2004) 529–547

The intuition behind this model is the following. Suppose that ut follows an ARMA process of form: ut ¼

p q X X fr utr þ et  yr etr ; r¼1

r¼1

where et is a white noise process and all zeros of fðLÞ ¼ ð1  f1 L   fp Lp Þ lie outside the unit circle and all zeros of yðLÞ ¼ ð1  y1 L   yq Lq Þ lie outside or on the unit circle. Clearly, the spectral density function of this process is then:  2 P s2  1  qr¼1 yr eirl  P ; (10) f ðl; tÞ ¼ 2p 1  pr¼1 fr eirl  where t corresponds to all AR and MA coefficients and s2 is the variance of et. Bloomfield (1973) showed that the logarithm of an estimated spectral density function is often found to be a fairly well-behaved function and can thus be approximated by a truncated Fourier series. He showed that (9) approximates to (10) well, where p and q are small values, which usually happens in economics. Like the stationary AR(p) model, the Bloomfield (1973) model has exponentially decaying autocorrelations and thus we can use a model like this for ut in (8). Formulae for Newton-type iteration for estimating the tr are very simple (involving no matrix inversion), updating formulae when m is increased is also simple, and ^ in Appendix A by the population quantity: we can replace A 1 X l¼mþ1

l2 ¼

m p2 X  l2 ; 6 l¼1

which indeed is constant with respect to the tr (unlike what happens in the AR case). The Bloomfield (1973) model, involving fractional integration has not been used very extensively in previous econometric models (though the Bloomfield model itself is a well-known model in other disciplines, e.g., Beran, 1993), and one by-product of the present work is the emergence of that model as a credible alternative to the fractional ARIMAs, which have become conventional in parametric modeling of long memory.8 The results based on the Bloomfield (1973) model (with m ¼ 1 in (9)) are displayed in Table 2. Other values for m were also tried and the results were very similar to those presented here. We see that monotonicity is achieved for all series and all values of d0. Starting with the case of no regressors (Table 2, panel i), we observe that we cannot reject the unit-root hypothesis for the per capita GDP in Taiwan nor for its difference with respect to the US, while we can reject this hypothesis (and thus, the non-convergence hypothesis) for the differences with respect to Japan. Looking at the confidence intervals, we observe that only for the differences with respect to Japan the unit root is excluded in the intervals. Including an intercept or an intercept and a linear time trend, the results are similar in all cases and the unit root is only rejected for the case of the differenced series with respect to Japan, suggesting once more the existence of real convergence between these two Asian economies. Furthermore, since the tests include the cases of no regressors, an intercept and an intercept and a 8 Amongst the few empirical applications found in the literature are Gil-Alana and Robinson (1997), Velasco and Robinson (2000), and more recently Gil-Alana (2001b).

J. Cunado et al. / Journal of Asian Economics 15 (2004) 529–547

541

Fig. 2. Estimates of d based on QMLE of Robinson (1995a) for a range of values of m ¼ 10, 40. The horizontal axes corresponds to the bandwidth parameter number m, while the vertical one refers to the order of integration.

linear trend, we can distinguish between long-run unconditional or conditional convergence or convergence as a catch-up (in the case of a linear trend). For the differences between per capita GDP in Japan and Taiwan, we obtain a significant intercept and a non-significant linear trend, suggesting thus the existence of conditional convergence.9 Next, we consider the possibility of a structural break at WWII. For this purpose, we first perform the same type of analysis as before but using data ending in 1945. The results though not reported in the paper, were very similar to those based on the whole sample. More interesting are the results based on post-war data. Tables 3 and 4 correspond, respectively, to Tables 1 and 2 but for the time period, 1945–1999. Starting with the case of white noise disturbances (Table 3) we observe that the unit root cannot be rejected in any of the cases, suggesting thus that real convergence does not take place. Moreover, a very similar picture is obtained in Table 4 where ut is allowed to be weakly autocorrelated. In view of this, we can conclude by saying that for the post-war period, there is no evidence of real convergence between Taiwan and the US and Japan. Next, we perform the semiparametric procedure described in Section 2.2. Fig. 2 reports the results based on the QMLE of Robinson (1995a), i.e., d^ given by (6), for a range of values of m (see Eq. (6)) from 10 to 40. Since the time series are nonstationary, the analysis will be carried out based on the first differenced data, adding then 1 to the estimated values of d to obtain the proper orders of integration of the series. We see that for the original series, the values are all strictly above 0, implying that the order of integration is strictly above 1, and the same happens with the differences with respect to the US. However, if we concentrate on the Japanese differences, the values are in practically all cases below 0, implying that mean reversion takes place for this series. Thus, the results here are consistent with those in Tables 1 and 2, supporting the real convergence hypothesis between Taiwan and Japan. However, performing this procedure on the post-war data (see Fig. 3), we observe that the estimated values of d are slightly lower for the case of the differences with respect to the 9 Note that the coefficients of the intercept and the linear trend are OLS, which are all based on the d0differenced model, which is short memory under the null. Thus, standard t-tests still remain valid.

542

J. Cunado et al. / Journal of Asian Economics 15 (2004) 529–547

Fig. 3. Estimates of d based on QMLE of Robinson (1995a) for a range of values of m ¼ 1, 25, based on data after World War II. The horizontal axes corresponds to the bandwidth parameter number m, while the vertical one refers to the order of integration.

US than for the original series, while are higher for the case of the differences with respect to Japan. The results are consistent with those in Tables 3 and 4, not supporting the convergence hypothesis for the post-war period. This final result may appear surprising, especially in view of Fig. 1, where a decreasing in the differenced series is observed after WWII. Several aspects should be however considered here. First, it is important to note that unit roots (and more generally I(d) models) require a sufficiently long span of data in order to make valid inference. Otherwise, the results are liable to be biased in favor of nonstationarities. Consider, for example, the US differenced series in Fig. 1 starting in 1945. In such a case, there is a continuous decreasing trend in the values, which may be consistent with a unit root. However, if we span the data backwards, the evidence of unit roots is then not so clear. We try to solve this problem by including a dummy variable in the regression model that takes into account a possible break at WWII. This may be done using Robinson’s (1994a) tests, since the inclusion of dummy variables in his set-up does not affect the standard null and local limit distributions of the tests. Denoting by Tb the time of the break (in our case, 1945) and following the parameterization of Perron (1989) and others, we will assume that b B 1 in (3) and zt contains an exogenous change in the intercept of the trend function (crash model), model A, i.e., zt ¼ m0 þ m1 DUt þ dt; 1 if t > Tb DUt ; 0 otherwise an exogenous change in the slope of the trend function (changing growth model, B), i.e. zt ¼ m þ d0 t þ d1 DTt ; t  Tb if t > Tb DTt ¼ ; 0 otherwise

J. Cunado et al. / Journal of Asian Economics 15 (2004) 529–547

543

and both effects in the model (C): zt ¼ m0 þ m1 DUt þ d0 t þ d1 DTt ; t if t > Tb  DTt ¼ ; 0 otherwise and test once more H0 (2) in (1) and (3) for the same d0-values as in the previous cases. Table 5 displays the results for the case of white noise disturbances, and we see that convergence is clearly satisfied for the case of Japan, with orders of integration smaller than 1 in all cases. However, for the US, the results strongly reject the hypothesis of convergence. If we permit autocorrelated (Bloomfield) disturbances, the results are displayed in Table 6. In this case, convergence is also satisfied for the US if we consider a slope change in the trend (models B and C).

4. Concluding remarks In this article, we have examined the real convergence hypothesis in Taiwan by means of fractional integration techniques. In particular, we have examined the order of integration of the log real GDP per capita series as well as its differences with respect to the US and Japan. For this purpose we have used a parametric testing procedure due to Robinson (1994a) and a semiparametric estimation method (QMLE; Robinson, 1995a). We have used these procedures, firstly because of the distinguishing features that make them particularly relevant in comparison with other methods. Thus, Robinson’s (1994a) tests allow us to consider unit and fractional root tests with no effect on its standard null limit distribution, which is also unaffected by the inclusion of deterministic trends and of different types of I(0) disturbances. In addition, the tests are the most efficient ones when directed against the appropriate (fractional) alternatives. The reason for using the QMLE of Robinson (1995b) is based on its computational simplicity along with the fact that it just requires a single bandwidth parameter, unlike other procedures where a trimming number is also required. A FORTRAN code with the programs is available from the authors upon request, and it can also be obtained throughout the Working Paper version of the article on the website (http:\www.unav.es). Using the parametric procedure of Robinson (1994a), the results support the view that the per capita GDP series may be specified in terms of unit root models. Performing the same tests on the differenced data, the results suggest the existence of mean reversion, and thus, real convergence only with respect to Japan. We also performed a semi-parametric procedure of Robinson, namely, the quasi maximum likelihood estimation (QMLE, Robinson, 1995a) method and the results here were consistent with the parametric ones in Robinson (1994a), finding thus conclusive evidence of real convergence with respect to Japan. Splitting the sample around WWII, the results substantially differ from one sub-sample to the other. Thus, using pre-war data, the results were in line with those based on the whole sample. However, when we analyze solely the post-war period, surprisingly we find no evidence of real convergence of the Taiwanese economy, nor with respect to Japan nor with respect to the US. This lack of convergence may be explained because of the small number

544

J. Cunado et al. / Journal of Asian Economics 15 (2004) 529–547

of observations used (the sample sizes are around 40 when splitting the sample around WWII). In order to solve this problem, we decided to use once more the test of Robinson (1994a), but this time incorporating a dummy variable to take into account the break. In doing so, evidence of convergence is found with respect to Japan if the underlying disturbances are white noise, and with respect to both (Japan and the US) with autocorrelated disturbances.

Acknowledgements The second named author gratefully acknowledges financial support from the Government of Navarra (‘‘Ayudas de Formacio´ n e Investigacio´ n y Desarrollo’’), Spain. Comments of two anonymous referees are also acknowledged.

Appendix A The functional form of the test statistic is: ^r ¼

T 1=2 ^ 1=2 ^ A a ^2 s

where T 1 T 1 2p X 2p X ^2 ¼ s2 ð^tÞ ¼ cðlj Þgðlj ; ^tÞ1 Iðlj Þ; s gðlj ; ^tÞ1 Iðlj Þ; T j¼1 T j¼1 0 1 !1 T 1 T 1 T 1 T 1 X X X X 2 ^ ¼ @ cðlj Þ2  ^eðlj Þ^eðlj Þ0 ^eðlj Þcðlj ÞA cðlj Þ^eðlj Þ0 A T j¼1 j¼1 j¼1 j¼1    lj  @ 2pj  log gðlj ; ^tÞ; lj ¼ ; ^t ¼ arg min s2 ðtÞ: cðlj Þ ¼ log2sin ; ^eðlj Þ ¼ @t T 2

^ a¼

I(lj) is the periodogram of ut evaluated under the null, i.e., ^0 wt ; ^ ut ¼ ð1  LÞd0 yt  b !1 T T X X ^¼ b wt w0 wt ð1  LÞd0 yt ; t

t¼1

wt ¼ ð1  LÞd0 zt ;

t¼1

and the function g above is a known function coming from the spectral density function of ut, f ðl; s2 ; tÞ ¼

s2 gðl; tÞ; 2p

p < l  p:

Note that these tests are purely parametric and therefore, they require specific modeling assumptions to be made regarding the short memory specification of ut. Thus, if ut is white

J. Cunado et al. / Journal of Asian Economics 15 (2004) 529–547

545

noise, g B 1, and if ut is an AR process of form f(L) ut ¼ et, g ¼ |f(eil)|2, with s2 ¼ V(et), so that the AR coefficients are functions of t.

References Adenstedt, R. K. (1974). On large sample estimation for the mean of a stationary random sequence. Annals of Statistics, 2, 259–272. Baillie, R. T., & Bollerslev, T. (1994). The long memory of the forward premium. Journal of International Money and Finance, 15, 565–571. Barro, R. (1991). Economic growth in a cross section of countries. Quarterly Journal of Economics, 106, 407– 443. Barro, R., & Sala-i-Martin, (1991). Convergence across states and regions. Brookings Papers of Economic Activity, 1, 107–182. Barro, R., & Sala-i-Martin, (1992). Convergence. Journal of Political Economy, 100, 223–251. Barro, R., & Sala-i-Martin. (1995). Economic growth. McGraw Hill. Baumol, W. J. (1986). Productivity growth, convergence, and welfare: What the long-run data show. American Economic Review, 76, 1072–1085. Beran, J. (1993). Fitting long memory models by generalized linear regressions. Biometrika, 80, 817–822. Beran, J. (1994). Statistics for long memory processes. New York: Chapman & Hall. Bernard, A. B. (1991). Empirical implications of the convergence hypothesis (Working paper). Center for Economic Policy Research, Stanford University. Bernard, A. B., & Durlauf, S. N. (1995). Convergence in international output. Journal of Applied Econometrics, 10, 97–108. Bloomfield, P. (1973). An exponential model in the spectrum of a scalar time series. Biometrika, 60, 217–226. Campbell, J. Y., & Mankiw, N. G. (1989). International evidence on the persistence of economic fluctuations. Journal of Monetary Economics, 23, 319–333. Carlino, G. A., & Mills, L. O. (1993). Are U.S. regional incomes converging? A Time Series Analysis. Journal of Monetary Economics, 32, 335–346. Cavaliere, G. (2001). Testing the unit root hypothesis using generalized range statistics. The Econometrics Journal, 4, 70–88. Chambers, M. (1998). Long memory and aggregation in macroeconomic time series. International Economic Review, 39, 1053–1072. Cogley, T. (1990). International evidence on the size of the random walk in output. Journal of Political Economy, 98, 501–518. Cioczek-Georges, R., & Mandelbrot, B. B. (1995). A class of micropulses and anti-persistent fractional Brownian motion. Stochastic Processes and their Applications, 60, 1–18. Cunado, J., Gil-Alana, L. A., & Pe´ rez de Gracia, F. (2004). Real convergence in some emerging countries. A long memory approach. Working Paper, Department of Economics, Universidad de Navarra, Pamplona, Spain. Dahlhaus, R. (1989). Efficient parameter estimation for self-similar process. Annals of Statistics, 17, 1749–1766. De Long, J. B. (1988). Productivity growth, convergence and welfare: A comment. American Economic Review, 78, 1138–1155. Dickey, D. A., & Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association, 74, 427–431. Diebold, F. X., & Rudebusch, G. D. (1989). Long memory and persistence in the aggregate output. Journal of Monetary Economics, 24, 189–209. Dowrick, S., & Nguyen, D. T. (1989). OECD comparative economic growth 1950–1985: Catch-up and convergence. American Economic Review, 79, 1010–1030. Easterly, W. (1995). Explaining miracles: Growth regressions meet the gang of four. In T. Ito & A. O. Krueger (Eds.), Growth theories in light of the East Asian experience (pp. 267–290). Chicago: University of Chicago Press.

546

J. Cunado et al. / Journal of Asian Economics 15 (2004) 529–547

Geweke, J., & Porter-Hudak, S. (1983). The estimation and application of long memory time series models. Journal of Time Series Analysis, 4, 221–238. Gil-Alana, L. A. (1999). Testing fractional integration with monthly data. Economic Modelling, 16, 613–629. Gil-Alana, L. A. (2000). Mean reversion in the real exchange rates. Economics Letters, 16, 285–288. Gil-Alana, L. A. (2001a). Testing stochastic cycles in macroeconomic time series. Journal of Time Series Analysis, 22, 411–430. Gil-Alana, L. A. (2001b). An exponential spectral model for the UK unemployment. Journal of Forecasting, 20, 329–340. Gil-Alana, L. A. (2002). Semiparametric estimation of the fractional differencing parameter in the UK unemployment. Computational Economics, 19, 323–339. Gil-Alana, L. A. (2004). Comparisons between semiparametric procedures for estimating the fractional differencing parameter. Working Paper, Department of Economics, Universidad de Navarra, Pamplona, Spain. Gil-Alana, L. A., & Robinson, P. M. (1997). Testing of unit roots and other nonstationary hypotheses in macroeconomic time series. Journal of Econometrics, 80, 241–268. Gil-Alana, L. A., & Robinson, P. M. (2001). Testing seasonal fractional integration in the UK and Japanese consumption and income. Journal of Applied Econometrics, 16, 95–114. Granger, C. W. J. (1980). Long memory relationships and the aggregation of dynamic models. Journal of Econometrics, 14, 227–238. Granger, C. W. J. (1981). Some properties of time series data and their use in econometric model specification. Journal of Econometrics, 16, 121–130. Granger, C. W. J., & Joyeux, R. (1980). An introduction to long memory time series and fractionally differencing. Journal of Time Series Analysis, 1, 15–29. Grier, K. B., & Tullock, G. (1989). An empirical analysis of cross-national economic growth, 1951–1980. Journal of Monetary Economics, 24, 259–276. Hsiao, F. S. T., & Hsiao, M. C. W. (2003). ‘‘Miracle Growth’’ in the twentieth century. International comparisons of East Asian development. World Development, 31, 2. Hosking, J. R. M. (1981). Fractional differencing. Biometrika, 68, 168–176. Kwiatkowski, D., Phillips, P. C. B., Schmidt, P., & Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics, 54, 159–178. Ku¨ nsch, H. (1986). Discrimination between monotonic trends and long-range dependence. Journal of Applied Probability, 23, 1025–1030. Lim, L. K., & McAleer, M. (2000). Convergence and catching up in South-East Asia: A comparative analysis. Econometric Society World Congress 2000, Contributed Paper 1844. Lippi, M., & Zaffaroni, P. (1999). Contemporaneous aggregation of linear dynamic models in large economies. Manuscript, Research Department, Bank of Italy. Lobato, I. (1999). A semiparametric two-step estimator for a multivariate long memory process. Journal of Econometrics, 73, 303–324. Lucas, R. (1988). On the mechanics of economic development. Journal of Monetary Economics, 22, 3–41. Maddison, A. (1995). Monitoring the world economy 1820–1992. OECD: Paris. Mankiw, G., Romer, P., & Weil, D. N. (1992). A contribution to the empirics of economic growth. Quarterly Journal of Economics, 107, 407–437. Michelacci, C., & Zaffaroni, P. (2000). (Fractional) beta convergence. Journal of Monetary Economics, 45, 129–153. Parke, W. R. (1999). What is fractional integration? The Review of Economics and Statistics, 81, 632–638. Perron, P. (1989). The great crash, the oil price shock and the unit root hypothesis. Econometrica, 57, 1361–1401. Phillips, P. C. B., & Perron, P. (1988). Testing for a unit root in a time series regression. Biometrika, 75, 335–346. Robinson, P. M. (1978). Statistical inference for a random coefficient autoregressive model. Scandinavian Journal of Statistics, 5, 163–168. Robinson, P. M. (1994a). Efficient tests of nonstationary hypotheses. Journal of the American Statistical Association, 89, 1420–1437.

J. Cunado et al. / Journal of Asian Economics 15 (2004) 529–547

547

Robinson, P. M. (1994b). Semiparametric analysis of long memory time series. Annals of Statistics, 22, 515–539. Robinson, P. M. (1995a). Gaussian semiparametric estimation of long range dependence. Annals of Statistics, 23, 1630–1661. Robinson, P. M. (1995b). Log-periodogram regression of time series with long range dependence. Annals of Statistics, 23, 1048–1072. Romer, P. (1986). Increasing returns and long-run growth. Journal of Political Economy, 94, 1002–1037. Schmidt, P., & Phillips, P. C. B. (1992). LM tests for a unit root in the presence of deterministic trends. Oxford Bulletin of Economics and Statistics, 54, 257–287. Silverberg, G., & Verspagen, B. (1999). Long memory in time series of economic growth and convergence (Working paper 99.8). Eindhoven Centre for Innovation Studies. Smith, A. A., Sowell, F., & Zin, E. (1997). Fractional integration with drift: Estimation in small samples. Empirical Economics, 22, 103–116. Solow, R. M. (1956). A contribution to the theory of economic growth. Quaterly Journal of Economics, 70, 65–94. Sowell, F. (1992). Maximum likelihood estimation of stationary univariate fractionally integrated time series models. Journal of Econometrics, 53, 165–188. Tanaka, K. (1999). The nonstationary fractional unit root. Econometric Theory, 15, 549–582. Taqqu, M.S. (1975). Weak convergence to fractional motion and to the Rosenblatt process. Zeitschrift fur Wahrscheinlichkeitstheorie Und Werwandte Gebiete, 31, 287–302. Taqqu, M. S., Willinger, W., & Sherman, R. (1997). Proof of a fundamental result in self-similar traffic modelling. Computer Communication Review, 27, 5–23. Velasco, C. (1999a). Nonstationary log-periodogram regression. Journal of Econometrics, 91, 299–323. Velasco, C. (1999b). Gaussian semiparametric estimation of nonstationary time series. Journal of Time Series Analysis, 20, 87–127. Velasco, C., & Robinson, P. M. (2000). Whittle pseudo-maximum likelihood estimation for nonstationary time series. Journal of the American Statistical Association, 95, 1229–1243. Young, A. (1992). A tale of two cities: Factor accumulation and technical change in Hong Kong and Singapore. NBER Macroeconomics Annual, 13–54. Young, A. (1995). The tyranny of numbers: Confronting the statistical realities of the East Asian growth experience. Quarterly Journal of Econometrics, 110, 641–680.