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3 Tests of Cross-Section Dependence

The formal statistical procedures designed to test for cross-sectional dependence in small-T, large-N panels are the Pesaran (2004) cross-sectional dependence (CD) test, Friedman's statistic (1937), and the test proposed by Frees (1995). The CD test is the Lagrange multiplier (LM) test developed by Breusch and Pagan [BP] (1980) often applied when the time-series dimension T of the panel is larger than the cross sectional dimension N as the case in our data. The Pesaran's CD statistic is a variant or an alternative to the BP test (LM statistic) which has a mean at exactly zero for fixed values of T and N, under a wide data space of panel-data models that include homogenous/heterogeneous dynamic models. When T is lesser than N, the LM test statistic exhibits substantial size distortions and loses the desirable statistical properties (Hoyos and Sarafidis 2006). The following subsections further summarize the statistical tests conditioned in the consideration of a standard panel data model:

(1)

Where yit is the observation on the dependent variable for individual i at time t, xit is a column vector of regressors with dimension K (K x 1 vector regressors), β is the corresponding parameter vector to be estimated, and γit is an individual-specific time-invariant nuisance parameter (unobserved effect), and υit under the null hypothesis is the error component that may be cross-sectionally correlated but assumed to be independent and identically distributed (i.i.d.) over periods and across cross-sectional units. Under the alternative, the error component possesses the assumption of no serial correlation. The null hypothesis according to Sarafidis and Wansbeek (2010) implies the following to be true:

(2)

as against

where the number of possible pairings (υit,υjt) increases with N and ρij is the product moment correlation coefficient of the disturbances and is given by

(3)

As further observed by Sarafidis and Wansbeek (2010), the cross-sectional dependence in the error term is a consequence of model misspecification. This connotes that if the model was specified correctly, cross-sectional dependence would have been taken into cognizance and the resulting disturbance uncorrelated across units.

3.1 Pesaran's CD Test

The Breusch and Pagan statistic (1980) tests the null hypothesis of zero correlation using an LM statistic, which holds for fixed N to T→∞ and is given by

Where p is the sample estimate of the pair-wise correlation of the residuals

and v2 is the estimate of vit in Eq. (1). The Breusch and Pagan (1980) statistic shows that under the null hypothesis of no cross sectional dependence, the CDLM statistic is asymptotically distributed as χ2 with N(N 1)/2 degrees of freedom with N fixed and T→∞. What is commonly experienced in empirical applications is a situation where T < N and the LM test statistic exhibits substantial size distortions, is biased and loses the desirable statistical properties. Pesaran (2004) subsequently proposed the following alternative CD test:

(4)

assuming under the null hypothesis of no cross-sectional dependence CD→N(0,1) for N→∞ and T sufficiently large.

3.2 Friedman's Test

The Friedman's test (1937) is nonparametric and based on Spearman's rank correlation coefficient. The coefficient is a regular product-moment correlation coefficient that accounts for the proportion of variability and computed from the ranks of the Spearman's rank correlation coefficient. The Friedman's statistic is given by:

(5)

Where

(6)

Equation (6) is the Spearman rank correlation by defining {rt, 1, .. .,ri,T} to be the ranks of {ut, 1, .. .,ui,T}, given the average rank to be{T + ½}. As for Eq. (5), r^ij is the sample estimate of the rank correlation coefficient of the residuals. Large values of Rave imply presence of nonzero cross-sectional correlations (Hoyos and Sarafidis 2006).

3.3 Levene Test

Levene (1960) proposed a test statistic which determines whether two or more groups are significantly different. Specifically, the Levene's test [LT] examines if k groups have equal variances. Where there is a significant difference in the variances of the groups, it indicates a strong evidence of that the groups are dissimilar and not selected from an identical population. In particular, Levene shows that if a variable X with sample size N divided into k subgroups (where Ni is the sample size of the ith subgroup), the LT equals

(7)

Where Zij = 1Xij Xei1 with Xi indicating the median of the ith subgroup, Zi are the group means of the Zij and Ze:: is the overall mean of the Zij (Cooray et al. 2013). The null hypothesis of the LT is that the groups have equal variances; and accepting the null hypothesis implies homogeneity.

3.4 Frees' Test

Frees' test (1995, 2004) is a power test aimed at detecting false null hypothesis even when there exits plenty of cross-sectional dependence left out in the disturbances. The Frees' Test (FT) is a statistic based on the sum of squared rank correlation coefficients and equals (Hoyos and Sarafidis 2006), and can be computed as:

(8)

and, and are independently χ2 random variables with T – 1 and T(T – 3)/2 degrees of freedom: and . If , where is quantile of the Q distribution.

3.5 Application of Cross-Sectional Dependence: The CD Test

In this section, we analyze the cross-dependence tests. Specifically, we estimate one of the four statistical tests; precisely the CD. The data set used in the analyses is obtained from the 2013 World Development Report and the 2013 African Development Report. Given a standard panel model:

(9)

where yit is GDP growth rate of country i; xit = vector of variables in Eq. (9) for unit i and time t; these are capital accumulation (capit, gross capital formation/investment), human capital development (hcdit health expenditure), inflation ( pit) and one period lagged GDP growth rate ðyit-1Þ. The βi and φi are respectively the intercept and slope coefficient which are allowed to be heterogeneous across i. Furthermore the intercept is allowed to vary across units.

The test for cross-sectional dependence is analyzed using Stata 12. The dataset consists of four countries (The Gambia, Ghana, Nigeria and Sierra Leone), each observed for 17 years (1995–2011) and a panel, declared as strongly balanced.

As reported in the estimated results presented in Table 2 once we account for country Fixed Effects (FE), human capital development (hcdit) has no effect upon the selected WAMZ economic growth. The inflation variable ( pit), and investment ( fcfit) reject the null hypothesis at 1 % significant level, since both regressors exhibit p-value of 0.000 and 0.003 respectively. Consequent upon this, only human capital development exhibits homogeneity among the four WAMZ countries.

The assumption implicit in estimating the null hypothesis of Eq. (4); and precisely the Pesaran's CD test is that the cross-sectional units are independent. The Pesaran CD test is reported as follows:

sectional dependence. Therefore, we can conclude there is strong evidence against

the hypothesis that the sampled WAMZ countries move together with respect to the group of growth enhancing variables. This suggests dissimilarity among the countries and as such, the selected WAMZ countries should be studied independently, thus arousing application of time-series estimation as against longitudinal data technique in the detection of growth enhancing variables. Second, arising from this result, though with caution, the formation of the group (WAMZ integration) may be better accomplished if there is 'sequencesation' of one economy after another; unless the political will to fuse 'real-time' is feasible.

Table 2 Test of cross-sectional dependence

3.6 Theoretical Model of Growth Enhancing Variables

The model of the individual economies is relatively small, open to rest of the world and consists of three sectors: government, banking sector and corporate with the household subsumed apparently more because there is lack of time series data. Furthermore, it is presumed that capital and labour are substitutable. The corporate sector produces non-capital goods using labour and imported physical capital as inputs. The government collects taxes from corporate sector and spends on consumption activities. The banking sector services the corporate and government sectors in domestic and international activities. All activities are geared to enhance growth; while prices as assumed, adjust to clear the goods and money markets.

In the empirical literature on economic growth process, several factors have been simulated for its enhancement. The notable pioneers are Kuznets (1955, 1966, 1971), Solow (1956) etc.; while in recent times, the works of Chenery and associates (e.g., Chenery et al. 1986), Barro (1989), Mankiw et al. (1992), Rao (2010), Rao and Hassan (2012) have also been prominent. The studies provide confirmation of the importance of some factors in growth determination; e.g. investment (often times broken down as private/public), trade openness, prices, human capital and government policies.

Some non-economic factors have been recognized in the literature to interact with economic growth process. The works of North and Thomas (1973) and North (1990) argue that strong political institutions and democracy enhance economic development. According to North (1994, p. 1), as cited in Kibritcioglu and Dibooglu (2001):

Institutions are the incentive structure of a society and therefore the rules, norms, and enforcement characteristics that make up the institutional foundations of a society direct the allocation of resources of that society and economy. Economic growth throughout history could only be realized by creating an institutional and organizational structure that would induce productivity enhancing activity – a supply side argument; and equally that the consequent tensions induced by the resulting societal transformation have resulted (and are continuing to result) in politically-induced fundamental changes in the institutional structure to mitigate these tensions – a demand side argument. Both the supply side and demand side institutional changes have been and continue to be fundamental influences on productivity change.

This quotation cascades the relation economic growth has with macroeconomic and institutional variables including democracy. However, one limitation of the model is the subsuming of labour and particularly the role of human capital particularly knowledge stock as a complementary input in the growth process.

 
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