Driscoll and Kraay Standard Errors

In this study, standard errors for standard fixed-effects technique (applied by “xtreg, fe” command in Stata) suffer from heteroscedasticity possibly due to serial

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correlation and cross-sectional dependence between the error terms. As shown in Table 5.1, results for the Wald test for group-wise heteroscedasticity strongly reject the null hypothesis that the variance is identical for all observations for models in previous chapters.

Table 5.1 Results for Wald Test for Group-wise Heteroscedasticity

H0: for all Chapter 2 (Table 2.2.b, Column 5) Chapter 3 (Table 3.2, Column 6) Chapter 4 (Table 4.2, Column 6) (67) 2372.24 12237.05 21337.75 Prob> 0.0000 0.0000 0.0000

Additionally, computing the dependent variable as the five-year forward moving geometric average of the growth rate in the second chapter, the five-year forward moving arithmetic average of the gross enrolment rate in the third chapter, and the five-year forward moving arithmetic average of the infant mortality rate in the fourth chapter introduces serial correlation to the error terms, as the dependent variables between and become correlated (Devarajan, Swaroop and Zou, 1996). The standard errors are corrected for serial correlation up to five lags as the dependent variables are computed as the five-year forward moving averages of the outcome variable. The pair-wise correlation coefficients for the error terms are provided in Table A.5.1.1 for the second chapter, in Table A.5.1.2 for the third chapter, and in Table A.5.1.3 for the fourth chapter. The statistics show that the correlation between the error terms decreases as the number of time periods increases between them, but the size of the correlation between error terms remains high over all years in all tables.

Additionally, standard errors obtained from the standard fixed-effects technique suffer from cross-sectional dependence. Pesaran’s, Friedman’s and Frees’ methods are employed for tests for cross-sectional dependence between error terms. For Stata, Breusch-Pagan test is the one that is commonly used for cross-sectional dependence for panel datasets, but this test cannot be used for panel datasets that have more panels ( ) than the number of time periods ( ). De Hoyos and Sarafidis (2006) provide tests for cross-sectional dependence in Stata using Pesaran’s, Friedman’s and Frees’ methods which can be applied to a panel dataset for which the number of panels is greater than the number of time periods. In Table 5.2 and Table 5.3, the results for all three tests are provided. Sarafidis and De Hoyos (2006) note that Pesaran’s and Friedman’s methods are not reliable when the sign of the

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correlation between error terms switches signs, as their computation requires the sum of the pair-wise correlation coefficients instead of the sum of squared correlation coefficients. Thus, negative and positive correlation coefficients would cancel out the effect of each other according to these methods, which may result in accepting the null hypothesis that there is no cross-sectional dependence. Frees’ test does not suffer from this drawback.

Table 5.2 Results for Pesaran and Friedman tests for Cross-Sectional Dependence

Correlation coefficient Chapter 2 (Table 2.2.b, Column 5 Chapter 3 (Table 3.2, Column 6) Chapter 4 (Table 4.2, Column 6) Pesaran -2.695 0.914 1.052 Probability 0.0070 0.3606 0.2926 Friedman 5.117 17.208 29.177 Probability 1.0000 1.0000 1.0000 Average Absolute Value of the Diagonal Elements in the Correlation Matrix

0.326 0.500 0.525

Table 5.3 Results for Frees’ Test for Cross-Sectional Dependence

Correlation coefficient Chapter 2 (Table 2.2.b, column 5) Chapter 3 (Table 3.2, column 6) Chapter 4 (Table 4.2, column 6) Frees 5.228 17.281 18.802

Critical Values from Frees’ Q distribution for T<30

=0.10 0.1231 0.1124 0.1124

=0.05 0.1611 0.1470 0.1470

=0.01 0.2338 0.2129 0.2129

Test results for the second, third and fourth chapters are provided in Tables 5.2 and 5.3. Friedman’s test accepts the null hypothesis that there is no cross- sectional dependence for all chapters, while Frees’ test rejects the null hypothesis for all chapters. Pesaran’s test accepts the null hypothesis only for the third and fourth chapters.

Here, the results according to Frees’ test are taken into account because of the drawback of Pesaran’s and Friedman’s tests explained above. Sarafas and Hoyos suggest checking the value of the average absolute value of the correlation coefficients in the correlation matrix for the reliability of these tests. The average absolute values for correlation coefficients in Table 4.3 are high for all chapters. Thus, there is evidence that indicates Pesaran’s and Friedman’s tests for cross- sectional dependence are not reliable.

The distribution of the residuals for each chapter is provided in Figures A.5.2.4 to A.5.2.9, which plot the residual terms from Table 2.2.b, column (5), Table

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3.2, column (6), and Table 4.2, column (6) against quartiles of normal distribution, and standard normal probability plot. While standard normal probability plot highlights the centre of distribution, quartile distribution plot underlines the tales of the distribution (stata.com, 2015). The histograms for the residuals can be seen in Figures A.5.2.1, A.5.2.2, and A.5.2.3 in Appendix- Chapter 5 in section A.5.2. It can be seen that the distribution of residuals for the second, third and fourth chapters resembles normal distribution; however, the residuals for the second and third chapters follow the quartiles of normal distribution and standardised normal probability plot more closely compared to the residuals for the fourth chapter. The difference can be seen in the histograms as well. For the second and the third chapters, the quartile distribution of residuals deviates from normal distribution plots at the tails.

A review of the methods to correct the standard errors for heteroscedasticity, autocorrelation and cross-sectional dependence in Stata can be found in Hoechle (2007). He explains that Newey and West standard errors are robust to heteroscedasticity and autocorrelation between error terms; however, Driscoll and Kraay (1998) state that this method’s asymptotic properties rely on large numbers of time periods in the data. Hoechle (2007) discusses Park and Kmenta standard errors and Beck and Katz standard errors that are computed to be robust to heteroscedasticity, serial correlation and cross-sectional dependence between error terms. However, these methods require the time dimension of the dataset to be greater than the panel dimension of the dataset, which is not the case in this study. Hoechle proposes Driscoll and Kraay’s (1998) method, which he makes available for Stata.

Driscoll and Kraay (1998, p.550) state that their method does not suffer from the restrictions regarding the size of the number of cross-sections in the data with respect to the number of time periods. The asymptotic approximation used in their method remains effective even in finite samples where the number of cross-sections is equal to or higher than the number of time periods. Thus, in this study, their method is chosen to produce robust standard errors.

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