Unit Root Tests

Table 5.5 Results for Hausman Specification Test

Chapter 2 (Table 2.2.b, Column 5) Chapter 3 (Table 3.2, Column 6) Chapter 4 (Table 4.2, Column 6) Hausman Specification Test F( 13, 66) = 116.40 Prob > F = 0.0000 F(10, 66)=704.89 Prob>F=0.0000 F(9,66)=212.70 Prob>F=0.0000

5.5 Unit Root Tests

In the analysis of panel data with high numbers of panels and time periods, whether the variables are stationary or non-stationary is a matter of concern. Standard econometric techniques for panel data are based on the assumption that the behaviour of indicators in time is stationary. There are several unit root tests available for panel data to examine the presence of unit roots, and Stata offers a command to use the majority of these. Hlouskova and Wagner (2006) call these tests

“first generation tests” which assume that the error terms between panels are not correlated.

Tests available in Stata are based on the asymptotic properties of data. They require data to have a sufficiently large number of time periods ( ) compared to the cross-sectional dimensions ( ). For example, in the Levin-Lin-Chu (LLC) test, for fixed-effects technique, the number of time periods (T) should approximate to infinity faster than the number of cross-sections (n), so that n/T tends to zero. Similarly, the Breitung test requires the number of time periods to tend to infinity

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faster than the number of panels (n). Hadri (LM) has the same asymptotic properties as the Breitung test: that the number of periods and cross-sections should approximate to infinity sequentially, T being the first (Stata Press, 2009). Hlouskova and Wagner’s (2006) findings indicate that these tests are not suitable for datasets that have a low number of time periods compared to the number of panels.

The size of the cross-sectional dimension in the dataset used for this study is greater than the number of time periods. Thus, to examine the presence of unit roots in panels, two tests are used: Harris-Tzavalis and Im-Pesaran-Shin (IPS). Both of these tests allow the number of time periods to be fixed while the number of panels tends to infinity, making them more suitable for this study. The Harris-Tzavalis test assumes that all panels have the same time-series properties, while the Im-Pesaran- Shin (IPS) test relaxes this assumption. For the Harris-Tzavalis test, the null hypothesis is that panels contain unit roots, while, for the Im-Pesaran-Shin (IPS) test, this is “all panels contain a unit root”. The alternative hypothesis for the Harris- Tzavalis test is that panels are stationary, while, for the Im-Pesaran-Shin (IPS) test, it is that some panels are stationary (Stata Press, 2009).

The results for both test statistics are provided in Table 5.6 according to the Harris-Tzavalis method and in Table 5.7 according to the Im-Pesaran-Shin method. The unit root tests for panel data test the null hypothesis that the correlation between the value of the variable at time and , equals 1, for all panels, against the alternative hypothesis that is less than . An alternative hypothesis gives the condition for an autoregressive process of order one [AR(1)] to be weakly dependent. This means that data become stationary as tends to infinity, because approximates to 0. Both tests allow for panel-specific means and trend in the model. It is also possible to subtract the cross-sectional mean to control for cross-sectional dependence. In Tables 5.6 and 5.7, the tests are carried out by controlling panel- specific means and the trend in the models, but cross-sectional means are not removed. This is because the results in the preceding chapters rely on data that are subtracted from their panel means, but not their cross-sectional means.

The Harris-Tzavalis unit root test reports the predicted value of , and uses the Z statistics, assuming all panels have a common . The Im-Pesaran-Shin (IPS) test for unit root runs the augmented Dickey-Fuller (ADF) test for unit root for each panel individually by allowing to change across panels. They provide three

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statistics: t-bar, t-tilde-bar and Z-tilde-bar. All statistics are reported for a fixed number of panels and time periods. While t-tilde-bar and Z-tilde-bar provide alternative but similar statistics to t-bar, t-tilde-bar is calculated by an estimator different than t-bar’s estimator. Both t-bar and t-tilde-bar statistics have non-standard sample distribution. Z-tilde-bar reports the test results for standardised t-bar statistics. Im, Pesaran and Shin (2003) state that the performances of Z-tilde-bar and t-bar statistics are equivalent for sufficiently large numbers of observations and remains similar for smaller sample size. The Im-Pesaran-Shin test for unit root reports “the exact critical values” at 1%, 5% and 10% significance level for t-bar and t-tilde bar, and p statistics for Z-tilde-bar. The critical values for t-bar and t-tilde-bar statistics are originally provided in Im, Pesaran and Shin (2003, Table 2) for a given number of time periods and panels, and for normal-distributed error terms; thus, critical values should be considered exact values only when the size of the panel data matches those provided in their paper. For this reason, critical values reported in Table 5.7 are considered to be approximate values. This is because Im, Pesaran and Shin (2003) do not report critical values for the exact numbers of , , but the statistics reported correspond to the critical values for the dimensions of panel data between and , and .

Tables 5.6 and 5.7 show that both tests reject the null hypothesis that the data for the second chapter contain unit roots. The results for population growth rate are somewhat contradictory between tests. Although the Harris and Tzavalis test for unit root indicate that the panels contain a unit root, Z-tilde-bar statistics in the Im- Pesaran-Shin test reject the null hypothesis that all panels contain unit roots. However, it must be noted that t-bar and t-tilde-bar statistics are lower than the critical values at 1%, 5% and 10% significance level. This is the case for the adult education indicator which is calculated using census data, for which both tests indicate that the panels contain unit roots5. Table 5.9 shows that the results for the third and fourth chapters do not change when the adult education indicator is removed from the regression.

In Table 5.8, the five-year forward arithmetic average of the gross enrolment rate and the five-year forward arithmetic average of the infant mortality rate do not appear to be stationary either; however, when the Im-Pesaran-Shin (IPS) test is

5

The unit roots for both the adult education indicator and population growth rate arise from the calculation of these variables.

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carried out by allowing for serial correlation in these variables in Table 5.8, the test rejects the null hypothesis that all panels contain a unit root 6 .

6

It should be noted that, in this case, the asymptotic properties of the test rely on the assumption that the numbers of time periods and panels tend to infinity, sequentially.

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Table 5.6 Unit Root Tests: Results According to the Harris-Tzavalis Test

Harris-Tzavalis unit-root test Ho: Panels contain unit roots Ha: Panels are stationary AR parameter: Common Asymptotics: N -> Infinity, T Fixed

Varıables Rho z p-value n T

Dependent Variable, Chapter 2 0.5832 -4.3766 0.0000 67 23

Public Agricultural Investment (% of GDP) 0.3299 -20.5589 0.0000 67 27

Public Mining Investment (% of GDP) 0.2858 -22.7645 0.0000 67 27

Public Manufacturing Investment (% of GDP) 0.4945 -12.3349 0.0000 67 27

Public Energy Investment (% of GDP) 0.5150 -11.3084 0.0000 67 27

Public Transportation Inv. (% of GDP) 0.3224 -20.9349 0.0000 67 27

Public Tourism Investment (% of GDP) -0.0292 -38.5023 0.0000 67 27

Public Housing Investment (% of GDP) 0.3041 -21.8499 0.0000 67 27

Public Education Inv. (% of GDP) 0.1766 -28.2173 0.0000 67 27

Public Health Inv. (% of GDP) 0.1616 -28.9665 0.0000 67 27

Public City Infra. & Sec. Inv. (% of GDP) 0.4073 -16.6930 0.0000 67 27

Public Investment (% of GDP) 0.5057 -11.7767 0.0000 67 27

Private Capital (% of GDP) 0.0345 -33.7388 0.0000 67 26

Population Growth Rate 0.7538 0.6187 0.7319 67 27

Dependent Variable, Chapter 3 0.9494 10.8645 1.0000 67 23

ln (GDP per capita) 0.6107 -6.5277 0.0000 67 27

Infant Mortality Rate 0.5521 -9.4567 0.0000 67 27

Adult Education Indicator 0.9873 12.2869 1.0000 67 27

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Table 5.7 Unit Root Tests: Results According to the Im-Pesaran-Shin Test

Im-Pesaran-Shin unit-root test Ho: All panels contain unit roots

Ha: Some panels are stationary AR parameter: Panel-specific Asymptotics: T,N -> Infinity sequentially

ADF regressions: No lags included

N-Fixed Exact Critical Values for t-bar and t-tilde-bar: 1%:-2.37 5%:-2.31 10%-2.28

Variables t-bar t-tilde-bar Z-t-tilde bar p-value n T

Dependent Variable, Chapter 2 -2.4289 -2.0928 -7.263 0.0000 67 23

Public Agricultural Investment (% of GDP) -3.2037 -2.7054 -13.3232 0.0000 67 27

Public Mining Investment (% of GDP) -* - - - 67 27

Public Manufacturing Investment (% of GDP) -3.3619 -2.7705 -14.0024 0.0000 67 27

Public Energy Investment (% of GDP) -3.0477 -2.5593 -11.7976 0.0000 67 27

Public Transportation Inv. (% of GDP) -3.2759 -2.6530 -12.7756 0.0000 67 27

Public Tourism Investment (% of GDP) -3.6377 -2.7705 -14.0024 0.0000 67 27

Public Housing Investment (% of GDP) -2.8487 -2.4169 -10.3099 0.0000 67 27

Public Education Inv. (% of GDP) -3.6102 -2.9080 -15.4383 0.0000 67 27

Public Health Inv. (% of GDP) -3.3806 -2.7092 -13.3619 0.0000 67 27

Public City Infra. & Sec. Inv. (% of GDP -3.0214 -2.5884 -12.1015 0.0000 67 27

Public Investment (% of GDP) -3.2219 -2.6727 -12.9808 0.0000 67 27

Private Capital (% of GDP) - - - - 67 26

Population Growth Rate -1.9068 -1.8325 -4.2088 0.0000 67 27

Dependent Variable, Chapter 3 -0.0169 -0.1059 13.7748 1.0000 67 23

ln (GDP per capita) -2.5324 -2.2482 -8.5492 0.0000 67 27

Infant Mortality Rate -3.1535 -2.6449 -12.6911 0.0000 67 27

Adult Education Indicator 2.4859 1.3597 29.1214 1.0000 67 27

Dependent Variable, Chapter 4 -1.6969 -1.4338 -0.2319 0.4083 67 23 *IPS test requires at least seven observations per panel. The share of public mining investment remained zero for Bilecik, Burdur, Sinop, and Sanliurfa for the years between 1975 and 2001.

Table 5.8 Im-Pesaran-Shin Unit Root Test, Asymptotics: T,N -> Infinity, Sequentially

W-t-bar p value ADF regressions

Dependent Variable, Chapter 3 -7.9324 0.0000 2.12 lags average (chosen by AIC) Dependent Variable, Chapter 4 -6.3419 0.0000 3.63 lags average (chosen by AIC)

AIC: Akaike Information Criterion

Dependent Variable, Chapter 3: The five-year forward moving arithmetic average of the gross enrolment rate Dependent Variable, Chapter 4: The five-year forward moving arithmetic average of the infant mortality rate Dependent Variable for Chapters 2: The five-year forward moving arithmetic average of the growth rate of real GDP per worker

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Table 5.9 Results for Chapter 3 and Chapter 4, Adult Education Indicator Excluded

Chapter 3 Chapter 4

Dependent Variable The Five-Year Forward

Moving Arithmetic Average of Gross Enrolment Rates

The Five-Year Forward Moving Arithmetic Average of Infant Mortality Rates

Public Energy Investment 0.001 0.001 -0.009 -0.009

Proportion of GDP (0.014) (0.013) (0.002)** (0.002)**

Public Transportation and Com. Investment -0.374 -0.426 -0.027 -0.026

Proportion of GDP (0.202) (0.218) (0.024) (0.024)

Public Education Investment 0.191 0.064 0.007 0.008

Proportion of GDP (0.187) (0.175) (0.028) (0.026)

Public Health Investment -0.171 -0.292 0.077 0.079

Proportion of GDP (0.525) (0.493) (0.045) (0.045)

Public City Infra. and Security Investment 1.034 0.771 -0.158 -0.155

Proportion of GDP (0.172)** (0.249)** (0.076)* (0.070)*

Population Growth Rate -1.022 -1.106 -0.047 -0.046

(0.244)** (0.281)** (0.027) (0.028) 1975 -0.139 -0.061 0.012 0.011 (0.022)** (0.010)** (0.001)** (0.001)** 1976 -0.139 -0.063 0.013 0.012 (0.021)** (0.009)** (0.002)** (0.000)** 1977 -0.135 -0.063 0.013 0.012 (0.020)** (0.007)** (0.001)** (0.000)** 1978 -0.131 -0.064 0.011 0.011 (0.020)** (0.009)** (0.001)** (0.001)** 1979 -0.113 -0.052 0.011 0.010 (0.019)** (0.009)** (0.001)** (0.001)** 1980 -0.108 -0.055 0.008 0.008 (0.019)** (0.011)** (0.001)** (0.002)** 1981 -0.094 -0.041 0.008 0.008 (0.018)** (0.011)** (0.001)** (0.001)** 1982 -0.084 -0.033 0.008 0.007 (0.018)** (0.011)** (0.001)** (0.001)** 1983 -0.074 -0.025 0.007 0.006 (0.017)** (0.010)* (0.001)** (0.001)** 1984 -0.060 -0.014 0.007 0.006 (0.015)** (0.010) (0.000)** (0.001)** 1985 -0.057 -0.013 0.007 0.007 (0.013)** (0.010) (0.001)** (0.001)** 1986 -0.066 -0.023 0.007 0.006 (0.013)** (0.008)** (0.001)** (0.000)** 1987 -0.068 -0.027 0.006 0.005 (0.012)** (0.007)** (0.001)** (0.000)** 1988 -0.069 -0.031 0.005 0.004 (0.012)** (0.007)** (0.001)** (0.000)** 1989 -0.070 -0.036 0.004 0.004 (0.011)** (0.007)** (0.001)** (0.000)** 1990 -0.075 -0.044 0.003 0.003 (0.010)** (0.006)** (0.001)** (0.000)** 1991 -0.079 -0.052 0.003 0.002 (0.009)** (0.006)** (0.000)** (0.000)** 1992 -0.083 -0.060 0.002 0.002 (0.009)** (0.005)** (0.000)** (0.000)** 1993 -0.086 -0.068 0.002 0.002 (0.007)** (0.005)** (0.000)** (0.000)** 1994 -0.071 -0.058 0.001 0.001 (0.007)** (0.005)** (0.000)** (0.000)** 1995 -0.048 -0.040 0.001 0.000 (0.006)** (0.005)** (0.000)** (0.000)* Ln (GDP per capita) 0.028 0.019 -0.002 -0.002 (0.018) (0.020) (0.001) (0.001)*

Infant Mortality Rate -0.297 -0.316

(0.064)** (0.094)**

Martial Law 0.019 0.020 0.003 0.003

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Adult Education Indicator -0.599 0.008

(0.169)** (0.013) Constant 0.457 0.467 0.032 0.032 (0.254) (0.277) (0.010)** (0.010)** Observations 1541 1541 1541 1541 Number of groups 67 67 67 67 F 434.84 177.81 181.16 70235.66 Within R-Squared 0.47 0.45 0.49 0.49

Standard Errors in parentheses. * Significant at 5%, ** significant at 1%

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5.6 Conclusion

In this chapter, post-estimation diagnostics have been provided for evidence that the results in the thesis are robust to heteroscedasticity, serial correlation, and cross-sectional dependence in error terms.

Both the Ramsey regression specification error test and the Lagrange multiplier test fail to reject the null hypothesis that the regression model is correctly specified for the second chapter. The data for the variables used in this chapter appear to be stationary, which increases the reliability of the results.

Unit root tests show that some panels for the adult education indicator in the third and fourth chapters are likely to be not stationary, which reduces the reliability of the results. Nevertheless, dropping the adult education indicator from estimates does not change the results.

While the Ramsey regression specification error test rejects the null hypothesis that the estimated model does not suffer from misspecification or inappropriate functional form for the third chapter, it rejects the null hypothesis for the fourth chapter. However, the Lagrange multiplier test provides evidence that the estimated models are correctly specified for all chapters.

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