The ARDL Estimation Approach

To test the long run relationship among our variables (equations (1)-(5) above), we use the robust econometric technique, Autoregressive Distributed Lag (ARDL or Bound testing) model originally introduced by Pesaran and Shin (1999) and further extended by Pesaran et al. (2001). We have decided to adopt this method for three reasons. Firstly, unlike other multivariate co- integration methods (e.g. Johansen and Juselius, 1990), the Bounds test is a simple technique that allows the co-integration relationship to estimated a single equation by OLS, once the lag order of the model is identified8. Secondly, in order to employ a valid standard co-integration

7 _{For more technical information see Clemente, Montañés and Reyes (1998).} 8

It is worth noting that with the ARDL. Variables may have a different optimal number of lags, while Johansen’s co-integration procedure does not allow it.

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testing (such as those carried out by Engle and Granger, or Johansen), we need to ensure that all of the series are integrated according to the same order9. The Bounds test allows a mixture of I(0) and I(1) variables as regressors, which means that the order of integration of appropriate variables may not necessarily be the same10. Therefore, the ARDL technique has the advantage that it does not require a specific identification of the order of the underlying data. The first step to take in any co-integration technique is to determine the degree of integration of each variable in the model. However, this depends on which unit root test one uses. Moreover, different unit root tests could lead to contradictory results. Interestingly, the ARDL approach is useful as it helps avoiding these problems. Thirdly, this technique is also suitable for small or finite sample sizes (Pesaran et al., 2001). The short and long-run coefficients of the model are estimated simultaneously and all variables of the model are assumed to be endogenous. Johansen’s co-integration technique (which avoids the ARDL approach) entails a large set of choices to be taken into consideration. For instance, it involves decisions such as the number of endogenous and exogenous variables to be included, the treatment of deterministic elements, as well as the order of VAR and the optimal number of lags to be used. Most importantly, the estimation procedures are very sensitive to the method used to make these choices and take these decisions (Pesaran and Smith, 1998).

According to Pesaran and Pesaran (1997), the ARDL approach requires the following two steps. In the first step, the existence of any long-run relationship among the variables of interest is determined using an F-test. The second step of the analysis is to estimate the coefficients of the long-run relationship and determine their values. This is followed by the estimation of the short-run elasticity of the variables with the error correction representation of the ARDL model.

The unrestricted error correction (UECM) versions of the ARDL model of the functional forms explained in the section above are given below:

∆𝑦𝑡 = 𝜌0+ 𝜌1𝑦𝑡−1+ 𝜌2𝑡𝑜𝑡_𝑒𝑥𝑝𝑡−1+ 𝜌3𝑘_𝑝𝑟𝑖𝑣𝑎𝑡𝑒𝑡−1+ 𝜌4𝑜𝑝𝑒𝑛𝑡−1+ 𝑚𝑖=1𝜌𝑖𝐸𝑥𝑝𝑡−1𝑖 +

𝜂1𝛥𝑦𝑡−𝑗 + 𝜂2𝛥𝑡𝑜𝑡_𝑒𝑥𝑝𝑡−𝑗 + 𝜂3𝛥𝑝𝑟𝑖𝑣𝑎𝑡𝑒_𝑘𝑡−𝑗 + 𝜂4𝛥𝑜𝑝𝑒𝑛𝑡−𝑗 + 𝑖∈𝜃𝜂𝑖∆𝐸𝑥𝑝𝑡−𝑗𝑖 + 𝜀𝑡 (20)

where 𝑗 is the optimal number of lags; all variables are as defined above and in the Appendix 1. 𝐸𝑥𝑝𝑖 _{represents 𝑚 different components of government expenditure (as defined above in}

section IV). Furthermore, 𝜌0 is drift component and 𝜀𝑡 represents the white noise.

9 _{Two or more variables are said to be integrated if they contain a stable long-run linkage. Greene (2003)}

elaborates co-integration as pre-test for the avoidance of spurious regression analysis, and explains that the integration order of all variables should be the same or greater than I(0); this also mean series should be non-stationary at level form.

10

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Equation (20) indicates that economic growth tends to be influenced and explained by its past values. The structural lags are established by using minimum Schwarz Bayesian Information Criteria (SBC)11.

After each regression, the Wald test (F-statistic) is computed to differentiate the long-run relationship between the concerned variables. The Wald test can be carried out by imposing restrictions on the estimated long-run coefficients. The null and alternative hypotheses (according to which all coefficients are jointly equal to zero) are as follows:

𝐻0: 𝜌1= 𝜌2= 𝜌3= 𝜌4= 𝜌𝑖 = 0 (21)

i.e., there is no co-integration among variables.

𝐻1: 𝜌1≠ 𝜌2≠ 𝜌3≠ 𝜌4≠ 𝜌𝑖≠ 0 (22)

i.e., there is co-integration among variables.

The asymptotic distribution of the Wald-test is non-standard under the null hypothesis of no co-integration among variables. Consequently, the computed F-statistic value will be evaluated with the critical values tabulated in Tables CI of Pesaran et al. (2001). These critical values are calculated for different regressors and whether the model contains an intercept and/or a trend. According to these authors, the lower bound critical values assume that the explanatory variables are integrated of order zero, or I(0), while the upper bound critical values assume that the same variables are integrated of order one, or I(1).

The two sets of critical values provide critical value bounds for all classifications of the regressors into purely I(1), purely I(0) or mutually cointegrated. However, these critical values are generated for sample size of 500 and 1000 observations and 20000 and 40000 replications respectively. Narayan (2004a) and Narayan (2005) argue that existing critical values, due to their large sample sizes, cannot be used for small sample sizes12. For this reason, we rely on those values reported in Narayan (2005), which calculated for sample sizes ranging from 30-80 observations.

Therefore, if the computed F-statistic is smaller than the lower bound value, then the null hypothesis is not rejected and we can conclude that there is no long-run relationship between economic growth and its determinants. Conversely, if the computed F-statistic is greater than

11

Pesaran and Smith (1998) argue that the SBC should be used in preference to other model specification criteria because it often has more parsimonious specifications.

12

For instance, Narayan (2005) compares the critical values generated with 31 observations and the critical values reported in Pesaran et al. (2001). He finds that the upper bound critical value at the 5% significance level for 31 observations with four regressors is 4.13 while the corresponding critical value for 1000 observations is 3.49, which is 18.3% lower than the critical value for 31 observations.

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the upper bound value, then capital expenditure and economic growth share a long-run level relationship. On the other hand, if the computed F-statistic falls between the lower and the upper bound values, then the results are inconclusive.

In order to find out the short run coefficients, we make use of the following equation (which represents the final version of the estimated model):

∆𝑦𝑡 = 𝜌0+ 𝜂1𝛥𝑦𝑡−𝑗 + 𝜂2𝛥𝑡𝑜𝑡_𝑒𝑥𝑝𝑡−𝑗 + 𝜂3𝛥𝑝𝑟𝑖𝑣𝑎𝑡𝑒_{𝑘 𝑡−𝑗} + 𝜂4𝛥𝑜𝑝𝑒𝑛𝑡−𝑗+

𝜂𝑖∆𝐸𝑥𝑝𝑡−𝑗𝑖 + 𝜔𝐸𝐶𝑇𝑡−1+ 𝜇𝑡

𝑖∈𝜃 (23)

where the 𝜂𝑠 are the short-run dynamic elasticities of the model's convergence to long-run equilibrium and 𝜔 is the speed of adjustment. Δ represents the first differences operator and the 𝐸𝐶𝑡−1 refers to the one period lagged error correction term. The coefficient of 𝐸𝐶𝑇

provides the speed with which variables returns to their equilibrium position in the long-run, in the event of shocks to the system. The sign of 𝐸𝐶𝑇 should be negative and statistically significant. In each equation, changes in the endogenous variables are caused not only by their lags, but also by the previous period’s disequilibrium in level.

In the final step, we apply Hendry and Ericson’s (1991) general-to-specific modelling technique to select the preferred ECM. This procedure first estimates the ECM with different lag lengths for the differences terms. Subsequently, it simplifies the representation by eliminating the lags with insignificant parameters. A correctly indicated ECM model has to pass a series of diagnostic tests. These include the Autoregressive LM (Lagrange Multiplier) test and/or the Durbin Watson test for serial correlation in residuals and the Jarcque-Bera test for normality distribution of residuals. In this study, we also apply the Ramsey RESET specification test. In summary, these tests have been conducted to ensure the reliability of results. Stability tests, such as CUSUM and CUSUMSQ are also employed to check the stability of the estimated coefficients over the time periods.