La sostenibilidad de los sistemas de protección social.

4.3.2.1 Johansen’s Cointegration Tests

Several cointegration tests can be used to see if there is cointegration between variables and to find the number of cointegrating relationships. Engle-Granger tests, which examine the stationarity of the residual of a regression, can be used to test the former issue. If the residuals do not have a unit root through these tests, it means that there is evidence supporting the existence of a cointegrating relationship between the variables, because the linear combination is stationary. Meanwhile, Johansen’s cointegration procedure can be used to find the number of cointegrating relationships between the variables.

In a first step, Engle-Granger (1987) tests59 can be applied to see if there is a cointegrating relationship between total government expenditure-to-GDP ratios and total government revenue-to-GDP ratios. This method is based on the OLS residuals from the cointegrating regression for each country and examines the null hypothesis of no cointegration. If there is a cointegrating relationship, that implies the consistency of the fiscal policy with the IBC (Prohl et al., 2009) 60. In the second step, Johansen cointegration tests can be conducted to examine the number of cointegrating relationships. Johansen (1988) and Johansen and Juselius (1992) suggest the method with which to test the number of cointegrating relationships and to determine whether a group of non-stationary time series are cointegrated or not. Johansen’s cointegration tests are widely used in the case of multivariate

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Engle-Granger tests are based on the premise that if two processes which are integrated of order 1 are cointegrated, then the residuals obtained from regressing one on the other should be stationary.

60 _{Alternatively, we can test the cointegrating relationship between the ratios of government debt to GDP and}

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analysis, because this test is an extended version of the Dickey-Fuller unit root test for the multivariate case. There are two types of cointegration rank test statistics - trace statistics and maximum eigenvalue statistics – which use procedures developed by Johansen.

4.3.2.2 Autoregressive Distributed Lag (ARDL) Bounds Tests

In the case of non-stationary variables, the standard OLS regression may cause a spurious regression problem61, which produces incorrect inferences. In this situation, it is necessary to check whether there is cointegrating relationship between the variables. Several cointegration tests exist; Engle and Granger’s (1987) two-step approach, Johansen’s (1991) VAR cointegration test approach, and Pesaran et al.’s (1996, 2001) Autoregressive

Distributed Lag (ARDL) bounds test approach.

Among these tests, the ARDL bounds test approach has advantages over the Johansen’s approach and Engle and Granger’s approach. Engle and Granger’s two-step approach can be used only when all the variables are integrated of the same order, I(1). If the variables are mixed with different orders of integration – e.g. I(1) and I(0), the ARDL bounds tests should be used because it can examine the cointegrating relationship between variables that have different orders of integration. Unlike Engle and Granger’s approach, the ARDL bounds tests do not have any endogeneity problem. The ARDL bounds tests also have an advantage over Johansen’s cointegration tests. First, the ARDL bounds tests make it possible

61 _{Spurious regressions can be called ‘nonsense correlations’. If two variables are trending over time, a}

regression of one on the other could have a high R2 even if the two are totally unrelated. Therefore, if standard regression techniques are applied to non-stationary data, the end result could be a regression that looks good against standard measures, but which is really valueless (Enders, 2010, p196).In regressions of independent random walk variables, the usual t-ratio does not possess a limiting distribution but diverges with increasing sample size, thus increasing the probability of incorrect inferences as the sample size increases (Phillips, 1986)

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to estimate consistent coefficients for I(0) variables, even in small samples, which Johansen’s cointegration tests do not. In this context, Mah (1999) and Narayan (2005) support the robustness and superiority of the ARDL bounds tests against Johansen’s cointegration tests. However, there is a weakness in the ARDL bounds test in that it can only be used in a single equation and on the assumption of one cointegrating relationship. In this point, the ARDL bounds test is less general than Johansen’s multivariate cointegration test.

Considering the advantages mentioned above, the ARDL bounds test is also applied in this chapter. The ARDL bounds test can be applied as follows. Above all, the following unrestricted error-correction model (UECM) has to be estimated to see if there is a long-run relationship among the variables (Pesaran et al., 2001).

(4.16)

where are long-run coefficients; is an intercept; t is a time trend; , , , and are short-run coefficients; and is an error term. The appropriate number of lag lengths is selected by the AIC.

To investigate the long-run relationship, following Pesaran et al. (2001), an F-test (Wald test) is used. Pesaran et al. (2001) provide an asymptotic critical values for two set bounds - the lower critical bound (LB) and the upper critical bound (UB). LB is based on the assumption that all the variables in the ARDL model are I(0), while UB is based on the assumption that all the variables in the ARDL model are I(1). If the F-statistic is higher than the UB critical value, the null hypothesis of no cointegration is rejected, which means the existence of a long-run equilibrium relationship between the variables. On the other hand, if

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the F-statistic is less than the LB critical value, the null hypothesis of no integration cannot be rejected. If the F-statistic is between the UB and LB critical values, it is difficult for the test to give a conclusion on the existence of a cointegrating relationship between the variables (Pesaran et al., 2001).

The null hypothesis of no cointegration is and the alternative one is . If the null hypothesis is rejected, this may be interpreted as showing that there is a cointegrating relationship between the variables; , , , and . The equation (4.16) is an error-correction version of the ARDL model of order (m, n, o, p). It is assumed that the numbers of lags of three independent variables are the same (i.e., n=o=p). If the null hypothesis of no cointegration is rejected, the long-run coefficients can be estimated by the following general ARDL model.

(4.17)