The long-run tendency of government expenditure: a semi-parametric modelling approach

Abstract

The empirical findings concerning the validity of Wagner’s Law seem to depend on both economic fundamentals and modelling choices. Using a semi-parametric model and data for a sample of five developed and five developing countries during the period 1960–2007, we find that, for the developing countries, Wagner’s Law holds regardless of whether the relationship between government expenditures and per capita income is modelled in a linear or a nonlinear fashion. For the developed countries, however, model specification matters: If a linear specification is a priori chosen, then we find support of Wagner’s Law, while a nonlinear specification implies on the contrary the opposite result. Hence, model misspecification could have led to incorrectly accept Wagner’s law for the developed countries.

Appendix

1.1 Unit root and cointegration tests

We use the Augmented Dickey–Fuller (ADF) test (1979) and the Phillips–Perron (PP) test (1988) to examine the time-series properties of the data. The ADF test is referred to the t-statistic of the $\theta _2$ coefficient on the following regression:

$$\begin{aligned} \Delta \chi _t =\theta _0 +\theta _1 t+\theta _2 \chi _{t-1} +\sum _{i=1}^k {\lambda _i \Delta \chi _{t-1} +\varepsilon _t } \end{aligned}$$

(8)

where $\chi =\left\{ {y,X,Z} \right\}$. The ADF regression tests for the existence of unit root of $\chi _t$, namely in the logarithm of all model variables at time t. The variable $\Delta \chi _{t-1}$ expresses the first differences with k lags, and $\varepsilon _t$ is the variable that adjusts the errors of autocorrelation. The coefficients $\theta _0,\theta _1, \theta _2$ and $\lambda _i$ are estimated. The null and the alternative hypotheses for the existence of unit root in variables $\chi _t$ are:

$$\begin{aligned} H_0 {:}\,\theta _2 =0 \quad H_1 {:}\,\theta _2 <0 \end{aligned}$$

The PP test is an extension of the Dickey–Fuller test, which corrects for autocorrelation and is more robust in the case of weakly autocorrelated and heteroscedastic regression residuals. Also it is more powerful than the ADF test for aggregate data.

Table 4 Unit root tests: sample size 1960–2007

The results are shown in Table 4. We find that the variables under examination are integrated of order 1 in all ten countries. Hence, we also perform a cointegration test. The null hypothesis of non-cointegration is tested against the alternative of cointegration. We follow the maximum likelihood procedure of Johansen (1988) and Johansen and Juselius (1990) where a p-dimensional $(p\times 1)$ vector autoregressive model with Gaussian errors can be expressed by its first differenced error correction form:

$$\begin{aligned} \Delta Y_t =\mu +\Gamma _1 \Delta Y_{t-1} +\Gamma _2 \Delta Y_{t-2} +\cdots +\Gamma _{p-1} \Delta Y_{t-p+1} +\Pi Y_{t-p} +u_t \end{aligned}$$

(9)

where $Y_t =[y\;\;\;X\;\;Z]$ is a $p\times 1$ vector containing all model variables; $\mu$ is the $p\times 1$ vector of constant terms; $\Gamma { }_i=-I+A_1 +A_2 +\cdots +A_i (i=1,2,\ldots ,p-1)$ is the$p\times p$matrix of coefficients; $\Pi =I-A_1 -A_2 -\cdots -A_p$ is the$p\times p$ matrix of coefficients; and $u_t$ is the $p\times 1$ vector of the disturbance term coefficients.

The $\Pi$ matrix conveys information about the long-run relationship between $Y_t$ variables, and the rank of $\Pi$ is the number of linearly independent and stationary linear combinations of the examined variables. Thus, testing for cointegration involves testing for the rank of $\Pi$ matrix by examining whether the eigenvalues of $\Pi$ are significantly different from zero. Johansen (1988) and Johansen and Juselius (1990) propose two test statistics for testing the number of cointegrating vectors (or the rank of $\Pi )$ in the VAR model. These are the trace test and the maximum eigenvalue test. The likelihood ratio statistic for the trace test is:

$$\begin{aligned} -2\ln Q=-T\sum _{i=r+1}^p {\ln (1-\hat{{\lambda }}_i )} \end{aligned}$$

(10)

where $\hat{{\lambda }}_{r+1} ,\ldots ,\hat{{\lambda }}_p$ are the $p-r$ smallest eigenvalues. The null hypothesis to be tested is that there are at most $r$ cointegrating vectors. That is, the number of cointegrating vectors is less than or equal to $r$, where $r$ is 0, 1, 2,..., and so forth. In each case, the null hypothesis is tested against the general alternative. Alternatively, the maximum eigenvalue statistic is:

$$\begin{aligned} -2\ln Q=-T\ln (1-\hat{{\lambda }}_{r+1} ) \end{aligned}$$

(11)

In this test, the null hypothesis of $r$ cointegrating vectors is tested against the alternative hypothesis of $r+1$ cointegrating vectors. Thus, the null hypothesis $r = 0$ is tested against the alternative $r = 1, r = 1$ against the alternative $r = 2$, and so forth. It is well known that the cointegration tests are very sensitive to the choice of lag length. We use the Schwartz criterion and the likelihood ratio test to select the number of lags required in the cointegration test (Table 5).

Table 5 Cointegration tests: sample size 1960–2007