An empirical investigation of the sustainability of the public deficit in Portugal

Abstract

In this paper, we investigate Portuguese government expenditures and revenues as an example for a long time series. Our hypothesis states that there may be periods when the deficit is sustainable and those when it is not. Usually, after a period of unsustainable deficits, a new regime takes over. These regime shifts call for an approach that takes into account a non-constant structure of the underlying data generating process. Consequently, we use different tests which we set up in a time-varying framework. We apply and compare the results of the Trace test, Breitung’s non-parametric test and the Bohn test. We identify several break points and find that the Trace test performs worst in this case while Breitung’s test and the Bohn test give similar results. Comparing the results with history, we find that the last two tests best reflect what happened historically.

Appendices

Appendix 1: Breitung’s nonparametric test for cointegration

Breitung’s (2002) unit root and cointegration test employs a variance ratio as a test statistic. This approach can eliminate the problems of having to specify the short-run dynamics and estimate nuisance parameters. Let $\left\{ {y_t } \right\}_1^T$ denote an observable process that can be decomposed as $y_t = \delta \prime d_t + x_t$, where $\delta \prime d_t$ is the deterministic part (d _t  = 1 or $\left[ {1,t} \right]\prime$) and x _t is the stochastic part. If we do not assume a deterministic part, then y _t is consistent with x _t . The null hypothesis is that x _t is I(1); if $T \to \infty$, $T^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}} x_{\left[ {aT} \right]} \Rightarrow \sigma W\left( a \right)$, where σ > 0 represents the constant long-run variance, W(a) denotes a Brownian motion, and [·] is the integer part. The expression of x _t allows for the application of a general data generating process. Asymptotically, to construct a consistent estimate which does not require the specification of short-run dynamics and an estimate of σ, Breitung proposed the following test statistic:

$$\hat\rho = \frac{{T^{ - 4} \sum\limits_{t = 1}^T {\hat U_t^2 } }}{{T^{ - 2} \sum\limits_{t = 1}^T {\hat u_t^2 } }},$$

(5.1)

where $\hat u_t$ is the OLS residual, i.e. $\hat u_t = y_t - \hat\delta \prime d_t$ , and $\hat U_t$ is the partial sum process $\hat U_t = \hat u_1 + \ldots + \hat u_t$ . If y _t is I(0), the test statistic $\hat\rho _T$ converges to 0. Using Monte Carlo simulations, Breitung shows that the variance ratio test has favourable small sample properties.

We can proceed and test for cointegration by generalizing the nonparametric unit root test on the assumption that the process can be decomposed into a q-dimensional vector of stochastic trend components ξ _t and an (n–q)-dimensional vector of transitory components of v _t where n is the number of variables. Asymptotically, ξ _t and v _t is $T^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}} \xi _{\left[ {aT} \right]} \Rightarrow W_q \left( a \right)$ and $T^{ - 2} \sum\nolimits_{t = 1}^T {v_t v_t^\prime = o_p \left( 1 \right)}$ , respectively, where W _q (a) denotes a q-dimensional Brownian motion with unit covariance matrix. The dimension of ξ _t is related to the cointegration rank. In addition, it assumes that the variance of ξ _t diverges at a faster rate than v _t instead of assuming the stationarity of v _t . From the assumption, the transitory component denoting the cointegration relationship can be generated by any process.

To test for the number of cointegrating vectors, Breitung proposed the following problem about the n x n matrix A _t , B _t :

$$\left| {\lambda _j B_T - A_T } \right| = 0$$

(5.2)

where $A_T = \sum\nolimits_{t = 1}^T {\hat u_t \hat u\prime } _t$ , $B_T = \sum\nolimits_{t = 1}^T {\hat U_t \hat U_t^\prime }$ , and $\hat U_t = \sum\nolimits_{j = 1}^t {\hat u_t }$ represent the n-dimensional partial sums concerning $\hat u_t$ . The problem is equivalent to solving for the eigenvalue of $R_T = A_T B_T^{ - 1}$ . The solution of Eq. 5.2 is $\lambda _j = {{\left( {\eta _j^\prime A_T \eta _j } \right)} \mathord{\left/ {\vphantom {{\left( {\eta _j^\prime A_T \eta _j } \right)} {\left( {\eta _j^\prime B_T \eta _j } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\eta _j^\prime B_T \eta _j } \right)}}$ where $\eta _j$ is the eigenvalue of $\lambda _j$ . If the vectors of the stochastic trends are less than q, T ² λ _j diverges to infinity. In that case, since stochastic trends are linked with each other, a cointegrating vector exists. Hence, the test statistic is the following:

$$\Lambda _q = T^2 \sum\limits_{j = 1}^q {\lambda _j } $$

(5.3)

where $\lambda _1 \leqslant \lambda _2 \leqslant ... \leqslant \lambda _n$ are the ordered eigenvalues of R _T . The idea of the cointegration rank behind the approach is similar to Johansen’s idea. The statistic tests whether a q-dimensional stochastic component is rejected at the significance level.

Appendix 2: Unit root test with structural break

If there is a shift in the time series, it should be taken into account when testing for a unit root because the ADF test may be distorted if the shift is simply ignored. Saikkonen and Lütkepohl (2002) and Lanne et al. (2002) proposed the following model:

$$y_t = \mu _0 + \mu _1 t + f_t \left( \theta \right)\prime \gamma + u_t $$

(6.1)

where θ and γ are unknown parameters or parameter vectors and the errors u _t are generated by an AR(p) process. The shift function, $f_t \left( \theta \right)\prime \gamma$ , could be (1) a simple shift dummy variable with shift data T _b , (2) based on the exponential distribution function which allows for a nonlinear gradual shift to a new level starting at time T _b and (iii) a rational function in the lag operator applied to a shift dummy. Saikkonen and Lütkepohl (2002) and Lanne et al. (2002) proposed unit root tests based on estimating the deterministic term by a generalised least squares (GLS) procedure and subtracting it from the original series. Then an ADF-type test is performed on the adjusted series. If the break date is unknown, the authors recommend, based on simulation results, choosing a reasonably large AR order in the first step and then picking up the break data that minimises the generalised sum-of-squares errors of the model in first differences.