RDP 8809: The Intertemporal Government Budget Constraint and Tests for Bubbles 4. Empirical Evidence

A description of the data including sources is included in Appendix 1. For our sample, the average nominal interest rate is exceeded by the rate of growth of real GDP plus average inflation. This means that in the first test use of the deficit including and excluding interest payments are equivalent. Also, in the second test a significant “bubble term” is not a finding of debt unsustainability but instead is compatible with the hypothesis that debt is being repatriated over the period.

a. The Cointegration Test

We apply the cointegration test described in the previous section to test the hypothesis that government spending (inclusive and exclusive of interest payments), taxation and monetary policy have been conducted in such a way that bonds are growing on an unstable path. It is first necessary to determine that both series are integrated of an equal order greater than one. An examination of the graphs for the series under consideration shows that each of the series has trended upwards with a large jump in the early seventies. This trend in the data indicates that it is likely that the series are non-stationary.

The results of Dickey-Fuller tests (Dickey and Fuller (1979) and (1981)), Perron and Phillips (1987) and Stock and Watson (1986) are given in Table 2^[6]. The government spending series (including and excluding interest payments) and the constructed taxation plus the seignorage series appear to be non-stationary around a linear time trend, i.e., there appears to be both a deterministic component and a stochastic unit root. The deterministic components of the series are not removed as it is the time path of the series we are interested in, and it is the equivalence of the non-stationary components between the series (both deterministic and stochastic) which is to be tested. The first differences of all series were tested for non-stationarity in the same manner as above. All tests reject non-stationarity indicating that all series under consideration are integrated of order one. We can now proceed with the cointegration tests.

The results for the cointegration test for the equivalence of the unit roots in the two series are shown on Table 3. The tests for cointegration are those proposed by Engle and Granger (1987), namely, their cointegrating regression Durbin Watson test statistic (CRDW) and the Dickey-Fuller statistic (DF). Both methods amount to the testing of the residuals of the cointegrating equation (equation (9)) for non-stationarity, the first uses the method of Bhargarva (1986) and the second uses the method of Dickey and Fuller. The null hypothesis of non cointegration of the series is accepted if non-stationarity of the residuals is accepted. The critical values for these tests are given in Engle and Yoo (1987) for the case of fifty observations. It has been shown that tests for non-stationarity such as these have low power when the alternative hypothesis is that rho is close to (but less than) one. The test of Stock and Watson (1986) is also applied. Phillips (1987) and Phillips and Perron (1986) have proposed non-parametric tests for non-stationarity which have greater power than those considered by Engle and Granger (1987) and these tests have been applied here.

Both the DF test and the CRDW tests indicate that the hypothesis of no cointegration can be rejected at the 95 per cent significance level for both constructions of government spending. These findings are backed up by the PP test. The cointegrating coefficient (α) is, at 0.934, close to the hypothesised value of unity whilst the constant is insignificantly different from zero. Dropping the insignificant constant also yields coefficients close to one^[7]. We know that our estimate for α is biased, although Banerjee, Dolado, Hendry and Smith (1986) have shown that this bias is proportional to the inverse of the R² for the cointegrating regression. With a high R² of 0.89 this bias is unlikely to be a problem here. The Stock and Watson (1986) tests both accept the alternate hypothesis of only one unit root at the 95% significance level.

Unfortunately, the standard errors calculated along with the regression results cannot be used to test the statistical significance of the departure of our estimate of the cointegrating coefficient from unity as the usually constructed t statistics are not t distributed. This problem is not insurmountable. One method, applied in Bewley and Elliott (1987), is to estimate an error correction model for a range of values of α and choose the alpha which maximises the log of the likelihood of the system. The plot of the log of likelihoods can be used to calculate critical bounds for the estimate of α. An alternative method which was foreshadowed in section III and applied here is to impose the hypothesised value for α and test the constructed variable for stationarity. If this variable (i.e. the primary deficit plus seignorage) is stationary, then we can accept the hypothesised value as a true cointegrating coefficient.

Tests for the stationarity of the deficit are given in Table 4. It can be seen that the null hypothesis of non-stationarity is rejected in both the Dickey Fuller tests and the more powerful Phillips test. The Stock and Watson (1986) tests indicate that for the series excluding interest payments stationarity can be rejected at the 99% level of significance whilst the series including interest payments rejects the null hypothesis at the 90% level of significance. On this evidence we accept that government spending and the taxation plus seignorage variables are cointegrated with a cointegrating coefficient of unity. As predicted, both specifications of the deficit (including and excluding interest payments) yield the same result.

b. The Restricted Flood Garber Test

The bubble test of Hamilton and Flavin (1986) has also been estimated. As mentioned before, the average nominal interest rate is exceeded by the rate of growth of real GDP plus average inflation so the bubble term is not a test for a rational bubble, i.e. a significant bubble term is compatible with the repatriation of debt over time. The true steady state interest rates should be used here, but data limitations confine us to the long run average rate over the sample period of our study.

The results of the estimation of equations (11) and (12) are presented in Table 5. For lags of the deficit term greater than one, nonlinear estimation techniques are necessary as there are cross equation restrictions which are nonlinear. Estimates for more than one lag of the deficit term show that the extra lags are insignificant (likelihood ratio tests were employed). The first point to note from these tests, given that we expect that the bond series and the bubble term are non-stationary, is that the equations should be considered in the framework of the cointegration literature. The low Durbin Watson coefficient of 0.17 shows that in fact the bubble term is not fully accounting for the non-stationarity in the bond series. An analysis of the residuals shows a strong negative deterministic trend which implies that the negative trend in the bubble term does not fully explain the determination of the level of bonds. This could be due to either a mis-specification of the bubble term or mis-specification of the proxy for expected deficits. The coefficients given in Table 5 are biased due to both non-stationarity and the mis-specification; very little trust can be put into their interpretation. It is interesting to note, however, that the bubble term is positive in sign. This is further, although very weak, evidence that inflation tax and GDP growth have contributed over the period to the sustainability of continuous primary government deficits.

Footnotes

All results refer to data deflated by GDP. [6]

The constant here has a significant interpretation. This is the mean deficit over the period (a small error comes from α not being exactly one). A significantly positive constant with stationarity of the residuals and an α of one implies a sustainable deficit over the sample period. [7]