THE PRICE LEVEL: TESTING FOR FISCAL DOMINANCE 2
Before estimating the VAR, we test for stationarity. Both Phillips-Perron and augmented Dickey-Fuller tests reject a unit root in Surplus/GDP at the 1% significance level. The evidence on the stationarity of Liabilities/GDP is somewhat mixed. The Phillips-Perron test rejects a unit root at the 5% level. The augmented Dickey-Fuller test, on the other hand, does not reject a unit root at conventional levels.
These results are consistent with those found in the literature. Kremers (1989), using data from the inter and post war periods (1923-1940, 1951-1985), found Augmented Dickey-Fuller tests reject a unit root in the debt to GDP ratio at about the 10% level. Bohn (1995), using a longer data set (1916-1989), found that ADF tests reject unit roots in the surplus to GNP ratio at the 1% level and in the debt to GNP ratio at about the 5% level. The general consensus from the literature seems to be that there is strong evidence that the surplus to GDP ratio is stationary while the evidence on the stationarity of the debt to GDP ratio is somewhat weaker.
We will adopt the stationarity of Surplus/GDP and Liabilities/GDP as a working hypothesis for two reasons. First, the results of the Phillips-Perron tests and the previous literature suggest both series are stationary. Second, the results for Surplus/GDP point unambiguously to stationarity and theory tells us that if Surplus/GDP is stationary then Liabilities/GDP, which is the expected present value of future surpluses, must also be stationary (provided the discount factor is constant).
Where might assuming stationarity create problems? The point estimates of the impulse response functions will still be valid. But the confidence intervals, which are calculated assuming the estimated parameters are asymptotically normal, may create problems. Fortunately, these problems do not prevent us from testing the main hypothesis of interest. The к-step ahead impulse response is calculated using only the first к lags in the estimated VAR. Sims, Stock, and Watson (1990) show that even if the VAR contains some 1(1) variables, subsets of the estimated lag coefficients will be asymptotically normal. In particular, the к-step ahead impulse response coefficients will be asymptotically normal provided the VAR has more than к lags. Our main interest is in the effect of a shock to st on wt+l. As long as our estimated VARs have more than one lag, the standard errors for the impulse response at t+1 will be asymptotically valid, even if some of the variables are not stationary.