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An earlier version of this paper was written when the author was in the North American Division. The author is grateful to Charles Adams for helpful suggestions and discussions; to Bankim Chadha, Liam Ebrill, Owen Evans, and Yusuke Horiguchi for comments; and to Fredesvinda Pham for research assistance.
Data are from the U.S. Department of Commerce, Bureau of Statistics (1975) and Homer (1963). Stocks returns are holding period yields calculated from dividend and price data as explained in the previous section.
Suppose a variable x can be modeled as an autoregressive process of order (p), the estimate of β, from an OLS regression of
is tested to be significantly different from zero, under the null hypothesis that X is 1(1). In the DF test all γi are set equal to zero while in the ADF test p-1 lags are included to ensure that the residual is empirical white noise.
The rule of thumb is:
where T is equal to the sample size and ℓ is the number of lags in the ADF regression. Thus 4 and 12 lags would be used for a sample of 100.
The sources for these data are U.S. Department of Commerce, Bureau of Census (1975) and Homer (1963).
In this data set, Treasury bill returns are based on an index of the shortest-term bills not less than a month in maturity. The long-term government and corporate bonds are constructed from a 20-year term bond portfolio. Equity returns are based on the Standard and Poor’s Composite Index with holding period yields calculated as in Section II above. Monthly returns are computed for each asset and geometric means are then calculated for different holding periods.
Dickey and Fuller have shown that asymptotically the unit root “t test” reported in these tables is not affected by estimation of higher order autoregressive parameters. Dickey and Fuller also proposed a test, T * β, which is more powerful against the alternative that (β-1) < 1 than the t test. This statistic would have to be scaled by a constant c where c is a function of the moving average parameter when the process is the general ARIMA model. Schwert’s simulation results suggest, however, that the t test is less sensitive to model misspecification and hence only the t results of the test are reported in the tables.
The ten-year rate in the Ibbotson and Sinquefield data covers 46 observations and hence does not justify the use of 12 lags according to the rule of thumb noted above. Moreover Schwert’s simulation results suggest that an ADF test performs better with a ℓ4 rule than with a ℓ12 rule when the sample size is below or at about 50. If 12 lags are included, however, the test statistics for ten-year real rates in the Ibbotson and Sinquefield data also fall below the critical value and the unit root hypothesis cannot be rejected for all long-term interest rates.
One-year and ten-year inflation rates are calculated as the geometric average of the increase in the CPI over one year and ten years, respectively.
If the processes were assumed to be purely autoregressive and lags to the ADF test added only until the Durbin Watson statistic indicated no autocorrelation, the ADF test would reject the unit root hypothesis for many more real interest rates and for the three-month and one-year inflation rate.
The stock of real federal government debt appeared to be 1(2). Its ratio relative to real GNP appeared to be 1(1), however, and was used instead. Correspondingly, private wealth including government debt appeared to be 1(2), but was 1(1) when government debt was excluded.
A continuous series was not available for the debt of state and local governments for the entire sample period.
This variable appeared to be borderline in terms of being I (1) in that the DF Statistic was −3.64 and the ADF Statistic at 8 lags was −2.66. The total balance was clearly 1(1). However, the primary balance appeared to be more appropriate for use as an exogenous variable since there would be some simultaneity between the total balance and the interest rate. In any event, the results were not significantly different when the total federal government balance was used instead. Some of the regression results reported in Tables 5-8 use the total rather than the primary balance.
The index of total factor productivity from 1960–88 was based on previous staff work; see Appendix XIII, SM/89/176.
These holding period yields were calculated as described in Section II. Unless otherwise noted, the cointegrating regression in Tables 5-8, use the univariate method to forecast dividends and stock prices. Unit root tests reported in Tables 3 and 4 indicate that three month holding period yields calculated under univariate and bivariate methods and ex post one year real holding period yields may be borderline in terms of being 1(1).
Barro and Sala i Martin use the change in the index; however, the index itself appeared to be 1(1) and the change in the index to be 1(0), and thus not appropriate for use in a cointegrating regression.
The critical values of these tests at a 5 percent significance level are approximately DF = 3.37; ADF = 3.13; and SB = 0.37. The SB test is simply the Durbin-Watson statistic under the null hypothesis that the autocorrelation parameter equals one rather than zero.
As noted above, fiscal policy could also affect real interest rates through its impact on stock market returns and world interest rates. In the case of regressions where stock returns are included as an explanatory variable, the coefficients on SG and B/GNP should be interpreted as the effect of deficits and debt on real interest rates given the real return on equity.
As discussed above, for a given capital stock, the rise in real rates due to a decrease in labor force participation may be offset to some extent by the rise in the capital-labor ratio.
Since these debt variables appeared to be 1(2), a debt to GDP ratio which was 1(1) was always used.