Yt [m1 y yP Pi y yyt1 st

The tests for these forms may be carried out in the same fashion. For the model with drift only, the center panels of Tables 20.4 and 20.5 are used. When the trend is included, the lower panel of each table is used.

Example 20.5 Tests for Unit Roots

In Section 19.6.8, we examined Cecchetti and Rich's study of the effect of recent monetary policy on the U.S. economy. The data used in their study were the following variables:

n = one period rate of inflation = the rate of change in the CPI y = log of real GDP

i = nominal interest rate = the quarterly average yield on a 90 day T-bill Am = change in the log of the money stock, M1 i - n = ex post real interest rate Am - n = real growth in the money stock.

TABLE 20.5 Critical Values for the Dickey-Fuller DFY Test

Sample Size

AR modela (random walk)

AR model with constant (random walk with drift)

AR model with constant and time trend (trend stationary)

aFrom Fuller (1976, p. 373 and 1996, Table 10.A.1).

Data used in their analysis were from the period 1959.1 to 1997.4. As part of their analysis, they checked each of these series for a unit root and suggested that the hypothesis of a unit root could only be rejected for the last two variables. We will reexamine these data for the longer interval, 1950.2 to 2000.4. The data are in Appendix Table F5.1. Figures 20.11 to 20.14 show the behavior of the last four variables. The first two are shown above in Figures 20.9 and 20.10. Only the real output figure shows a strong trend, so we will use the random walk with drift for all the variables except this one.

The Dickey-Fuller tests are carried out in Table 20.6. There are 202 observations used in each one. The first observation is lost when computing the rate of inflation and the change in the money stock, and one more is lost for the difference term in the regression. The critical values from interpolating to the second row, last column in each panel for 95 percent significance and a one tailed test are -3.70 and -24.2, respectively for DFT and DFy for the output equation, which contains the time trend and -3.14 and -16.8 for the other equations which contain a constant but no trend. For the output equation (y), the test statistics are