Lecture 23 – The Dickey-Fuller TestWe have seen that- the dynamic behavior of I(1) processes is quite different from the behavior of I(0) processes- the way we go about defining and estimating the trend and cyclical components of a time series may depend on whether we assume the seriesis (trend) stationary or difference stationary.- regressions with difference stationary variables need special care.For these reasons we might be interested in testing the null hypothesis of a unit root against the stationary or trend-stationary alternative.Consider the following AR(1) model for yt:yt = ρyt-1 + εt , εt ~ iid (0,σ2)-1 < ρ < 1If ρ < 1, yt ~ I(0), mean 0 and var σ2/(1-ρ2)If ρ = 1, yt ~ I(1), a random walk- The OLS estimator of ρ is consistent for all ρ; it is super-consistent when ρ = 1.- The OLS t-statistic )ˆ.(.ˆeswhere )ˆ(.esis the OLS s.e. of ρ-hat,is asymptotically standard normal whenρ < 1; it has a non-standard asymptoticdistribution when ρ = 1, being skewed tothe left.Therefore we cannot apply the standard t-test to test the unit root null. This test willreject the null too often; the actual test size will be greater than the nominal test size.White (1956) showed that under H0:ρ=1, τ does have a stable limiting distribution. Dickey (1976) and Dickey and Fuller (1979)tabulated the percentiles of the this distribution. (How?) The percentiles of the Dickey-Fuller distribution are available in many time series textbooks (including the Enders and Hamilton texts). See TableNotes –- the unit root test is a one-sided test. Reject H0 if τ is “too negative”, i.e., if ρ-hat is too much less than one.- the median of the DF distribution is about -0.5.- the 0.025 percentile of the standard normal is -1.96. The 0.025 percentile of the DF distribution is -2.23. The 0.05 percentile of the standard normal is -1.65. The 0.05 of theDF is -1.95- The asymptotic distribution seems to be appropriate even if T is as small as 25. - an equivalent test: Regress Δyt on yt and use)ˆ.(.ˆes as the test statistic.- Under H0 the test-statistic )1ˆ( Talso converges in distribution. The percentiles of this distribution were also tabulated by Dickey and Fuller. Unit root tests based on this test statistic seem to be less powerful and, so, lesswidely used.Suppose we modify the model and null hypothesis as follows:yt = α + ρyt-1 + εt , εt ~ iid (0,σ2), -1 < ρ < 1H0: ρ = 1 and α = 0HA: ρ < 1The difference between this and the previouscase? Under the I(0) alternative, yt can have a non-zero mean. But, in neither case does ythave a deterministic trend component. (Once we allow the ε’s to be serially correlated, this case could be appropriate fortesting for a unit root in the unemployment or inflation rates.)Under the null hypothesis, the t-statistic )ˆ(1ˆsefrom the regression of yt on 1, yt-1, or, equivalently, the t-statistic )ˆ(ˆsefrom the regression of Δyt on 1,yt-1converges in distribution to a DF distribution, though the limiting distribution of τμ is different from the limiting distribution of τ.See TableNotes –- the distribution of τμ is more highly skewed than the distribution of τ. (Using a standard normal distribution to test H0:ρ=1 would be even more misleading in this case.)- Dickey and Fuller (1981) tabulated thepercentiles for the asymptotic distribution of the F statistic associatedwith H0:ρ=1 and α=0. The F-test of thejoint hypothesis might seem more natural to apply in this setting than the t-test of H0:ρ=1. In practice, however, the t-test is much more commonly used. Then, if H0:ρ=1 is not rejected it is assumed that ρ =1 and α=0.The τ and τμ tests are appropriate unit root tests for non-trending zero mean (τ) or non-zero mean (τμ) series. Consider the following model yt = α + βt + ρyt-1 + εt ,εt ~ iid (0,σ2), -1 < ρ < 1and consider H0: ρ = 1 and β = 0 , yt is a rw with driftHA: ρ < 1 , yt is trend-stationaryUnder the null hypothesis, the t-statistic )ˆ(1ˆsetfrom the regression of yt on 1, t, yt-1, or, equivalently, the t-statistic )ˆ(ˆsetfrom the regression of Δyt on 1,t, yt-1converges in distribution to a DF distribution. The limiting distribution of τt is different from the limiting distributions of τ and τμ.See TableNotes –- the distribution of τt is more highly skewed than the distribution of τμ . - Dickey and Fuller (1981) tabulated thepercentiles for the asymptotic distribution of the F statistic associatedwith H0:ρ=1 and β=0. The F-test of thejoint hypothesis might seem more natural to apply in this setting than the t-test of H0:ρ=1. In practice, however, the t-test is much more commonly used. Then, if H0:ρ=1 is not rejected it is assumed that ρ =1 and β=0.Allowing for serial correlation in the DF error terms –I. Augmented Dickey-Fuller TestII. Phillips-Perron