ISU ECON 674 - The Dickey-Fuller Test

Unformatted text preview:

Lecture 23 – The Dickey-Fuller Test We 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 series is (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 < 1 If 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.(.ˆρρρτes−= where )ˆ(.ρesis the OLS s.e. of ρ-hat, is asymptotically standard normal when ρ < 1; it has a non-standard asymptotic distribution when ρ = 1, being skewed to the left.Therefore we cannot apply the standard t-test to test the unit root null. This test will reject 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 the DF is -1.95 • The asymptotic distribution seems to be appropriate even if T is as small as 25. • an equivalent test: Regress ∆yt on yt and use )ˆ.(.ˆρρτes= as the test statistic. • Under H0 the test-statistic )1ˆ( −ρT also 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, less widely used.Suppose we modify the model and null hypothesis as follows: yt = α + ρyt-1 + εt , εt ~ iid (0,σ2), -1 < ρ < 1 H0: ρ = 1 and α = 0 HA: ρ < 1 The difference between this and the previous case? Under the I(0) alternative, yt can have a non-zero mean. But, in neither case does yt have a deterministic trend component. (Once we allow the ε’s to be serially correlated, this case could be appropriate for testing for a unit root in the unemployment or inflation rates.)Under the null hypothesis, the t-statistic )ˆ(1ˆρρτµse−= from the regression of yt on 1, yt-1, or, equivalently, the t-statistic )ˆ(ˆρρτµse= from the regression of ∆yt 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 the percentiles for the asymptotic distribution of the F statistic associated with H0:ρ=1 and α=0. The F-test of the joint 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 < 1 and consider H0: ρ = 1 and β = 0 , yt is a rw with drift HA: ρ < 1 , yt is trend-stationaryUnder the null hypothesis, the t-statistic )ˆ(1ˆρρτset−= from the regression of yt on 1, t, yt-1, or, equivalently, the t-statistic )ˆ(ˆρρτset= from the regression of ∆yt 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 the percentiles for the asymptotic distribution of the F statistic associated with H0:ρ=1 and β=0. The F-test of the joint 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 Test II. Phillips-Perron


View Full Document

ISU ECON 674 - The Dickey-Fuller Test

Download The Dickey-Fuller Test
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view The Dickey-Fuller Test and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view The Dickey-Fuller Test 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?