Dickey Fuller Test

#1
I am studying the Dickey Fuller test. The book of reference is Introductory Econometrics for Finance by C. Brooks. I firstly consider the zero mean Dickey Fuller test that uses the "random walk" type of regression:

ΔY t =δY t−1 +u t

The null hypothesis is that δ is zero and the alternative is that δ is not zero so I run the regression and compare the t-value with the critical values tabulated by Dickey and Fuller. Though the results are valid only if u t is white noise. According to the book previously mentioned, if the dependent variable of the regression (the first difference of the series) is autocorrelated then the u t would also be autocorrelated. Why is that, how this can be proved?

Also if the dependent variable is not autocorrrelated can we infer that the u t of the regression would also be uncorrelated?

In general can we draw coclusions about whether the error term is autocorrelated or not based on whether the dependent variable is autocorrelated or not?

If yes, can the same inferenece be made in the case of the two other forms of the Dickey Fuller test regressions (1) the equation above with a constant term (drift) or single mean, 2) The equation above with drift and time trend).



Thanks in advance,


Andreas
 

vinux

Dark Knight
#2
According to the book previously mentioned, if the dependent variable of the regression (the first difference of the series) is autocorrelated then the u t would also be autocorrelated. Why is that, how this can be proved?

Also if the dependent variable is not autocorrrelated can we infer that the u t of the regression would also be uncorrelated?

In general can we draw coclusions about whether the error term is autocorrelated or not based on whether the dependent variable is autocorrelated or not?

Thanks in advance,
Andreas
1. Regarding proving part, take any ARIMA ( Eg: ARIMA(1,1,0) and fit the following model
ΔY t =δY t−1 +u t
You can see that ΔYt is autocorrelated and ut is also autocorrelated.( it is a function of ar coefficient).

2. True. ut as per above model.

3. Yes as per the above model. Normally people do ADF test than df test. philip peron test is nonparametric and there ut should be stationary with some restriction on moments (not required to be white noise).