Dickey Fuller test

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:

$\Delta Y_t = \delta Y_{t-1} + u_t$

The null hypothesis is that $\delta$ is zero and the alternative is that $\delta$ 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?

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).

• I think we are missing some items of information: where are the dependent variables in the above equation if $\Delta Y_t=Y_t-Y_{t-1}$? – Xi'an Nov 23 '11 at 16:09