# Talk:Augmented Dickey–Fuller test

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Hi! The bottom line of the article says that

if the test statistic is greater [...] than the critical value, then the null hypothesis of γ = 0 is rejected and no unit root is present.

Under Examples, it says

This is more negative than the tabulated critical value of −3.50, so at the 95 per cent level the null hypothesis of a unit root will be rejected.

So, in the main article, the null is rejected if the test statistic is greater than the critical value, in the example it is rejected if it is smaller than the critical value. I think that the example is right... —Preceding unsigned comment added by 193.103.207.10 (talk) 14:27, 30 September 2009 (UTC)

why are elihu's ADF test numbers positive then?

The crucial factor is not whether the numbers are positive or negative but whether they are greater or smaller than the relevant tabulated critical values. Karina.l.k 12:44, 26 April 2007 (UTC)

In the example, the page currently says:

A sample of 50 observations and a model which includes a contant and a time trend yields the DFτ statistic of -2.57. This is greater than the tabulated critical value of -3.50 at the 95 per cent level so a unit root is present and the null hypothesis cannot be rejected.

In the last sentence, it's true that the null hypothosis cannot be rejected, but it is not necessarily true that a unit root is present. Failure to reject the null does not imply that the null is true.

I stuck to Greene but I agree that this formulation is sloppy, since one can also wrongly fail to reject the null or wrongly reject the null (Type I and Type II errors). I've changed it now, hope that it's better. Feel free to edit further as I think this example section could and should be expanded.Karina.l.k 08:05, 27 May 2007 (UTC)

Guest:, should the NULL Hypothesis be ${\displaystyle {\hat {\gamma }}=0}$ ? Please advise.

[Guest2] I think null should be gamma = 0, have seen this from two other sources —Preceding unsigned comment added by Steveok64 (talkcontribs) 19:11, 16 February 2008 (UTC)

Yes you are right, if the model is written with variables in levels, the null hypothese is gamma=1 but if the model is written in difference (as in this article), then the null hypothese becomes gamma=0.EtudiantEco —Preceding comment was added at 22:38, 17 February 2008 (UTC)
This issue (and a few others) now tidied up. —Preceding unsigned comment added by 203.20.253.5 (talk) 01:52, 23 April 2008 (UTC)

## error in formula

The following formula is wrong (LaTex):

\Delta y_t = \alpha + \beta t + \gamma y_{t-1} + \delta_1 \Delta y_{t-1} + \cdots + \delta_{p-1} \Delta y_{t-p+1} + \varepsilon_t,

It should be:

\Delta y_t = \alpha + \beta + \gamma y_{t-1} + \delta_1 \Delta y_{t-1} + \cdots + \delta_{p-1} \Delta y_{t-p+1} + \varepsilon_t,

You can google it or compare it with the definition of the Wiener process with drift μ — Preceding unsigned comment added by 84.173.125.84 (talk) 16:10, 15 April 2012 (UTC)

The orig formula is correct. That at Wiener_process#Related_processes Wiener process with drift μ is not equivalent to what you left and has an extra assumption that the process starts at zero to t=0. Melcombe (talk) 21:59, 25 April 2012 (UTC)
if the delta y_t is what it is supposed to be (y_t - y_t-1 ) then my suggestion is correct and the formula wrong. You can find an explanation there. To be correct in the more general case you can also edit the beta * t into beta * delta t .
The link you gave was flawed and I have corrected it. The formula in the article agrees essentially (apart from signs) with equations 8-11 etc in the pdf you linked, so what is the problem? Moreover the formula in the article is an obvious expansion of the formulae in the Dickey–Fuller test article. Melcombe (talk) 17:03, 26 April 2012 (UTC)

## A missing in the article

Does ADF test regression (whose lag length (that is optimal) obtained via SBC or AIC after performing ADF test when a given max lag length) remove all autcorrelation? (Do we really know with 100% certainty that ADF-resulted-regression has no autocorrelation?)

In detail:

The goal of ADF is to test whether we have unit root or not.

DF (Dickey-Fuller) test equation (regression equation) may include autocorrelation, and its result is not so reliable.

In ADF, additional lags of the differenced variable are added to the right of the ADF regression equation. These added lags of the differenced variable reduces autocorrelation.

Now, assume that we are performing ADF test when max lag length (say 14) is given (or assume max lag length (say 14) is the obtained from Schwetz's formula; as in Eviews).

If ADF test results in optimal lag length of 5 (via AIC or SBC criterion) in ADF test regression equation, then do we really know that the ADF equation with optimal lag length of 5 includes no autocorrelation in residuals of the ADF regression (with 5 lengths)?

OR, do we have to check that the ADF test regression equation with optimal lag length of 5 (that we found via SBC or AIC during the execution of ADF test) includes no autocorrelation by the help of some autocorrelation tests?88.226.65.125 (talk) 10:05, 28 December 2014 (UTC)

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