# strategy for using stationarity tests

I have a long (~20,000 points) time series that I tested for stationarity. I am following this strategy:

1. I started by plotting the series and determining visually whether a drift / trend exists or not, so as to chose the right alternative hypothesis in next step.
2. Next I performed Augmented Dickey-Fuller test with no drift/trend, with 1-50 lags.
3. Now I examined the BIC for each model corresponding to lags 1-50 and chose the one with minimum BIC. This corresponded to large lag, >25. Note: it was suggested in another thread (here) that one should chose lag by examining the residuals, choosing minimum lag that do not have serial correlation in residual. If I do that I find that there is no serial correlation (as confirmed by Durbin–Watson test) for all tested lags and thus I am tempted to choose lag 1.
4. I found that the ADF rejected the null hypothesis for lower lags, but failed to reject at the BIC-chosen lag (at 1% confidence level).
5. To make sure I am not sensitive to the lag value I did a Phillip-Parron test as a confirmation. I found that the null hypothesis is indeed rejected, at same confidence level, but for all lags.
6. Examining the coefficient value for the ADF test, I found that it is larger than 0.99 for these large lags. I know that the ADF test has low power at such values close to 1.
7. Thus I resorted to a variance ratio test as a final judgement. Random walk should have a ratio of 1, but at the BIC-lag I got a value of less than 0.1 (and even at lag 1, the ratio is < 0.5).
8. I thus concluded that the series is stationary, although probably (very) close to a random walk.

Is this line of reasoning sound?

## 1 Answer

Your selection of the number of lags to include in the ADF test based on BIC may be leading you astray. The ADF test is sensitive to the number of lags included. You essentially want to include the smallest possible number of lags, which will get rid of the serial correlation in the residuals, but you also want the number of lags such that adding an extra lag will not lead to a substantial increase in AIC/BIC (because you want the dynamics to be properly modelled). The correct number of lags needs to balance these 2 factors.

One way to select the number of lags is the Ng and Perron approach Ng and Perron approach. That technique suggests starting with a maximum number of lags and using a criterion to iteratively decide whether the number of lags can be reduced by 1. The maximum number of lags can be set using Schwert's rule, which in your case of ~20,000 observations would suggest starting with a maximum of 45. This technique, however runs the risk of including too many lags. Including too many lags will reduce the power of the test. This technique seems overly complicated and risk-prone (risk of including too many lags). You'd be better off finding the smallest number of lags which get rid of serial correlation, and then start adding 1 lag incrementally and observe the increase in AIC/BIC, finding the number of lags, such that adding an additional lag won't increase the AIC/BIC substantially.

The Phillips-Perron test is essentially the ADF test run with errors that are robust to serial correlation and heteroskedasticity. For this reason, it is much more straightforward to apply, but it runs the risk of not modelling the dynamic fully.

• "You essentially want to include the smallest possible number of lags, which will get rid of the serial correlation in the residuals" Even if this lag is just '1'? – student1 Jan 29 at 18:48
• Yes. What is your hesitation with including only 1 lag? – ColorStatistics Jan 29 at 18:49
• Because lag 1 is far from being the lag with minimum BIC. – student1 Jan 29 at 18:50
• Here is a more reputable resource than me describing the lag selection in the ADF test from the perspective of removing the serial correlation faculty.washington.edu/ezivot/econ584/notes/… ADF removes the serial correlation by adding however many lags it takes to do so... The PP test uses robust standard errors. – ColorStatistics Jan 29 at 18:57
• You are correct. Having consulted Woolridge, he warns against including too few lags as the dynamics may not be properly modelled. I misspoke when I said that you want to include the fewest number of lags that gets rid of serial correlation. That is a good start then one probably wants to observe the marginal increase in BIC as additional lags are added. If adding the second lag leads to a very large increase in BIC but the addition of the third lag leads to a tiny increase in BIC, then 2 lags is probably what you want. – ColorStatistics Jan 29 at 19:09