# Correct Unit Root Testing

I have a few time series of variables with each 40 monthly observations. Now I want to test each variable for Unit Root (non-stationarity).

My Question: How to choose the optimal lag length when testing for unit root (lags=x)?

Do I just go ahead with the urca package and

ur.df(table$variable1, type = c("const"), selectlags = c("AIC"))?  I realize that some of the variables then have a really long lag length chosen by AIC or BIC. Is it better to apply VARselect(table$variable1,lag=6,type="const")


first and go with the indicators returned here? In this case, lag=6 is chosen as some sources say 4 or 6 lags would be a good value for monthly data of this observation size.

Cheers, Chris

PS: Found the same question unanswered here: Link

Background

Several methods have been proposed in the literature to choose the appropriate lag order in unit root tests. Some of them are based on Akaike's and Schwarz's information criteria, others follow a top-down strategy where lags are tested for significance starting from a maximum lag order, others follow a bottom-up approach. (Ng and Perron, 1995; Hylleberg, 1995).

Less than four years of data is a small sample size to test for unit roots. Bootstrapping the test statistics may result in a more accurate result. I would recommend a model-based bootstrap in the same vein as the bootstrap method for seasonal unit root tests discussed in Burridge and Taylor (2003).

The computation of the p-value based on response surfaces may be a useful alternative as well (MacKinnon, 1994; Cheung and Lai, 1995).

Practical recommendation

For your application, I would recommend you selecting the lag order using the BIC criterion. Then try changing the truncation parameter to values close to that chosen by BIC and check in the conclusions remain similar. If not, the data may not bee informative enough to reach a conclusion, in this case bootstrap or response surface based p-values may be helpful.

If you are interested in the literature you may be interested in this review.

References

Cheung, Y.-W. and Lai, K. S. (1995). Lag order and critical values of the augmented Dickey-Fuller test. Journal of Business and Economic Statistics 13(3), pp. 277-280. Link from one of the author's university website: klai/KLPaper/JBES95Jy.pdf.

Hylleberg S. (1995). Tests for Seasonal Unit Roots General to Specific or Specific to General? Journal of Econometrics 69(1), pp. 5-25. DOI: 10.1016/0304-4076(94)01660-R.

MacKinnon J. G. (1994). Approximate Asymptotic Distribution Functions for Unit-Root and Cointegration Tests. Journal of Business and Economic Statistics 12(2), pp. 167-176. DOI: 10.1080/07350015.1994.10510005.

Ng S., Perron P. (1995). Unit Root Tests in ARMA Models with Data-Dependent Methods for the Selection of the Truncation Lag. Journal of the American Statistical Association 90 (429), pp. 268-281. DOI: 10.1080/01621459.1995.10476510.