# VAR lag length vs Johansen cointegration test outcome?

First puzzle: I am taught that the lag order of VECM does not affect the cointegration rank because the lag order is for the differenced regressors. But, I see the contrary: I experimented with sample data of 4 variables, using lag orders between 1 and 12. At low lag orders the cointegration rank is low, but as I increase the lag length, cointegration rank goes up. Does anyone know of a paper or source discussing this issue? Any suggestions as to what to do in this situation?

Second puzzle: If I go by what information criteria indicate as an optimal lag length, establish the coinegration rank and then set up a VECM model, the residuals of the model are heteroscedastic and serially correlated. The info criteria based lag order is 2. To circumvent the issue, I have two options:

1. Go back, increase the lag order and set up a new VECM and do diagnostic tests

2. Just keep the cointegration rank as it is and increase the lag order until the residuals are homoscedastic and free of autocorrelation

What would be the optimal course of action?

This is a usual problem with the two steps procedure, where one selects first the lag, then the cointegration rank depending on the lag chose in the first step.

Puzzle 1: The claim that the lag order does not matter for the cointegration test depends on the type of test:

• VAR based test: if you use a Johansen test, the lag matters!

• Residual-based test (cf Engle-Granger, Phulips-Ouliaris), then yes, the lag order does not matter

Puzzle 2: My favourite approach is the simultaneous-selection approach, where you choose the lag and rank based on a simlutaneous criterion selection (This is implemented in package tsDyn in R for example, see http://www.inside-r.org/packages/cran/tsDyn/docs/rank.select). See references below.

Now what should you do if the ideal criterion selects a model where there is still some auto-correlation in errors? Great question, and I am not aware of papers tackling this directly. So one option would be to increase lags till you get a model with good residuals. Another option would be to stick with the AIC/BIC choice, and use heteroskedasticity and autocorrelation robust estimators (HAC) for any test you do (t-test, Granger causality, etc). Cheng an Philips (2009) show that their AIC/BIC lag/rank procedure is robust even in presence of auto-correlation (hence the name of semi-parametric). The issue is probably that few softwares will allow you do to this for all tests, in particular, IRF, usually computed with bootstrap, might not include a bootstrap scheme mimicking auto-correlation.

• Aznar A and Salvador M (2002). Selecting The Rank Of The Cointegration Space And The Form Of The Intercept Using An Information Criterion. Econometric Theory, 18(04), pp. 926-947. .

• Cheng X and Phillips PCB (2009). Semiparametric cointegrating rank selection. Econometrics Journal , 12(s1), pp. S83-S104.

• Thanks. But, what if the info criterion suggested lag order does not remove serial correlation? Should we increase the lag order whilst keeping the rank order from previous step? – mr.rox May 30 '16 at 16:33
• true, I did not discuss this directly. I edited my text for this point! – Matifou Jun 4 '16 at 18:42