# Multicollinearity, variable selection for cointegration testing in ARDL and VECM/VAR frameworks

I have 15 variables some of which are highly correlated. I want to run a cointegration test in the ARDL and VAR/VECM frameworks. Due to the correlation multicollinearity is a big problem; however, I do not want to omit variables as I want to test all of them.

I am only interested in the long/short run relationship between a variable $y$ and $x_1,x_2,\dotsc,x_{14}$ and not $x_1$ on $y$, $x_2,\dotsc,x_{14}$ etc.

Q: Is it possible to divide the variables into groups by, for example, category and correlation to avoid multicollinearity while still getting relevant results?

E.g. ARDL on $y,x_1,x_2,x_3,x_4,x_5,x_6,x_7$
and ARDL on $y,x_8,x_9,x_{10},x_{11},x_{12},x_{13},x_{14}$

• ARDL with F-bounds test is not the same framework as VAR/VECM with either Johansen or Engle-Granger test. Consider using something like "ARDL and VAR/VECM frameworks" instead of "ARDL/VECM/VAR framework". – Richard Hardy Feb 9 '16 at 18:28
• If your variables are cointegrated, they will be highly correlated. That is to be expected and should not be perceived as problematic. Could you elaborate more on the issue of multicollinearity? – Richard Hardy Feb 9 '16 at 19:41
• Good point, however if the variables (independent) are correlated the corresponding p-values will be biased? However as you said below correlation may not be meaningful as some variables are non stationary. I been reading alot of research papers regarding this topic now and none seem to bother about the correlation, only arch-effects, heterosked. and serial correlation. It makes sense from the answer you provided. – Parash Dejmar Feb 10 '16 at 7:57
• The $p$-values from a decent model provide a fair evaluation of what the data has to tell. Calling them biased is questionable. Also, I completely agree that multicollinearity is hardly ever mentioned in time series literature, and that has bothered me a little bit. – Richard Hardy Feb 10 '16 at 8:23

Cointegration analysis of subsets of variables in place of using all the variables at once will generally be problematic. For example, if three variables $x_1$, $x_2$ and $x_3$ have only one cointegrating relationship (and thus two stochastic trends), presence of cointegration will not be revealed by examining subsets of variables: neither $(x_1,x_2)$ nor $(x_1,x_3)$ or $(x_2,x_3)$ will be cointegrated. Cointegration will only be revealed by examining the full three-variable system.
However, in some special cases the problem will be less severe. For example, if the same three variables have two cointegrating relationships (and thus only one stochastic trend), you will discover cointegration in all possible pairs of variables ($(x_1,x_2)$, $(x_1,x_3)$ and $(x_2,x_3)$). In such a case, you could obtain two error correction terms from running bivariate regressions, e.g. $x_1$ on $x_2$ and $x_1$ on $x_3$ (like the first stage of the Engle-Granger procedure) and then use them in a vector error correction model (VECM) to estimate the coefficients on the error correction terms and on other regressors (lags of first differences of $x_1$, $x_2$ and $x_3$). (When obtaining standard errors, I think you would have to account for the fact that the error correction terms are estimated rather than known precisely.)