I am working on a six-variable VAR model which contains endogenous variables that are all stationary at levels at least at the 5% significance level (ADF test). However, some of these variables are trend stationary. Therefore, in the VAR model, is it right to include trends as an exogenous variable in the equations of the endogenous variables that are trend stationary?
I'd very much appreciate comments and reasons on why I should or should not include the trend.
Dear Eli, you should do that because those TS variables will then provide more stability (and stationarity) to the whole VAR model. In order to get more validity for such strategy, you can also take a look to the ADF tests for each individual variable with trend, lags of "deltas" AND lags of the other variables considered in the VAR (all together as explanatory variables). There is literature on this strategy for "A-ADF" tests for uroots "with covariates", which provide improvements over "ADF" tests and easily justify.
Carlos R. Barrera-Chaupis , thank you so much for sparing your time to answer my question. Much appreciated.
Well, you know, everything depends on everything :) In the context of Eli's question, nonstationary VAR models are not quite relevant because we are dealing with variables whose LEVELS are trend-stationary, so they are stationary after including a trend (and this may also be the case after including other covariates). Ashley & Verbrugge (2009) address one specific question, to differentiate or not (they do not address TS variables as far as I am reading their paper). One important point in their specific results is that the explanatory variables should appear in differenced form: as far as I understand, this implies that (in the context of Eli's question) the VAR system should be estimated with additional equations (for transforming those explanatory variables into a differenced form) but note the explanatory variables can also be in detrended form (this issue is not addressed by Ashley & Verbrugge, 2009). Since Eli's system may include endogenous variables that must be in differenced form, those specific variables will not need additional equations and their lags will naturally enter as explanatory variables of other endogenous variables. Eli's question does not address the issue of cointegrated variables, an issue that appears to be well covered by Ashley & Verbrugge (2009) because they use already known results in the literature (at least as far as Toda & Yamamoto, 1995). I should mention that their ideas about using HP-filtered data are not new either.