# determinstic trend in VAR-models

I'm asking myself the following question. I want to build a VAR-Model with 6 time series A, B, C, D, E and F. I analysed every series univariate and I found out that A, D, E and F are stationary and B has a linear and C a quadratic deterministic trend. Analyzing the VAR seems like I need 2 Lags to include. Now I allow for estimating a linear trend for B and a quadratic trend for C in the VAR model, too - that makes sense for me. But I'm wondering if it is ok to build the following model (to allow only these specific trend terms for the associated variable, t-values are shown):

... or if I allow a quadratic trend for series C and since every series is effecting the other ones in a VAR-model do I then have to allow quadratic trends for ALL the other veriables, too? I think no, because if I would let the model estimate a deterministic trend for every series it will be significant for several series even if they don't have a quadratic (or linear) trend in fact.

Would be glad if someone can help me discussing this...

thanks, eski19

• Some thoughts: suppose $y_t$ has no trend while $x_t$ has a linear trend. If I build a model where $y_t$ depends linearly on $x_t$ without any terms that would counterbalance the linear trend brought in by $x_t$, such a model would not make sense. $y_t$ and $x_t$ should diverge due to the linear trend in $x_t$. So even if $y_t$ itself does not have a linear trend, I would include a linear trend in the linear model in which $y_t$ depends on $x_t$ so as to counterbalance the effect of trend inside $x_t$. Alternatively, $x_t$ could be de-trended before entering the model. – Richard Hardy Mar 19 '15 at 14:14
• That said, it is no surprise you would find the trend significant even in the equations where the dependent variable has no trend. It is because the trend counterbalances the effect of the trending regressors. – Richard Hardy Mar 19 '15 at 14:17
• Thank you very much!!! So I would first find out the highest degree of deterministic trend (t, t², ...) for all the series I want to include in my VAR. E.g. if it is a cubic trend t³ then I allow t, t² and t³ in EVERY single equation of the VAR. If you detrend one series before entering the model you do not compare the original series any longer - the interpretation seems a little bit weired to me. It is the same if one series has a unit root and you difference only this series and put it in the VAR with the other original variables. Do I not compare apples and oranges then? – eski Mar 19 '15 at 14:33
• Sometimes de-trending can have a natural interpretation. You might be interested in how the fluctuations of $x_t$ around its trend affect $y_t$. Since the trend terms in the model equations will effectively de-trend $x_t$ anyway (since $y_t$ has no trend), this will be a nuisance. By looking at estimated equations, it might seem that there is some trend that $y_t$ is following, while actually the trend will be just counterbalancing the trend inside $x_t$. So using de-trended $x_t$ may in the end be more transparent and less deceptive when it comes to analyzing how $x_t$ affects $y_t$. – Richard Hardy Mar 19 '15 at 14:48
• everything sounds reasonable, thanks! do you have a bibliographical reference for citation where they recommend to detrend single variables before entering the model? In this case can I despite not comparing original variables go on with interpreting in this way that "if x changes like this then y changes like this" (and take the values from the VAR system)? – eski Mar 19 '15 at 15:10