# Vector autoregression with exogenous variables

Im dealing with a VAR model where I also want to include exogenous variables. Based on my sampling, the exogenous variables in $t$ are independent from my other variables in $t$, but highly dependent on the other variables in $t-1$.

Could there be some serious error from an estimation point, like simultaneous equations bias?

And how are the impulse response function affected by the exogenous variables?

Assuming I would only include lagged values of the exogenous variables with the same lag length as the endogenous, then there should not be a difference in the estimated parameters (for the "original endogenous" variables), compared to including them as endogenous as well?

edit (the model): The real model consists of 4 or 5 variables and has 10 lags included. But assuming I include one lag of the endogenous variables and for the exogenous variable I include the instantaneous and one lag.

\begin{array} $\begin{bmatrix} y_{1t} \\ y_{2t} \\ \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{bmatrix} \begin{bmatrix} y_{1t} \\ y_{2t} \\ \end{bmatrix} + \begin{bmatrix} v_{1} & b_{01} & b_{11} \\ v_{2} & b_{02} & b_{12} \\ \end{bmatrix} \begin{bmatrix} 1 \\ x_{t} \\ x_{t-1} \end{bmatrix} +\begin{bmatrix} u_{1t} \\ u_{2t} \\ \end{bmatrix} \end{array}$x_t$is highly dependent of$y_{t-1}$, but independent of$y_t$. This comes from the sampling of my variables. The$y_t$'s are aggregated values (over an time interval) of irregulary spaced time series data, while the$x_t$'s are recorded (like a snapshot) in the beginning the time time interval. So$x_t$are independent of$y_t$, but they are approximately determined by$x_t\approx x_{t-1}+y_{t-1}$and should be independent of the error term$u_t$. I also estimated another model, where$x_t$was also included in in the system. So, basically a commom VAR model, with 3 variables. \begin{array}$ \begin{bmatrix} y_{1t} \\ y_{2t} \\ y_{3t} \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{13} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix} \begin{bmatrix} y_{1t} \\ y_{2t} \\ y_{3t} \\ \end{bmatrix} +\begin{bmatrix} u_{1t} \\ u_{2t} \\ u_{2t} \end{bmatrix} \end{array} where $y_{3t}=x_t$.

I have very complicated acf's and ccf's for my variables. Every variable is highly dependent on the other variables (the ccf's are also pretty slowly dying out). Also the acf's are highly persistent, especially the variable I want to thread as exogenous. "An almost unit root process". I tested for Granger Causality (with model 2) for noncausality of the $x_t$ variable and the hypothesis got clearly rejected (as expected).

Any suggestions regarding this problem?

• What are those other variables in $t$ which you are talking about? For VAR with exogenous variables, on the right hand sides you have lags of response variable and exogenous variables. So there is high correlation between the lags and exogenous variables? – mpiktas Dec 19 '13 at 8:36
• With other variables I mean the endogenous variables, which are determined through the system. Exactly, there is a high correlation between the exogenous variables in $t$ and the lagged endogenous variables in $t-1$. On the right hand side the exogenous variables are included from $x_{t-j}$, j=0,..s, so there is also the instantaneous effect of $x_t$ included. – user34714 Dec 19 '13 at 12:09
• Could you write down the model and then point out which variables are which? I suspect you are trying to estimate structural VAR. If so, then you would need to use instrumental variable estimation and then correlation between different time periods only matters in instrument selection. – mpiktas Dec 19 '13 at 13:53