How to formally write a VAR model including an exogenous variable? I fitted a Vector Auto-Regressive models of order 2 VAR(2) with two variables "V1" and "V2" plus an exogenous one "EX". How can I formally write the equations of the model, including the exogenous variable?
 A: Consider a (reduced-form) VARX($p$) model, i.e. a vector autoregression of order $p$.
Let $y_t$ be a $k$-dimensional vector representing the current (time $t$) values of the $k$-variate time series of endogenous variables.
Let $x_t$ be an $\ell$-dimensional vector representing the current (time $t$) values of the $\ell$-variate time series of exogenous variables.
Let $\varepsilon_t$ be a $k$-dimensional vector representing the current (time $t$) values of the $k$-variate time series of (reduced-form) innovations.

In matrix notation the model reads:
$$
y_t=A_0+A_1 y_{t-1}+\dots+A_p y_{t-p}+Bx_t+\varepsilon_t
$$
where $A_0$ is a $k$-dimensional vector of intercepts, $A_1$ to $A_p$ are $k\times k$ square matrices of coefficients, and $B$ is an $\ell\times k$ matrix of coefficients. (The time index for $x$ could be different depending on whether you are including contemporaneous or lagged values.)

In non-matrix notation, the simplest case of a bivariate VARX(1) with a univariate $x$ would be
\begin{aligned}
y_{1,t} &= a_{0,1} + a_{1,1,1}y_{1,t-1} + a_{1,1,2}y_{2,t-1} + b_{1}x_{t} + \varepsilon_{1,t}, \\
y_{2,t} &= a_{0,2} + a_{1,2,1}y_{1,t-1} + a_{1,2,2}y_{2,t-1} + b_{2}x_{t} + \varepsilon_{2,t}. \\\end{aligned}
where all coefficients are scalars.

In your particular case, $k=2$ and $\ell=1$. You would use V1 in place of $y_1$, V2 in place of $y_2$ and EX in place of $x$.
