Let us consider a vector $Y_t$ of $k$ time series variables. For example, $Y_t = ( x_t , y_t )'$, in which case $k = 2$. We assume that $Y_t$ follows a stochastic process that can be well approximated by a linear VAR process of the form:

$Y_t = \mu + A_1 Y_{t-1} + \dots + A_p Y_{t-p} + u_t \quad (1)$

where $\mu$ is a $k \times 1$ vector of constants, $A_i$ ($i = 1, \dots, p$) is a $k \times k$ matrix and $u_t$ is a $k \times 1$ vector of white noise, whose elements are referred to as reduced-form residuals. Each element of $u_t$ is in turn assumed to be a linear combination of latent structural shocks, varepsilon_{1t}, dots, varepsilon_{kt}, which are the sources of variation of the system. A usual assumption is that varepsilon_{1t}, dots, varepsilon_{kt} are mutually independent, although orthogonality is sufficient in many applications. Thus we have:

u_t = B varepsilon_t quad (2)

where $B$ is a $k \times k$ invertible matrix (the impact or mixing matrix) and varepsilon_t = ( varepsilon_{1t}, dots, varepsilon_{kt} )' is a vector of independent shocks. Let $W$ be $B^{-1}$. Then we get the structural VAR form:

W Y_t = mu' + Gamma_1 Y_{t-1} + dots + Gamma_p Y_{t-p} + varepsilon_t quad (3)

where $\mu' = W \mu$ and $\Gamma_i = W A_i$ for $i = 1, \dots, p$.

The idea of VAR analysis is to follow a two-step procedure:

• First, Eq. (1) is estimated through standard regression methods to obtain an estimate of the reduced-form residuals $u_t$.
• Second, the parameters of Eq. (3) can be recovered by analyzing the relationships among the elements of $u_t$. Notice that, having estimated (1), knowing $B$ is sufficient for identifying (3).