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} + \cdots + A_p Y_{t-p} + u_t (1)$where$μ$is a k × 1 vector of constants,$A_i ( i = 1, …, p )$is a k × k matrix and$u_t$ is a k × 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, $ε_{1 t} , …, ε_{kt}$ which are the sources of variation of the system. An usual assumptions is that $ε_{1 t} , …, ε_{kt}$ are mutually independent, although orthogonality is sufficient in many applications. Thus we have: $u_t = B ε_t (2)$ where B is a k × k invertible matrix (the impact or mixing matrix) and $ε_t = ( ε_{1 t} , …, ε_{kt} )'$ is a vector of independent shocks. Let W be $B^{−1}$ . Then we get the structural VAR form: $WY_t = \mu'+Γ_1 Y_{t-1} + \cdots + Γ_p Y_{t-p} + ε_t (3)$ where $μ' = Wμ$ and $Γ_i = WA_i$ for $i = 1, …, p$.

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).