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 formY_t = \mu +A_1 Y_{t-1} + \cdots + A_p Y_{t-p} + u_t (1)$where$μis a k × 1 vector of constants,A_i ( i = 1, …, p )is a k × k matrix andu_tis a k × 1 vector of white noise, whose elements are referred to as reduced-form residuals .

Each element ofu_tis 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 thatε_{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) andε_t = ( ε_{1 t} , …, ε_{kt} )'is a vector of independent shocks. Let W beB^{−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 residualsu_t.
• Second, the parameters of Eq. (3) can be recovered by analyzing the relationships among the elements ofu_t$. Notice that, having estimated (1), knowing B is sufficient for identifying (3).