4. Structural Vector Autoregressive (SVAR) ModelΒΆ

A SVAR has the following form

\[B_0 y_t = B_1 y_{t-1} + \dots + B_p y_{t-p} + \omega_t\]

where \(\omega_t\) has mean zero and is serially uncorrelated. \(B_0\) governs the contemporaneous interactions between variables. Compactly written, we get

\[B(L) y_t = \omega_t\]

The variance-covariance matrix is normalized so that

\[E[\omega_t \omega'_t] = \Sigma_\omega = I_K.\]

Normally, we have

  1. As many natural shocks as variables in the model
  2. Structural shocks are by definition uncorrelated which means that \(\Sigma_\omega\) is a diagonal matrix.
  3. W.l.o.g. the varianced is normalized to unity

If we want to represent \(y_t\) as a function of its lagged terms only, we use the reduced form representation. For that, we premultiply with \(B^{-1}_0\) which leads to

\[\begin{split}B_0 y_t &= B_1 y_{t-1} + \dots + B_p y_{t-p} + \omega_t\\ B^{-1}_0 B_0 y_t &= B^{-1}_0 B_1 y_{t-1} + \dots + B^{-1}_0 B_p y_{t-p} + \omega_t\\ y_t &= A_\end{split}\]