Generalised Impulse Response Functions
_Ken Nyholm, 22 March 2016_

  

Generalised impulse response functions (GIRFs) are calculated from the
moving average representation of the VAR model, as the difference between the
a conditional and unconditional forecast, where the conditioning information
set is the shock to the j'th variable (koop et al (1996)).

Let $D_{t-1}$ represent the data available at time $t-1$, let $h=\{1,2,\ldots\}$
be the forecast horizon, let $i=\{1,2,\ldots,i_N\}$ count the variables in included
in the VAR model, which are collected in $Z$, such that $Z^i$ refers to the
i'th variable, and denote by $\delta_j$ the shock to the j'th variable in the
VAR, where $j=\{1,2,\ldots,j_N\}$, with $i_N=j_N$.

The GIRF is then defined in the following way:

$$GIRF(i,t+h,\delta_{j,t},D_{t-1}) = E\left(Z^i_{t+h} \vert u_{j,t}=\delta_{j,t},
D_{t-1} \right) - E\left(Z^j_{t+h} \vert D_{t-1} \right)$$

for the VAR model:

$$Z_t = k + F \cdot Z_{t-1} + u_t, $$    $$u \sim N(0,\Upsilon)$$

Following Peseran and Shin (1998) the orthogonalised and generalised IRFs
can be calculated in the following way, respectively, for a shock to the j'th
equation:

$$\psi^o_j(h)=A_hPe_j,$$   $$n=0,1,2,\ldots$$

$$\psi^g_j(h)=\upsilon_{(j,j)}^{-\frac{1}{2}} \; A_h \, \Upsilon \, e_j$$   
$$n=0,1,2,\ldots$$

where $A_h$ is the h'th coefficient-matrix from the moveing average representation
of the VAR model, P is the lower-triangular Cholesky decomposition $\Upsilon$,
$e_j$ is a vector having unity at the j'th position and zeros elsewhere, and
$\upsilon_{(j,j)}$ is the (j,j) element of $\Upsilon$.

The VAR model is written here as a VAR(1) model without loss of generality
due to the companion form.

$$\begin{array}{ll}
\underset{\boldsymbol{\Theta} \succ \mathbf{0}}{\textsf{maximize}} & \log \det \boldsymbol{\Theta}
- \mathrm{tr}(\mathbf{S}\boldsymbol{\Theta}) - \alpha \Vert \boldsymbol{\Theta}\Vert_{1, \text{off}},
\end{array} $$

$$
\begin{array}{ll}
&\underset{\boldsymbol{\Theta}}{\textsf{maximize}} & \log \mathrm{gdet} \boldsymbol{\Theta}
- \mathrm{tr}(\mathbf{S}\boldsymbol{\Theta}) - \alpha \Vert \boldsymbol{\Theta}\Vert_{1, \text{off}},\\
&\textsf{subject to} & \boldsymbol{\Theta} \in \mathcal{S}_{\mathcal{L}},
\end{array}
$$

$$
\begin{array}{ll}
\underset{{\mathbf{w}},{\boldsymbol{\psi}},{\mathbf{V}}}{\textsf{minimize}} &
\begin{array}{c}
- \log \det (\mathcal{L} \mathbf{w}+\frac{1}{p}\mathbf{11}^{T})+\text{tr}({\mathbf{S}\mathcal{L} \mathbf{w}})+
  \alpha \Vert\mathcal{L}\mathbf{w}\Vert_{1}+
  \frac{\gamma}{2}\Vert \mathcal{A} \mathbf{w}-\mathbf{V} {\sf Diag}(\boldsymbol{\psi}) \mathbf{V}^T \Vert_F^2,
\end{array}\\
\text{subject to} & \begin{array}[t]{l}
\mathbf{w} \geq 0, \ \boldsymbol{\psi} \in \mathcal{S}_{\boldsymbol{\psi}}, \ \text{and} \ \mathbf{V}^T\mathbf{V}=\mathbf{I},
\end{array}
\end{array}$$

$$
p(x) = \frac{\lambda^x e^{-\lambda}}{x!}}$$
$$
p(x) = \lambda^x exp(-\lambda)/x!
$$