Non-Gaussian SVAR: Identification Problem

The identification problem

This page explores the central challenge in SVAR analysis: the identification problem. We'll discuss why identifying structural shocks from reduced-form shocks is difficult and how non-Gaussian properties can help.

Structural Shocks and Innovations

Let's fix the rotation at $\phi_0=0.5$ such that the reduced form shocks are equal to one specific rotation of the structural shocks:

However, in a given application, we neither know the rotation matrix $B(\phi_0)$ nor do we observe the structural shocks $\epsilon_t$. Instead, we only observe the reduced-form shocks $u_t$.i Consequently, the question is how can we identify $B(\phi_0)$ and recover the structural shocks $\epsilon_t$ from the reduced-form shocks $u_t$?

We can choose any angle $\phi$ and compute $B(\phi)$:

$ B(\phi) = \begin{bmatrix} \cos(\phi) & -\sin(\phi) \\ \sin(\phi) & \cos(\phi) \end{bmatrix} $
and get the corresponding innovations:
$ e_t(B(\phi)) = B(\phi)^{-1} u_t $
These innovations represent the shocks that would yield $u_t$ if the true angle of the data-generating process $\phi_0$ were equal to the chosen angle $\phi$. Specifically, for $\phi=\phi_0$ we get $e_t(B(\phi_0)) = \epsilon_t$.

Interactive Simulation

Move the $\phi$ slider above to see how $B(\phi)$ leads to different innovations $e_t$ equal to rotations of $u_t$:

$ \begin{bmatrix} e_{1,t} \\ e_{2,t} \end{bmatrix} = \begin{bmatrix} \cos(\phi) & \sin(\phi) \\ -\sin(\phi) & \cos(\phi) \end{bmatrix} \begin{bmatrix} u_{1,t} \\ u_{2,t} \end{bmatrix} $

The question is: how do we know whether we picked $\phi=\phi_0$ and whether our innovations are equal to the structural shocks?

Next Steps

In the following sections, we'll explore two complementary approaches to solving the identification problem using non-Gaussian properties:

  1. Minimizing the dependencies between the recovered shocks.
  2. Maximizing the non-Gaussianity of the recovered shocks.

Proceed to Minimizing Dependencies to learn about the first approach.