Introduction

Welcome to an interactive guide on Non-Gaussian Structural Vector Autoregression (SVAR) models. This website aims to visualize and explain the identification approach in Keweloh (2021).

The structure of the guide

The guide is structured as follows:

Setup: introduces the SVAR model, assumptions, and notation.
Reduced Form: illustrates the reduced form.
Identification Problem: discusses the identification problem.
Min. Dependencies: introduces the minimum dependence approach.
Max. non-Gaussianity: introduces the maximum non-Gaussianity approach.
Non-Gaussianity: explains the impact of the degree of non-Gaussianity on identification.

The SVAR model

In an SVAR, a vector of time series $$ y_t = [y_{1t},\cdots,y_{nt}]' $$ is explained by its own past and a vector of structural shocks $$\epsilon_t = [\epsilon_{1t},\cdots,\epsilon_{nt}]' $$ using

$$ y_t = A_1 y_{t-1} + \cdots + A_p y_{t-p} + u_t $$ $$ \quad u_t = B_0 \epsilon_t. $$

The following assumptions simplify the visualization of the identification problem.

Simplifying assumptions

The VAR(p)

$ y_t = A_1 y_{t-1} + \cdots + A_p y_{t-p} + u_t $

can be estimated by OLS. To simplify, consider an SVAR(0) such that the reduced-form shocks $u_t$ can be viewed as observable random variables equal to an unknown linear combination (via $B_0$) of structural shocks $\epsilon_t$.

Assumption 1: No Lags

The SVAR has zero lags:

$$ y_t = u_t \quad \text{and} \quad u_t = B_0 \epsilon_t $$

To further simplify, let's consider an SVAR with only two variables and shocks such that each of the two reduced-form shocks is equal to an unknown linear combination of the two structural shocks.

Assumption 2: Bivariate SVAR

The SVAR contains only two variables and shocks:

$ \begin{bmatrix} u_{1,t} \\ u_{2,t} \end{bmatrix} = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} \begin{bmatrix} \epsilon_{1,t} \\ \epsilon_{2,t} \end{bmatrix} $

Finally, assume that $B_0$ is orthogonal.i Without the orthogonality assumption the matrix $B_0$ contains four unknown parameters. However, an orthogonal matrix $B_0$ essentially only depends on a single unknown parameter, which simplifies the visualizations.

Assumption 3: Orthogonal $B_0$

The matrix $B_0$ is orthogonal and can be written as:

$B_0 = B(\phi_0) = \begin{bmatrix} \cos(\phi_0) & -\sin(\phi_0) \\ \sin(\phi_0) & \cos(\phi_0) \end{bmatrix} $

where $\phi_0$ is the unknown rotation angle.

Next Steps

The next page illustrates the relation between reduced-form shocks and structural shocks. Proceed to Reduced Form.