Lesson 4.4: Structural VAR (SVAR) Models

Imposing economic theory on VAR models to identify and interpret causal economic shocks.

Part 1: The Problem with Standard VARs

A standard VAR model is a "reduced form" model. It's great for forecasting, but its error terms, $\bm{\epsilon}_t$ , are composites of the "true" underlying economic shocks and are contemporaneously correlated.

This makes Impulse Response Functions hard to interpret. A shock to the VIX might be correlated with a simultaneous shock to stock returns. We can't isolate the effect of a "pure" volatility shock.

Part 2: The SVAR Solution

The Structural VAR

An SVAR model imposes restrictions based on economic theory to disentangle the reduced form errors ( $\bm{\epsilon}_t$ ) into a set of uncorrelated, "structural" shocks ( $\mathbf{u}_t$ ).

We assume a relationship like: $\mathbf{A}\bm{\epsilon}_t = \mathbf{B}\mathbf{u}_t$ where $\mathbf{u}_t$ has a diagonal covariance matrix.

The restrictions are placed on the matrices $\mathbf{A}$ and $\mathbf{B}$ . For example, a common "short-run" restriction is to assume a Cholesky ordering, where some shocks cannot affect other variables contemporaneously.

By identifying these structural shocks, an economist can trace out the effect of a "pure monetary policy shock" or a "pure technology shock" on the economy, leading to causal interpretations.

What's Next? A Practical Capstone

We have now built a complete multivariate toolkit, from VARs for short-run dynamics to VECMs for long-run equilibrium.

In our final lesson of this module, we will put this all together in a capstone project: building a **Pairs Trading Strategy** based on the principles of cointegration.

State Space Models & Kalman Filters

Capstone: A Pairs Trading Strategy