Lesson 4.1: Vector Autoregression (VAR) Models

Modeling the dynamic interplay between multiple time series simultaneously.

The Core Idea

A VAR model is a system of equations where each variable is regressed on its own lagged values and the lagged values of all other variables in the system.

It generalizes the univariate AR model to multiple time series, allowing us to capture feedback loops and cross-variable dynamics (e.g., how a shock to interest rates affects stock returns, and how that change in stock returns then feeds back to influence future interest rates).

The VAR(p) Model Specification

The VAR(p) Model in Matrix Form

Let $\mathbf{y}_t$ be a vector of $K$ variables at time $t$ . A VAR(p) model is:

\mathbf{y}_t = \mathbf{c} + \mathbf{\Phi}_1 \mathbf{y}_{t-1} + \dots + \mathbf{\Phi}_p \mathbf{y}_{t-p} + \bm{\epsilon}_t

$\mathbf{y}_t$ : The $K \times 1$ vector of variables.
$\mathbf{\Phi}_i$ : The $K \times K$ coefficient matrices for lag `i`.
$\bm{\epsilon}_t$ : The $K \times 1$ vector of white noise error terms.

All variables in a VAR model must be stationary. If they are not, you must difference them first.

Analysis Tools for VAR Models

1. Granger Causality

A statistical test to see if past values of one variable are useful for forecasting another variable. It answers, "Does X Granger-cause Y?"

2. Impulse Response Functions (IRFs)

The most important tool for VAR analysis. An IRF traces out the dynamic impact of a one-time shock to one of the variables on the future paths of all variables in the system. It lets us see how shocks propagate through the system over time.

What's Next? Long-Run Relationships

VAR models are powerful tools for stationary data. But what if our non-stationary variables (like stock prices) are bound by a long-run equilibrium relationship? Differencing them would destroy this valuable information.

In the next lesson, we will introduce **Cointegration**, the statistical concept for finding these stable, long-run relationships.

Capstone: Modeling S&P 500 Volatility

Cointegration and Error Correction Models (VECM)