Lesson 3.2: ARCH Models

Introducing the Autoregressive Conditional Heteroskedasticity (ARCH) model, the first formal model to capture volatility clustering.

The Core Idea

Developed by Robert Engle, the ARCH model suggests that we can model the **conditional variance**—the variance at time $t$ given past information—as a function of past shocks.

An ARCH(q) model models today's variance as a weighted average of the magnitude of the last `q` surprises or shocks.

The ARCH(q) Model Specification

The ARCH(q) Model

Let $\epsilon_t$ be the error term (shock) from the mean equation. The conditional variance, $\sigma_t^2$ , is modeled as:

\sigma_t^2 = \alpha_0 + \sum_{i=1}^q \alpha_i \epsilon_{t-i}^2

$\alpha_0$ : A constant term for the long-run average variance.
$\epsilon_{t-i}^2$ : The **squared shocks** from previous periods. The model uses the magnitude of past surprises to forecast today's variance.
$\alpha_i$ : The ARCH coefficients, measuring the influence of past shocks.

Limitations

Requires many lags (`q`) to capture persistent volatility.
Symmetric: A large positive shock has the same effect as a large negative shock. This contradicts the "leverage effect" seen in markets.

What's Next? A More Powerful Model

The ARCH model was revolutionary, but its limitations led to a more powerful and parsimonious extension.

In the next lesson, we'll explore the **GARCH model**, which adds a 'memory' of past variance itself, becoming the industry standard for volatility forecasting.

Introduction to Volatility Modeling

GARCH Models