GARCH Command

The GARCH command estimates a GARCH(1,1) model for a time series and analyzes how volatility evolves through time. GARCH models are widely used in finance because market volatility tends to cluster: periods of calm are often followed by periods of calm, while periods of turbulence tend to be followed by further turbulence.

Main idea: estimate conditional volatility and determine how strongly market shocks influence future risk.

Example


slideshow(garch(sp500))

This example estimates a GARCH(1,1) model using S&P 500 log returns and examines the resulting volatility process.

Output Panels

Log Return Series – The return series used in the estimation.
Parameter Interpretation – Estimated ω, α, and β parameters together with stability diagnostics.
Conditional Volatility Distribution – Histogram of estimated conditional volatility values.

Syntax


garch(series)

The input series is automatically converted to log returns if necessary.

Mathematical Definition

The GARCH(1,1) model assumes:


σ²(t) = ω
      + α ε²(t-1)
      + β σ²(t-1)

where:

ω (Omega) – Long-run baseline variance.
α (Alpha) – Sensitivity to recent market shocks.
β (Beta) – Persistence of volatility through time.

Interpretation

The parameter α measures how strongly new information affects volatility. Large values indicate that market shocks have an immediate impact on risk.

The parameter β measures volatility persistence. When β is large, periods of high volatility tend to remain elevated for extended periods.

A commonly used stability condition is:


α + β < 1

α + β < 1 – Mean-reverting volatility process.
α + β ≈ 1 – Highly persistent volatility.
α + β > 1 – Potentially unstable volatility process.

Why GARCH Matters

Financial markets rarely exhibit constant volatility. Periods such as the 2008 Financial Crisis, the COVID crash, and other market disruptions demonstrate that risk changes dramatically over time.

The GARCH model provides a framework for estimating these changing volatility levels and is widely used in:

Risk management
Value-at-Risk (VaR) estimation
Portfolio construction
Derivative pricing
Volatility forecasting