GARCH (Generalized Autoregressive Conditional Heteroskedasticity)

GARCH, short for generalized autoregressive conditional heteroskedasticity, is a family of statistical models that describe how the volatility of a time series changes over time. In a GARCH model the variance of the next observation is not fixed. It is forecast from recent data, rising after large moves and decaying gradually when the series is calm.

The model is the standard econometric description of financial volatility. Returns on shares, currencies and commodities show volatility clustering: large changes tend to follow large changes and small follow small, in either direction. GARCH captures this pattern with very few parameters. It is used to forecast volatility, to measure risk and to price options. The long name describes it exactly: heteroskedasticity means changing variance, conditional means given past information, and autoregressive means depending on its own past values.

Robert Engle introduced the ARCH model in 1982, in a study of UK inflation. It made the variance of the current error a function of past squared errors. In 1986 Tim Bollerslev generalized it by adding past variances to the equation, producing GARCH. The generalization allowed a long memory in volatility to be captured with only one or two terms. Engle received the Nobel Memorial Prize in Economic Sciences in 2003 for methods of analyzing time series with time-varying volatility.

The most widely used version is GARCH(1,1):

• $\sigma_t^2$ is the variance forecast for period $t$.
• $\varepsilon_{t-1}$ is the previous period's return surprise, the return minus its expected value.
• $\sigma_{t-1}^2$ is the previous period's variance forecast.
• $\omega$, $\alpha$ and $\beta$ are constants that are zero or positive. $\alpha$ sets the reaction to news and $\beta$ sets the persistence of volatility.

When $\alpha + \beta$ is less than one, the variance reverts to a long-run level:

Suppose a model fitted to daily returns has $\omega = 0.000002$, $\alpha = 0.10$ and $\beta = 0.88$. The long-run variance is 0.000002 ÷ (1 − 0.98) = 0.0001, a daily volatility of 1 percent, or about 16 percent a year.

Yesterday's variance forecast was at that long-run level, 0.0001, but the actual return was −3 percent, so the squared surprise is 0.0009. Today's forecast is 0.000002 + (0.10 × 0.0009) + (0.88 × 0.0001) = 0.000002 + 0.00009 + 0.000088 = 0.00018. The square root is 1.34 percent. One large move has raised the volatility forecast by a third.

If quiet days follow, the forecast drifts back toward 1 percent. The sum $\alpha + \beta = 0.98$ governs the speed. The gap from the long-run level shrinks by 2 percent each day, giving a half-life of ln(0.5) ÷ ln(0.98), or about 34 trading days.

The parameters are estimated by maximum likelihood, usually on daily returns. For most financial series $\alpha$ is small, $\beta$ is large, and their sum is close to one, which means shocks to volatility die out slowly. The GARCH equation applies the idea of autoregression to the variance, and it is commonly paired with an ARIMA model for the mean of the series.

A GARCH model with normally distributed shocks still produces returns with fat tails, because mixing calm and turbulent periods creates more extreme values than a single normal distribution would. Real returns are more extreme still, so practitioners often assume shocks follow Student's t-distribution.

• EGARCH and GJR-GARCH. In equity markets, a fall in price raises volatility more than a rise of the same size. These models add a term that lets negative and positive shocks have different effects.
• Integrated GARCH. When $\alpha + \beta$ equals one, shocks to volatility never fade. The exponentially weighted moving average popularized by the RiskMetrics system is a special case with no constant.
• Multivariate GARCH. These models let correlations between assets change through time as well as volatilities, for use with portfolios.

The main use is volatility forecasting. A risk manager computing value at risk needs tomorrow's volatility, not the average of the past year, and a GARCH forecast responds to current conditions. Portfolio strategies based on volatility targeting scale exposure down when the forecast is high and up when it is low.

Option traders compare GARCH forecasts with implied volatility to judge whether options look cheap or expensive, and option pricing models built on GARCH dynamics reproduce features that constant-volatility models miss. In econometrics, allowing for GARCH effects gives more reliable standard errors when testing hypotheses about returns.

GARCH forecasts volatility from past daily returns alone, and a single daily return is a noisy signal of that day's true volatility. Models that use realized volatility computed from intraday prices generally forecast better where such data is available. A GARCH forecast reacts only after a large move has occurred. It cannot anticipate a scheduled event, which option markets price in advance.

The model assumes its parameters are fixed. A structural change in the market can make the estimated long-run level wrong, and apparent persistence close to one is sometimes a sign of such breaks and not of true long memory. Forecasts beyond a few weeks converge on the long-run average and carry little information. Like any volatility model, GARCH describes the typical size of moves and says nothing about their direction.

• The Nobel Prize: The Prize in Economic Sciences 2003