GARCHinMean

GARCH-in-Mean (GARCH-M) is an extension of GARCH models in which the conditional variance (volatility) enters the conditional mean equation of a time series. This specification allows the model to capture a risk–return trade-off: higher risk, as measured by conditional volatility, can imply higher expected returns.

A common form specifies the conditional mean of the return r_t as a function of the conditional variance h_t = Var(r_t | F_{t-1}) and possibly other terms. One standard specification is:

r_t = μ + δ h_t + ε_t,

where ε_t = sqrt(h_t) z_t, z_t are i.i.d. with zero mean and unit variance, and h_t follows a GARCH(1,1) process:

h_t = ω + α ε_{t-1}^2 + β h_{t-1}.

Here δ is the risk premium parameter: a positive δ indicates that higher conditional variance raises the expected return.

Some formulations use the conditional variance directly in the mean as r_t = μ + δ σ_t^2 + ε_t, with σ_t^2 = h_t. The model may also be extended to include exogenous variables or to use alternative error distributions (normal, Student-t, etc.).

Estimation and interpretation: GARCH-M models are typically estimated by maximum likelihood (or quasi-maximum likelihood) under a chosen error distribution. Stationarity requires ω > 0, α ≥ 0, β ≥ 0, and α + β < 1. The sign and magnitude of δ quantify the presence and strength of the risk premium associated with volatility.

Applications and limitations: GARCH-M is used to assess whether investors demand a premium for risk in returns, across stocks, currencies, and commodities. Results can be sensitive to sample, distributional assumptions, and model specification; misspecification can lead to biased inferences about the risk–return relationship.

Extensions include GARCH-M-X (with external regressors) and multivariate GARCH-M models.