Conditional Variance, GARCH, and Squared Residuals

Summary

The note distinguishes the conditional variance modeled by a GARCH process from the squared residual observed after estimating an ARMA conditional mean. Given information through the prior time step, the conditional variance of the series equals the conditional expectation of the squared raw error. A single squared residual is therefore a noisy proxy for that conditional variance, not the variance itself.

The note also separates conditional variance from unconditional, long-run variance. The latter depends on the full ARMA-GARCH model and cannot generally be read from a residual at one point in time. The explanation is conceptual rather than an empirical comparison: it gives no data, estimation procedure, or discussion of assumptions behind a particular specification. Its practical lesson is to avoid interpreting each realized squared residual as the model's variance estimate.

Key ideas

• A GARCH component models conditional variance, while an ARMA component models the conditional mean.
• The conditional variance equals the conditional expectation of the squared raw error.
• A realized squared residual is a noisy proxy for conditional variance, not an exact observation of it.
• Long-run variance depends on the full model and is not identified by a single period's residual.

Tags

# Squared Residuals equal Variance of Dependent Variable (ARMA-GARCH)


# Squared Residuals equal Variance of Dependent Variable (ARMA-GARCH)












My understanding of ARMA-GARCH models for a variable $X$ is as follows: I estimate a conditional mean of a variable $X$ by use of the ARMA part of the model. I estimate the conditional variance of variable $X$ by use of the GARCH part of the model.

And as far as I understand those models, this means that the variance of $X$ is simply interpreted as the squared residual of the mean model at a specific point in time.

Is my understanding correct?

## Answer by Richard Hardy (score 1, accepted)

https://quant.stackexchange.com/a/59947

Not quite. The conditional variance of $X_t$, conditional on the information up to and including time $t-1$, equals the conditional variance of the squared error: $$ \text{Var}(X_t|I_{t-1})=\mathbb{E}(u_t^2|I_{t-1}) $$ where $u_t$ is the raw error. The squared residual $\hat u_t^2$ is a rather noisy proxy for it / estimate thereof.

The unconditional / long-run variance of $X$ depends on the ARMA-GARCH model but in any case is not equal to the square residual at any period, except when/if it is numerically equal, which is by chance.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.