# 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?

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.
• @shenflow, (1) is by definition of $u_t$; $X_t=\mu_t+u_t$ where $\mu_t:=\mathbb{E}(X_t\mid I_{t-1})$ and so $\mathbb{E}(u_t\mid I_{t-1})=0$. (2) It just is, even for correctly specified conditional mean models; see Andersen & Bollerslev "Answering the Skeptics: Yes, Standard Volatility Models do Provide Accurate Forecasts" (1998) p. 889 and onwards. The magnitude of correlation between squared errors and true underlying volatilities may be pretty low (say, 0.5, 0.4, 0.3 or even 0.2) for plausible data generating processes. – Richard Hardy Dec 14 '20 at 19:29