# When using the GARCH model, should you subtract the mean (if it's none-zero) from your observations?

My understanding is the squared observation is a proxy for variance in GARCH models. If the data itself has a non-zero mean, does it make sense to transform the data beforehand by subtracting the mean from each point before hand?

Consider a GARCH(1,1) model for a time series $x_t$ with conditional mean $\mu_t$ and conditional variance $\sigma_t^2$:

\begin{aligned} x_t &= \mu_t+u_t, \\ u_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \beta_1 \sigma_{t-1}^2, \\ \varepsilon_t &\sim i.i.d.(0,1). \end{aligned}

The model includes a specification of the conditional mean and the conditional variance.

My understanding is the squared observation is a proxy for variance in GARCH models.

Unless you explicitly assume the data to have zero mean, squared observations $x_t^2$ are not proxies for variance; squared model residuals $u_t^2$ are (you find them in the equation for $\sigma_t^2$).

If the data itself has a non-zero mean, does it make sense to transform the data beforehand by subtracting the mean from each point before hand?

No, you do not need to do that. You do not need to preprocess the data to remove the mean since you can specify the mean equation within the model. In your case, it would be $\mu_t=\mu$ (a constant).

• I guess I got confused because when we test to see if an ARCH model fits the data, we square the data and look at ACF plot and Ljung-Box test. That's why I thought squared observation is a proxy for variance Commented May 4, 2018 at 16:36
• @user6472523, even then, when testing for ARCH effects, conditional mean should be subtracted from the data. When testing the adequacy of a GARCH model, we examine the standardized residuals (fitted values of $\varepsilon_t$ above) and their squares. So in fact, we do not examine square data unless we are assuming the mean to be zero. Commented May 4, 2018 at 16:47
• conditional mean would be the mean derived from an ARIMA model? so we would never just square the data and test for ARCH effects, correct? Commented May 4, 2018 at 16:56
• @user6472523, either an ARIMA model or a constant or some other model (say, with exogenous regressors). We would not square the data themselves unless we assume their conditional mean is zero. This could sometimes be assumed, e.g. for stock returns or the like. Commented May 4, 2018 at 16:59
• ah that makes sense. my class does tend to work with stock returns so that's why we square the data itself sometimes. thank you! Commented May 4, 2018 at 17:24