I am going to use the ARMA-GARCH model for financial time series and was wondering whether the series should be stationary before applying the said model.

I know to apply ARMA model the series should be stationary, however I’m not sure for ARMA-GARCH since I’m including GARCH errors which imply volatility clustering and non-constant variance and hence non-stationary series no matter what transformation I do.Are financial time series usually stationary or non-stationary?

I tried applying ADF test to a few volatile series and got p-value<0.01 which seems to indicate stationarity but the principle of volatile series itself tells us that the series isn’t stationary.Can somebody clear that up for me?I’m getting really confused

**Answer**

Copying from the abstract of Engle’s original paper:

“These are mean zero, serially uncorrelated processes with nonconstant variances conditional on the past, but constant unconditional variances. For such processes, the recent past gives information about the one-period forecast variance”.

Continuing with the references, as the author who introduced GARCH shows (Bollerslev, Tim (1986). “Generalized Autoregressive Conditional Heteroskedasticity“, Journal of Econometrics, 31:307-327)

for the GARCH(1,1) process, it suffices that $\alpha_1 + \beta_1 <1$ for 2nd-order stationarity.

Stationarity (the one needed for estimation procedures), is defined relative to the *unconditional* distribution and moments.

**ADDENDUM**

To summarize here discussion in the comments, the GARCH modeling approach is an ingenious way to model suspected heteroskedasticity over time, i.e. of some form of *heterogeneity* of the process (which would render the process non-stationary) as an observed feature that comes from the existence of *memory* of the process, in essence *inducing* stationarity at the unconditional level.

In other words, we took our two “great opponents” in stochastic process analysis (heterogeneity and memory), and used the one to neutralize the other -and this is indeed an inspired strategy.

**Attribution***Source : Link , Question Author : ankc , Answer Author : Alecos Papadopoulos*