Does applying ARMA-GARCH require stationarity? 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
 A: Yes the the series should be stationary. GARCH models are actually white noise processes with not trivial dependence structure. Classical GARCH(1,1) model is defined as
$$r_t=\sigma_t\varepsilon_t,$$
with 
$$\sigma_t^2=\alpha_0+\alpha_1\varepsilon_{t-1}^2+\beta_1\sigma_{t-1}^2,$$
where $\varepsilon_t$ are independent standard normal variables with unit variance.
Then
$$Er_t=EE(r_t|\varepsilon_{t-1},\varepsilon_{t-2},...)=E\sigma_tE(\varepsilon_t|\varepsilon_{t-1},\varepsilon_{t-2},...)=0$$
and
$$Er_tr_{t-h}=EE(r_tr_{t-h}|\varepsilon_{t-1},\varepsilon_{t-2},...)=Er_{t-h}\sigma_{t}E(\varepsilon_t|\varepsilon_{t-1},\varepsilon_{t-2},...)=0$$
for $h>0$. Hence $r_t$ is a white noise process. However it is possible to show that $r_t^2$ is actually a $ARMA(1,1)$ process. So GARCH(1,1) is stationary process, yet has non-constant conditional variance.
A: For anyone who is wondering about this question still, i will clarify - Volatility clustering does not at all imply that the series is non-stationary. It would suggest that there is a shifting conditional variance regime - which may still satisfy constancy of the unconditional distribution.
The GARCH(1,1) model of Bollerslev is not weakly stationary when $\alpha_{1}+\beta>1$, however it is actually still stricktly stationary for a much larger range, Nelson 1990. Further Rahbek & Jensen 2004 (Asymptotic inference in the non-stationary GARCH), showed that the ML estimator of $\alpha_{1}$ and $\beta$ is consistent and asymptotically normal for any parameter specification that ensure the model is non-stationary. Combining this with the results of Nelson 1990 (all weak or strict stationary GARCH(1,1) models have MLE estimator as consistent and asymptotically normal), suggests that any parameter combination whatsoever of $\alpha_{1}$ and $\beta>1$ will have consistent and Asymptotically normal estimators.
It is important to note however that if the GARCH(1,1) model is non stationary, the constant term in the conditional variance is not estimated consistently.
Regardless, this suggests that you do not have to worry about stationarity before estimating the GARCH model. You do however have to wonder whether it seems to have a symmetric distribution, and whether the series has high persistence, as this is not allowed in the classical GARCH(1,1) model.
When you have estimated the model it is of interest to test whether $\alpha_{1}+\beta=1$ if you are working with financial timeseries, since this would imply a trending conditional variance which is hard to immagine being a behavioral tendency amongst investors. Testing this however can be done with a normal LR test.
Stationarity is fairly misunderstood, and is only partially connected to whether the variance or mean seems to be ocationally changing - as this can still ocour while the process maintains a constant unconditional distribution.
The reason you may think that the seeming shifts in variance may cause a departure from stationarity, is because such a thing as permanent levelshift in the variance equation (or the mean equation) would by definition break stationarity. But if the changes are caused by the dynamic specification of the model, it may still be stationary even though the mean is impossible to identify and the volatility constantly changes. Another Beautiful example of this is the DAR(1,1) model introduced by Ling in 2002.
A: 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.
A: Stationarity is a theoretical concept which is then modified to other forms like Weak Sense Stationarity which can be tested easily. Most of the tests like adf test as you have mentioned test for linear conditions only. the ARCH effects are made for series which do not have autocorrelation in the first order but there is dependence in the squared series. 
The ARMA-GARCH process you talk about, here the second order dependence is removed using the GARCH part and then any dependence in the linear terms is captured by the ARMA process. 
The way to go about is to check for the autocorrelation of the squared series, if there is dependence, then apply the GARCH models and check the residuals for any linear time series properties which can then be modelled using ARMA processes.
