I am slightly irritated about weak stationarity in connection to ARCH/GARCH models. I do not know the answer and I am not sure about it:
The basic question is:
Do we have to test weak stationarity before applying an ARMA-GARCH model?
Further on it can be said:
ADF and others test the mean equation, but this is not for the volatility equation, so what test do we have to use for the autocovariance-stationarity?
Standard ARMA models assume the unconditional mean and unconditional variance to be constant. For ARMA-GARCH models this is also the case: The unconditional mean and unconditional variance need to be constant, whereas in case of the ARMA-GARCH models the conditional variance does not need to be constant.
It is correct that for the mean equation we may have to think about using a trend-stationary or difference-stationary model. But this is only concerning the mean equation, yes.
Conditional variance can be tested by testing for ARCH effects (Box-Ljung, Lagrange Multiplier).
So for ARMA-GARCH models we still need weak stationarity, since the unconditional mean and unconditional variance need to be constant. So I am not sure, but we have to test for weak stationarity before applying an ARMA-GARCH model? And especially with financial returns, do we also have to test for it? And which test do we use (and which command is implemented in R, so what command can you suggest?)
I know that if the unconditional variance is nonstationary (not finite and then it is also not constant) an integrated GARCH may be appropriate. But just because it is not constant I cannot say I use an integrated GARCH model?
I also know that for ARMA-GARCH processes all the "characteristical" roots lie outside the unit circle. So in case of a ARMA-GARCH(1,1) $\alpha_1+\beta_1<1$ is necessary. But I only see this after estimation? This is not a test for covariance stationarity?
EDIT: It basically pins down to (see the comments): How can I test the unconditional variance to be constant? I mean in order to apply a GARCH model I have to make sure that I have constant unconditional mean (ADF test and so) and I have to test for constant unconditional variance (how?). I know that I have to further make sure that I have nonconstant conditional variance for GARCH processes, otherwise having a constant conditional variance ARMA is sufficient (test for ARCH effects).
EDIT 2: There is a Wavelet Spectrum test in the locits package, what about this test?