In short, you should select models using AIC and/or out-of-sample fit criteria and view the rejected hypothesis as a suggestion to consider other types of models.
When using this class of time series models researchers are usually interested in accurate prediction\forecasting. Since AIC measures how well a model predicts the data in-sample, it operates as a fair means of model selection in this case (you may also want to test how well the models fit out-of-sample…more on that below).
However, just because a particular model has the lowest AIC does not mean that that model is correctly specified or that it approximates the true data generating process well. It could be that all the models you proposed were poor choices, or that the true process FTSE follows is so complex that practically every reasonable model will be rejected given enough data. AIC provides no information on this point which is where hypothesis testing can come in.
Under the assumptions of standard ARMA-GARCH, the residuals should be homoscedastic and more generally iid normal. Your hypothesis test suggests that your residuals are not homoscedastic and, in turn, that your ARMA-GARCH model may be miss specified. On this note you may want to consider alternative specifications for the volatility process including other variants of GARCH models, i.e. EGARCH, GJR-GARCH, TGARCH, AVGARCH, NGARCH, GARCH-M, etc. and/or stochastic volatility models. It is highly likely that one of these models will offer a lower AIC value and produce residuals which cannot be rejected for homoscedasticity.
One important thing to note though is that no model will be perfect, especially for something like the FTSE 100. The true data generating process driving a large financial index like this is impossibly complex, so pretty much every model you propose will be false. For this reason, it can be argued that any meaningful hypothesis you do not reject is a reflection of insufficient data or lack of statistical power rather than evidence supporting one model over others.
One way to partially resolve this dilemma is to use out-of-sample fit as opposed to or in conjunction with AIC. A simple example would be to fit the model using only the first 80% or 90% of the data and using the resulting coefficient estimates to obtain a log-likelihood for the remaining 20%-10% portion of the data. The model with the highest log-likelihood would be preferred. If the ARMA-GARCH model is truly misspecified in a way that impairs its forecasting performance, then an out-of-sample fit will help expose it.
- Even though you cannot specify an ARIMA model for the conditional mean directly in function
ugarchspec, you can do this indirectly by differencing your data a desired number of times before feeding into estimation via ugarchfit. So if the desired model for series x is ARIMA$(p,d,q)$, then specify ARMA$(p,q)$ in ugarchspec and feed diff(x,d) instead of x to the function ugarchfit.
- If I understand your question correctly, you are asking whether you can fit an ARMA-GARCH model on differenced data -- presumably instead of fitting an ARIMA-GARCH model on the original data. Yes, this is fine, and this is exactly what I suggest in part 1. If there was an option to specify ARIMA-GARCH with an integration order greater than zero, the function would start with differencing your data the specified number of times ($d$) and then proceed as with an ARMA-GARCH model.
Note that there does not seem to be an option to use SARMA models in the "rugarch" package, so you will have to let the "S" part go. But if there is a seasonal pattern (and that is quite likely when it comes to tourist arrivals), you will have to account for it somehow. Consider using exogenous seasonal variables (dummies or Fourier terms) in the conditional mean model via the argument external.regressors inside the argument mean.model in function ugarchspec. Alternatively, note that a SARMA model corresponds to a restricted ARMA model. An approximation of SARMA could thus be an ARMA with the appropriate lag order but without the SARMA-specific parameter restrictions (since those might not be available in "rugarch").
Also consider whether a GARCH model for the conditional variance is relevant. I do not know the tourist business well but I am not immediately convinced that tourist arrivals will have a GARCH pattern. You could check for the need of GARCH-type of conditional variance model by testing the residuals from the SARIMA model using ARCH-LM test or some other test for (G)ARCH effects.
Best Answer
My experience with equities suggested that if you are confined to garch(p,q), then garch(1,1) is what you will want. Using a components model (Lee and Engle) is better -- it is sort of like a garch(2,2) but not quite the same.
When modeling multivariate garch (where there was a lot of choice in parameterization), it seemed to be that BIC was defnitely better than AIC. BIC has a larger penalty and so suggests smaller models. It looked like the penalty should be even bigger than in BIC -- that the BIC models were still too big.