How to assess adequacy of fitted GARCH model?

I am curious how to check whether an already fitted GARCH or ARIMA model with GARCH variance adequately captures the data it is fitted to, not whether one GARCH model is a better fit than another.

For example, after fitting an ARIMA model with constant variance to a time series Y, we can use the infer function with the data set Y and the fitted ARIMA model to retrieve the residuals, which, if the model is an adequate fit, should be a white noise process.

I am unsure how, after fitting a GARCH model with the estimate function, I can check the adequacy of the fit by, say, looking at the residual series. When the infer function is used with a GARCH model, it does not return a residual series so it seems similar analysis to that above may not be able to be performed. Please offer suggestions on how a fitted GARCH model may be tested to see whether it adequately describes (is fit to) a dataset. Thank you.

Answer 1

[Roger Wohlwend]

When you fit an ARIMA model to a time series the residuals should exhibit no heteroscedasticity. If they do you fit an ARIMA/GARCH model. The model is adequate if the residuals have no significant autoregressive conditional hetereoscedasticity once they have been standardized by their conditional volatility. That is how you assess the adequacy of a GARCH model according to one of my textbooks. In addition you can do a visual inspection. Plot the residuals and the GARCH volatility and check if the model is capable of capturing volatility clustering.

Answer 2

[Danny]

Hi Roger,

Thank you for the reply. What is the name of the textbook you are referencing?

[Roger Wohlwend]

I am referencing "Market Models" by Carol Alexander.