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I've estimated an ARMA(1,2)-GARCH(1,1) model fitted on financial data. It is very satisfactory in modeling the autocorrelation and the volatility in my data, however, the qq-plot empirical quantiles vs t-student quantiles for standardized residuals is not totally satisfactory (see the first figure below). I've tried ALL the possible innovations (normal, skewed t-student, ged etc.) contained in the r function "garchFit" but all the model are worse than the t, what do you suggest? From the q-q plot do you think t innovations are sufficient? In my opion the departure from the t-distribution is not very severe, I thought that with a skewed t the problem could be fixed but actually worsen (see the second figure). Data of course are stationary. enter image description here

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  • $\begingroup$ Can you also put a picture of the ACF and PACF of the returns? You could also include the process in which you conclude the data is stationary. It may be a good idea to include your data and script as it makes testing ideas easier. $\endgroup$
    – ABIM
    Aug 26, 2018 at 15:03
  • $\begingroup$ Data are easily accessible from yahoo finance, s&p 500 index adjusted close and the series goes from 31 December 2007 to 24 July 2018 (included). I take the logs and than differentiate (log returns), the two models are fitted over the log returns series from 2 January 2008 (included) to 24 July 2018 (included) for a total of 2659 observations. If needed I will add the acf and pacf but they are almost perfect and the series is 100% stationary (tested with adf , Kpss, phillips Perron). I cannot understand why the qq plot is not that good. $\endgroup$ Aug 26, 2018 at 16:53
  • $\begingroup$ I forgot that data are on daily basis. $\endgroup$ Aug 26, 2018 at 17:21
  • $\begingroup$ Fisrt of all, I believe you should run a grid search for model selection. I do not see any reason for your model to be the best. Second, I suggest using GJR-Garch as it treats the left tail and the right tail somehow separately. You can see a discussion here. I think it will work better since you seem to be okay for the right tail. $\endgroup$
    – ABIM
    Aug 26, 2018 at 17:40
  • $\begingroup$ Move this OP to Quantitative Finance (QF) at StackExchange. $\endgroup$
    – user32398
    Aug 26, 2018 at 17:41

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