I am currently working on a model for exchange rates, and I want to use an ARIMA-GARCH specification. More precisely, I work on the log-returns series.
First of all, I perform multiple KPSS and ADF tests to get the conclusion that the series is stationnary up to a trend. I then use the auto.arima function to select the best model based on AIC or BIC series. My problem arises when I want to check the model adequacy by looking at the residuals. Since I work with data from January 1999 to December 2012, I have a time series of 3428 observations. When I look at the p-values of the Ljung-Box test, I get the conclusion that the different lags are not significant up to lag 23, and become significant from this lag onwards.
Which conclusion should I make ? If I take the usual $\log(n)$ rule, I have to look around lag 8 and I thus have no problem. But sometimes in the literature, it is asked to look around $\sqrt(n)$ which is around lag 59, and I can not accept the model. Moreover, I want next to simulate trajectories for one year, that is, 250 points, and we can find advices saying it would be better to look around the lag corresponding to the projection horizon. Thus here, I would have a big problem, since I won't find a model which Ljung-Box p-values significative up to lag 250.
Would you have any advice for me ? Thanks in advance.