# Residuals of ARIMA model not normally distributed, can I still use an ARCH\GARCH Model?

I have been trying to model the value of bitcoin log returns using R and my main objective is to apply an ARCH-GARCH model to the data set and produce some forecast from this. I think it might be applicable to note that this is for academic purposes and apologies if I have misinterpreted any of the concepts explained below.

So, in essence, I have run my dataset and obtained the log returns. I then used an ADF test to confirm stationarity thus I could begin to apply an ARIMA model to the data set. I used the AIC values to determine that the ARIMA(7,0,5) is the best-suited model for the log returns of bitcoin. However, when analysing the residuals for the Arima models the ACF gave evidence for discrete white noise but the QQ plots indicated that the residuals are not normally distributed and are in fact heavy-tailed (possible t-distribution?). This immediately raised some concerns for me as 1. The AIC values which use maximum likelihood function assumes asymptotic normality and thus imply that the model selection is not viable? And 2. R uses ML to determine the model.

        Box-Ljung test


data: resid(arima705) X-squared = 24.286, df = 20, p-value = 0.2301

    Jarque Bera Test


data: resid(arima705) X-squared = 12840, df = 2, p-value < 2.2e-16

So then my question arises, can I still use a GARCH on the ARIMA(7,0,5) model if the residuals from the ARIMA model are not normally distributed? if it is ok could you explain why? Or am I better off using it on the log data data itself?

I noticed with the GARCH function you can fit a T-distribution given a much better fit on the QQ plot, I was wondering if there is a package within R that lets me fit a distribution to the ARIMA model such as the GARCH function.

• The relevant question when fitting a GARCH model is not whether or not the error term is Gaussian - it is whether the error term scaled with the time-varying heteroskedasticity is Gaussian. A GARCH model is able to match the observed leptokurtic behaviour, but sometimes a more heavy tailed distribution is also needed. – Johan Stax Jakobsen Jan 24 '18 at 13:23
• I addition, if you are looking at GARCH effects - check the squared residuals – Johan Stax Jakobsen Jan 24 '18 at 13:24
• Thanks for the response, the ACF of the squared residuals of the Arima Model suggest serial correlation and hence provide evidence for a GARCH effect. I have then fitted a GARCH(1,1) model following a t distribution and both the ACF of the residuals and squared residuals give evidence for discrete white noise. However the QQ plot is based on the T-distribution not a Gaussian. – J Ossa Jan 24 '18 at 13:51
• Could you please expand on the following " it is whether the error term scaled with the time-varying heteroskedasticity is Gaussian", i'm not sure i'm quite following or how i would check this? – J Ossa Jan 24 '18 at 13:52
• The garch model is basically modelling the heteroskedasticity of the error term $\varepsilon_t = \sigma_t z_t$. Typically, $z_t$ is simply assumed to be Gaussian. Then, $\varepsilon_t$ is Gaussian conditional on the information at time $t-1$. However, due to the time-varying heteroskedasticity, the "unconditional" distribution will not be Gaussian. The fitted $\hat{z}_t$ obtained as $\varepsilon_t/\sigma_t$ should however be Gaussian. If not, then the distributional assumption for $z_t$ is wrong - could be changed to e.g. a t distribution. Hope it is ok - sorry for any shortcomings. – Johan Stax Jakobsen Jan 24 '18 at 14:03