Error distribution assumption in a simple ARIMA model

why in an ARIMA-GARCH structure I have to assume an error distribution to run the estimation while in a simple ARIMA model it is not required?

Thank you

• What estimation method are you using for each? To run maximum likelihood, you need a likelihood function that comes from your error distribution. To run ordinary least squares (or minimize other penalty functions of the error term), you don't need an explicit likelihood function. – Matthew Gunn Aug 22 '18 at 13:58
• I use standard E-Views tool with ML method, but I cannot choose among the different distribution assumptions. Thank you – LeoAn Aug 23 '18 at 8:06

Simple ARIMA model can be estimated using OLS methods. MA models (or ARMA model) can be estimated using iterative OLS, which provide similar results if MLE is used, assuming that error terms follows normal distribution with constant variance.

Since, GARCH model assumes that conditional variance is not constant. Such dynamic behaviour in volatility can not be accommodated into OLS method. Therefore, we use MLE for GARCH models, which mandatorily require assumption about distribution of error terms.

• Ok, Thank you. And if I leave to assume a normally distributed error in this step (parameters estimation) and then, once fitted the model, I assume for example a t-student's error distribution to evaluate the risks (VaR models for example), am I making any logical mistake? It must necessarily be consistency between the estimation step error assumption and the final distribution assumption for the computation of the VaR? – LeoAn Aug 23 '18 at 8:55
• @LeoAn It means you are using Normal distribution for the GARCH model, and t-distribution for calculating VaR (in parametric models of VaR). If this is the case, then definitely you are making a serious mistake. If your actual data fit more closely to the t-distribution, then it does not make any sense to use normal distribution for the GARCH model. It means you are making two different assumption about the distribution of the same data just for your convenience, which is not accepted in the research. – Neeraj Aug 23 '18 at 19:20
• And what if I cannot choose the error distribution, as in the simple ARIMA setup in Eviews? It means I’m forced to assume a normally distributed error both in the estimation step and in the final computation of the risk (VaR step). It seems unlikely that Eviews doesn’t allow to set the coorect error distibution (in a simple arima estimation with no garch). Do you know how to do that? Thank you again. – LeoAn Aug 23 '18 at 19:46
• @LeoAn I have no idea about what EViews provide. But yes If you know about R, then there are many libraries that provide flexibility in choosing distribution and at the same time compute VaR too. Most popular library is "Rugarch". You can check it once. – Neeraj Aug 23 '18 at 19:57
• ARIMA models cannot be estimated directly by OLS (only if the MA part is absent, conditional least squares can be used). GARCH models cannot be estimated directly by OLS. There are workarounds for ARIMA estimation with iterative methods based on least squares. I suppose we agree on these points. Now my question is, are you sure there are no least-squares based methods for GARCH estimation? Your answer seems to contrast ARIMA to GARCH with respect to whether OLS estimation can be applied, but I am not sure this is correct. I suspect there are iterative least-squares based methods for GARCH. – Richard Hardy Sep 1 '18 at 19:35

I would imagine this depends on the package you are using to estimate the ARIMA model and the GARCH model. Often, there is an implicit error distribution used in the fitting process. Many packages will not allow different error distributions, and will assume a Normal error distribution.

So, for example, if you look at the MATLAB ARIMA time series modelling, you can see that you can specify an error distribution (see halfway down https://uk.mathworks.com/help/econ/arima-class.html where you can choose 'Gaussian' or 'T').

• I use standard E-Views tool, where I can choose among the different methods of estimation (ols, ml...) but I can't choose the error distribution assumption when I run a simple ARIMA. So, If I assume a normal error distribution in this step (estimation parameters) and then, once fitted the model, I use the t-student's distribution to evaluate the VaR, am I making any logical mistake? Thank you – LeoAn Aug 23 '18 at 8:19