# Interpreting QLIKE and MSE Loss function (Patton 2011)

In Patton (2011) the author finds that both the MSE and the QLIKE loss function are robust when used to compare rivalling volatility forecasting models, which means that using a proxy for volatility gives the same ranking as using the true (unobservable) volatility of an asset.

In my current project I am comparing a family of GARCH/AGARCH models and while the MSE suggests that nothing outperforms a GARCH(1,1), the QLIKE statistic suggests that an APARCH(1,1) model performs significantly better.

Is this caused by the two loss functions penalising deviations differently? Specifically, what do the two loss functions place the highest penalty on, i.e. how do I interpet this?

I am hoping this is not down to some trivial coding error.

#MSE
MSE<-function(sigmafc,RV){
MSE=1/length(sigmafc)*sum((sigmafc^2-RV)^2)
return(MSE)
}

#QLIKE
QLIKE<-function(sigmafc,RV){
varfc=sigmafc^2
QLIKE=sum(
(RV/varfc-log(RV/varfc)-1)
)
return(QLIKE)
}


I gather that the MSE depends on forecast errors, while QLIKE depends on standardised errors, but how would I interpret this?

• Could you give a full reference for Patton (2011)? Also, where exactly in the paper is QLIKE defined? I would like to check whether your code matches that definition? (I have found some paper by Patton from 2011 and I don't think you got QLIKE right, but that is based on just a quick look at the paper.) Jul 27 '16 at 11:52
• Patton (2011) does indeed have a different definition of the QLIKE loss function. However, in later papers (and in my lectures) the function was given as $QLIKE=\frac{\hat{\sigma}^2}{h}-\log{\frac{\hat{\sigma}^2}{h}}-1$ with $\hat{sigma}^2$ being the term for the volatility proxy and $h$ the forecast for a given period. The reasoning, if I recall correctly, is that this gives a QLIKE value of 0, if the forecast and the volatility proxy are identical. Jul 27 '16 at 13:58
• The paper I reference is listed below econpapers.repec.org/paper/utsrpaper/175.htm Jul 27 '16 at 14:05
• After a bit more searching, the reasoning is presumably that the MSE is more sensitive to outliers/extreme values Jul 27 '16 at 15:14