# Effect of adding orthogonal variables to the model under ARCH/GARCH errors

I am trying to estimate a restricted and unrestricted version using the ARCH/GARCH framework. Specifically, I am using a GARCH(1,1) model and am assuming a normal distribution. While I am not really interested in the the conditional variance, I am interested in achieving coefficient efficiency. Two of my variables are orthogonal to all other variables in the model. The resticted version on the model is as follows:

My second unrestricted version is:

In the unrestricted model, ex_j203tr_res and dl_exmsci_w_res are orthogonal to the other factors. However, when I include these factors, the estimated coefficients change. Under OLS, this does not happen - the addition of the orthogonal factors does not impact any of the coefficient estimates.

Why is this happening under ARCH/GARCH estimation?
I am under the impression that when factors that are orthogonal are, there is be no impact on existing estimated coefficients? This seams to hold under OLS but not maximum likelihood?

• I have posted an answer. Let me know if anything is unclear. Otherwise be aware that satisfactory answers may be accepted by clicking a tick mark to the left of the answer - this is how Cross Validated works. But poor and unclear answers of course need not be accepted. Commented Aug 25, 2017 at 8:05

GARCH effectively introduces weighting of observations in the conditional mean model. The ones with higher estimated conditional variance get downweighted while the ones with lower estimated conditional variance get upweighted. The factors that are orthogonal (and/or uncorrelated) in absence of weighting will easily become non-orthogonal (and/or correlated) when weighting is applied. This is an effect of conditional heteroskedasticity (GARCH) rather than estimation technique (OLS vs. maximum likelihood).

For example, you could have two series that are orthogonal: $x=(-1,0,1)$ and $y=(0.5,1,0.5)$. Their inner product is zero (and their correlation is zero). But if you weight the first observation twice as heavily as the other two observations, you effectively get $x'=(-1,-1,0,1)$ and $y'=(0.5,0.5,1,0.5)$ which are not orthogonal. Their inner product is $-0.5$ (and their correlation is $0.1741$).