# 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. Aug 25 '17 at 8:05

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 $0.1741$).