Re: st: error variance increases when we add a variable in a GARCH model

[vwiggins@stata.com (Vince Wiggins, StataCorp)]

John Levendis <johnlevendis@business.loyno.edu> is surprised that when
he adds regressors to an ARCH model, the variance of the residuals is
not reduced,

> I'm trying to estimate a GARCH(1,1) model, with and without an
> exogenous variable. My dependent variable is usually some financial
> stock's daily returns. Sometimes, when I add the exogenous variable
> to the model, the error variance actually increases; i.e. the model
> fits *worse* when I include an extra variable. I was under the
> impression that goodness-of-fit never goes down when you add a
> variable. Any ideas why I'm getting this strange result? Is it just
> because the models are estimated iteratively, and the likelihood
> isn't quite maxed out?

John provides the example,

> fetchyahooquotes UTX SPY, freq(d) chg(ln) start(01jan2000)
>
> gen day=_n 
> tsset day 
>
> arch ln_UTX L(1/2).ln_UTX, arch(1) garch(1)
> predict double resid_restricted, resid 
>
> arch ln_UTX L(1/2).ln_UTX L(1/2).ln_SPY, arch(1) garch(1)
> predict double resid_unrestricted, resid 
>
> sum resid*

Which indeed shows smaller residual variance for the more restricted
first model.

The problem lies with the expectation that the residual variance will
necessarily go down when adding terms to an ARCH or GARCH model.
Recall that these models are attempting to model the conditional
variance of those residuals.  What that means is that all residuals are
not created equal when fitting these models.  Some contribute more than
others because they have smaller conditional variance.

The likelihood for the model, once you have conditioned on the
regressors and ARCH/GARCH process is particularly simple.  Each
observation's contribution to the likelihood is,

                                 
                                              2
                                    2      e_t
    ln(L_t) = -1/2 { ln(2*pi*sigma_t ) + --------- }
                                                 2
                                          sigma_t
                                
        where:
                e_t       are the errors, and
                       2
                sigma_t   are the conditional variances of the errors

                (see page 49 of [TS] for a typeset version)

We see that the errors are scaled by our estimates of the conditional
variances (sigma_t^2).  The errors always enter the likelihood scaled
by the variances.  Said another way, minimizing error variance would
not provide maximum likelihood estimates of ARCH/GARCH models.

So, we must find ways to compare the models that do not use residuals.
I ran John's models, and while they take several iterations to converge
(typical of ARCH/GARCH models), they look to be well converged.
Moreover, the model with covariates has a larger likelihood, as
expected.  The models are nested, so we can perform a likelihood ratio
test.  I did, and with these models, the chi2(2) = 4.31 with 
a Prob > chi2 = .1157.

On a side note I had forgotten about -fetchyahooquotes- (available from
the Stata Journal downloads).  I am going to try and forget it again.
It is far too much fun.


-- Vince 
   vwiggins@stata.com

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