B.D. McCullough 1295Table 28.2 Multivariate GARCH results
Package μc μf c 1 a 1 b 1 c 2 a 2 b 2 c 3 a 3 b 3
Package A 0.064 0.064 0.377 0.128 0.411 0.566 0.145 0.365 0.474 0.128 0.348
Package B 0.062 0.069 0.012 0.041 0.946 0.012 0.034 0.956 0.011 0.035 0.953
Package C 0.061 0.037 0.010 0.037 0.952 0.010 0.031 0.961 0.009 0.032 0.959
Package D 0.073 0.082 0.076 0.112 0.798 0.125 0.134 0.762 0.099 0.120 0.773
maximum likelihood (FIML) estimates for Klein’s Model I. Clearly, not all three sets
of answers can be correct. McCullough and Renfro (1999) published seven different
estimates for the parameters of the same generalized autoregressive conditional
heteroskedasticity (GARCH) model. What does this make one think of the GARCH
results published in the literature? As far as GARCH estimation is concerned, recall
that any nonlinear estimation procedure requires starting values which need to be
carefully chosen if the solver is to have much of a chance to find an extremum.
Some GARCH procedures in some packages do not allow the user to set the starting
values! What does one think of the developers of such packages? (What do the
developers of such packages think of their users?) Brooks, Burke and Persand (2003)
did the same thing for multivariate GARCH models, and it is instructive to present
the model they estimated; the results from their Table II are given in our Table 28.2.
The multivariate GARCH model estimated by Brooks, Burke and Persand was:
st=μs+ (^) s,t
ft=μf+ (^) f,t
hs,t=c 1 +a 1
s^2 ,t− 1 +b 1 hs,t− 1
hf,t=c 2 +a 2
f^2 ,t− 1 +b 2 hf,t− 1
hs,f,t=c 3 +a 3
s,t− 1
f,t− 1 +b 3 hs,f,t− 1.
Study the coefficients in Table 28.2. Each package estimates completely different
parameters for the same model. What does this say about multivariate GARCH
results published in the literature? Perhaps the multivariate GARCH likelihood is
complicated and difficult to optimize, and may have multiple optima. Not that
these numerical difficulties are an excuse for inaccuracy, but maybe a user could
have more faith in a simple nonlinear estimation problem for a convex likelihood
like the probit model? Surely nothing could go wrong with that? Stokes (2004)
gave the same probit problem to six packages and got six different answers. What
is particularly interesting is that, for this particular probit problem, no solution
exists. The problem, as posed, exhibited what is called “complete separability” and
so there is no set of parameters that maximizes the likelihood. This phenomenon is
ignored by most econometrics texts that present the probit model (but see Davidson
and Mackinnon, 2004, pp. 458–9, for an exception). This minor impediment did
not stop the six packages from reporting that they had found solutions. A seventh