Doug asks:
>I hate to break up the flow of discussion on samuelson, summers, etal, but
>I have a short economietrics question I need help with. I know there are
>some metrics experts lurking out there. If you don't want to give yourselves
>away, you can reply to me directly.
>
does this imply some shame should be attributed to being an econometrics person
on a left list? i feel no shame.
>Lets say you have a specification that exibits some
>heteroscedacticity.
how have you detected for this? which test? which sample? what type of data?
>I realize that there are several techniques to try to
>remove it.
yes, but also some which incorporate it (garch and arch-m, for example)
>My question is this. I seem to remember that ANYTIME you add
>another variable to the model, if there is any correlation at all, the measured
>amount of heteros. will be reduced somewhat. Is this true?
well simple ways of dealing with heteroscedascity is to model a process which
exhibits volatility in its variance as
y(t) = e(t)X(t)
where e(t) is an error term and X(t) is a variable (your added variable Doug)
which can be observed and which is meant to help predict the hetero.
so a conditional prediction for y(t+1) is
y(t+1) = e(t+1)X(t)
if x(t) = x(t-1) = x(t-2) = some constant, then the {y(t)} sequence is clearly
just a normal white noise process with constant variance.
the interesting case (your case Doug) is when X is not a constant, and then the
variance of y(t+1) is conditional on x(t) and is:
var[Y(t+1)|x(t)] = X(t)^2s^2
where s^2 is the variance of e(t) and is constant.
So an estimating equation might be in logs:
y(t) = a1 + a2y(t-1) + a3X(t-1) + e(t)
and so if a3 is significant, you should expect some of the volatility in y(t)
to be absorbed by the introduction of the X variable. btw, if successive x
values are serially correlated then the conditional variance of y(t) will also
be s.c. If the size of X^2 is large, then the volatility of y(t) will depart
sizably from s^2 and vice versa.
The problem with this approach to heterosced. is that you have to assume there
is some specific cause of the volatility and be able to find the variable
which is relevant (the X).
It is often the case that there is no real guiding light variable to be used
and many likely suspects exist.
moreover, the transformation above relies on the {e(t)} sequence being white
with constant variance. if it is not then another approach is needed.
>Also, is there
>any test that indicates whether the reduction in heteros is the result of just
>adding another variable vs. reducing the original misspecification of the
>model?
>
well there is a battery of tests you should use. you can get an idea of whether
the addition of X(t) is the saviour by running an arch test (it is an LM test
of the TR^2 variety taken off the auxiliary regression of the residuals on
squared values of the residuals plus) and also some diagnostics on the
residuals (s.c, reset, normality, predictive error).
if the arch(p) is fine, and the residuals appear to be white (no sc, etc) then
your approach is okay......contingent of-course on the addition of the variable
making some economic sense in the context of your model.
however, you should also be aware of the pre-testing problems and what they do
to the difference b/tw nominal and true significance levels of your tests.. in
other words, i do not advocate a wild search for X(t) and a million regressions
to find it.
better by far to use an arch or garch model and proceed more circumspectly.
Okay!
so now send me the flames for being a technocrat, or maybe i have to say
something about trade unions to get them.
kind regards
bill
--
#### ## William F. Mitchell
####### #### Head of Economics Department
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