> There are two other situations: (3) X1 and X2 are inter-related with
> each other, but there is no clear direction of the relationship. This
> means there is no clear theory to identify which factor causes which.
> (4) X1 and X2 are not related theoretically, although statistically
> correlated.
>
> My questions are: (1) How would we deal with the third situation?
You have to determine whether a variable is an intervening variable
or a confounding variable in order to decide whether or not to
include a variable. So you really need to get that clear. A possibly
very complicated problem is when x1 influences x2 _and_ x2 influences
x1. In that case you can either a priori decide which direction is
the more important / dominant one, or you'll have to use additional
information to try to disentangle those (e.g. instrumental variables
or panel data)
> (2) The fourth situation should be a multicollinearity problem, and
> what shall we do if findings from correlation and VIF tests are not
> consistent?
You should only control for variables you think are confounding
variables, so if you believe that the two should not be related, then
you believe that x2 is not a confounding variable, so it should not
be in your model.
Multicollinearity it is just the phenonemenon that you loose power
when two explanatory variables are correlated. This is not a problem,
this is exactly as it should be. When we enter two (or more) variables
in a model, then we have to be able to distinguish between the two
variables. If two variables are highly correlated, then we have a lot
of trouble distinguishing between the two, so we become more uncertain
about their individual effects, so we should get less power. In that
sense VIF and correlation are useful to understand your model, but
they do not help you in deciding which model to choose.
Hope this helps,
Maarten
--------------------------
Maarten L. Buis
Institut fuer Soziologie
Universitaet Tuebingen
Wilhelmstrasse 36
72074 Tuebingen
Germany
http://www.maartenbuis.nl
--------------------------
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