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st: probit with interaction dummies (significance and marginal effects)

From   "Stephen P. Jenkins" <>
To   <>
Subject   st: probit with interaction dummies (significance and marginal effects)
Date   Mon, 28 Jul 2008 09:49:57 +0100

> Date: Sun, 27 Jul 2008 05:44:28 -0400
> From: "Erasmo Giambona" <>
> Subject: Re: st: probit with interaction dummies 
> (significance and marginal effects)
> Dear Tony, Stephen and Allan,
> Thanks very much for your insightful contributions. I am still
> on how to intepret the simple coefficient on the interaction of two
> continuos variables after logit in relation to the marginal effects.
> Any thoughts on this would be appreciated.
> Thanks,
> Erasmo

You appear to be seeking a one number quick fix. There isn't one.
I recommend going back to first principles. A "marginal effect" (ME)
can be defined in a number of ways according to taste, but one way of
thinking about it in context of binary depvar models is:

  ME = change in Pr(y=1) given a one unit change in x (one of the RHS

with appropriate redefinition in the case of binary/categorical x.

You tell us that x is interacted with another variable, call it z.

If the model is a probit

			Pr(y=1|x, z, ...) = Normal(a + b*x + c*xz +

then you can calculate the predicted probabilities

			Pr(y=1|x = x_1, z = z_1, ...) = Normal(a +
b*x_1 + c*(x_1)*(z_1) + ....)
			Pr(y=1|x = x_2, z = z_1, ...) = Normal(a +
b*x_2 + c*(x_2)*(z_1) + ....)

where x_2 = x_1 + 1, and the parameters are now understood to be
estimated values. Change "Normal()" to -invlogit- for the Logit model.

So, we can calculate:

	ME = Pr(y=1|x = x_2, z = z_1, ...) - Pr(y=1|x = x_1, z = z_1,

and this depends on the value of z_1, and also on the values of the
other covariates ...

Observe that there is no single unique "marginal effect" in non-linear
models (logit, probit, poisson, etc.) regardless of whether
interaction terms are included or not as covariates. Whatever ME you
calculate depends on the values of the covariates. This is also true
when there are interaction effects.  Indeed, in the expression above,
you would get another ME estimate were z held at a different value
from z_1.

Expressions like ME = Pr(y=1|x = x_1, z = z_1, ...) - Pr(y=1|x = x_2,
z = z_1, ...) can be calculated, together with associated standard
errors, using -nlcom-.  The Norton et al. Stata Journal article that
you cited simply canned some of these calculations.  (Scott Long's
NASUG presentation that I referred to last message is a clear
demonstration of the predicted probability approach, together with
associated graphs to illustrate the results. (He uses some of his own
programs, but understand the principles.))

To me, the principal lessons of the Norton et al. article are a
reminder of the following about non-linear models:

* Some researchers wish to conclude that an "interaction effect is
significant" by eyeballing the absolute t-ratio of the coefficient on
an interaction term in a non-linear model (e.g. checking whether the
|t| on coefficient c in the model above is greater than 1.96)
* This is problematic because we are usually interested in concepts
like MEs rather than a coefficient per se. 
* The value of an ME depends on the values of the covariates at which
evaluated and, moreover, so too does the SE of the ME.
* In particular an ME may not differ significantly from zero, even in
cases where the coefficient on the interaction term is statistically
* So, interpretation of estimates from non-linear models can be
trickier than interpretation of estimates from linear models

Professor Stephen P. Jenkins <>
Director, Institute for Social and Economic Research
University of Essex, Colchester CO4 3SQ, U.K.
Tel: +44 1206 873374.  Fax: +44 1206 873151.  
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