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

 From "Erasmo Giambona" To statalist@hsphsun2.harvard.edu Subject Re: st: probit with interaction dummies (significance and marginal effects) Date Mon, 28 Jul 2008 07:56:05 -0400

```This is a tremendous help. You are right. I was trying to have a one
number quick fix. This is because I have a panel of firms and
therefore I am using clogit to control for firm fixed effects. As far
as I understand, marginal effects are problematic with conditional
logit mainly because I would not be able to invoke the invlogit that
you suggest below to calculate predicted probabilities for x_1 and
x_2. I have noticed that people suggest using the sample average
predicted probability obtained with pc1 in this case. But then would
one be able to really calculate the change in the probability as x_1
changes to x_2?
Erasmo

On Mon, Jul 28, 2008 at 4:49 AM, Stephen P. Jenkins
<stephenj@essex.ac.uk> wrote:
>>
> ----------------------------------------------------------------------
>> Date: Sun, 27 Jul 2008 05:44:28 -0400
>> From: "Erasmo Giambona" <e.giambona@gmail.com>
>> 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
> unclear
>> 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
> vbles)
>
> 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) + ....)
> and
>                        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
> significant.
> * So, interpretation of estimates from non-linear models can be
> trickier than interpretation of estimates from linear models
>
>
> Stephen
> -------------------------------------------------------------
> Professor Stephen P. Jenkins <stephenj@essex.ac.uk>
> Director, Institute for Social and Economic Research
> University of Essex, Colchester CO4 3SQ, U.K.
> Tel: +44 1206 873374.  Fax: +44 1206 873151.
> http://www.iser.essex.ac.uk
> Survival Analysis using Stata:
> http://www.iser.essex.ac.uk/teaching/degree/stephenj/ec968/
>
> Learn about the UK's new household panel survey, the United Kingdom
> Household Longitudinal Study: http://www.iser.essex.ac.uk/ukhls/
>
>
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