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st: Re: Interaction and squared effects in a probit (pa) model


From   Laura Gomez-Mera <lauragmera@gmail.com>
To   Maarten buis <maartenbuis@yahoo.co.uk>
Subject   st: Re: Interaction and squared effects in a probit (pa) model
Date   Tue, 12 Apr 2011 09:51:15 -0400

> Maarten,
> 
> Thank you for your very helpful reply.
> 
> We are still not sure how to deal with the marginal effects of interacted terms (between two continuous variables) in our xtgee model. Norton's 'inteff' command seems to work for probit models with interaction terms but can it be used when we are working with panel data? 
> 
> Thanks again,
> 
> Laura and Andrea

On Apr 11, 2011, at 5:56 AM, Maarten buis wrote:

> 
> --- Andrea and Laura wrote me privately:
>> We´re working on a model and, when trying to solve some econometrical 
>> issues, we found your name on the statalist and thought you may be 
>> able to help us.
> 
> The rule is that you ask questions to statalist and not to individual
> members:
> <http://www.stata.com/support/faqs/res/statalist.html#private>
> 
>> We´re estimating a panel data model with 90 cross-section observations 
>> (country pairs) and 10 time series observations (years), using the 
>> -xtgee- command with the  -family(bin) link(probit) corr(ar1) robust 
>> force- options.
>> 
>> We recently included interactive terms in our model and we´re finding 
>> difficulties in estimating the correspondent marginal effects, as the 
>> command -margins- is not suitable for nonlinear estimations, 
>> especially when our variables of interest are combined with each other.
>> 
>> To make things even more complex, we´ve also got variables interacted 
>> with themselves (quadratic terms).
>> 
>> Searching for a command that would be suitable in our case, we found 
>> -inteff- (http://www.stata-journal.com/sjpdf.html?articlenum=st0063), 
>> but are a little confused because of the mention of the squared 
>> variables.
> 
> -margins- is actually exactly right when you want the marginal effect
> after a model that includes square terms. As you can see in the first
> part of the example below, the marginal effect returned by -margins-
> corresponds exactly with the marginal effect computed by hand.
> 
> The real question should be: Do you really want marginal effects? 
> Marginal effects can be thought of as a linear model on top of your
> previous model. In the graph below we can see that the predicted
> probabilities follow a strong non-linear pattern. This begs the 
> question: Do you believe that there can be a single straight line 
> that can meaningfully summarize the pattern in the predicted 
> probabilities?
> 
> For the example below my answer would: no, that pattern is just too
> non-linear. This should come as no surprise. The quadratic term 
> was added because we believed there to be substantial non-linearity.
> So either we believe that a linear line is a good-enough 
> approximation, in which case we can use marginal effects but it
> raises the question why we added the quadratic term. Or we believe
> that the non-linearity is substantial, which means that the quadratic
> term may be justified, but now marginal effects loose their meaning.
> 
> If you are in the latter case I would add a footnote to the table
> of marginal effects saying that the effect is just too non-linear to 
> be meaningfully summarized by marginal effects and leave that cell 
> them empty in the table. Than I would add a graph of the predicted 
> probability against that variable.
> 
> *-------------------- begin example -------------------------
> sysuse auto, clear
> recode rep78 1/2=3
> probit foreign c.mpg##c.mpg i.rep78
> 
> // do it with margins
> margins, dydx(*) at(mpg=20 rep78=4)
> 
> // do it by hand
> tempname xb
> scalar `xb' = _b[_cons] + _b[mpg]*20 + _b[c.mpg#c.mpg]*400 + ///
>              _b[4.rep78]
> di normalden(`xb')* (_b[mpg] + 2*20*_b[c.mpg#c.mpg])
> 
> //============================= do you really want marginal effects?
> 
> // create predicted probabilities by repair status
> predict pr
> separate pr, by(rep78)
> 
> // create the "regression lines" implied by marginal effects
> scalar `xb' = _b[_cons] + _b[mpg]*20 + _b[c.mpg#c.mpg]*400 
> 
> local b3 = normalden(`xb') * ///
>           (_b[mpg] + 2*20*_b[c.mpg#c.mpg])
> local c3 = normal(`xb')-20*`b3'
> sum mpg if rep78 == 3, meanonly
> local l3 = r(min)
> local u3 = r(max)
> 
> local b4 = normalden(`xb' + _b[4.rep78])*  ///
>           (_b[mpg] + 2*20*_b[c.mpg#c.mpg])
> local c4 = normal(`xb' + _b[4.rep78])-20*`b4'
> sum mpg if rep78 == 4, meanonly
> local l4 = r(min)
> local u4 = r(max)
> 
> local b5 = normalden(`xb' + _b[5.rep78])*  ///
>           (_b[mpg] + 2*20*_b[c.mpg#c.mpg])
> local c5 = normal(`xb' + _b[5.rep78])-20*`b5'
> sum mpg if rep78 == 5, meanonly
> local l5 = r(min)
> local u5 = r(max)
> 
> 
> // display them in a graph
> twoway line pr3 mpg, sort lpattern(solid) lcolor(black) || ///
>       function y = `c3' + `b3'*x,                         ///
>       range(`l3' `u3') lpattern(solid)  lcolor(gs8)    || ///
>       line pr4 mpg, sort lpattern(dash) lcolor(black)  || ///
>       function y = `c4' + `b4'*x,                         ///
>       range(`l4' `u4') lpattern(dash) lcolor(gs8)      || ///
>       line pr5 mpg, sort lpattern(shortdash)              ///
>            lcolor(black)                               || ///
>       function y = `c5' + `b5'*x,                         ///
>       range(`l5' `u5') lpattern(shortdash) lcolor(gs8)    ///
>       ytitle(predicted probability) xline(20)             ///
>       xtitle(miles per gallon)                            ///
>       legend( cols(1) pos(4)                              /// 
>              order( - "probit predictions"                ///
>                     1 "rep78=3"                           ///
>                     3 "rep87=4"                           ///
>                     5 "rep87=5"                           ///
>                     - "marginal effects"                  ///
>                       `""predictions""'                   ///
>                     2 "rep78=3"                           ///
>                     4 "rep87=4"                           ///
>                     6 "rep87=5"  ))    
> *------------------ end example ------------------------
> (For more on examples I sent to the Statalist see: 
> http://www.maartenbuis.nl/example_faq )
> 
> 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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