{smcl}
{* 25jun2009}{...}
{cmd:help inteff3}{right: ({browse "http://www.stata-journal.com/article.html?article=st0178":SJ9-4: st0178})}
{hline}

{title:Title}

{p2colset 5 16 18 2}{...}
{p2col :{hi:inteff3} {hline 2}}Compute partial effects in a probit or logit
model with a triple dummy-variable interaction term{p_end}
{p2colreset}{...}


{title:Syntax}

{p 8 15 2}
{cmd:inteff3}
{ifin}, [{cmd:average} {cmd:at(}{it:numlist}{cmd:)} {cmd:post} 
{cmd:varx1(}{it:varname}{cmd:)} {cmd:varx2(}{it:varname}{cmd:)}
{cmd:varx3(}{it:varname}{cmd:)} {cmd:varx1x2(}{it:varname}{cmd:)}
{cmd:varx1x3(}{it:varname}{cmd:)} {cmd:varx2x3(}{it:varname}{cmd:)}
{cmd:varx1x2x3(}{it:varname}{cmd:)}  
{cmd:pex1(}{it:varname}{cmd:)} {cmd:pex2(}{it:varname}{cmd:)}
{cmd:pex3(}{it:varname}{cmd:)} {cmd:pex1x2(}{it:varname}{cmd:)}
{cmd:pex1x3(}{it:varname}{cmd:)} {cmd:pex2x3(}{it:varname}{cmd:)}
{cmd:pex1x2x3(}{it:varname}{cmd:)} 
{cmd:sx1(}{it:varname}{cmd:)} {cmd:sx2(}{it:varname}{cmd:)}
{cmd:sx3(}{it:varname}{cmd:)} {cmd:sx1x2(}{it:varname}{cmd:)}
{cmd:sx1x3(}{it:varname}{cmd:)} {cmd:sx2x3(}{it:varname}{cmd:)}
{cmd:sx1x2x3(}{it:varname}{cmd:)}]

{p 4 6 2}{cmd:inteff3} is for use after a probit or logit model has been
estimated. The model must include the three single dummies, all double
interactions, and the triple interaction term.


{title:Description}

{pstd} Norton, Wang, and Ai (2004) derived the formulas for partial
effects of interaction terms of two variables in logit and probit models,
implemented in the {helpb inteff} command.

{pstd}Using the same logic, {cmd:inteff3} computes partial effects in a probit or logit model with a triple dummy-variable interaction term. 
The default is to compute the partial effects at means.

{pstd} The model may be applied when the effect of a binary regressor on a
binary dependent variable is allowed to vary over combinations of two
subgroups.  For example, one may be interested in the way a university degree
and the presence of children affect the gender difference in labor-market
participation. To this effect, one may run a probit model of labor-market
participation including dummies for female, university degree, and presence of
children, as well as their pairwise and triple interaction terms.

{pstd} A difference-in-difference-in-differences estimator with a binary
dependent variable (see, for example, Gruber and Poterba [1994]) may also seem to
be a suitable application. However, Puhani (2008) argues that the treatment
effect in nonlinear difference-in-differences models is not given by the
interaction effect of Ai and Norton (2003). In fact, computing the interaction
effect of Ai and Norton (2003) would not ensure that the
difference-in-differences treatment effect is bound between 0 and 1.

{pstd} See Cornelissen and Sonderhof (2009) for the formulas of the partial
effects of the interaction terms in the probit model with triple dummy-variable interactions. {cmd:inteff3} uses the delta method to compute the
standard errors of the partial effects and partly relies on the built-in Stata
commands {helpb nlcom} and {helpb predictnl} for this purpose.


{title:Options} 

{phang}
{cmd:average}, rather than computing the partial effect at certain values
(e.g., means), computes the effect for each observation and then averages
over all observations. The default is to compute the partial effects at means.

{phang}
{cmd:at(}{it:numlist}{cmd:)} specifies that partial effects be computed
at the values passed in this option.  The first three values are those for the
three dummy variables followed by the values for the control variables
(excluding the double and triple dummy-variable interactions). The default is to
compute the partial effects at means.

{phang}
{cmd:post} saves the vector of partial effects and the diagonal elements of
the variance-covariance matrix to {cmd:e()}. Important:  {cmd:inteff3} 
computes only the variances of the partial effects needed for simple tests on the
partial effects.  {cmd:inteff3} does not compute the covariances. The
covariances are posted in {cmd:e(V)} as {cmd:zero}, which are not the true
covariances!

{pmore}
The {cmd:post} option overwrites the saved estimates of the preceding probit
model, and {cmd:inteff3} cannot be recalled unless another probit model is
estimated.

{phang}
{cmd:varx1(}{it:varname}{cmd:)}, {cmd:varx2(}{it:varname}{cmd:)}
{cmd:varx3(}{it:varname}{cmd:)}, {cmd:varx1x2(}{it:varname}{cmd:)},
{cmd:varx1x3(}{it:varname}{cmd:)}, {cmd:varx2x3(}{it:varname}{cmd:)}, and
{cmd:varx1x2x3(}{it:varname}{cmd:)} specify the variable names
of the three single dummies (x1, x2, x3), all double interactions (x1*x2,
x1*x3, x2*x3), and the triple interaction term (x1*x2*x3). By default,
{cmd:inteff3} assumes (and checks) that they are given by the first seven
regressors in the following order: x1 x2 x3 x1*x2 x1*x3 x2*x3 x1*x2*x3.

{phang}
{cmd:pex1(}{it:varname}{cmd:)}, {cmd:pex2(}{it:varname}{cmd:)},
{cmd:pex3(}{it:varname}{cmd:)}, {cmd:pex1x2(}{it:varname}{cmd:)},
{cmd:pex1x3(}{it:varname}{cmd:)}, {cmd:pex2x3(}{it:varname}{cmd:)}, and
{cmd:pex1x2x3(}{it:varname}{cmd:)} specify names of new variables in which
the respective partial effects are to be stored. These options
must be
combined with the {cmd:average} option.

{phang}
{cmd:sx1(}{it:varname}{cmd:)}, {cmd:sx2(}{it:varname}{cmd:)},
{cmd:sx3(}{it:varname}{cmd:)}, {cmd:sx1x2(}{it:varname}{cmd:)},
{cmd:sx1x3(}{it:varname}{cmd:)}, {cmd:sx2x3(}{it:varname}{cmd:)}, and
{cmd:sx1x2x3(}{it:varname}{cmd:)} specify names of new variables in which
the standard errors of the partial effects are to be stored. These options
must be
combined with the {cmd:average} option.


{title:Examples}

{pstd}{inp:. probit y x1 x2 x3 x1x2 x1x3 x2x3 x1x2x3 z1 z2 z3 z4}{p_end}
{pstd}{inp:. inteff3 }{p_end}

{pstd}{inp:. probit y x1 x2 x3 x1x2 x1x3 x2x3 x1x2x3 z1 z2 z3 z4}{p_end}
{pstd}{inp:. inteff3, average pex1x2x3(pe) sx1x2x3(se) }{p_end}

{pstd}{inp:. probit y x1 x2 x3 x1x2 x1x3 x2x3 x1x2x3 z1 z2 z3 z4}{p_end}
{pstd}{inp:. inteff3, at(0.5 0 1 1.2 -0.3 1 0.7 1)}{p_end}


{title:References}

{phang}Ai, C., and E. C. Norton. 2003. Interaction terms in logit and probit
models. {it:Economics Letters} 80: 123-129.

{phang}Cornelissen, T., and K. Sonderhof. 2009. INTEFF3: Stata module to
compute partial effects in a probit or logit
model with a triple dummy variable interaction
term. Statistical Software Components, Boston College Department of Economics. http://ideas.repec.org/c/boc/bocode/s456903.html.

{phang} Gruber, J., and J. Poterba. 1994. Tax incentives and the decision to
purchase health insurance: 
Evidence from the self-employed. {it:Quarterly Journal of Economics} 109: 701-733.

{phang} Norton, E. C., H. Wang, and C. Ai. 2004. Computing interaction effects and standard errors in logit and probit 
models. {it:Stata Journal} 4: 154-167.

{phang}Puhani, P. A. 2008. The treatment effect, the cross difference, and the
interaction term in nonlinear "difference-in-differences" models. Discussion
Paper No. 3478, Institute for the Study of Labor (IZA). 
http://ideas.repec.org/p/iza/izadps/dp3478.html.


{title:Authors}

{pstd}Thomas Corneli{c ss}en{p_end}
{pstd}Department of Economics{p_end}
{pstd}University College London, UK{p_end}
{pstd}t.cornelissen@ucl.ac.uk{p_end}

{pstd}Katja Sonderhof{p_end}
{pstd}Leibniz Universit{c a:}t Hannover, Germany{p_end}
{pstd}sonderhof@aoek.uni-hannover.de{p_end}


{title:Also see}

{psee}
Article: {it:Stata Journal}, volume 9, number 4: {browse "http://www.stata-journal.com/article.html?article=st0178":st0178}

{psee}
Online:  {helpb probit}, {helpb logit}, {helpb inteff} (if
installed){p_end}