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From |
Maarten Buis <maartenlbuis@gmail.com> |

To |
statalist@hsphsun2.harvard.edu |

Subject |
Re: st: Marginal effects in Probit |

Date |
Tue, 7 Aug 2012 10:13:34 +0200 |

On Tue, Aug 7, 2012 at 4:45 AM, Shikha Sinha wrote: > I am running a probit model as my dependent variable is binary. For > ease of interpretation, I am estimating marginal effects at mean. > > My question: What is the relationship between the magnitudes of (a) > probit coefficient and (b) marginal effects at mean? Is (a) always > bigger than (b) or vice versa? Is it possible to say a prioro, on what > factors these magnitudes would depend? Yes, the marginal effect is normalden(xb)*b, where xb is the linear predictor and b is the coefficient of the variable of interest. So, the marginal effect will always be smaller than the probit coefficient as the maximum value of the density function of a standard normal distribution is a bit less than .4 (normalden(0) to be precise). In the first part of the example below I show that this is indeed the formula that Stata is using. In the second part I illustrate that the marginal effect is not constant across individuals. This is a logical consequence of fitting a non-linear model like -probit-; if the marginal effect were constant than we would be fitting a linear model. Typically, people don't want to report many different marginal effects for the same variable, so they instead report a summary measure of it, like the average marginal effect. However, this leads to an inconsistency in their argument. By only reporting (or only discussing) the average marginal effects they are in effect turning their non-linear model into a linear model. Either the non-linearity was not so important but than why estimate your linear model in such a round-about two-step manner? You'd than be much better off fitting a linear probability model, that is a much more direct an honest way of presenting results from a linear probability model. Or you think that the non-linearity in the probit model is crucial, but than you cannot suffice with one-number summaries of marginal effects because than you would undo that crucial non-linearity. I tend to prefer to choose the preferred (and, given the data, possible ) metric of the effect and choose the model such that it immediately returns that. If you don't do that --- and consequently use things like marginal effects to force your preferred metric on a model for whom it is not the natural metric --- than you'll always run into the kind of "friction" or inconsistency discussed above. If you have a binary dependent variable and you want to ensure that the predictions always remain between 0 and 1 than I tend to prefer odds ratios, and thus a -logit- model instead of a -probit-. Odds and odds ratios have an undeserved reputation of being hard to interpret, so you need to be a bit careful about how you are going to present your results. An example of one possibility of how to do that is given in this Stata tip: M.L. Buis (2012) "Stata tip 107: The baseline is now reported", The Stata Journal, 12(1), pp. 165-166. *---------------------- begin example --------------------------- sysuse nlsw88, clear gen byte high_occ = occupation < 3 if occupation < . probit union grade high_occ i.race // collect the mean tempname mgrade xb sum grade if e(sample), meanonly scalar `mgrade' = r(mean) scalar `xb' = _b[_cons] + _b[grade]*`mgrade' + _b[high_occ]*0 /// + _b[2.race]*0 + _b[3.race]*0 // see the marginal effect di normalden(`xb')*_b[grade] // compare with the results computed by stata margins, dydx(grade) atmeans at(high_occ=0 race=1) // see how variable the marginal effect // is across observations predictnl marg = normalden(xb())*_b[grade] if e(sample) twoway scatter marg grade // so typically we report the average marginal effect margins, dydx(grade) // but in that case we are better of estimating a // linear probability model instead reg union grade high_occ i.race, vce(robust) *----------------------- 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 WZB Reichpietschufer 50 10785 Berlin Germany http://www.maartenbuis.nl --------------------------------- * * For searches and help try: * http://www.stata.com/help.cgi?search * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

**References**:**st: Marginal effects in Probit***From:*Shikha Sinha <shikha.sinha414@gmail.com>

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