Probit model with parametric and non-parametric components

[Alexandros Polycarpou]

Hi, I am trying to estimate a probit model with the following form:

prob( y = 1 )= F( Xb + V(z) ) where

F( . ) is the cumulative normal distribution function, Xb is a parametric part and V(z) is a non-parametric part; V(.) is an unknown function of the variable z .

I search for a command or procedure in Stata which can estimate such models but with no luck. Can you help me?

Thanks for your help
Alexandros

[Maarten Buis]

fp tends to work failrly well. The major advantage is that it works for a large number of models, including probit.

// open example data
sysuse nlsw88, clear

// prepare the data
gen byte occat = cond(occupation < 3, 1,                    ///
                 cond(inlist(occupation,5, 6, 8, 9), 2, 3)) ///
                 if !missing(occupation)
label variable occat "occupational category"
label define occat 1 "white collar" ///
                   2 "skilled"      ///
                   3 "unskilled"
label value occat occat

// estimate the model
fp <grade>, scale : probit union <grade> i.south i.race i.occat

// graph the result
replace south = 0
replace race = 1
replace occat = 2
predict pr
twoway line pr grade, sort

(For more on examples I sent to the Statalist see:
http://www.maartenbuis.nl/example_faq )

[Alexandros Polycarpou]

Dear Maarten thank you for the reply and help.

I was looking for a semi-parametric technique for estimating the probit model but I guess fractional polynomial regression is a good alternative.

V(.) is an unknown function of the vector z (and not of a variable as I miswrite in the first post) so I guess I should use the multi-variable fractional polynomial models “mfp” command.

Any alternative semi-parametric commands or procedures in Stata which can estimate the particular pobit model are more than welcome.

Thanks a lot for your help
Alexandros

[Stephen Jenkins]

search semi-nonparametric, all will lead you to some Stata implementations of models that have been used for binary (or ordered categorical) models. These models are typically of the form

 Prob(y = 1) = G(Xb + e)

where G is a nonparametric CDF (i.e. not normal). Do you have full literature references for the sort of model specification that you refer to?

See also the wonderful review by Vincent Verradi of related models, at http://www.stata.com/meeting/uk13/ab...13_verardi.pdf

[Alexandros Polycarpou]

Dear Prof. Jenkins thank you for your reply and suggestions.

I have already review the paper of Vincent Verradi on “Semiparametric regressions in Stata” but unfortunately it is no help for my model. The “snp” and “sml” Stata procedures which give semiparametric binary choice estimates solve the nonparametric CDF problem.

My work is based on the work of Rodríguez‐Póo, Sperlich and Fernández. http://onlinelibrary.wiley.com/doi/1...e.796/abstract

The hours equation is given by the semi-parametric equation

h= Xb + V(w) + u

where Xb is the parametric part and V(w) is a non-parametric part; V(.) is an unknown function of the wage rate w; and u is the error term.

The wage equation is given by

h= Zγ + v

where Z is a vector of characteristics and γ the vector of parameters and v is the error term.

Due to the known problems of missing wages for non-working and endogeneity of wages I use a heckman-correction model to estimate the wage rates of non-working and to also correct for endogeneity.

So my problem is that the participation equation is of the following form

prob( h>0 )= F( Xb + V(z) )

i.e. F( . ) is the cumulative normal distribution function and it has a parametric (Xb) and a non-parametric part (V(z)) and consequently I have to use semi-parametric estimation techniques in the estimation of the probit model.

Any suggestion on how to estimate probit model of such form in Stata are more than welcome.

Many thanks

Alexandros