## Faster estimation of fixed-effect probit

Popularity of binary choice models makes their applications more and more sophisticated, reaching beyond the simple low-dimension applications. The implications of having plenty of independent variables may be severe, making the likelihood function highly irregular or even impossible to be solved. Even if the local maximum can be found, it is likely to be unstable or it may take a lot of time and computing power, especially if the independent variables are simple dummies.

Stataprobitfunction tries to address some of the above these issues. For instance, the algorithm will check which observations do not bring extra information to the model and leave them out from the estimation. However, this step can be taking too long for some of the Big Data problems. A quick solution is to tell Stata in advance which observations to exclude when running the probit. Here is a simple plug and play code snippet. Firstly, I define a generic setup with one dependent variable

Yvar, some independent variables given by

Xvars, and a set of fixed effects spanned by variable

strata.

/* general setup the variables must be specified in global macros as Yvar - dependent variable (binary) Xvars - independent vars strata - factor variable (like country-sector-year) */ global Yvar VARNAME global Xvars VARNAME(S) global strata VARNAMESecondly, I run the standard

probitestimation and allow Stata to find and treat fixed effects. I will later compare the timing of that procedure to the one proposed below.

*standard estimation timer on 1 xi: probit \$Yvar \$Xvar i.\$strata timer off 1The basic strategy is to exclude the observations which are not used in the estimation before running the

probitcommand. It includes overall three steps, which adjust the sample selection variable

modelSelect. It will be equal to 0 if the observations are to be excluded and 1 otherwise. Firstly, remove the observations which have either missing

Yvaror

Xvars. Secondly, exclude the stratas for which there is no variability in the dependent variable. For this I check if the standard deviation of

Yvaris 0. Lastly, create a vector of adjusted fixed effects for there is non-zero variability in

Yvar. The number of adjusted fixed effects is smaller or equal to their original number. This is where the efficiency gains come from.

*efficient estimation timer on 2 *missing Yvar gen modelSelect = !missing(\$Yvar) *missing Xvar foreach var of global \$Xvars { replace modelSelect = 0 if missing(`var`) } *variability bys \$strata: egen sd_y = sd(\$Yvar) if modelSelect == 1 replace modelSelect = 0 if sd_y == 0 drop sd_y *exclude redundant factors levelsof \$strata if modelSelect==1, local(slevs) egen match = anymatch(\$strata), values(1 `slevs`) gen sub_\$strata = match * \$strata replace sub_\$strata = . if sub_\$strata == 0 drop match xi I.sub_\$strata, prefix(_E2) probit \$Yvar \$Xvars _E* if modelSelect == 1 timer off 2As the last step I compare the execution times of both methods.

timer list 1 2

As a quick demonstration I take a sample of European firms making investment decision, which is a dependent dummy variable. There is a total of ten independent variables and some 400 strata variables, comprising 407k observations. The overall file size was around 1gb, including variables not used in the model. (Unfortunately, I cannot disclose the data points.)

This simple trick reduced the computational time from 465.88 to 58.58 seconds, so by a factor of 8. Clearly, the efficiency gains are substantial and they increase with the data size and complexity of the setup.