With the new cfregress and cfprobit commands, fit control-function linear and probit models, which provide a flexible alternative to traditional instrumental-variables (IV) methods for models with endogenous variables. Include continuous, binary, fractional, and count endogenous variables. And easily test for endogeneity.

Control-function models allow researchers to estimate causal relationships even when some explanatory variables are endogenous. First-stage models are fit for all endogenous variables, and the residuals are then used to form control functions that are included in the main outcome model to account for endogeneity.

Researchers often use control-function methods when traditional IV methods cannot accommodate desired model features such as flexible handling of interacted endogenous variables or modeling endogenous binary, fractional, and count variables. The cfregress and cfprobit commands fit control-function models, allow for great flexibility in the interaction and modeling of endogenous variables, and provide standard errors that account for the inclusion of estimated control functions. After fitting the model, you can easily perform tests of endogeneity.

Let's see it work

Say we are interested in modeling the average rental rate (rent) in each US state as a function of average housing values (hsngval) and the proportion of the population living in urban areas (pcturban). Because average housing values are likely to be endogenous, we include a measure of median family income, faminc, and indicator variables for the region in which each state is located, i.region, as instruments for hsngval. For convenience, we rescale hsngval and faminc to be on a scale similar to rent.

With cfregress, we could reproduce the estimates of a two-stage least-squares (2SLS) IV regression. Whereas 2SLS replaces the endogenous variable in the main regression with fitted values from a first-stage regression, control-function regression keeps the endogenous variable and includes the first-stage residual as a regressor called a control function.

. webuse hsng
(1980 Census housing data)

. replace hsngval = hsngval/1000
variable hsngval was long now double
(50 real changes made)

. replace faminc = faminc/1000
variable faminc was long now double
(50 real changes made)

. cfregress rent pcturban (hsngval = faminc i.region)

Control-function linear regression                     Number of obs =      50
                                                       Wald chi2(2)  =   90.76
                                                       Prob > chi2   =  0.0000
                                                       R-squared     =  0.5989
                                                       Root MSE      = 22.1656
Endogenous variable model:
    Linear: hsngval



        rent 
 
 Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
 
 
 
rent         
 
                                                                
     hsngval 
 
   2.239833   .3284392     6.82   0.000     1.596104    2.883562
    pcturban 
 
    .081516   .2987652     0.27   0.785     -.504053     .667085
       _cons 
 
   120.7065   15.22839     7.93   0.000     90.85942    150.5536

e.rent       
 
                                                                
  cf(hsngval)
 
  -1.588908   .4333422    -3.67   0.000    -2.438243   -.7395726

The control function is shown as cf(hsngval) because it is the control function generated from the first-stage model of the endogenous variable hsngval. Control functions enter the main equation, but they are listed under e.rent because we consider them part of the model for the error term.

We might suspect that the endogeneity in the model depends not just on the control function but on its interaction with faminc. We include this interaction using the interact() option.

. cfregress rent pcturban (hsngval = faminc i.region, interact(faminc))


Control-function linear regression                     Number of obs =      50
                                                       Wald chi2(2)  =   95.16
                                                       Prob > chi2   =  0.0000
                                                       R-squared     =  0.5945
                                                       Root MSE      = 22.2851
Endogenous variable model:
    Linear: hsngval



        rent 
 
 Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
 
 
 
rent         
 
                                                                
     hsngval 
 
   2.155381   .3437284     6.27   0.000     1.481686    2.829076
    pcturban 
 
   .4794597   .2362242     2.03   0.042     .0164688    .9424506
       _cons 
 
   98.15909   13.86958     7.08   0.000     70.97521     125.343

e.rent       
 
                                                                
  cf(hsngval)
 
   10.66765   3.619442     2.95   0.003     3.573673    17.76163
             
 
                                                                
  cf(hsngval)
 
                                                                
      faminc 
 
  -.5610651   .1743049    -3.22   0.001    -.9026965   -.2194338

Instruments for hsngval: faminc 2.region 3.region 4.region

The control-function interaction is shown as cf(hsngval)#faminc.

Relative to the first model, there are several changes. We have evidence that the coefficient on pcturban is different from 0, while the coefficient on hsngval is slightly smaller. We also have evidence that the iteraction has a coefficient different from 0, and so should be included in the model.

A joint test of cf(hsngval) and cf(hsngval)#faminc amounts to a test of endogeneity, and we can perform this test by using the postestimation command estat endogenous.

. estat endogenous

Tests of endogeneity
H0: Variables are exogenous

 ( 1)  [e.rent]cf(hsngval) = 0
 ( 2)  [e.rent]cf(hsngval)#c.faminc = 0

           chi2(  2) =   15.30
         Prob > chi2 =    0.0005

This gives strong evidence for endogeneity. Here we used conventional standard errors, but estat endogenous will conduct an appropriate test after estimation even with robust, cluster–robust, or heteroskedasticity- and autocorrelation-consistent standard errors.

Stata's control-function regression commands allow users to specify nonlinear first-stage models for endogenous binary, fractional, or count variables.

For example, we can estimate the effect of having health insurance (ins) on the log of prescription drug expenditure (lndrug) using marital status (married) and employment status (work) as instruments.

. webuse drugexp
(Presciption drug expenditures)

. cfregress lndrug age lninc (ins = married work, probit interact(ins)),
     mainonly(chron) vce(robust)


Control-function linear regression                     Number of obs =   6,000
                                                       Wald chi2(4)  = 1973.78
                                                       Prob > chi2   =  0.0000
                                                       R-squared     =  0.2432
                                                       Root MSE      =  1.2172
Endogenous variable model:
    Probit: 1.ins



              
 
               Robust                                           
       lndrug 
 
 Coefficient  std. err.      z    P>|z|     [95% conf. interval]
 
 
 
lndrug        
 
                                                                
        1.ins 
 
  -.8598836   .3483648    -2.47   0.014    -1.542666   -.1771011
        chron 
 
   .4671725   .0319731    14.61   0.000     .4045064    .5298387
          age 
 
   .1021359     .00292    34.98   0.000     .0964128    .1078589
        lninc 
 
   .0550672   .0225036     2.45   0.014     .0109609    .0991735
        _cons 
 
   1.665539   .2527527     6.59   0.000     1.170153    2.160925

e.lndrug      
 
                                                                
      cf(ins) 
 
   .5252243    .226367     2.32   0.020     .0815532    .9688954
              
 
                                                                
  cf(ins)#ins 
 
   .2702095   .2585099     1.05   0.296    -.2364605    .7768796

Instruments for 1.ins: married work

Here we used the probit option within the parentheses to specify a probit model for our first-stage regression. (Note that if we had multiple sets of parentheses, each first-stage regression could have its own model.) We have again included a control-function interaction, and we have also included an indicator for a chronic condition, chron, in the main regression but not the first stage by using the mainonly() option. We requested heteroskedasticity-robust standard errors by specifying the vce(robust) option.

This regression is equivalent to fitting an endogenous treatment-effects model (see Example 2 in [CAUSAL] etregress). What if the outcome is binary? The cfprobit command fits control-function models in the same way except that the model for the main equation is a probit model. Both cfregress and cfprobit allow users the flexibility to specify a large class of models where one or more explanatory variables are endogenous.