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How do I impose the restriction that rho is zero using the heckman command with full ml?

Short answer:

In this particular model, heckman does not estimate the parameter rho directly, but estimates a transformation:

          atanh_rho = 1/2*ln[(1+rho)]/(1−rho)]

It is estimated in a constant-only equation athrho. Thus we need to constrain the constant term of equation athrho to be 0 (rho=0 implies atanh_rho=0).

. constraint define 1 _b[/athrho]=0
        
. heckman ..., select(...) constraint(1)

Long answer:

Now let’s extend the answer to more general cases: how to define constraints on parameters of a model in Stata. The syntax is generally

       constraint define # [ exp=exp | coefficientlist ]

When we want to fix a parameter at a certain value, it becomes

       constraint define # [equation_name]coefficient_name = #

The equation_name may not be necessary for a single-equation model such as OLS. It is easy to apply this rule to the coefficient of a covariate.

       constraint define # [equation_name]covariate_name = #

One can find the equation_name easily from the output. Often it is just the name of the dependent variable.

But how about other parameters in the model, such as rho in heckman? This needs some understanding on how Stata estimates those parameters. In ML estimation, Stata always defines them in separate equations, i.e., one equation for one parameter. Those equations are constant-only, and the estimated constants will be the estimated parameters. Often, some transformations are needed to fit the parameter spaces. For instance, the standard deviation sigma of a normal distribution should be always greater than 0, so a log-transformation will be used to allow the estimation (ln(sigma)) from −infinity to +infinity. One can check the Methods and Formulas section of the estimation command to find out if any transformation is applied.

Now let’s go back to the question in heckman. As described in the short answer, heckman does use a transformation to estimate rho.

       atanh_rho = 1/2*ln[(1+rho)]/(1−rho)]	(p.976 of [R] heckman)

Using the example in the manual

. use http://www.stata-press.com/data/r14/womenwk, clear

. heckman wage educ age, select(married children educ age)

Iteration 0:  Log likelihood = -5178.7009
Iteration 1:  Log likelihood = -5178.3049
Iteration 2:  Log likelihood = -5178.3045

Heckman selection model                         Number of obs     =      2,000
(regression model with sample selection)              Selected    =      1,343
                                                      Nonselected =        657

                                                Wald chi2(2)      =     508.44
Log likelihood = -5178.304                      Prob > chi2       =     0.0000



        wage 
 
 Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
 
 
 
wage         
 
 
   education 
 
   .9899537   .0532565    18.59   0.000     .8855729    1.094334
         age 
 
   .2131294   .0206031    10.34   0.000     .1727481    .2535108
       _cons 
 
   .4857752   1.077037     0.45   0.652    -1.625179     2.59673
 
 
 
select       
 
 
     married 
 
   .4451721   .0673954     6.61   0.000     .3130794    .5772647
    children 
 
   .4387068   .0277828    15.79   0.000     .3842534    .4931601
   education 
 
   .0557318   .0107349     5.19   0.000     .0346917    .0767718
         age 
 
   .0365098   .0041533     8.79   0.000     .0283694    .0446502
       _cons 
 
  -2.491015   .1893402   -13.16   0.000    -2.862115   -2.119915
 
 
 
     /athrho 
 
   .8742086   .1014225     8.62   0.000     .6754241    1.072993
    /lnsigma 
 
   1.792559    .027598    64.95   0.000     1.738468     1.84665
 
 
 
         rho 
 
   .7035061   .0512264                      .5885365    .7905862
       sigma 
 
   6.004797   .1657202                       5.68862    6.338548
      lambda 
 
   4.224412   .3992265                      3.441942    5.006881

LR test of indep. eqns. (rho = 0): chi2(1) = 61.20        Prob > chi2 = 0.0000

Here are four equations: wage and select equation for those covariates, athrho for atanh_rho, and lnsigma for ln(sigma). The constant-only equation for a parameter is often displayed as /equation_name in the output table. The last row of the table displays the estimated value for rho, sigma, and lambda, which are transformed back from the estimation results.

Now we can impose the constraint on rho, which is actually on the constant term of equation athrho.

 use http://www.stata-press.com/data/r14/womenwk, clear

. local athrho=1/2*ln((1+0)/(1-0))
    
. constraint define 1 _b[/athrho]=0
    
. heckman wage educ age, select(married children educ age) constraint(1)
 
Iteration 0:   log likelihood = -5283.1781  
Iteration 1:   log likelihood = -5230.2173  
Iteration 2:   log likelihood = -5208.9358  
Iteration 3:   log likelihood = -5208.9038  
Iteration 4:   log likelihood = -5208.9038  

Heckman selection model                         Number of obs      =      2000
(regression model with sample selection)        Censored obs       =       657
                                                Uncensored obs     =      1343

                                                Wald chi2(2)       =    456.00
Log likelihood = -5208.904                      Prob > chi2        =    0.0000

 ( 1)  [athrho]_cons = 0


        wage 
 
 Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
 
 
 
wage         
 
 
   education 
 
   .8965829   .0497504    18.02   0.000     .7990738     .994092
         age 
 
   .1465739   .0186926     7.84   0.000     .1099371    .1832106
       _cons 
 
   6.084875   .8886241     6.85   0.000     4.343204    7.826546
 
 
 
select       
 
 
     married 
 
   .4308575    .074208     5.81   0.000     .2854125    .5763025
    children 
 
   .4473249   .0287417    15.56   0.000     .3909922    .5036576
   education 
 
   .0583645   .0109742     5.32   0.000     .0368555    .0798735
         age 
 
   .0347211   .0042293     8.21   0.000     .0264318    .0430105
       _cons 
 
  -2.467365   .1925635   -12.81   0.000    -2.844782   -2.089948
 
 
 
     /athrho 
 
          0  (constrained)
    /lnsigma 
 
   1.694868   .0192951    87.84   0.000      1.65705    1.732686
 
 
 
         rho 
 
          0  (omitted)
       sigma 
 
   5.445927   .1050797                      5.243821    5.655824
      lambda 
 
          0  (omitted)

Wald test of indep. eqns. (rho = 0): chi2(1) = .          Prob > chi2 =      .

The output shows that the constraint is applied correctly.