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 [athrho]_cons=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.556 of [R] heckman)

Using the example in the manual

 . use http://www.stata-press/data/r12/womenwk, clear

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

 (output omitted)
 ------------------------------------------------------------------------------
              |      Coef.   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/data/r12/womenwk, clear

 . local athrho=1/2*ln((1+0)/(1-0))
    
 . constraint define 1 [athrho]_cons=`athrho'
    
 . 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 |      Coef.   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  (omitted)
     /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)
 ------------------------------------------------------------------------------
 LR test of indep. eqns. (rho = 0):   chi2(1) =     0.00   Prob > chi2 = 1.0000

The output shows that the constraint is applied correctly.