Search
   >> Home >> Resources & support >> FAQs >> Define constraints for parameters

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

Title   Define constraints for parameters
Author Weihua Guan, StataCorp
Date November 2001; minor revisions July 2013

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/r13/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/r13/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.

The Stata Blog: Not Elsewhere Classified Find us on Facebook Follow us on Twitter LinkedIn Google+ Watch us on YouTube