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## Maximum likelihood without programming

### Highlights

• Specify log-likelihood function interactively
• Optionally specify first derivatives
• Robust SEs to relax distributional assumptions
• Cluster–robust SEs for correlated data
• Linear and nonlinear postestimation hypotheses tests

### Show me

Maximization of user-specified likelihood functions has long been a hallmark of Stata, but you have had to write a program to calculate the log-likelihood function. Now it is even easier. The only requirements are that you be able to write the log likelihood for individual observations and that the log likelihood for the entire sample be the sum of the individual values.

Stata can fit probit models, but let’s write our own.

The log-likelihood function for probit is

                LL(y) = ln(normal(x'b))   if  y==1
= ln(normal(-xb))       y==0


To fit a model of outcome on age and weight, we type

. mlexp (cond(outcome==1, ln(normal({xb:age weight} + {b0})), ln(normal(-1*({xb:} + {b0}))) )) initial: log likelihood = -51.292891 final: log likelihood = -51.292891 rescale: log likelihood = -51.292891 Iteration 0: log likelihood = -51.292891 Iteration 1: log likelihood = -25.436922 Iteration 2: log likelihood = -25.303978 Iteration 3: log likelihood = -25.303693 Iteration 4: log likelihood = -25.303693 Maximum likelihood estimation Log likelihood = -25.303693 Number of obs = 74
 Coef. Std. Err. z P>|z| [95% Conf. Interval] /xb_age .2279405 .0887648 2.57 0.010 .0539648 .4019163 /xb_weight .01195 .0094324 1.27 0.205 -.0065372 .0304373 /b0 -9.765827 2.656796 -3.68 0.000 -14.97305 -4.558604

Those results are exactly the same as those produced by Stata’s probit.

### Show me more

See the manual entry.

It’s hard to beat the simplicity of mlexp, especially for educational purposes.

mlexp is an easy-to-use interface into Stata’s more advanced maximum-likelihood programming tool that can handle far more complex problems; see the documentation for ml.

ml itself is an easier-to-use interface into Stata’s most advanced optimization programs found in Stata’s matrix language; see the documentation for mopmitize(), optimize(), solvenl(), and deriv().

If you want to fit models via the generalized method of moments (GMM), see the documentation for Stata’s gmm.

See New in Stata 17 to learn about what was added in Stata 17.