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## Maximum likelihood without programming

### 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


### 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

To fit a model of outcome foreign on mpg and price, we type

. sysuse auto
(1978 automobile data)

. mlexp (cond(foreign==1, ln(normal({xb:mpg price} + {b0})),
ln(normal(-1*({xb:} + {b0})))))

initial:       log likelihood = -51.292891
alternative:   log likelihood = -51.724316
rescale:       log likelihood = -46.186316
rescale eq:    log likelihood = -44.952041
Iteration 0:   log likelihood = -44.952041
Iteration 1:   log likelihood = -36.332989
Iteration 2:   log likelihood = -36.266131
Iteration 3:   log likelihood = -36.266068
Iteration 4:   log likelihood = -36.266068

Maximum likelihood estimation

Log likelihood = -36.266068                                 Number of obs = 74

Coefficient  Std. err.      z    P>|z|     [95% conf. interval]

xb
mpg     .1404876   .0373599     3.76   0.000     .0672635    .2137117
price     .0001571   .0000641     2.45   0.014     .0000315    .0002827

/b0    -4.592058   1.115921    -4.12   0.000    -6.779223   -2.404893



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

### Show me more

See the manual entry. Read In the spotlight: mlexp

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 moptimize(), optimize(), solvenl(), and deriv().

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