How do I estimate a nonlinear model using ml?

Consider the model

y = f(x) + e

where y is the outcome, f(x) is a nonlinear form of covariate x, and e is the random error. The command nl will estimate the parameters of f(x) by using least squares. Here is an example:

. sysuse auto, clear
1978 Automobile Data)

. nl (rep78 = {b0}*(1-exp(-{b1}*headroom))), initial(b0 1 b1 0.1) nolog
(obs = 69)
 
      Source |       SS       df       MS
-------------+------------------------------         Number of obs =        69
       Model |  800.673786     2  400.336893         R-squared     =    0.9235
    Residual |  66.3262137    67  .989943487         Adj R-squared =    0.9212
-------------+------------------------------         Root MSE      =   .994959
       Total |         867    69  12.5652174         Res. dev.     =  193.0865

------------------------------------------------------------------------------
       rep78 |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         /b0 |   3.435331   .1604479    21.41   0.000     3.115075    3.755586
         /b1 |   1.947203    1.76742     1.10   0.275    -1.580583    5.474988
------------------------------------------------------------------------------

We can write an nl program to fit the same model. We may want to do this if we need to be able to fit the model with different variables.

program nlnexpgr, rclass

	version 10
	syntax varlist(min=2 max=2) if
	local lhs : word 1 of `varlist'
	local rhs : word 2 of `varlist'

	return local eq "`lhs'={b0=1}*(1-exp(-{b1=0.1}*`rhs'))"

end

This program will give the same results as we obtained above:

. sysuse auto, clear
(1978 Automobile Data)

. nl nexpgr : rep78 headroom, nolog
(obs = 69)

      Source |       SS       df       MS
-------------+------------------------------         Number of obs =        69
       Model |  800.673786     2  400.336893         R-squared     =    0.9235
    Residual |  66.3262137    67  .989943487         Adj R-squared =    0.9212
-------------+------------------------------         Root MSE      =   .994959
       Total |         867    69  12.5652174         Res. dev.     =  193.0865

------------------------------------------------------------------------------
       rep78 |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         /b0 |   3.435331   .1604479    21.41   0.000     3.115075    3.755586
         /b1 |   1.947203    1.76742     1.10   0.275    -1.580583    5.474988
------------------------------------------------------------------------------

Now let’s consider how we would fit the same model with ml, which uses maximum likelihood. We can change the equation into

  e = y − f(x)

If we assume that the errors are distributed as N(0,sigma), the density function is as follows:

Then, the log-likelihood function will be

  lnL = -0.5*ln(2*_pi) - ln(sigma) - 0.5*e^2/sigma^2
          = -0.5*ln(2*_pi) - ln(sigma) - 0.5*(y-f(x))^2/sigma^2

In our example, we have

  f(x) = b0*(1-exp(-b1*x))

Considering the formula, we may treat b0 as a parameter and b1*x as the linear combination xb, which has no constant term. We also need to estimate another parameter, “sigma”, the standard deviation of the normal distribution.

program mlnexpgr
	version 10
	args lnf b1x b0 sigma
	tempvar res
	quietly gen double `res' = $ML_y1 - `b0'*(1-exp(-`b1x'))
	quietly replace `lnf' = -0.5*ln(2*_pi)-ln(`sigma')-0.5*`res'^2/`sigma'^2
end

. ml model lf mlnexpgr (b1: rep78 = headroom, nocons) (b0:) (sigma:)
  
. ml max
(output omitted)

                                                  Number of obs   =         69
                                                  Wald chi2(1)    =       2.85
Log likelihood = -96.543274                       Prob > chi2     =     0.0912

------------------------------------------------------------------------------
       rep78 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
b1           |
    headroom |   1.947201   1.152959     1.69   0.091    -.3125565    4.206959
-------------+----------------------------------------------------------------
b0           |
       _cons |   3.435331    .137529    24.98   0.000     3.165779    3.704883
-------------+----------------------------------------------------------------
sigma        |
       _cons |   .9804333     .08346    11.75   0.000     .8168547    1.144012
------------------------------------------------------------------------------

The estimates we obtained from ml are close to those from the nl program. The estimated standard errors are a little different, but we know that they are from different algorithms.

We could extend the ml program to estimate robust variances with the vce(robust) option or with the vce(cluster clustvar) option.

. ml model lf mlnexpgr (b1:rep78=headroom,nocons) (b0:) (sigma:), vce(robust)

. ml max
 (output omitted)

                                                  Number of obs   =         69
                                                  Wald chi2(1)    =       3.98
Log pseudolikelihood = -96.543274                 Prob > chi2     =     0.0462

------------------------------------------------------------------------------
             |               Robust
       rep78 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
b1           |
    headroom |   1.947201   .9765839     1.99   0.046     .0331318     3.86127
-------------+----------------------------------------------------------------
b0           |
       _cons |   3.435331   .1145499    29.99   0.000     3.210817    3.659844
-------------+----------------------------------------------------------------
sigma        |
       _cons |   .9804333   .0743649    13.18   0.000     .8346808    1.126186
------------------------------------------------------------------------------

The robust estimation gives the same estimated coefficients, but it gives adjusted standard errors for the three parameters in this model. We may also apply this program for clustered data:

 . ml model lf mlnexpgr (b1:rep78=headroom,nocons) (b0:) (sigma:), vce(cluster turn)