We develop FIML code for estimating a Poisson Count data model with lognormal unobserved heterogeneity and an endogenous dummy variable as proposed by Terza (1998). Gauss-Hermite quadrature is used for calculating the log-likelihood and a -ml d0- method is employed. We present an example and discuss the problems found during the development of the code.

Alfonso Miranda Caso Luengo

English

Research Papers in Economics

FIML estimation of an endogenous
switching model for count data in Stata 7
Alfonso Miranda
University of Warwick
Email: Alfonso.Miranda-Caso-Lunego@warwick.ac.uk
9th Stata UK Users' Group Meeting
May 19, 2003Outline
•  Introduction
•  The Model
•  esp Syntax
•  Example
•  Final commentsIntroduction
• esp  estimates a FIML endogenous switching Poisson
count model (Terza, 1998).
• Cross-section Data.
• Alternative: Two-Stage method of moments (TSM) –not
available in Stata.
• ml d0 method is used –but a lf method might be
implemented.THE MODEL
• Conditional on a vector of variables xi, a dummy di, and
a random term ei, the count yi has a Poisson distribution
[ ][ ]
!
) ' exp( ) ' exp( exp
) | (
i
y
i i i i
i i y
x x
y f
i e b e b
e
+ + -
=
• Given a vetor of variables zi, di is characterised by
￿
￿
￿ > +
=
otherwise
v z if
d
i i
i 0
0 ' 1 a
.
• Finally, ei and vi are jointly normal with mean zero and
covariance matrix
￿ ￿
ł
￿
￿ ￿
Ł
￿
= S
1 sr
sr s
.• Conditional on ei, di and yi are independent. Hence we
can write
  { ￿
¥
¥ -
+ = = = ) | 1 Pr( ) , 1 | ( ) , ( i i i i i i i d d y df d y f e e
} i i i i i i i d f d d y f d e e e e ) ( ) | 0 Pr( ) , 0 | ( ) 1 ( = = -
• Let wi be all exogenous variables and consider a change
of variable,
.
2 s
e
h
i
i =After some algebra the joint conditional pdf of  yi and  di
becomes
{ { ￿
¥
¥ -
+ F = ) 2 ( ) 2 , , | (
1
) | , (
*
i i i i i i i i i d w d y f w d y f h s h s
p
}} , ) exp( ) 2 ( ) 1 (
2 *
i i i i d d h h sh - - F - +
where,
￿
￿
ł
￿
￿
￿
Ł
￿
-
+
F = F
2
*
1
2 '
) 2 (
r
rh a
sh
i i z
.
• Gauss-Hermite quadrature is then used to approximate
the integral. Finally, the log-likelihood is simply
  { } ￿
=
=
n
i
i i i w d y f LogL
1
| , ( ln
.SYNTAX
esp  depvar [varlist] [if exp] [in range],
edummy(varname)   [  switch(varlist)
quadrature(#)  rho0(#) sigma0(#)  mlopts]
• depvar declares the dependent variable
• varlist specifies covariates for the Poisson process
• edummy declares endogenous dummy
• switch specifies covariates for the switching variable
• quadrature indicates the number of quadrature points
• rho0  sets initial value for r -default 0.01-
• lnsigma0 sets initial value for s  -default 1.0-EXAMPLE
• Fertility data from the National Survey of Demographic
Dynamics 1997 (ENADID), Mexico.
• Dependent variable: Number of children ever born,
children ( includes children who died a few hours or
days after birth).
• Endogenous dummy: Completed Primary School, edu12
(switch variable).Descriptive Statistics
    Variable |     Obs        Mean   Std. Dev.       Min        Max
-------------+-----------------------------------------------------
         age |   19559    45.93481   4.210048         40         54
    children |   19559    4.428652   2.753055          0         18
    catholic |   19559    .8943709   .3073702          0          1
    indspker |   19559    .0940232   .2918685          0          1
     after49 |   19559    .6184365   .4857827          0          1
       edu12 |   19559    .5064676   .4999709          0          1Comparison model: Poisson count with lognormal
heterogeneity
. lnhp children catholic indspker after49 edu12,  q(16)
Getting Initial Values:
Fitting Full model:
Iteration 0:   log likelihood = -50093.352
<<omitted output>>
Iteration 8:   log likelihood = -44742.419
Poisson Regression--Lognormal Heterogeneity
(16 quadrature points)
                                                  Number of obs   =      19559
                                                  Wald chi2(4)    =    4067.23
Log likelihood = -44742.419                       Prob > chi2     =     0.0000
------------------------------------------------------------------------------
    children |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
children     |
    catholic |   -.037471   .0129797    -2.89   0.004    -.0629108   -.0120312
    indspker |   .0259195   .0134347     1.93   0.054     -.000412    .0522509
     after49 |  -.1498286   .0082753   -18.11   0.000    -.1660478   -.1336093
       edu12 |  -.4819543   .0084677   -56.92   0.000    -.4985507   -.4653579
       _cons |   1.778055   .0137364   129.44   0.000     1.751132    1.804978
-------------+----------------------------------------------------------------
       sigma |   .2965293   .0054761    54.15   0.000     .2859882     .307459
------------------------------------------------------------------------------Example: esp
. esp children catholic indspker after49 edu12, ed(edu12) s(indspker after49) q(16)
Getting Initial Values:
Fitting Full model:
Iteration 0:   log likelihood = -63211.383  (not concave)
<<omitted output>>
Iteration 10:  log likelihood = -57578.991
       
Endogenous-Switch Poisson Regression
(16 quadrature points)
                                                  Number of obs   =      19559
                                                  Wald chi2(4)    =    2285.21
Log likelihood = -57578.991                       Prob > chi2     =     0.0000
------------------------------------------------------------------------------
    children |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
children     |
    catholic |  -.0355656   .0129769    -2.74   0.006    -.0609999   -.0101313
    indspker |  -.2627088   .0187704   -14.00   0.000    -.2994981   -.2259194
     after49 |  -.0722156   .0099639    -7.25   0.000    -.0917445   -.0526868
       edu12 |  -1.202968   .0268409   -44.82   0.000    -1.255575   -1.150361
       _cons |   2.118148   .0191674   110.51   0.000     2.080581    2.155716
-------------+----------------------------------------------------------------
switch       |
    indspker |   -1.09937   .0384229   -28.61   0.000    -1.174678   -1.024063
     after49 |   .2901192   .0218779    13.26   0.000     .2472393    .3329992
       _cons |   -.076211   .0169429    -4.50   0.000    -.1094186   -.0430035
-------------+----------------------------------------------------------------
       sigma |   .4632146   .0118455    39.10   0.000     .4405701     .487023
         rho |   .9539425   .0102185    93.35   0.000      .929037    .9702418
------------------------------------------------------------------------------Final Comments
• Robust option can be implemented using ml lf method.
Speed would also improve.
• FIML estimation of sample selection models for count
data might be implemented with relatively minor
modifications.
• Zero-inflated models are as well feasible.

FIML estimation of an endogenous switching model for count data

http://fmwww.bc.edu/repec/usug2003/esp_usug.pdf

FIML estimation of an endogenous switching model for count data

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Research Papers in Economics