Some Inquiry on
Microeconometric Issues
Wen-Jen Tsay, Peng-Hsuan Ke
March 8, 2010
Agenda

Case 1 : Bivariate Normal Cumulative Distribution
Function (BNCDF)

Case 2 : Random Effects Probit Model

Bivariate Normal Cumulative Distribution Function
(BNCDF)
BNCDF

APPLICATIONS:

1. Pearson (1901, 1903) employs bivariate normal distribution for
studying biometric data. Kotz et al. (2000) provides an overview about
the references concerning the historical development of bivariate
normal distribution. Genz (2004) reviews new methods for the
numerical computation of bivariate normal probabilities.

2. Bivariate normal distribution also has been extensively used in
social science. Greene (2000) claims that achieving a fast and accurate
method of computing the BNCDF is a longstanding challenge in
applied econometrics.
BNCDF

APPLICATIONS:

3. Boyes et al. (1989) apply this sample selection probit model to
the problem of bank credit scoring.

4. The BNCDF is necessary for the binary endogenous regressor
probit (BERP) model which is popularly used for computing the
average treatment effect between the controlled and the treated
groups. Evans and Schwab (1995) apply the BERP model to
evaluate the causal effect of attending Catholic school on finishing
high school and starting college.
BNCDF
 MAIN
1.
2.
3.
RESULTS:
BNCDF

MAIN RESULTS:

4. Alternative quadrature methods have been proposed to
evaluate Φ2 (.) based on a single integral presentation of
Φ2 (.) :
The above single integral representation leads to the
possibility of Gauss-Legendre numerical integration.
BNCDF

MAIN RESULTS:

5. We now address the main idea of this paper. First of all, the BNCDF
in (4) is rewritten as:
5.1.
BNCDF
5.2.
BNCDF
5.3.

5.3.1. The cornerstone of this paper is to recognize that the value of erf
(x) in (8) is well approximated by a function,

5.3.2. This strategy has never been mentioned in the reviews of Kotz et
al. (2000) and Genz (2004). It is first employed by Tsay et al. (2009) in
deriving the cdf of a composite random variable which is important
for the stochastic frontier analysis pioneered in Aigner et al. (1977).
BNCDF
BNCDF

6. There are several advantages of using Fapp in approximating Φ2 .

6.1 The implementation of Fapp does not depend on the range
of ρ as long as it is not equal to ± 1, the boundary points.

6.2 The computation of Fapp involves the function erf (.)
which can be directly calculated with a standard statistic
package.

6.3
can be generalized to evaluate the
trivariate and higher-dimensional normal integrals.
BNCDF

NUMERICAL STUDIES :

1. The quality of Fapp in approximating Φ2
BNCDF
BNCDF

NUMERICAL STUDIES :

2.1. To further demonstrate the ability of Fapp in approximating Φ2 , by
adapting the design in Table 46.1 of Kotz et al. (2000, p.276), we compare the
values generated from Fapp with those from Φ2 in Table 1.
BNCDF
BNCDF

2.1. Table 2 shows that the worst difference between Fapp and the value generated from the exact method
contained in Table 1 of Albers and Kallenberg (1994).
BNCDF
BNCDF

SIMULATIONS:

1. Because both the outcome measures and the treatment variable are
dichotomous under the set-up of Evans and Schwab (1995), this
study falls into the framework of the probit model with binary
endogenous regressor:
BNCDF

SIMULATIONS:

2. Wooldridge (2002, p.478) documents that MLE is nontrivial, we
show that the parameters can be easily and accurately estimated with the
approximated likelihood function suggested in this paper. For ease of
exposition, we represent all the exogenous variables contained in x1 and
z2 as w. With the independently and identically distributed (iid)
observations from i = 1,... , n, the log-likelihood function for the BERP
model consists of four parts:
BNCDF
2.1.
2.2.
BNCDF
2.3.
BNCDF
2.4.
2.5.
2.6.
BNCDF

SIMULATIONS:

3. The design of Monte Carlo experiment follows closely with that in Freedman and
Sekhon (2008). We consider a set of experiments as follows:
BNCDF

SIMULATIONS:

4. In order to create a realistic simulation scheme, the inverse function
of the preceding transformation function calculated at the true
parameter value pIus an extra (6 × 1) random vector generated from
N(0, 1/3) is used as the initial values for the MLE procedure, i.e., the
initial value of
is:
BNCDF

SIMULATIONS:

5. The optimization algorithm used to implement the MLE is the quasiNewton algorithm of Broyden, Fletcher, Goldfarb, and Shanno (BFGS)
contained in the GAUSS MAXLIK library. Table 3 and Table 4 clearly
show that our method is computationally efficient in that it handles the
simulations with a large sample size of 8000 and 500 replications easily.
BNCDF
BNCDF
BNCDF

CONCLUSIONS:




1. Our method is better than the approaches considered in Cox and Wermuth
(1991) and Lin (1995) under various configurations considered in this paper’s
Table 1, because the worst error of our method is found to four decimal places
only.
2. The worst difference between the value generated from the exact method
contained in Table I of Albers and Kallenberg (1994) and that generated from
our method is not greater than 0.0001 for all configurations considered in that
table.
3. We also apply the proposed method to approximate the likelihood function
of the probit model with binary endogenous regressor in order to demonstrate
that the BERP model can be easily and accurately estimated with the
approximated likelihood function suggested in this paper .
4. The simulations clearly reveal that the bias and MSE of the maximum
likelihood estimator based on our method are very much similar to those
obtained from using the exact method in the GAUSS package.
 Random
Effects Probit Model
Random Effects Probit Model
 APPLICATIONS:

1. The first application of random effects probit model is that of
Heckman and Willis (1976). It is well known that the likelihood
function of the random effects probit model does not have a closedform even when the time span ( T ) of the data is only 2.

2. Dynamic probit models with an unobserved effect also require the
implementation of the Gaussian quadrature procedure when we intend
to integrate out the unobserved effect as discussed in Wooldridge
(2005).
Random Effects Probit Model
 APPLICATIONS:

3. The simulation results in Guilkey and Murphy (1993) where the
probit estimator is found to be comparable to the random effects probit
estimator (MLE), even though the data-generating process (DGP) is
truly the random effects probit model.

4. Due to the computational demanding nature of the MLE, Guilkey
and Murphy (1993, p. 316) recommend that if only two points (T = 2)
are available, then one may as well use the probit estimator.

5. The preceding results induce Rabe-Hesketh et al. (2005) to suggest
an adaptive quadrature for numerical integral and their method is
promising, even though their method is computational demanding as
well.
Random Effects Probit Model

1.
ESTIMATION USING AN ANALYTIC FORMULA:
Random Effects Probit Model
ESTIMATION USING AN ANALYTIC FORMULA:

2.
3.
Random Effects Probit Model

ESTIMATION USING AN ANALYTIC FORMULA:
4.
5.
Random Effects Probit Model

6.
ESTIMATION USING AN ANALYTIC FORMULA:
Random Effects Probit Model

SIMULATION – MONTE CARLO EXPERIMENT:

1. The design of the Monte Carlo experiment follows closely with that in Guilkey and
Murphy (1993) as follows:
The true parameter values considered for the simulations are:
Random Effects Probit Model

SIMULATION – MONTE CARLO EXPERIMENT:

2. In order to create a realistic simulation scheme, the inverse function
of the preceding transformation function calculated at the true
parameter value pIus an extra (3 × 1) random vector generated from
N(0, 1/3) is used as the initial values for the MLE procedure, i.e., the
initial value of
is:

3. Similarly, the initial values of
and
for the probit estimator are
computed at the true parameter value plus an extra (2 × 1) random
vector generated from N (0, 1/3).
Random Effects Probit Model

SIMULATION – MONTE CARLO EXPERIMENT:

4. Tables 1, 2, and 3 clearly show that our method is computationally
efficient in that it easily handles the simulations with a large sample
size of 1600 and 500 replications. In fact, there is only one failure of
normal convergence when conducting MLE for the 13,500 replications
conducted in the first 3 tables.
Random Effects Probit Model

SIMULATION – MONTE CARLO EXPERIMENT:

5. To clearly demonstrate the advantage of MLE over the probit
estimator in estimating , we compute the RMSE ratio between the
MLE and probit estimators in Table 4 as RMSE (probit) / RMSE
(MLE). If RMSE (probit) / RMSE (MLE) > 1, then the MLE is more
efficient than the probit estimator.
Random Effects Probit Model
Random Effects Probit Model

SIMULATION – MONTE CARLO EXPERIMENT:

6. Table 5 clearly shows that the MLE of the random effects probit
model can be easily and accurately carried out with the analytic
approximation Lapp even when the sample size is as large as 12,800
and the value of ρ is 0.9. It is also clear from Tables 5 and 6 that the
above observation that “the larger the value of ρ is, the larger the
RMSE ratio between the probit and MLE estimator is” remains intact
from the new experiments.
Random Effects Probit Model
Random Effects Probit Model

CONCLUSIONS:

1. We propose a simple approximation method for the random
effects probit model based on the error function with T = 2. This
approach does not involve a numerical integral as suggested by
Butler and Moffitt (1982) and can be easily implemented with
standard statistics packages.

2. The simulations reveal that the random effects probit model
can be accurately estimated with the approximated likelihood
function, even though the sample size is over 10,000 and the
correlation coefficient within each unit is close to its upper limit.
Furthermore, our methodology can be extended to the cases
where T ≥ 3.
Random Effects Probit Model

CONCLUSIONS:

3. One useful message from our investigation is that,
against the recommendation made in Guilkey and Murphy
(1993) that one may as well use the probit estimator if only
two points (T = 2) are available, the MLE is preferred to
the probit estimator as long as the possible bias from the
numerical integral can be avoided. The rationale is that the
MLE utilizes the error component structure inherent in the
random effects probit model, while the probit method does
not take this information into account.
Q
&A