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Estimation of sample selection bias models by the maximum likelihood estimator and Heckman's two-step estimator
Economics Letters 0165-1765/94/$07.00
45 (1994) 33-40 0 1994 Elsevier
33 Science
B.V. All rights
reserved
Estimation of sample selection bias models by the maximum likelihood estimator and Heckman’s two-step estimator Kazumitsu Nawata”’ Department of Social and International Relations, University of Tokyo, 3-8-l Komaba, Meguro-ku, Department of Economics, University of Western Australia, Nedlands, Perth WA 6009, Australia
Tokyo 1.53, Japan
Received 28 June 1993 Final revision 3 November 1993 Accepted 8 November 1993
Abstract In this paper, methods of estimating models with sample selection biases are analyzed. Finite sample properties of the maximum likelihood estimator (MLE) and Heckman’s two-step estimator are compared using Monte Carlo experiments. JEL
classification: C34
1. Introduction Econometric models with sample selection biases (type II Tobit models) are widely used in various fields of economics. Unlike other types of econometric models, the maximum likelihood estimator (MLE) is seldom used because of its computational difficulty. Heckman (1976, 1979) proposed a simple two-step estimator. However, Heckman’s estimator sometimes performs poorly [Wales and Woodland (1980), Nelson (1984), Paarsch (1984), and Nawata (1994)]. In this paper, a method of calculating the MLE is considered, and the finite sample properties of the MLE and Heckman’s two-step estimator are compared using Monte Carlo experiments. Although Heckman’s two-step estimator performs well when there is not a high degree of multicollinearity between the hazard ratio and explanatory variables, it performs poorly when there is a high degree of multicollinearity between them.
* Correspondence to: Kazumitsu Nawata, Department of Social and International Komaba, Meguro-ku, Tokyo 153, Japan. ’ I wish to thank Michael McAIeer and Paul Miller for their helpful comments. calculating the MLE, is available through the author. SSDI
0165-1765(93)00369-Y
Relations,
University
The program
used
of Tokyo, in this paper,
3-8-l for
34
K. Nuwata I Economics
Letters 4.5 (1994) 33-40
2. Models The model
considered
in this paper
is
y,; =.x1$ + ui ) Yz,=W:~+U,,
(2.1)
d,=l(y,,>O),
i-l,2
)...)
N,
where l(q) is an indicator function such that l(.) = 1 if. is true and 0 otherwise. yzi is not observed and only the sign of y,; (i.e. d,) is observed. y,, is observed when d, = 1. (u,, u,) are jointly normal with mean zero, variances (a:, l), and covariance ui2. (Since the sign of yzi does not change if we multiply it by a positive constant value, it can be assumed that the variance of ui is 1 without loss of generality.)
3. Heckman’s
two-step
The conditional
expected
value
1di = 1) = x;p
E(y,, G)
estimator
= $(z)/@(z)
of y,, given
+ ai2A(w~:a)
dj = 1 is
,
9
(3.1)
where z is a real number over (--w, w), and C#Jand @ are the density and distribution the standard normal distribution, respectively. It is possible to write yil =x,:/3 + a,,h(w$)
functions
+ E,
of
(3.2)
for i such that dj = 1. Heckman’s two-step estimator is based on (3.2). First, (Y is estimated by the probit maximum likelihood method, and then cx is replaced by the probit MLE, ~5. Finally, p is estimated by the ordinary least squares (OLS) method.
4. The maximum
likelihood
estimator
Unlike other econometric models, computational difficulty. Setting p = logL=i
the MLE
a,,/a, , the
is seldom used in these models because log-likelihood function is given by
((-d,)log[l-@(w;a)]+d,[log@{[w;a
+&(Y,,-.#](I
of its
-P~))“~]
i=l -loga,
+1og4[a,
ml(Yli
pxiP)ll) ’
(4.1)
If the MLE is calculated by the standard methods, such as the one used in LIMDEP, the following problems exist. (i) The procedure often does not converge, especially when pO, which is the true value of p, is close to tl. (ii) Because of the existence of local maxima [Olsen (1982)], the result may not be correct even if the procedure converges.
K. Nawata
I Economics
35
Letters 45 (1994) 33-40
-380 -400 -420 -440 -440 -480 -500 -520 -540 -560 -1 Fig. 1. Log-likelihood
function
-0.5
0
0.5
1
for p. = 0.0.
In this paper, the scanning method is modified and the following procedure is used to calculate the MLE. (i) Choose M, equidistant points from (-1,l). Let 6 be the distance between any two points. (ii) Let p = 0 and calculate &,, &, and &,, which maximize the conditional likelihood function by the Newton-Raphson method. Note that this estimator is the same as the OLS estimator and the probit MLE*of the first and second equations in (2.1). (iii) Let Gi, pi and ci be the jth estimators. Increase p by 6, choose the initial values of the iteration as bi, pj and Gj, and calculate the (j + l)th estimator. Since the likelihood function is a continuous function of p, the previous estimators are considered to be in the neighborhood of the maximum values. (iv) Continue (iii) and calculate estimators up to the largest value of p determined in (i). (v) In the same way, calculate estimators from 0 to the smallest value of p. (vi) Choose the value of p that maximizes the conditional likelihood function, (vii) If necessary, choose M, points in the neighborhood of cl, as determined in (vi), and repeat the procedure. (viii) Determine the final estimators &, b, & and 6. The degree of difficulty in calculating 6 depends on the true parameter value, p,,. If p. is close to 0 (which means the effect of the selection bias is small), the log-likelihood function is flat near p = 0 and behaves ‘well’ in wide ranges of p, as shown in Fig. 1, so that it is easy to compute b. However, if p,, is close to +l, the behavior of the log-likelihood function is asymmetric and it is difficult to determine 6 (see Fig. 2). This means that we should choose S in (i) sufficiently small to calculate the MLE.
5. Monte Carlo experiments
In this section some Monte Carlo results are presented for the MLE and Heckman’s two-step estimator. The framework of the model experiments is similar to that of Paarsch (1984). The basic model is given by
K. Nawata I Economics
36
-340
Letters 45 (1994) 33-40
r
-360 -380 -400 -420 -440 -460 -480 -500 -1 Fig. 2. Log-likelihood
Yi =
function
0
0.5
1,2,3,
.,N .
-0.5
1
1.5
for p. = 0.8.
PO+ PlXi + u, >
d, = l(a”
+ cqw;
+ u; > 0) ,
i =
if di = 1. The following items are considered in the Monte Carlo YL is observable (i) The effect of t h e correlation of X, and wi. (ii) The effect of the correlation of U, and u,. The values of the exogenous variables, xi and wj, are determined as follows:
(5.1) study:
wi = 51, 7 xi = [7gli + (1 -
?T).&]/@q-q
.
(5.2)
{5,,) and {Ll are i.i.d. random variables distributed uniformly on (0,201. rr is the correlation coefficient of wi and xi, and rr = 0, 0.5, 0.8, 0.9, 0.95 and 1.0 are considered. {(u,, u;)} are jointly normal and determined as follows:
u, = PIJO.El1+ Cl-
/40/P; + (1 -POT.
(5.3)
{eIi} are i.i.d. normal random variables with mean zero and variance 1. {Q} are i.i.d. normal random variables with mean zero and variance 100. {Q} and {eZi} are independently distributed. p0 is the correlation coefficient of uj and ui, and p0 = 0, 0.4 and 0.8 are considered. The true values of the parameters are (Y”= -1.0,
cy, =O.l,
PO = -10.0
and
/3, = 1.0 ,
The degree of censoring is 50%, the sample size is 200 for all cases, and the number in all cases. The MLE is calculated as follows. (i) Choose p from [-0.99, 0.991 with an interval of 0.01.
of trials is 200
K. Nawata I Economics Table 1 Heckman’s
Heckman P” = 0.0 p,, = 0.4 p,, = 0.8
MLE P” = 0.0
p” = 0.4
p” = 0.8
Letters 45 (1994) 33-40
37
two-step estimator and the MLE (r = 0.0) Mean
STD
Qt
Median
-10.01 1.03 -9.88 1.01 -10.21 1.01
2.85 0.19 2.89 0.14 2.95 0.13
-11.80 0.88 -11.73 0.92 - 12.03 0.91
- 10.04 1.03 -9.83 1.00 - 10.07 1.02
-8.02 1.16 -7.82 1.09 -8.28 1.08
0.41 -13.89 0.98 0.87 -13.58 0.99 0.97 - 10.23 1.00
0.73 7.16 0.23 0.07 2.71 0.19 0.04 1.70 0.10
-0.66 -19.06 0.87 0.84 -15.70 0.88 0.98 -11.29 0.93
0.80 -16.19 0.96 0.88 - 13.46 0.98 0.98 - 10.24 1.01
0.89 -7.29 1.12 0.92 -11.55 1.13 0.99 -9.24 1.08
Q3
(ii) Calculate the maximum value of the conditional likelihood function for each p by the Newton-Raphson method as described in section 4. (iii) Choose 6, which maximizes the likelihood function. (iv) Choose p in the neighborhood of lj, with an interval of 0.001, and calculate the maximum values of the conditional likelihood function. (v) Determine the final estimates. Table 2 Heckman’s two-step estimator and the MLE (r = 0.5) Mean Heckman p” = 0.0 p” = 0.4 p0 = 0.8
MLE p. = 0.0
p” = 0.4
p” = 0.8
STD
Q1
Median
Q3
- 10.79 1.04 -10.19 1.02 -10.48 1.01
6.55 0.27 5.41 0.19 5.33 0.18
- 15.89 0.83 - 13.30 0.87 - 13.73 0.88
-11.09 1.04 - 10.28 1.02 -9.93 1.02
-6.11 1.25 -6.66 1.12 -7.22 1.11
0.22 - 14.48 1.13 0.83 -16.50 1.18 0.95 -9.66 0.99
0.77 13.12 0.41 0.21 4.95 0.23 0.06 3.15 0.16
-0.75 -25.75 0.75 0.82 -19.97 1.06 0.97 -11.77 0.90
0.77 -21.09 1.26 0.89 - 17.02 1.19 0.98 -10.20 1.00
0.87 0.00 1.42 0.93 - 14.22 1.33 0.99 -8.06 1.09
38 Table 3 Heckman’s
K. Nawata
PC,= 0.4 PO = 0.8
MLE P” = 0.0
p” = 0.4
p0 = 0.8
Letters 45 (1994) 33-40
two-step estimator and the MLE (n = 0.8) Mean
Heckman p,, = 0.0
J Economics
STD
Q1
Median
-9.24 0.96 -9.12 0.94 -8.07 0.91
17.37 0.68 17.13 0.64 12.04 0.45
-22.35 0.46 -20.90 0.44 -17.54 0.56
-9.89 0.99 - 10.59 0.99 -6.90 0.91
2.32 1.47 3.74 1.36 0.91 1.26
0.17 -13.38 1.11 0.60 -11.83 1.05 0.91 -8.18 0.91
0.68 13.35 0.55 0.35 7.51 0.34 0.07 4.28 0.21
-0.55 -26.20 0.58 0.30 -17.58 0.80 0.84 -11.63 0.74
0.41 - 13.50 1.14 0.76 -13.11 1.07 0.92 -8.48 0.88
0.83 0.00 1.59 0.88 -4.24 1.30 0.99 -4.33 1.08
Q3
The results are presented in Tables 1-6. Note that the following notation is used in the tables: STD = standard deviation, Ql = first quartile, and Q3 = third quartile. Heckman’s two-step estimator performs quite well when r is small, and is even better than the MLE for rr = 0. However, when rr is large and close to 1, Heckman’s estimator performs quite
Table 4 Heckman’s
Heckman p” = 0.0 p” = 0.4 p,, = 0.8
MLE p,, = 0.0
p. = 0.4
p0 = 0.8
two-step estimator and the MLE (r = 0.9) Mean
STD
Ql
Median
-10.43 0.98 - 10.52 1.00 -11.96 1.09
32.39 1.30 25.97 1.02 21.58 0.80
-31.74 0.09 -28.22 0.29 -26.94 0.54
-9.65 0.94 -11.84 1.09 -11.69 1.11
8.72 1.74 8.19 1.76 0.81 1.59
0.16 -13.71 1.16 0.57 - 12.02 1.09 0.91 -8.35 0.93
0.70 13.87 0.59 0.35 7.98 0.37 0.08 4.52 0.23
-0.60 -26.84 0.62 0.25 -19.01 0.80 0.83 -11.80 0.74
0.46 - 15.77 1.24 0.76 - 13.59 1.05 0.95 -8.59 0.91
0.84 0.01 1.66 0.89 -4.09 1.37 0.99 -4.55 1.08
Q3
K. Nawata I Economics
Letters 45 (1994) 33-40
39
Table 5 Heckman’s two-step estimator and the MLE (r = 0.95)
Heckman PO= 0.0 PO= 0.4 p” = 0.8
MLE p0 = 0.0
p. = 0.4
p,, = 0.8
Mean
STD
Q1
-6.76 0.90 - 10.05 1.01 - 10.42 0.98
40.46 1.64 35.25 1.42 30.05 1.19
-35.65 -0.24 -30.60 0.14 -30.19 0.09
-8.49 0.95 -8.95 0.94 -5.84 0.88
18.92 1.99 10.48 1.78 10.44 1.79
0.13 -12.82 1.13 0.55 -11.46 1.07 0.91 -8.51 0.92
0.70 13.49 0.60 0.37 7.39 0.36 0.08 3.67 0.22
-0.69 -24.78 0.57 0.23 -17.83 0.74 0.83 -11.60 0.76
0.40 -15.68 1.25 0.75 -12.14 1.04 0.95 -9.36 0.95
0.81 0.04 1.62 0.89 -3.87 1.32 0.99 -4.92 1.07
Q3
Median
Q3
Table 6 Heckman’s two-step estimator and the MLE (P = 1.0) Mean
STD
Ql
Median
-7.32 0.85 -8.33 0.92 -11.31 1.05
48.67 1.94 67.60 2.37 40.74 1.60
-31.36 -0.57 -37.00 -0.40 -36.37 -0.13
-4.51 0.74 -9.45 0.96 -14.85 1.16
25.53 1.93 26.08 2.12 16.30 2.09
0.08 -12.07 1.10 0.55 -11.66 1.08 0.91 -8.48 0.93
0.68 11.97 0.53 0.37 7.10 0.34 0.08 3.64 0.21
-0.63 -23.56 0.64 0.23 - 17.86 0.83 0.83 -11.52 0.76
0.23 -12.55 1.12 0.71 -11.76 1.05 0.94 -9.18 0.95
0.77 -1.20 1.57 0.87 -4.61 1.39 0.98 -4.81 1.08
Heckman PO = 0.0
P”
p” = 0.4
;:
p0 = 0.8
;: P,
MLE p,=o.o
p0 = 0.4
po=0.8
p 2 i” PI p
poorly. If T L 0.9, the standard errors and quartile ranges of estimates become several times as large as those of the MLE, and the estimates of p1 even become negative in some trials. On the other hand, the MLE behaves reasonably well in all cases, especially when p,, is large. One interesting finding is that the estimates of p are biased toward 1 for p0 > 0.
K. Nawata
40
6. Concluding
I Economics
Letters
45 (1994) X-40
remarks
In this paper the finite sample performance of the MLE and Heckman’s two-step estimator Heckman’s estimator is widely used in the were examined using Monte Carlo experiments. estimation of models with selection biases. Since the hazard ratio, A(z), is closely approximated by a linear function of z, Heckman’s estimator is expected to perform poorly when there is a high degree of multicollinearity between w’ i; and x, [Nawata (1994)]. Estimation of the MLE in this model can be difficult, especially when pO is close to +l. However, the MLE is much more efficient than Heckman’s estimator when there is a high degree by the MLE in that of multicollinearity between w: G and x,, and the model should be calculated case.
References Heckman, J., 1974, Shadow prices, market wages and labor supply, Econometrica 42, 679-694. Heckman, J., 1976, The common structure of statistical models of truncation, sample selection bias and limited dependent variables and a simple estimator for such models, Annals of Econometric and Social Measurement 5, 475-492. Heckman, J., 1979, Sample selection bias as a specification error, Econometrica 47, 153-161. Nawata, K., 1994, A note on estimation of the models with sample-selection biases, Economics Letters, forthcoming. Nelson, F.D., 1984, Efficiency of the two-step estimator for models with endogenous sample selection, Journal of Econometrics 24, 181-196. Olsen, R.J., 1982, Distributional tests for selectivity bias and a more robust likelihood estimator, International Economic Review 23. 223-240. Paarsch, H.J., 1984, A Monte Carlo comparison of estimators for censored regression models, Journal of Econometrics 24, 197-213. Wales, T.J. and A.D. Woodland, 1980, Sample selectivity and the estimation of labor supply functions, International Economic Review 21. 437-468.
45 (1994) 33-40 0 1994 Elsevier
33 Science
B.V. All rights
reserved
Estimation of sample selection bias models by the maximum likelihood estimator and Heckman’s two-step estimator Kazumitsu Nawata”’ Department of Social and International Relations, University of Tokyo, 3-8-l Komaba, Meguro-ku, Department of Economics, University of Western Australia, Nedlands, Perth WA 6009, Australia
Tokyo 1.53, Japan
Received 28 June 1993 Final revision 3 November 1993 Accepted 8 November 1993
Abstract In this paper, methods of estimating models with sample selection biases are analyzed. Finite sample properties of the maximum likelihood estimator (MLE) and Heckman’s two-step estimator are compared using Monte Carlo experiments. JEL
classification: C34
1. Introduction Econometric models with sample selection biases (type II Tobit models) are widely used in various fields of economics. Unlike other types of econometric models, the maximum likelihood estimator (MLE) is seldom used because of its computational difficulty. Heckman (1976, 1979) proposed a simple two-step estimator. However, Heckman’s estimator sometimes performs poorly [Wales and Woodland (1980), Nelson (1984), Paarsch (1984), and Nawata (1994)]. In this paper, a method of calculating the MLE is considered, and the finite sample properties of the MLE and Heckman’s two-step estimator are compared using Monte Carlo experiments. Although Heckman’s two-step estimator performs well when there is not a high degree of multicollinearity between the hazard ratio and explanatory variables, it performs poorly when there is a high degree of multicollinearity between them.
* Correspondence to: Kazumitsu Nawata, Department of Social and International Komaba, Meguro-ku, Tokyo 153, Japan. ’ I wish to thank Michael McAIeer and Paul Miller for their helpful comments. calculating the MLE, is available through the author. SSDI
0165-1765(93)00369-Y
Relations,
University
The program
used
of Tokyo, in this paper,
3-8-l for
34
K. Nuwata I Economics
Letters 4.5 (1994) 33-40
2. Models The model
considered
in this paper
is
y,; =.x1$ + ui ) Yz,=W:~+U,,
(2.1)
d,=l(y,,>O),
i-l,2
)...)
N,
where l(q) is an indicator function such that l(.) = 1 if. is true and 0 otherwise. yzi is not observed and only the sign of y,; (i.e. d,) is observed. y,, is observed when d, = 1. (u,, u,) are jointly normal with mean zero, variances (a:, l), and covariance ui2. (Since the sign of yzi does not change if we multiply it by a positive constant value, it can be assumed that the variance of ui is 1 without loss of generality.)
3. Heckman’s
two-step
The conditional
expected
value
1di = 1) = x;p
E(y,, G)
estimator
= $(z)/@(z)
of y,, given
+ ai2A(w~:a)
dj = 1 is
,
9
(3.1)
where z is a real number over (--w, w), and C#Jand @ are the density and distribution the standard normal distribution, respectively. It is possible to write yil =x,:/3 + a,,h(w$)
functions
+ E,
of
(3.2)
for i such that dj = 1. Heckman’s two-step estimator is based on (3.2). First, (Y is estimated by the probit maximum likelihood method, and then cx is replaced by the probit MLE, ~5. Finally, p is estimated by the ordinary least squares (OLS) method.
4. The maximum
likelihood
estimator
Unlike other econometric models, computational difficulty. Setting p = logL=i
the MLE
a,,/a, , the
is seldom used in these models because log-likelihood function is given by
((-d,)log[l-@(w;a)]+d,[log@{[w;a
+&(Y,,-.#](I
of its
-P~))“~]
i=l -loga,
+1og4[a,
ml(Yli
pxiP)ll) ’
(4.1)
If the MLE is calculated by the standard methods, such as the one used in LIMDEP, the following problems exist. (i) The procedure often does not converge, especially when pO, which is the true value of p, is close to tl. (ii) Because of the existence of local maxima [Olsen (1982)], the result may not be correct even if the procedure converges.
K. Nawata
I Economics
35
Letters 45 (1994) 33-40
-380 -400 -420 -440 -440 -480 -500 -520 -540 -560 -1 Fig. 1. Log-likelihood
function
-0.5
0
0.5
1
for p. = 0.0.
In this paper, the scanning method is modified and the following procedure is used to calculate the MLE. (i) Choose M, equidistant points from (-1,l). Let 6 be the distance between any two points. (ii) Let p = 0 and calculate &,, &, and &,, which maximize the conditional likelihood function by the Newton-Raphson method. Note that this estimator is the same as the OLS estimator and the probit MLE*of the first and second equations in (2.1). (iii) Let Gi, pi and ci be the jth estimators. Increase p by 6, choose the initial values of the iteration as bi, pj and Gj, and calculate the (j + l)th estimator. Since the likelihood function is a continuous function of p, the previous estimators are considered to be in the neighborhood of the maximum values. (iv) Continue (iii) and calculate estimators up to the largest value of p determined in (i). (v) In the same way, calculate estimators from 0 to the smallest value of p. (vi) Choose the value of p that maximizes the conditional likelihood function, (vii) If necessary, choose M, points in the neighborhood of cl, as determined in (vi), and repeat the procedure. (viii) Determine the final estimators &, b, & and 6. The degree of difficulty in calculating 6 depends on the true parameter value, p,,. If p. is close to 0 (which means the effect of the selection bias is small), the log-likelihood function is flat near p = 0 and behaves ‘well’ in wide ranges of p, as shown in Fig. 1, so that it is easy to compute b. However, if p,, is close to +l, the behavior of the log-likelihood function is asymmetric and it is difficult to determine 6 (see Fig. 2). This means that we should choose S in (i) sufficiently small to calculate the MLE.
5. Monte Carlo experiments
In this section some Monte Carlo results are presented for the MLE and Heckman’s two-step estimator. The framework of the model experiments is similar to that of Paarsch (1984). The basic model is given by
K. Nawata I Economics
36
-340
Letters 45 (1994) 33-40
r
-360 -380 -400 -420 -440 -460 -480 -500 -1 Fig. 2. Log-likelihood
Yi =
function
0
0.5
1,2,3,
.,N .
-0.5
1
1.5
for p. = 0.8.
PO+ PlXi + u, >
d, = l(a”
+ cqw;
+ u; > 0) ,
i =
if di = 1. The following items are considered in the Monte Carlo YL is observable (i) The effect of t h e correlation of X, and wi. (ii) The effect of the correlation of U, and u,. The values of the exogenous variables, xi and wj, are determined as follows:
(5.1) study:
wi = 51, 7 xi = [7gli + (1 -
?T).&]/@q-q
.
(5.2)
{5,,) and {Ll are i.i.d. random variables distributed uniformly on (0,201. rr is the correlation coefficient of wi and xi, and rr = 0, 0.5, 0.8, 0.9, 0.95 and 1.0 are considered. {(u,, u;)} are jointly normal and determined as follows:
u, = PIJO.El1+ Cl-
/40/P; + (1 -POT.
(5.3)
{eIi} are i.i.d. normal random variables with mean zero and variance 1. {Q} are i.i.d. normal random variables with mean zero and variance 100. {Q} and {eZi} are independently distributed. p0 is the correlation coefficient of uj and ui, and p0 = 0, 0.4 and 0.8 are considered. The true values of the parameters are (Y”= -1.0,
cy, =O.l,
PO = -10.0
and
/3, = 1.0 ,
The degree of censoring is 50%, the sample size is 200 for all cases, and the number in all cases. The MLE is calculated as follows. (i) Choose p from [-0.99, 0.991 with an interval of 0.01.
of trials is 200
K. Nawata I Economics Table 1 Heckman’s
Heckman P” = 0.0 p,, = 0.4 p,, = 0.8
MLE P” = 0.0
p” = 0.4
p” = 0.8
Letters 45 (1994) 33-40
37
two-step estimator and the MLE (r = 0.0) Mean
STD
Qt
Median
-10.01 1.03 -9.88 1.01 -10.21 1.01
2.85 0.19 2.89 0.14 2.95 0.13
-11.80 0.88 -11.73 0.92 - 12.03 0.91
- 10.04 1.03 -9.83 1.00 - 10.07 1.02
-8.02 1.16 -7.82 1.09 -8.28 1.08
0.41 -13.89 0.98 0.87 -13.58 0.99 0.97 - 10.23 1.00
0.73 7.16 0.23 0.07 2.71 0.19 0.04 1.70 0.10
-0.66 -19.06 0.87 0.84 -15.70 0.88 0.98 -11.29 0.93
0.80 -16.19 0.96 0.88 - 13.46 0.98 0.98 - 10.24 1.01
0.89 -7.29 1.12 0.92 -11.55 1.13 0.99 -9.24 1.08
Q3
(ii) Calculate the maximum value of the conditional likelihood function for each p by the Newton-Raphson method as described in section 4. (iii) Choose 6, which maximizes the likelihood function. (iv) Choose p in the neighborhood of lj, with an interval of 0.001, and calculate the maximum values of the conditional likelihood function. (v) Determine the final estimates. Table 2 Heckman’s two-step estimator and the MLE (r = 0.5) Mean Heckman p” = 0.0 p” = 0.4 p0 = 0.8
MLE p. = 0.0
p” = 0.4
p” = 0.8
STD
Q1
Median
Q3
- 10.79 1.04 -10.19 1.02 -10.48 1.01
6.55 0.27 5.41 0.19 5.33 0.18
- 15.89 0.83 - 13.30 0.87 - 13.73 0.88
-11.09 1.04 - 10.28 1.02 -9.93 1.02
-6.11 1.25 -6.66 1.12 -7.22 1.11
0.22 - 14.48 1.13 0.83 -16.50 1.18 0.95 -9.66 0.99
0.77 13.12 0.41 0.21 4.95 0.23 0.06 3.15 0.16
-0.75 -25.75 0.75 0.82 -19.97 1.06 0.97 -11.77 0.90
0.77 -21.09 1.26 0.89 - 17.02 1.19 0.98 -10.20 1.00
0.87 0.00 1.42 0.93 - 14.22 1.33 0.99 -8.06 1.09
38 Table 3 Heckman’s
K. Nawata
PC,= 0.4 PO = 0.8
MLE P” = 0.0
p” = 0.4
p0 = 0.8
Letters 45 (1994) 33-40
two-step estimator and the MLE (n = 0.8) Mean
Heckman p,, = 0.0
J Economics
STD
Q1
Median
-9.24 0.96 -9.12 0.94 -8.07 0.91
17.37 0.68 17.13 0.64 12.04 0.45
-22.35 0.46 -20.90 0.44 -17.54 0.56
-9.89 0.99 - 10.59 0.99 -6.90 0.91
2.32 1.47 3.74 1.36 0.91 1.26
0.17 -13.38 1.11 0.60 -11.83 1.05 0.91 -8.18 0.91
0.68 13.35 0.55 0.35 7.51 0.34 0.07 4.28 0.21
-0.55 -26.20 0.58 0.30 -17.58 0.80 0.84 -11.63 0.74
0.41 - 13.50 1.14 0.76 -13.11 1.07 0.92 -8.48 0.88
0.83 0.00 1.59 0.88 -4.24 1.30 0.99 -4.33 1.08
Q3
The results are presented in Tables 1-6. Note that the following notation is used in the tables: STD = standard deviation, Ql = first quartile, and Q3 = third quartile. Heckman’s two-step estimator performs quite well when r is small, and is even better than the MLE for rr = 0. However, when rr is large and close to 1, Heckman’s estimator performs quite
Table 4 Heckman’s
Heckman p” = 0.0 p” = 0.4 p,, = 0.8
MLE p,, = 0.0
p. = 0.4
p0 = 0.8
two-step estimator and the MLE (r = 0.9) Mean
STD
Ql
Median
-10.43 0.98 - 10.52 1.00 -11.96 1.09
32.39 1.30 25.97 1.02 21.58 0.80
-31.74 0.09 -28.22 0.29 -26.94 0.54
-9.65 0.94 -11.84 1.09 -11.69 1.11
8.72 1.74 8.19 1.76 0.81 1.59
0.16 -13.71 1.16 0.57 - 12.02 1.09 0.91 -8.35 0.93
0.70 13.87 0.59 0.35 7.98 0.37 0.08 4.52 0.23
-0.60 -26.84 0.62 0.25 -19.01 0.80 0.83 -11.80 0.74
0.46 - 15.77 1.24 0.76 - 13.59 1.05 0.95 -8.59 0.91
0.84 0.01 1.66 0.89 -4.09 1.37 0.99 -4.55 1.08
Q3
K. Nawata I Economics
Letters 45 (1994) 33-40
39
Table 5 Heckman’s two-step estimator and the MLE (r = 0.95)
Heckman PO= 0.0 PO= 0.4 p” = 0.8
MLE p0 = 0.0
p. = 0.4
p,, = 0.8
Mean
STD
Q1
-6.76 0.90 - 10.05 1.01 - 10.42 0.98
40.46 1.64 35.25 1.42 30.05 1.19
-35.65 -0.24 -30.60 0.14 -30.19 0.09
-8.49 0.95 -8.95 0.94 -5.84 0.88
18.92 1.99 10.48 1.78 10.44 1.79
0.13 -12.82 1.13 0.55 -11.46 1.07 0.91 -8.51 0.92
0.70 13.49 0.60 0.37 7.39 0.36 0.08 3.67 0.22
-0.69 -24.78 0.57 0.23 -17.83 0.74 0.83 -11.60 0.76
0.40 -15.68 1.25 0.75 -12.14 1.04 0.95 -9.36 0.95
0.81 0.04 1.62 0.89 -3.87 1.32 0.99 -4.92 1.07
Q3
Median
Q3
Table 6 Heckman’s two-step estimator and the MLE (P = 1.0) Mean
STD
Ql
Median
-7.32 0.85 -8.33 0.92 -11.31 1.05
48.67 1.94 67.60 2.37 40.74 1.60
-31.36 -0.57 -37.00 -0.40 -36.37 -0.13
-4.51 0.74 -9.45 0.96 -14.85 1.16
25.53 1.93 26.08 2.12 16.30 2.09
0.08 -12.07 1.10 0.55 -11.66 1.08 0.91 -8.48 0.93
0.68 11.97 0.53 0.37 7.10 0.34 0.08 3.64 0.21
-0.63 -23.56 0.64 0.23 - 17.86 0.83 0.83 -11.52 0.76
0.23 -12.55 1.12 0.71 -11.76 1.05 0.94 -9.18 0.95
0.77 -1.20 1.57 0.87 -4.61 1.39 0.98 -4.81 1.08
Heckman PO = 0.0
P”
p” = 0.4
;:
p0 = 0.8
;: P,
MLE p,=o.o
p0 = 0.4
po=0.8
p 2 i” PI p
poorly. If T L 0.9, the standard errors and quartile ranges of estimates become several times as large as those of the MLE, and the estimates of p1 even become negative in some trials. On the other hand, the MLE behaves reasonably well in all cases, especially when p,, is large. One interesting finding is that the estimates of p are biased toward 1 for p0 > 0.
K. Nawata
40
6. Concluding
I Economics
Letters
45 (1994) X-40
remarks
In this paper the finite sample performance of the MLE and Heckman’s two-step estimator Heckman’s estimator is widely used in the were examined using Monte Carlo experiments. estimation of models with selection biases. Since the hazard ratio, A(z), is closely approximated by a linear function of z, Heckman’s estimator is expected to perform poorly when there is a high degree of multicollinearity between w’ i; and x, [Nawata (1994)]. Estimation of the MLE in this model can be difficult, especially when pO is close to +l. However, the MLE is much more efficient than Heckman’s estimator when there is a high degree by the MLE in that of multicollinearity between w: G and x,, and the model should be calculated case.
References Heckman, J., 1974, Shadow prices, market wages and labor supply, Econometrica 42, 679-694. Heckman, J., 1976, The common structure of statistical models of truncation, sample selection bias and limited dependent variables and a simple estimator for such models, Annals of Econometric and Social Measurement 5, 475-492. Heckman, J., 1979, Sample selection bias as a specification error, Econometrica 47, 153-161. Nawata, K., 1994, A note on estimation of the models with sample-selection biases, Economics Letters, forthcoming. Nelson, F.D., 1984, Efficiency of the two-step estimator for models with endogenous sample selection, Journal of Econometrics 24, 181-196. Olsen, R.J., 1982, Distributional tests for selectivity bias and a more robust likelihood estimator, International Economic Review 23. 223-240. Paarsch, H.J., 1984, A Monte Carlo comparison of estimators for censored regression models, Journal of Econometrics 24, 197-213. Wales, T.J. and A.D. Woodland, 1980, Sample selectivity and the estimation of labor supply functions, International Economic Review 21. 437-468.