Economics Chapter 19 Homework The log likelihood for the truncated regression with only a constant term is

Chapter 19

Limited Dependent Variables—Truncation,

Censoring, and Sample Selection

◼ Exercises

1. The sample mean of all 20 observations is 4.18222. For the 14 nonzero observations, the mean is

(20/14)4.18222 = 5.9746. Both of these should overestimate . In the first case, all negative values

2. The log likelihood for the Tobit model is given in (19-13). With only a constant term, this is

The necessary conditions for maximizing this with respect to  and  are

3. The log likelihood for the truncated regression with only a constant term is

lnL = (−n/2)[ln(2) + ln2] − (1/(22))1(yi − )2 − iln(/).

Once again transforming to  and , this is

Chapter 19 Limited Dependent Variables—Truncation, Censoring, and Sample Selection 147

4. Using Theorem 19.5, we have 1 − (z) = 14/35 = 0.4, z = −1(0.6) = 0.253, (z) = 0.9659,

5. The conditional mean function is E[y|x] = (xi /i)xi + i(xi /i) using the equation before

(19-12). Suppose that i =  exp(xi) for the same vector xi . (We will relax that assumption shortly.)

Now, differentiate this expression with respect to x. We differentiate the two parts, first with respect

to x then with respect to i.

6. The transformed log-likelihood function is

logL = y > 0 (−1/2)[log2 − log2 + (y − x)2] + y = 0 log[1 − (x)].

It will be convenient to define ai = xi. Note also that 1 − (ai ) = (−ai). The first derivatives and

Hessian in the transformed parameters are

148 Greene • Econometric Analysis, Seventh Edition

The second derivatives can be collected in a matrix format:

where i is the last scalar term in 2logL/ . By Theorem 19.2 [see (19-4)], we know that i is

Applications

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Application 19.1 Tobit Model for Psychology Today Data

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Ordinary least squares regression …………

LHS=Y Mean = 1.45591

——–+——————————————————————–

| Standard Prob. 95% Confidence

Y| Coefficient Error z |z|>Z* Interval

——–+——————————————————————–

Constant| 5.87201*** 1.13750 5.16 .0000 3.64256 8.10146

Z1| .05409 .30049 .18 .8572 -.53486 .64303

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Limited Dependent Variable Model – CENSORED

Dependent variable Y

Log-likelihood function -704.73107

Chapter 19 Limited Dependent Variables—Truncation, Censoring, and Sample Selection 149

Lower= .0000 Upper=+infinity

LM test [df] for tobit= 10.513[ 9]

——–+——————————————————————–

| Standard Prob. 95% Confidence

Y| Coefficient Error z |z|>Z* Interval

——–+——————————————————————–

|Primary Index Equation for Model

Constant| 7.60849* 3.90599 1.95 .0514 -.04711 15.26408

|Disturbance standard deviation

——–+——————————————————————–

Note: ***, **, * ==> Significance at 1%, 5%, 10% level.

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Partial derivatives of expected val. with

——–+——————————————————————–

| Partial Standard Prob. 95% Confidence

Y| Effect Error z |z|>Z* Interval

——–+——————————————————————–

Z2| -.04475** .01874 -2.39 .0170 -.08148 -.00802

Z4| .23666 .29626 .80 .4244 -.34399 .81732

Z6| .00589 .05286 .11 .9113 -.09772 .10950

——–+——————————————————————–

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Partial Effects Analysis for Tobit (Censored) Regression Function

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Effects on function with respect to Z2

150 Greene • Econometric Analysis, Seventh Edition

(Delta method) Effect Error |t| 95% Confidence Interval

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df/dZ3 Partial Standard

(Delta method) Effect Error |t| 95% Confidence Interval

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Partial effect .13616 .03805 3.58 .06158 .21075

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Effects on function with respect to Z4

Partial effects for binary var Z4 computed by first difference

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Effects on function with respect to Z5

Partial effects for continuous Z5 computed by differentiation

Effect is computed as derivative = df(.)/dx

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Effects on function with respect to Z6

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Effects on function with respect to Z7

Partial effects for continuous Z7 computed by differentiation

———————————————————————

df/dZ7 Partial Standard

(Delta method) Effect Error |t| 95% Confidence Interval

Chapter 19 Limited Dependent Variables—Truncation, Censoring, and Sample Selection 151

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Application 19.2 Mroz Labor Supply Data

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Limited Dependent Variable Model – CENSORED

Dependent variable WHRS

——–+——————————————————————–

| Standard Prob. 95% Confidence

WHRS| Coefficient Error z |z|>Z* Interval

——–+——————————————————————–

|Primary Index Equation for Model

Constant| 1690.88*** 507.7730 3.33 .0009 695.66 2686.10

|Disturbance standard deviation

—————————————————————————–

Partial derivatives of expected val. with

respect to the vector of characteristics.

——–+——————————————————————–

| Partial Standard Prob. 95% Confidence

WHRS| Effect Error z |z|>Z* Interval

——–+——————————————————————–

KL6| -642.905*** 73.70149 -8.72 .0000 -787.357 -498.453

K618| -75.2897*** 25.26098 -2.98 .0029 -124.8003 -25.7791

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152 Greene • Econometric Analysis, Seventh Edition

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Application 19.3 We Break the Tobit Model into a Two-Part Model

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The first equation is for labor force participation. The second is for hours given LFP = 1. The two

equations are a probit model for LFP and a truncated regression for positive hours. The chi-squared

statistic is 61.1 with 9 degrees of freedom. The critical value is 15.51, so the null hypothesis of the tobit

model is rejected in favor of a model that provides for different participation and hours equations.

–> calc ; lc = logl $

–> namelist ; x = one,kl6,k618,wa,we,ha,he,cit$

–> probit;lhs=lfp;rhs=x$

Normal exit: 5 iterations. Status=0, F= 461.2908

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Binomial Probit Model

Dependent variable LFP

Log-likelihood function -461.29077

——–+——————————————————————–

| Standard Prob. 95% Confidence

LFP| Coefficient Error z |z|>Z* Interval

——–+——————————————————————–

|Index Function for Probability

Constant| .98816** .50094 1.97 .0485 .00634 1.96998

KL6| -.90345*** .11423 -7.91 .0000 -1.12735 -.67956

—————————————————————————–

–> calc ; lp=logl$

–> reject ; lfp=0$

–> trunc ; lhs=whrs;rhs=x $

Normal exit: 5 iterations. Status=0, F= 3407.819

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Limited Dependent Variable Model – TRUNCATE

Dependent variable WHRS

Chapter 19 Limited Dependent Variables—Truncation, Censoring, and Sample Selection 153

——–+——————————————————————–

| Standard Prob. 95% Confidence

WHRS| Coefficient Error z |z|>Z* Interval

——–+——————————————————————–

|Primary Index Equation for Model

Constant| 2524.20*** 502.0844 5.03 .0000 1540.13 3508.27

–> calc ; lt=logl$

–> calc ; list ; lrtest = 2*((lp+lt)-lc)$

[CALC] LRTEST = 61.9122923

–> calc ; list ; ctb(.95,kreg)$

[CALC] *Result*= 15.5073131

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Application 19.4

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create ; lc = log(cost/pf)$

create ; lq = log(q) ; lq2 = lq*lq $

create ; lpk=log(pk/pf) $

create ; lpl=log(pl/pf)$

sample ; 1-123 $

frontier ; lhs = lc ; cost ; rhs = one,lq,lq2,lpk,lpl ; eff=ucost $

dstat ; rhs = ucost $

kernel ; rhs = ucost $

plot ; lhs = q ; rhs = ucost ; grid $

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Limited Dependent Variable Model – FRONTIER

Dependent variable LC

——–+——————————————————————–

| Standard Prob. 95% Confidence

LC| Coefficient Error z |z|>Z* Interval

——–+——————————————————————–

154 Greene • Econometric Analysis, Seventh Edition

—————————————————————————–

–> dstat ; rhs = ucost $

Descriptive Statistics

——–+———————————————————————

Variable| Mean Std.Dev. Minimum Maximum Cases Missing

——–+———————————————————————

UCOST| .118840 .614993E-01 .298216E-01 .378595 123 0

——–+———————————————————————

–> kernel ; rhs = ucost $

+—————————————+

| Kernel Density Estimator for UCOST |

| Observations = 123 |

| Points plotted = 123 |

Chapter 19 Limited Dependent Variables—Truncation, Censoring, and Sample Selection 155