Wooldridge J., Introductory Econometrics - A Modern Approach (Instructors Manual)

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= 7.26 + 1.81 log(wage) − .129 educ − .012 age



log( )hours

(1.02) (0.50) (.087) (.011)

− .543 kidslt6 − .019 nwifeinc

(.211) (.009)

n = 428.

The biggest effect is to reduce the size of the coefficient on educ as well as its statistical

significance. The labor supply elasticity is only moderately smaller.

(iii) After obtaining the 2SLS residuals, , from the estimation in part (ii), we regress these

on age, kidslt6, nwifeinc, exper, exper

, motheduc, and fatheduc. The n-R-squared statistic is

408(.0010) = .428. We have two overidentifying restrictions, so the p-value is roughly

> .43) .81. There is no evidence against the exogeneity of the IVs.

≈

16.11 (i) The OLS estimates are

= 25.23 − .215 open

inf

(4.10) (.093)

n = 114, R

= .045.

The IV estimates are

= 29.61 − .333 open

inf

(5.66) (.140)

n = 114, R

= .032.

The OLS coefficient is the same, to three decimal places, when log(pcinc) is included in the

model. The IV estimate with log(pcinc) in the equation is −.337, which is very close to −.333.

Therefore, dropping log(pcinc) makes little difference.

(ii) Subject to the requirement that an IV be exogenous, we want an IV that is as highly

correlated as possible with the endogenous explanatory variable. If we regress open on land we

obtain R

= .095. The simple regression of open on log(land) gives R

= .448. Therefore,

log(land) is much more highly correlated with open. Further, if we regress open on log(land)

and land we get

= 129.22 − 8.40 log(land) + .0000043 land



open

(10.47) (0.98) (.0000031)

n = 114, R

= .457.

153

While log(land) is very significant, land is not, so we might as well use only log(land) as the IV

for open.

[Instructor’s Note: You might ask students whether it is better to use log(land) as the single IV

for open or to use both land and land

. In fact, log(land) explains much more variation in open.]

(iii) When we add oil to the original model, and assume oil is exogenous, the IV estimates

are

= 24.01 − .337 open + .803 log(pcinc) − 6.56 oil

inf

(16.04) (.144) (2.12) (9.80)

n = 114, R

= .035.

Being an oil producer is estimated to reduce average annual inflation by over 6.5 percentage

points, but the effect is not statistically significant. This is not too surprising, as there are only

seven oil producers in the sample.

16.12 (i) The usual form of the test assumes no serial correlation under H

, and this appears to be

the case. We also assume homoskedasticity. After estimating (16.35), we obtain the 2SLS

residuals, . We then run the regression on gc

t-1

, gy

t-1

, and r3

t-1

. The n-R-squared statistic is

35(.0613)

≈

2.15. With one df the (asymptotic) p-value is P(

> 2.15)

≈

.143, and so the

instruments pass the overidentification test at the 10% level.

(ii) If we estimate (16.35) but with gc

t-2

, gy

t-2

, and r3

t-2

as the IVs, we obtain, with n = 34,

−.0054 + 1.204 gy



− .00043 r3

(.0274) (1.272) (.00196)

The coefficient on gy

has doubled in size compared with equation (16.35), but it is not

statistically significant. The coefficient on r3

is still small and statistically insignificant.

(iii) If we regress gy

on gc

t-2

, gy

t-2

, and r3

t-2

we obtain

= .021

− .070 gc



t-2

+ .094 gy

t-2

+ .00074 r3

t-2

(.007) (.469) (.330) (.00166)

n = 34, R

= .0137.

The F statistic for joint significance of all explanatory variables yields p-value .94, and so

there is no correlation between gy

≈

and the proposed IVs, gc

t-2

, gy

t-2

, and r3

t-2

. Therefore, we

never should have done the IV estimation in part (ii) in the first place.

154

[Instructor’s Note: There may be serial correlation in this regression, in which case the F

statistic is not valid. But the point remains that gy

is not at all correlated with two lags of all

variables.]

16.l3 This is an open-ended question without a unique answer. Even if we settle on extending

the data through a particular year, we might want to change the disposable income and

nondurable consumption numbers in earlier years, as these are often recalculated. For example,

the value for real disposable personal income in 1995, as reported in Table B-29 of the 1997

Economic Report of the President (ERP), is $4,945.8 billions. In the 1999 ERP, this value has

been changed to $4,906.0 billions (see Table B-31). All series can be updated using the latest

edition of the ERP. The key is to use real values and make them per capita by dividing by

population. Make sure that you use nondurable consumption.

16.14 (i) If we estimate the inverse supply function by OLS we obtain (with the coefficients on

the monthly dummies suppressed)

= .0144

− .0443 gcem



gprc

+ .0628 gprcpet

+ K

(.0032) (.0091) (.0153)

n = 298, R

= .386.

Several of the monthly dummy variables are very statistically significant, but their coefficients

are not of direct interest here. The estimated supply curve slopes down, not up, and the

coefficient on gcem

is very statistically significant (t statistic

≈

−4.87).

(ii) We need grdefs

to have a nonzero coefficient in the reduced form for gcem

. More

precisely, if we write

gcem

grdefs

gprcpet

feb

+ K +

dec

+ v

then identification requires

≠ 0. When we run this regression,

= −1.054 with a t statistic of

about –0.294. Therefore, we cannot reject H

= 0 at any reasonable significance level, and

we conclude that grdefs

is not a useful IV for gcem

(even if grdefs

is exogenous in the supply

equation).

(iii) Now the reduced form for gcem is

gcem

grres

grnon

gprcpet

feb

+ K +

dec

+ v

and we need at least one of

and

to be different from zero. In fact,

= .136, t(

) = .984

and

= 1.15, t(

) = 5.47. So grnon

is very significant in the reduced form for gcem

, and we

can proceed with IV estimation.

155

(iv) We use both grres

and grnon

as IVs for gcem

and apply 2SLS, even though the former

is not significant in the RF. The estimated labor supply function (with seasonal dummy

coefficients suppressed) is now

= .0228

− .0106 gcem



gprc

+ .0605 gprcpet

+ K

(.0073) (.0277) (.0157)

n = 298, R

= .356.

While the coefficient on gcem

is still negative, it is only about one-fourth the size of the OLS

coefficient, and it is now very insignificant. At this point we would conclude that the static

supply function is horizontal (with gprc on the vertical axis, as usual). Shea (1993) adds many

lags of gcem

and estimates a finite distributed lag model by IV, using leads as well as lags of

grres

and grnon

as IVs. He estimates a positive long run propensity.

16.15 (i) If county administrators can predict when crime rates will increase, they may hire more

police to counteract crime. This would explain the estimated positive relationship between

Δlog(crmrte) and Δlog(polpc) in equation (13.33).

(ii) This may be reasonable, although tax collections depend in part on income and sales

taxes, and revenues from these depend on the state of the economy, which can also influence

crime rates.

(iii) The reduced form for

Δlog(polpc

), for each i and t, is

Δlog(polpc

) =

d83

d84

d85

d86

d87

Δlog(prbarr

) +

Δlog(prbconv

) +

Δlog(prbpris

)

Δlog (avgsen

) +

Δlog(taxpc

) + v

We need

≠ 0 for Δlog(taxpc

) to be a reasonable IV candidate for Δlog(polpc

). When we

estimate this equation by pooled OLS (N = 90, T = 6 for n = 540), we obtain

= .0052 with a t

statistic of only .080. Therefore,

Δlog(taxpc

) is not a good IV for Δlog(polpc

(iv) If the grants were awarded randomly, then the grant amounts, say grant

for the dollar

amount for county i and year t, will be uncorrelated with

Δu

, the changes in unobservables that

affect county crime rates. By definition, grant

should be correlated with Δlog(polpc

) across i

and t. This means we have an exogenous variable that can be omitted from the crime equation

and that is (partially) correlated with the endogenous explanatory variable. We could reestimate

(13.33) by IV.

16.16 (i) To estimate the demand equations, we need at least one exogenous variable that appears

in the supply equation.

156

(ii) For wave2

and wave3

to be valid IVs for log(avgprc

), we need two assumptions. The

first is that these can be properly excluded from the demand equation. This may not be entirely

reasonable, and wave heights are determined partly by weather, and demand at a local fish

market could depend on demand. The second assumption is that at least one of wave2

and

wave3

appears in the supply equation. There is indirect evidence of this in part three, as the two

variables are jointly significant in the reduced form for log(avgprc

(iii) The OLS estimates of the reduced form are

−1.02 − .012 mon



log( )

avgprc =

− .0090 tues

+ .051 wed

+ .124 thurs

(.14) (.114) (.1119) (.112) (.111)

+ .094 wave2

+ .053 wave3

(.021) (.020)

n = 97, R

= .304

The variables wave2

and wave3

are jointly very significant: F = 19.1, p-value = zero to four

decimal places.

(iv) The 2SLS estimates of the demand function are

8.16

− .816 log(avgprc



log( )

totqty =

) − .307 mon

− .685 tues

(.18) (.327) (.229) (.226)

− .521 wed

+ .095 thurs

(.224) (.225)

n = 97, R

= .193

The 95% confidence interval for the demand elasticity is roughly

−1.47 to −.17. The point

estimate,

−.82, seems reasonable: a 10 percent increase in price reduces quantity demanded by

about 8.2%.

(v) The coefficient on is about .294 (se = .103), so there is strong evidence of positive

serial correlation, although the estimate of

is not huge. One could compute a Newey-West

standard error for 2SLS in place of the usual standard error.

−

(vi) To estimate the supply elasticity, we would have to assume that the day-of-the-week

dummies do not appear in the supply equation, but they do appear in the demand equation. Part

(iii) provides evidence that there are day-of-the-week effects in the demand function. But we

cannot know about the supply function.

157

(vii) Unfortunately, in the estimation of the reduced form for log(avgprc

) in part (iii), the

variables mon, tues, wed, and thurs are jointly insignificant [F(4,90) = .53, p-value = .71.] This

means that, while some of these dummies seem to show up in the demand equation, things cancel

out in a way that they do not affect equilibrium price, once wave2 and wave3 are in the equation.

So, without more information, we have no hope of estimating the supply equation.

[Instructor’s Note: You could have the students try part (vii), anyway, to see what happens.

Also, you could have them estimate the demand function by OLS, and compare the estimates

with the 2SLS estimates in part (iv). You could also have them compute the test of the single

overidentification condition.]

158

CHAPTER 17

TEACHING NOTES

I emphasize to the students that, first and foremost, the reason we use the probit and logit models

is to obtain more reasonable functional forms for the response probability. Once we move to a

nonlinear model with a fully specified conditional distribution, it makes sense to use the efficient

estimation procedure, maximum likelihood. It is important to spend some time on interpreting

probit and logit estimates. In particular, the students should know the rules-of-thumb for

comparing probit, logit, and LPM estimates. Beginners sometimes mistakenly think that,

because the probit and especially the logit estimates are much larger than the LPM estimates, the

explanatory variables now have larger estimated effects on the response probabilities than in the

LPM case. This may or may not be true.

I view the Tobit model, when properly applied, as improving functional form for corner solution

outcomes. In most cases it is wrong to view a Tobit application as a data-censoring problem

(unless there is true data censoring in collecting the data or because of institutional constraints).

For example, in using survey data to estimate the demand for a new product, say a safer pesticide

to be used in farming, some farmers will demand zero at the going price, while some will

demand positive pounds per acre. There is no data censoring here; some farmers find it optimal

to use none of the new pesticide. The Tobit model provides more realistic functional forms for

E(y|x) and E(y|y > 0,x) than a linear model for y. With the Tobit model, students may be tempted

to compare the Tobit estimates with those from the linear model and conclude that the Tobit

estimates imply larger effects for the independent variables. But, as with probit and logit, the

Tobit estimates must be scaled down to be comparable with OLS estimates in a linear model.

(See Equation (17.27); for an example, see Computer Exercise 17.10.)

Poisson regression with an exponential conditional mean is used primarily to improve over a

linear functional form for E(y|x). The parameters are easy to interpret as semi-elasticities or

elasticities. If the Poisson distributional assumption is correct, we can use the Poisson

distribution compute probabilities, too. But overdispersion is often present in count regression

models, and standard errors and likelihood ratio statistics should be adjusted to reflect this.

Some reviewers of the first edition complained about either the inclusion of this material or its

location within the chapter. I think applications of count data models are on the rise: in

microeconometric fields such as criminology, health economics, and industrial organization,

many interesting response variables come in the form of counts. One suggestion was that

Poisson regression should not come between the Tobit model in Section 17.2 and Section 17.4,

on censored and truncated regression. In fact, I put the Poisson regression model between these

two topics on purpose: I hope it helps emphasize that the material in Section 17.2 is purely about

functional form, as is Poisson regression. Sections 17.4 and 17.5 deal with underlying linear

models, but where there is a data-observability problem.

Censored regression, truncated regression, and incidental truncation are used for missing data

problems. Censored and truncated data sets usually result from sample design, as in duration

analysis. Incidental truncation often arises from self-selection into a certain state, such as

employment or participating in a training program. It is important to emphasize to students that

159

the underlying models are classical linear models; if not for the missing data or sample selection

problem, OLS would be the efficient estimation procedure.

160

SOLUTIONS TO PROBLEMS

17.1 (i) Let m

denote the number (not the percent) correctly predicted when y

= 0 (so the

prediction is also zero) and let m

be the number correctly predicted when y

= 1. Then the

proportion correctly predicted is (m

+ m

)/n, where n is the sample size. By simple algebra, we

can write this as (n

/n)(m

) + (n

/n)(m

) = (1 −

)(m

) +

), where we have used

the fact that

= n

/n (the proportion of the sample with y

= 1) and 1 −

= n

/n (the proportion

of the sample with y

= 0). But m

is the proportion correctly predicted when y

= 0, and m

is the proportion correctly predicted when y

= 1. Therefore, we have

+ m

)/n = (1 −

)(m

) +

If we multiply through by 100 we obtain

= (1 −

) +

⋅

where we use the fact that, by definition,

= 100[(m

+ m

)/n], = 100(m

), and =

100(m

(ii) We just use the formula from part (i):

= .30(80) + .70(40) = 52. Therefore, overall we

correctly predict only 52% of the outcomes. This is because, while 80% of the time we correctly

predict y = 0, y

= 0 accounts for only 30 percent of the outcomes. More weight (.70) is given to

the predictions when y

= 1, and we do much less well predicting that outcome (getting it right

only 40% of the time).

17.2 We need to compute the estimated probability first at hsGPA = 3.0, SAT = 1,200, and

study = 10 and subtract this from the estimated probability with hsGPA = 3.0, SAT = 1,200, and

study = 5. To obtain the first probability, we start by computing the linear function inside

Λ(⋅):

−1.77 + .24(3.0) + .00058(1,200) + .073(10) = .376. Next, we plug this into the logit function:

exp(.376)/[1 + exp(.376)] .593. This is the estimated probability that a student-athlete with

the given characteristics graduates in five years.

≈

For the student-athlete who attended study hall five hours a week, we compute –

1.77 + .24(3.0) + .00058(1,200) + .073(5) = .011. Evaluating the logit function at this value

gives exp(.011)/[1 + exp(.011)]

≈

.503. Therefore, the difference in estimated probabilities

is .593

− .503 = .090, or just under .10. [Note how far off the calculation would be if we simply

use the coefficient on study to conclude that the difference in probabilities is .073(10 – 5) = .365.]

17.3 (i) We use the chain rule and equation (17.23). In particular, let x

≡ log(z

). Then, by the

chain rule,

1111

(| 0,) (| 0,) (| 0,)1

Eyy Eyy Eyy

zxzx

∂

∂> ∂> ∂>

=⋅=⋅

∂∂∂∂

where we use the fact that the derivative of log(z

) is 1/z

. When we plug in (17.23) for

161

∂E(y|y > 0,x)/ ∂x

, we obtain the answer.

(ii) As in part (i), we use the chain rule, which is now more complicated:

1112

( | 0, ) ( | 0, ) ( | 0, )

xEyy Eyy Eyy

zxzx

∂

∂∂> ∂> ∂>

=⋅+

∂∂∂∂

xx x

⋅

∂

where x

= z

and x

. But ∂E(y|y > 0,x)/ ∂x

{1 −

(xβ/

)[xβ/

(xβ/

)]}, ∂E(y|y >

0,x)/δx

{1 −

(xβ/

)[xβ/

(xβ/

)]}, ∂x

/∂z

= 1, and ∂x

/∂z

= 2z

. Plugging these into

the first formula and rearranging gives the answer.

17.4 Since log(⋅) is an increasing function – that is, for positive w

and w

, w

> w

if and only if

log(w

) > log(w

) – it follows that, for each i, mvp

> minwage

if and only if log(mvp

) >

log(minwage

). Therefore, log(wage

) = max[log(mvp

), log(minwage

)].

17.5 (i) patents is a count variable, and so the Poisson regression model is appropriate.

(ii) Because

is the coefficient on log(sales),

is the elasticity of patents with respect to

sales. (More precisely,

is the elasticity of E(patents|sales,RD) with respect to sales.)

(iii) We use the chain rule to obtain the partial derivative of exp[

log(sales) +

RD +

] with respect to RD:

(|,E patents sales RD)

∂

= (

+ 2

RD)exp[

log(sales) +

RD +

A simpler way to interpret this model is to take the log and then differentiate with respect to RD:

this gives

+ 2

RD, which shows that the semi-elasticity of patents with respect to RD is

100(

+ 2

RD).

17.6 (i) OLS will be unbiased, because we are choosing the sample on the basis of an exogenous

explanatory variable. The population regression function for sav is the same as the regression

function in the subpopulation with age > 25.

(ii) Assuming that marital status and number of children affect sav only through household

size (hhsize), this is another example of exogenous sample selection. But, in the subpopulation

of married people without children, hhsize = 2. Because there is no variation in hhsize in the

subpopulation, we would not be able to estimate

; effectively, the intercept in the

subpopulation becomes

+ 2

, and that is all we can estimate. But, assuming there is variation

in inc, educ, and age among married people without children (and that we have a sufficiently

varied sample from this subpopulation), we can still estimate

, and

162