Final S15_Answers.pdf

About you: Points:

Question 1 _________




Total _________

Name: Answer key

Student ID#:

Econ 141, U.C. Berkeley, Spring 2015

Final Exam time 2 hours and 45 minutes

This final contains four questions for a total of 100 points. The points for every question and sub-question are listed

on the exam. After each sub-question there is a box that you can use to fill in your answer. Apart from the answers

that you are explicitly asked to fill in in a table or draw in a figure, the content of the answer boxes is all that will

be graded. Hand in the exam at the end of the 2 hours and forty five minutes. Please do not forget to fill in your

name and student ID on this front page! THIS IS A CLOSED BOOK EXAM AND THE USE OF CALCULATORS

IS NOT ALLOWED. BRIEFLY EXPLAIN ALL YOUR ANSWERS. ANSWERS WITHOUT

EXPLANATIONS WILL NOT GET CREDIT.

Question 1: Seemingly Unrelated Regressions and Panel OLS (25 points)

In this problem, we consider the estimation of the following regression model, written in matrix

notation,

= + , where (1)

= [1 2

], = [1 2

] , = [1 2

], and = [1 2

] (2)

Here, 1 and 1 are (1 1)-vectors, 1 is an (1 ( + 1))-matrix and, equivalently, 2 and 2

are (2 1)-vectors, 2 is an (2 ( + 1))-matrix. The matrix is the null-matrix, filled with all

zeros. The parameter vectors 1 and 2 are both of dimension (( + 1) 1). Moreover, the

covariance matrix of the residuals equals [|] = 21+2. The normal OLS assumptions hold

in this model.

U.C. BERKELEY, ECON 141, SPRING 2015, FINAL

2

a) (5 points) Show that the OLS estimator of amounts to OLS per equation. That is, 1 is the OLS

estimator obtained by regressing 1 on 1 and 2 is the OLS estimator obtained by regressing 2

on 2.

Hint: Use the fact that the inverse of a block diagonal matrix satisfies [

]1

=[1 1

].

The OLS estimator in this case is given by

= ()1() Filling in the definitions of and above, yields

= ([1 2

]

[1 2

])1

[1 2

]

[12

] = [1

1 2

2]

1

[1

1 2

2]

Using the hint about the inverse of a block diagonal matrix provided in the question, we can write

= [(1

1)1

(2 2)

1] [1

1 2

2] = [

(1 1)

11 1

(2 2)

12 2

] = [12

].

This is what you were asked to show.

b) (2 points) Suppose, instead, that the covariance matrix of the residuals equals

[|] = 21+2 (3)

Would the OLS estimator from part a) still be consistent and unbiased in this case?

Yes (well, formally yes potentially), this general form of heteroskedasticity does not in all cases

affect the consistency and unbiasedness of the OLS estimator, but it would make the estimator

inefficient.


3

c) (5 points) Under the assumption that [|] = 21+2, what is the efficient estimator

(in terms of being the Best Linear Unbiased Estimator) of ?

Under this assumption that BLUE is the GLS estimator, which is the OLS estimator for the

transformed homoskedastic linear system

1/2 = 1/2 + 1/2 and yields

= (1)11

Because it is the BLUE in this transformed system, it is the BLUE for the linear regression model

that we consider in this subquestion.

d) (5 points) Suppose again that [|] = 21+2. Show that the restricted OLS estimator, ,

under the restriction that 1 = 2 equals

= 11 + 22, where = (1 1 + 2

2)1(

), for = 1,2,

where 1 and 2 are the OLS estimates from part (a).

Hint: Redefine the matrix to reflect that both parameter vectors are the same.

In principle, one could apply restricted least squares here, but that turns out to be rather

cumbersome. The easiest way to show this is to rewrite the model under the restriction. In that case

the model can be written as

= [12

] + ,

such that

= ([1 2

] [12

])1

[1 2

] [12

] = (1 1 + 2

2)1(1

1 + 2 2).

We can write this as

= (1 1 + 2

2)1(1

1)(1 1)

11 1+(1

1 + 2 2)

1(2 2)(2

2)12

1, which is

= (1 1 + 2

2)1(1

1)1+(1 1 + 2

2)1(2

2)2 This is what we are asked to show.


4

e) (5 points) How are the matrices, 1 and 2, related to the estimated variance-covariance

matrices of the OLS estimators 1 and 2?

The covariance matrices of 1 and 2 are given by 1 = 2(1

1)1 and 2 =

2(2 2)

1,

respectively. The covariance matrix of is given by = 2(1

1 + 2 2)

1.

So, this means that

= 1, for = 1,2.

f) (3 points) Give an intuitive interpretation of the result in parts (d) and (e). In what way can the

matrices 1 and 2 be interpreted as weighting matrices and as a weighted average of

1 and 2?

The above result suggests that is a weighted average of 1 and 2, where the relative weights are inversely related to the covariance matrix of the estimator. That is, the more precisely estimated

of the two gets the highest weight in the restricted estimator 2.


5

Question 2: Instrumental variables with a simple random treatment (35 points)

In this problem we the study application of an instrumental variable in the simple linear regression

model of the form

= 0 + 1 + . (4)

The sample consists of independent, identically distributed observations but the problem is that

[|] 0 for all and OLS would yield an inconsistent estimate of 1. Fortunately, there is an

instrumental variable, {0,1}, which is an indicator variable and has the properties that

[| = 1] [| = 0] and that [| = 1] = [| = 0] = 0.

Throughout the rest of this problem we consider the case of a sample = 1, , , where

1 = =1 and 0 = 1. (5)

Moreover, besides the standard sample means, and , we consider the conditional sample means

0 =1

0 (1 )

=1 , 1 =

1

1

=1 , 0 =

1

0 (1 )

=1 , and 1 =

1

1

=1 . (6)

This allows us to write the sample means of and as

=00+11

, and =

00+11

. (7)

Using this notation, we derive the two-stage least squares estimator of 1 in this problem.

First-stage regression The first step of the two-stage least squares procedure in this case would be

to estimate the first-stage regression

= 0 + 1 + (8)

a) (5 points) Show that the OLS estimator, 1, in this first-stage regression equals 1 = 1 0.

The OLS estimator, 1, in this case is given by

1 =

1

( )( )=1

1

( )2=1

=

1

=1

1

2

=1 2

=

1 1

1

2=

1 1

1

(1 )

Continued on next page


6

Question 2, part a), continued

Here in the second step in the denominator, we have used that because (0,1), we have =

2, such that

1

2=1 =

1

=1 = . This now allows us to write

1 =

1 1

1

(1 )=

1 (1

1 ) 1

1

0 0

1 (1

1 )

=

1

0 1

1

0 0

1

0

= 1 0

This is the result we are asked to derive.


7

b) (3 points) Show that the OLS estimator of 0 in the first-stage regression is 0 = 0.

The OLS estimator 0 equals

0 = 1 = (0

0 +1

1) 1

(1 0) = (0

+1

) 0 = 0

c) (2 points) Show that the fitted value of based on the first-stage regression equals

= {0 if = 0

1 if = 1 (9)

The fitted value of the first-stage regression equals

= 0 + 1

Thus, if = 1, then

= 0 + 11 = 0 + (1 0) = 1

and if = 0, then

= 0 + 10 = 0

There you are! Note that this implies that = . This will come in handy later on.


8

Second-stage regression The next step is to run the second-stage regression by estimating 0 and 1

using OLS in the following regression model

= 0 + 1 + (10)

d) (5 points) What happens in this regression if 0 = 1? Why would we expect this not to be the

case as gets large?

If 0 = 1 then is constant across all observations . Consequently, the explanatory variable and

constant term are linearly dependent and the regression suffers from multicollinearity.

We do not expect this to happen as , because from the LLN, we know that

plim

0 = [| = 0] [| = 1] = plim

1.

Thus, as the conditional sample means 0 and 1 will with certainty converge to different

numbers and thus not be equal.

e) (10 points) Show that the 2SLS estimator of 1, that is obtained from estimating 1 using OLS in

this second-stage regression equals

1,2 =10

10 (11)

First, note that = . This will come in handy later on.

The 2SLS estimator in this case equals

1,2 =

1

( )( )=1

1

( )2

=1

=

1

=1

1

2

=1 2

Lets first take a closer look at the denominator

Continued on the next page

Question 2, part e) continued


9

1

2=1

2 =0

0

2 +1

1

2 2 =0

0

2 +1

1

2 (0

0 +

1

1)

2,

Which can be simplified by FOILING away. That is, this equals

0

(1

0

) 0

2 +1

(1

1

) 1

2 20

1

01 =

0

1

(0 1)

2.

Now, lets focus on the numerator. We can write

1

=1 =

0

00 +1

11 (0

0 +1

1) (0

0 +1

1)

This is a similar FOIL, but slightly more complicated than that for the denominator. The above

equals

0

(1

0

) 00 +

1

(1

1

) 11

0

1

(01 + 10) =

0

1

(0 1)(0 1).

Combining the two results above, we obtain that

1,2 =

1

=1

1

2

=1 2

=

0

1

(0 1)(0 1)

0

1

(0 1)2=

(0 1)

(0 1)


10

f) (5 points) Give an intuitive interpretation of the result you derived in part e)

One can think of as whether an observation gets treated or not and 1 as the average of the

explanatory variable under treatment and 0 as the average in the absence of treatment. Thus,

(1 0) is the average difference in the effect of treatment on the explanatory variable. Think of

this as the intensity of the treatment. (1 0) is the average difference in outcomes between the

treated and untreated. Thus, the 2SLS estimator of the marginal effect of an increase in the intensity

of treatment is the difference in outcomes divided by the difference in the intensity of treatment.

g) (5 points) Suppose that the instrumental variable is a fully random treatment. This means that

for any set of potentially omitted variables, , in the regression equation we estimate, where

= 0 + 1 + + , (12)

the instruments have the property that [| = 1] = [| = 0] = . Would the

instrumental variable estimator we derived in parts a) through e) still be consistent when we ignore

these omitted variables? Briefly explain, without mathematical derivations, why or why not.

If [| = 1] = [| = 0] = , then the omitted variables are uncorrelated with the instrument . That is, the instrument satisfies the exogeneity condition with respect to . Thus, for the purposes of the instrumental variables regression, has the same property as and it is thus effectively another residual in this regression.

Thus, because the instrument is exogenous with respect to the omitted variables means that the

2SLS estimator we derived would still be consistent.


11

Question 3: Cause and Defect (20 points)

In August 2009, the Economist Magazine published the following article about the use of instrumental

variables in the economics profession. Read the article and answer the questions about it below.

Instrumental variables help to isolate causal relationships. But they can be taken too far

Aug 13th 2009 | Economist Magazine

Like elaborately plumed birdswe preen and strut and display our t-values. That was Edward Leamer's uncharitable description of his profession in 1983. Mr Leamer, an economist at the

University of California in Los Angeles, was frustrated by empirical economists' emphasis on

measures of correlation over underlying questions of cause and effect, such as whether people who

spend more years in school go on to earn more in later life. Hardly anyone, he wrote gloomily,

takes anyone else's data analyses seriously. To make his point, Mr Leamer showed how different (but apparently reasonable) choices about which variables to include in an analysis of the effect of

capital punishment on murder rates could lead to the conclusion that the death penalty led to more

murders, fewer murders, or had no effect at all.

In the years since, economists have focused much more explicitly on improving the analysis of

cause and effect, giving rise to what Guido Imbens of Harvard University calls the causal literature. The techniques at the heart of this literaturein particular, the use of so-called instrumental variableshave yielded insights into everything from the link between abortion and crime to the economic return from education. But these methods are themselves now coming under

attack.

Instrumental variables have become popular in part because they allow economists to deal with one

of the main obstacles to the accurate estimation of causal effectsthe impossibility of controlling for every last influence. Mr Leamer's work on capital punishment demonstrated that the choice of

controls matters hugely. Putting too many variables into a model ends up degrading the results.

Worst of all, some relevant variables may simply not be observable. For example, the time someone

stays in school is probably influenced by his innate scholastic ability, but this is very hard to

measure. Leaving such variables out can easily lead econometricians astray. What is more, the

direction of causation is not always clear. Working out whether deploying more policemen reduces

crime, for example, is confused by the fact that more policemen are allocated to areas with higher

crime rates.

Instrumental variables are helpful in all these situations. Often derived from a quirk in the

environment or in public policy, they affect the outcome (a person's earnings, say, to return to the

original example) only through their influence on the input variable (in this case, the number of

years of schooling) while at the same time being uncorrelated with what is left out (scholastic

ability). The job of instrumental variables is to ensure that the omission of factors from an

analysisin this example, the impact of scholastic ability on the amount of schoolingdoes not end up producing inaccurate results.


12

In an influential early example of this sort of study, Joshua Angrist of the Massachusetts Institute of

Technology (MIT) and Alan Krueger of Princeton University used America's education laws to

create an instrumental variable based on years of schooling. These laws mean that children born

earlier in the year are older when they start school than those born later in the year, which means

they have received less schooling by the time they reach the legal leaving-age. Since a child's birth

date is unrelated to intrinsic ability, it is a good instrument for teasing out schooling's true effect on

wages. Over time, uses of such instrumental variables have become a standard part of economists'

set of tools. Freakonomics, the 2005 bestseller by Steven Levitt and Stephen Dubner, provides a

popular treatment of many of the techniques. Mr Levitt's analysis of crime during American election

cycles, when police numbers rise for reasons unconnected to crime rates, is a celebrated example of

an instrumental variable.

Two recent papersone by James Heckman of Chicago University and Sergio Urzua of Northwestern University, and another by Angus Deaton of Princetonare sharply critical of this approach. The authors argue that the causal effects that instrumental strategies identify are

uninteresting because such techniques often give answers to narrow questions. The results from the

quarter-of-birth study, for example, do not say much about the returns from education for college

graduates, whose choices were unlikely to have been affected by when they were legally eligible to

drop out of school. According to Mr Deaton, using such instruments to estimate causal parameters

is like choosing to let light fall where it may, and then proclaim[ing] that whatever it illuminates is what we were looking for all along.

IV leagues

This is too harsh. It is no doubt possible to use instrumental variables to estimate effects on

uninteresting subgroups of the population. But the quarter-of-birth study, for example, shone light

on something that was both interesting and significant. The instrumental variable in this instance

allows a clear, credible estimate of the return from extra schooling for those most inclined to drop

out from school early. These are precisely the people whom a policy that sought to prolong the

amount of education would target. Proponents of instrumental variables also argue that accurate

answers to narrower questions are more useful than unreliable answers to wider questions.

A more legitimate fear is that important questions for which no good instrumental variables can be

found are getting short shrift because of economists' obsession with solving statistical problems. Mr

Deaton says that instrumental variables encourage economists to avoid thinking about how and why things work. Striking a balance between accuracy of result and importance of issue is tricky. If economists end up going too far in emphasising accuracy, they may succeed in taking the con out of econometrics, as Mr Leamer urged them toonly to leave more pressing questions on the shelf.


13

a) (5 points) According to Mr Deaton, using such instruments to estimate causal parameters is like

choosing to let light fall where it may, and then proclaim[ing] that whatever it illuminates is

what we were looking for all along.. Does Angus Deaton criticize internal or external validity of

many instrumental variable studies in this quote? Explain your answer.

Angus Deaton criticizes the external validity of instrumental variable studies. He does so by

suggesting that the results that IV studies come up with shine light on issues that are not necessarily

helpful in answering broader research questions economists are interested in. He is critical of the

claim by researchers that do IV regressions that their results are applicable beyond what they

illuminate, i.e. of their claims of external validity beyond their narrow illuminated results.

b) (5 points) Mr Leamer showed how different (but apparently reasonable) choices about which

variables to include in an analysis of the effect of capital punishment on murder rates could lead

to the conclusion that the death penalty led to more murders, fewer murders, or had no effect at

all.. What type of bias in OLS estimates did Ed Leamer illustrate with this example about the

effect of capital punishment on murder rates?

Mr. Leamer illustrated omitted variable bias, where the estimated coefficient of interest is biased

because of it partially picking up effects of variables that are not included in the regression and that

are correlated with the explanatory variable of interest.


14

c) (5 points) A more legitimate fear is that important questions for which no good instrumental

variables can be found are getting short shrift because of economists' obsession with solving

statistical problems. What two main properties does a good instrumental variable have to

have?

A good instrumental variable needs to be (i) relevant in that it is (highly) correlated with the

endogenous variable of interest that it instruments for, and (ii) exogenous in that it is uncorrelated

with the residuals in the regression that the endogenous explanatory variable of interest is correlated

with.

In a comment on this article online a reader remarked

Why do you cite Levitt(1997) as the celebrated example of IV?

That paper is wrong due to computer programming error. Election cycle is not a valid instrument.

d) (5 points) What does the reader mean by That paper is wrong due to computer programming

error?

Justin McCrary points out, in a replication study, that Levitt has inverted the weights in his

weighted least squares regression in which he aimed to correct for heteroskedasticity. With the correct

weights the results he emphasizes are not significant anymore.


15

Question 4: Fertility and the labor supply decisions of married women (20 points)

In the paper Children and Their Parents' Labor Supply: Evidence from Exogenous Variation in

Family Size Joshua Angrist and Bill Evans are interested in estimating the effect of number of

children on a married womans labor supply decision.

Let be a measure of the labor supply and be a measure of fertility, then Angrist and Evans are

interested in the causal effect of fertility on labor supply. The particular measure of fertility that

Angrist and Evans use is whether or not a woman has more than two children. That is, they would

like an unbiased estimate of the parameter in the simple regression model

= 0 + 1 + + , (13)

where the vector contains a set of demographic variables.

They use four different measures of labor supply: (i) whether the woman worked for pay, (ii) the

number of weeks she worked in the previous year, (iii) the number of hours per week she worked,

and (iv) her labor income. They also look at two samples: The 1980 and 1990 PUMS data.

a) (4 points) Explain why omitted variable bias is a likely source of violations of internal validity

that biases the OLS estimate of 1 in the above regression.

Whether or not a woman has more than one child depends on many variables that also affect her

labor market opportunities. For example, fertility rates are generally declining in education level

while labor market opportunities, and the induced observed labor supply, are increasing.

Just like the example for education, most potential omitted variables in this case tend to have an

opposite effect on fertility as on labor market opportunities and labor supply.

Such variables would bias tend to bias the estimated coefficient 1 in the above regression downwards. This would be a violation of internal validity.

Continued on next page


16

Question 3, part a), continued

Angrist and Evans then consider whether the likelihood of a woman having another child depends on

the gender composition of children she already has. They do so because the gender of children can be

considered as random. Their aim is to use this gender composition as an instrumental variable for

their fertility variable . They report the table below.


17

The top part of the table reports the likelihood of a woman having another child, after having one

child already, conditional on the gender of the first child. The bottom half reports the likelihood of a

woman having another child, after she already had two children, conditional on the gender of the first

two.

b) (4 points) Based on this table, explain why Angrist and Evans do not use whether or not a woman

has more than one child as the fertility measure, , in their regression.

The top of the table suggests that the gender of the first child is not correlated with the probability

that a woman has a second child. That is, the estimated probabilities that a woman has a second child

are not significantly different when we condition on the gender of the first child.

This means that the gender of the first child is not a relevant instrument for the endogenous fertility

variable that a woman has a second child. So, for that endogenous variable Angrist and Evans do not

have a good instrument.

Angrist and Evans then run the first-stage regression, where they estimate

= +

+ (14)

The set of instruments that they include in the vector are about the gender composition of the first

two children. That is includes (i) whether first child was a boy, (ii) whether the second child is a

boy, (iii) whether the first two children are of the same gender, (iv) whether the first two children are

boys, and (v) whether the first two children are girls.

Of course, (iii), (iv), and (v) are multicollinear and are never jointly included in any specification.

In fact, Angrist and Evans either include (iii) or (iv) and (v). The table on the next page shows the

first-stage regression results.


18

c) (4 points) If you would like to advise Angrist and Evans to make the case that the instruments are

relevant more directly, which additional test-statistic would you suggest them to include in this

table? Why? What information would it add to what is already reported.

What is of interest is whether the potential instrument explain a significant part of the variation in

the fertility variable. The best way to test for this an F-test for the joint hypothesis that all coefficients

on the instruments in this first-stage regression are zero. They do not report such a test and since they

include other covariates, it is not implied by the reported 2.

Continued on the next page


19

Question 4, part c) continued.

The table below shows the results of the second-stage regression from Angrist and Evans. They run

their second-stage regression with the labor supply variables for unmarried women, married women,

as well as the husbands of married women.


20

d) (4 points) What does the fact that the estimated labor supply response reduces when estimated

with 2SLS rather than OLS tell you about properties of the omitted variables that you discussed

in part a)?

If the 2SLS estimate is consistent and the OLS estimate is biased, then this suggests that the OLS

coefficient has a negative bias and that omitted variables that tend to increase labor supply tend to

reduce fertility, as discussed in my answer to part a).

In the discussion of their results Angrist and Evans note

e) (4 points) How does this paragraph address the criticism of Angus Deaton from part a) of Question

3?

Angrist and Evans use this paragraph to make the point that the fertility effect on labor supply

that this study analyzes illuminates a part of individuals labor supply decisions that accounts for a

substantial part of the aggregate movements in labor supply. Thus, though, to some extent the light

fell randomly on more-than-two children, this turns out to be a substantial part of the overall labor

supply effect between 1970 and 1990.

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