homework2 2 Econometrics Havard

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Harvard University
Department of Economics
Ec1126: Homework #2
Due 10/08
It is fine to discuss the questions with others, but you should write up your own solutions.
1. This part is based on a sample of N = 815 observations from the Young Mens Cohort of
the National Longitudinal Survey. The variables are usual weekly earnings in 1980 (uwe),
age in 1980 (age80), years of schooling completed (educ), fathers education (fed), mothers
education (med), and an IQ score (iq). There is an additional test score (kww) that will be
discussed below. Once you are in Matlab, type the command (you can use other programming
languages)
load nls
Now the series uwe, age80, educ, fed, med, iq, kww will be available. We will be using a
(potential) labor market experience variable defined as age - schooling - 6.
(a) Calculate the least squares regression of log(earnings) on a constant, schooling, experi-
ence, and experience squared. (Multiply the coefficients by 100 to make them easier to
read.)
(b) Now add IQ to the regression and calculate the coefficients. Show how the results so far
are sufficient to calculate the coefficient on schooling in a regression of IQ on a constant,
schooling, experience, and experience squared. Then run this third regression to check
your answer.
(c) The IQ coefficient in 1b can be obtained from a simple regression of log(earnings) on a
single variable w. What is w? Construct w and run the regression to check your answer.
(d) Calculate the coefficients in a regression of log(earnings) on a constant, schooling, ex-
perience, experience squared, IQ, fathers education, and mothers education. Discuss
the magnitudes of the coefficients. (The IQ scores are constructed to have a normal
distribution with a mean of 100 and a standard deviation (=
√
V (IQ)) of 15.)
(e) Compare the schooling coefficient in 1d with the schooling coefficient from the regression
in 1a, which did not include family background variables or IQ. Discuss the magnitude
of the change in the coefficient.
(f) The sample individuals were first surveyed in 1966 at ages 14-24, and resurveyed at one
or two year intervals thereafter. A test on Knowledge of the World of Work (kww) was
administered as part of the initial survey in 1966. We shall interpret the IQ score, which
was obtained from school records, as a measure of early ability that is not affected by
the amount of schooling completed. (Think of the IQ test as being administered in the
third grade, with everyone in the sample completing at least the sixth grade.) The kww
test, however, could be influenced by the amount of schooling that the individual had
as of 1966, when the test was administered.
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Harvard University
Department of Economics
Ec1126: Homework #2
Due 10/08
i. Add kww to the regression in 1d and calculate the coefficients.
ii. Suppose that an individual wants to use our results to determine how additional
school- ing would affect his earnings. Which of the various regressions we have run
would be most relevant? Explain your reasoning.
2. Refer to Tables 1, 2, and 3 in Card and Krueger, “Does School Quality Matter? Returns
to Education and the Characteristics of Public Schools in the United States,” Journal of
Political Economy, 1992, 140. The tables and the article are posted on the class website in
Homework 2 folder. Consider the following regression function:
E(Y |S,EX,B,C) =
3∑
k=1
α1kEX + α2kEX2 + 49∑
j=1
γjk1(B = j) +
49∑
j=1
λjkS.1(B = j)
 1(C = k)
The variables are Y = log of earnings, S = years of education, EX = years of potential work
experience, B = state of birth, and C = birth cohort. The notation 1(B = j) is for an indicator
function that equals 1 if B = j and equals zero otherwise.
(a) Explain how to interpret differences like (λ52 − λ12) and (λ53 − λ52).
(b) Explain how to express this regression function as a linear predictor:
E(Y |S,EX,B,C) = β1X1 + . . .+ βKXK
What are the variables X1, . . . , XK?
(c) Consider obtaining estimates b1, ..., bK of β1, . . . , bK from a least-squares fit. Explain
how to set up the data matrices corresponding to Y,X1, . . . , XK .
(d) How would you obtain the estimated returns to education, as in Table 2, from these
estimates?
3. Midterm Spring 2015 Interpreting Coefficient in Linear Regressions.
Earnings functions attempt to find the determinants of earnings, using both continuous and
binary variables. One of the central questions analyzed in this relationship is the returns to
education.
(a) (6pts) To shed light on this question, you think of running the following regression to
examine this predictive relationship:
E∗[logEarn|1, Educ] = β0 + β1Educ
where Earn is average hourly earnings and Educ is years of education. What is the
(predictive) effect of an increase in 1 year of education on earnings?
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Harvard University
Department of Economics
Ec1126: Homework #2
Due 10/08
(b) (6pts) Now, suppose that you are now interested in the relationship between education
and earnings for males and females. So, your research assitant suggests running the
following regression:
E∗[logEarn|1, Educ] = β0 + β1Educ+ β2S
where S is a dummy variable that takes the value of 1 for females and zero otherwise.
How would you interpret β1 in this regression compared to β1 in part (a). Be brief.
(c) (6pts) Now, your data contains information from which you get the following
̂log(Earn) = .54
.14
+ .083
.011
× Educ
(Throughout this question, the standard errors are given underneath the coefficient
estimates)
What is the (predictive) effect of an additional year of schooling? If you had a strong
belief that years of high school education were different from college education, how
would you modify the equation?
Statistical significance is an indication whether the confidence interval for a coefficient
contain zero. So, Educ is statistically significant at the 95% level if the confidence interval
for the coefficient on Educ, β1 does not contains zero. Construct such a confidence
interval (assume that the sample size is large enough so that you can use LLN and CLT
approximations). Is Educ statistically significant?
(d) (6pts) You read in the literature that on-the-job training matters in predicting earnings.
To approximate on-the-job training, researchers often use the so called Mincer experience
variable, which is defined as Exper = Age− Educ− 6. You incorporate the experience
variable into your original regression
̂ln(Earn) = −.01
.16
+ .101
.012
× Educ+ .033
.006
× Exper − .0005
.0001
× Exper2
Carefully interpret the coefficient on Education. What is the effect of an additional year
of experience for a person who is 40 years old and had 12 years of education? What
about for a person who is 60 years old with the same education background? How does
this return to Educ changes between this regression and the one in (d) above?
(e) (6pts) You want to find the effect of introducing two variables, gender and marital status.
Accordingly you specify a binary variable that takes on the value of one for females and
zero otherwise (Female), and another binary variable that is one if the worker is married
but is zero otherwise (Married). Adding these variables to the regressors results in:
̂ln(Earn) = .21
.16
+ .093
.012
× Educ+ .032
.006
× Exper − .0005
.0001
× Exper2
− .289
.049
× Female+ .062
.056
Married
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Harvard University
Department of Economics
Ec1126: Homework #2
Due 10/08
Is the Married statistically significant? what about Female? Can you comment on this.
(f) (10pts) In your final specification, you allow for the binary variables to interact. The
results are as follows:
ln(Earn) = .14
.16
+ .093
.011
× Educ+ .032
.006× Exper − .0005
.0001
× Exper2
− .158
.075
× Female+ .173
.080
Married− .218
.097
(Female×Married)
Interpret the coefficient on Female × Married. Is Female × Married statistically
significant? Comment.
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