homework1 Econometrics Havard

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Harvard University
Department of Economics
Ec1126: Homework #1
Due 09/24
It is fine to discuss the questions with others, but you should write up your own solutions.
1. What is the best predictor of Y 2 given X = x under square loss?
2. Computer Problem: Download the dataset Mroz.dat from the website (there is also a
description of this data there). These data were used by Mroz in his study of the women’s
labor supply1.
(a) Read the data into Matlab.
(b) We are interested in the log of earnings for women who work (LFP=1). So, create a new
variable call it W = logWW which is the log of wage for women with positive market
wage. We will restrict our analysis below to this subpopulation.
(c) Provide summary statistics for both W and WE, the wife’s educational attainment
(here include, ten quantiles, standard deviations, and correlation). Try and program
these commands from scratch.
(d) Plot the best predictor of W given WE and another plot for W given WE and WE2
(here, plot these two functions on the same graph for WE = 5, . . . 17.
3. Suppose that Z1 and Z2 are binary random variables: Z1 takes on only the values 0 and 1,
and Z2 takes on only the values 0 and 1. Consider the (population) linear predictor of Y
given 1, Z1, Z2, Z1Z2 :
E∗(Y |1, Z1, Z2, Z1.Z2) = β0 + β1Z1 + β2Z2 + β3Z1Z2
(a) Does
E[Y |Z1, Z2] = E∗(Y |1, Z1, Z2, Z1.Z2)
Explain
(b) Suppose that data (yi, zi1, zi2) are available from a random sample of i = 1, ..., n indi-
viduals. The following four sample means have been tabulated:
y¯00, y¯01, y¯1,0, y¯1,1
where
y¯lm =
∑n
i yi1{zi1 = l, zi2 = m}∑n
i 1{zi1 = l, zi2 = m}
Use these means to provide an estimator for β3.
1“The Sensitivity of an Empirical Model of Married Women’s Hours of Work to Economic and Statistical Assump-
tions,” Thomas A. Mroz, Econometrica Vol. 55, No. 4 (Jul., 1987), pp. 765-799.
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Harvard University
Department of Economics
Ec1126: Homework #1
Due 09/24
4. Let Y = logearnings, Z1 = years−of −education, Z2 = years−of −work−experience. A
random sample of n individuals has provided the data (yi, zi1, zi2) for i = 1, ..., n. This data
has been used to obtain the following least squares fit:
yˆi = b0 + b1z1i + b2z2i + b3z
2
i1 + b4zi1zi2 + b5z
2
i2
(a) We are interested in the following partial (predictive) effect of education on log earnings:
θ = E(Y |Z1 = 16, Z2 = 20)− E(Y |Z1 = 12, Z2 = 20).
Explain how to use the least-squares estimates (the bs) to obtain an estimate of θ.
(b) Now consider the average partial effect:
γ = E[E(Y |Z1 = 16, Z2)E(Y |Z1 = 12, Z2)],
where the outer expectation is over the marginal distribution of Z2. Explain how to use
the least-squares estimates and the z data to obtain an estimate of γ.
5. Consider the following regression function:
E(Y |Z1, Z2) = θZ1 + g(Z2),
where the function g() is an unknown function, which is not restricted. This regression
function does impose the restriction that Z1 enters linearly and there is no interaction between
Z1 and Z2. The random variable Z2 is discrete and takes on only the values δ1, δ2, δ3. A
random sample provides the data (yi, zi1, zi2) for i = 1, ..., n. Suggest an estimator for θ,
based on a least-squares fit of y on x1, ..., xK , where
y =

y1
...
yn
 xi =

x1j
...
xnj
 j = 1, . . . ,K
Be explicit on how the xj are constructed from the data on (zi1, zi2) for i = 1, ..., n.
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