Econ 166
Joseph G. Altonji
Spring 2007
Problem Set 1
Due Tuesday, January 30th, in class.
Collaboration: You are free to discuss the problem set with one other
student. However, you have to do your own computing in STATA and write up
your own answers.
Do not hand in reams of undigested computer logs.
Notation: I sometimes use lower case b rather than
ˆ
β
to denote OLS
coefficient estimates or estimators.
I sometimes use lower case e to denote
residuals.
1. The distinction between the stochastic error term (u
i
) and the residual
(e
i
) is a very important one.
a. Define both in words.
b. Usually we can’t observe the error term, but we can get around this
difficulty if we assume values for the true coefficients. Suppose that the
true
β
0
= 2 and the true
β
1
equals 1.5, so that the population regression
function is
E(y
i
x
i
) = 2 + 1.5x
i
Suppose also that the sample estimate of
β
0
is 0.0 and the sample estimate of
β
1
is 1.
b.1) Write down the sample regression function.
b.2 Fill in the following table, which is for the first four observations.
y
i
x
i
E(y
i
x
i
)
Y_hat
i
β
0
b
0
+
(
β
1
b
1
)x
u
i
e
i
1
2
3
3
2.5
1
6
4
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2
b.3 Verify for the first two data points that the residual e
i
is the sum
of the true error u
i
and the estimation error
β
0
b
0
+ (
β
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 Spring '07
 JosephAltonji
 Econometrics, Regression Analysis, sample regression function, population regression function, OLS coefficient estimates, STATA command Banks

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