18 May 2021
Econ 607 Assignment 1: Linear Regression models
Due Date: 1 June 2021
This assignment carries 25 points (25% of the total score). If you are submitting handwritten answers, please write legibly, if not illegible answers will not be marked. Note that,
plagiarism is a serious academic o¤ence which may result in nulli…cation of your scores. If
you copy from one another, both who copied and who let his paper get copied will be given
a score of zero.
1). Consider the model
yi =
where
N ID (0;
i
2
1 xi1
+
2 xi2
+
3 xi2
+
i
) ; yi observable random variable and the xij ’s, j = 1; 2; 3, are
observable non-stochastic variables.
The data that follows is based on a sample of size n = 120 observations and gives the
sums of squares and cross-products of the indicated variables.
y
x1
x2
x3
y
195
30
10
10
x1
30
20
0
5
x2
10
0
20
5
x3
10
5
5
5
a). Compute the best linear unbiased estimates of the coe¢ cients.
(3 points)
b). Give a 95% con…dence interval for
1
2.
(2 points)
c). Test the hypothesis that H0 :
2
= 2 and
1
+
2
+
hypothesis that HA : not H0 at 95% con…dence level.
3
= 1 against the alternative
(4 points)
2). Let x1 ; x2 ; : : : ; xn be a random sample from a normal distribution with mean
variance
2
.
1
and
a). Find the maximum likelihood estimator of
2
= 2.
(4 points)
b). Find the asymptotic distribution of the maximum likelihood estimator of obtained
in part (a).
(4 points)
3). Consider the following saving function,
si =
where ui =
p
1
+
yi i ; E ( i ) = 0 and var ( i ) =
2 yi
+ ui
2
a). Show that E (ui j yi ) = 0
(2 points)
b). Show that (conditional) homoscedasticity assumption is violated.
(2 points)
c). Provide an expression for E (u2i j yi ).
(2 points)
d). Discuss the "economic" meaning of the expression you got in (iii).
(2 points)
Bonus: Question 4 is a bonus and you are not required to do it. However, if
you provide a full answer to this question, you will be given a bonus of 5 points.
No point for an attempt!
4). Consider a K variables linear regression model in which the variance of the MLE b of
the true parameter vector
0
is var b and the information matrix
I ( 0) =
E
@ 2 ln L ( 0 )
@ @ 0
Show that
in the sense that var b
var b
I ( 0)
1
I ( 0)
1
0,
psd
is a positive semi-de…nite (psd) matrix. (5 points)
2