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Introduction to Econometrics (Econ 107) Signature:
Drake University, Spring 2008
William M. Boal Printed name:
MIDTERM EXAMINATION #1 VERSION A
“Introduction and Statistics Review”
February 14, 2008
INSTRUCTIONS: This exam is closed-book, closed-notes. You may use a calculator on this
exam, but not a graphing calculator or a calculator with alphabetical keys. Point values for each
question are noted in brackets. A table of the t-distribution is attached.
I. MULTIPLE CHOICE: Circle the one best answer to each question. Feel free to use
margins for scratch work [3 pts each—42 pts total]
(1) Which of the following is not necessarily
true?
a. ^ ^.
i i
x x
b. ^ ^ .
(^2 ) x x x nx i i
c. (^) ^ ^ ^ ^ ^ . i i i i
xy x y
d. 0.
x x i
e. x nx. i
(2) ^ ^
n
i
i
x
1
2 5
a. ^
n
i
i
x
1
b.
n
i
i
x
1
c.
n
i 1
d. ^
n
i
i
x
1
2 5 (^).
e.
n
i
i
x
1
(3) Suppose we wish to fit the equation
y = 1
2 x to data by the method of least
squares. This method minimizes which
function of the data?
a. ^ ,^ ^ ^ .
1 2 ^12
i i
f y x
b. ^ ,^ ^ ^ ^ .
2
1 2
2
1 2
i i
f y x
c. ^ ,^ ^ ^ ^.
2
1 2 ^12
i i
f y x
d. ^ ,^ ^.
1 2 1 2
i i
f y x
e. ^ ,^ ^ ^ ^.
2
1 2 1 2
i
f x
The next two questions assume the
following. Suppose X is a Bernoulli
random variable, with Prob{X=1} = 0.2 and
Prob{X=0} = 0.8.
(4) The mean or expected value of X is
a. zero.
b. 0..
c. 0..
d. 0..
e. one.
Drake University, Spring 2008 Page 2 of 7
(5) The variance of X is
a. zero.
b. 0..
c. 0..
d. 0..
e. one.
(6) Which of the following distributions
does not have a symmetric bell-shaped
density function?
a. normal distribution.
b. t distribution.
c. chi-square distribution.
d. all of the above have symmetric bell-
shaped density functions.
e. none of the above have symmetric bell-
shaped density functions.
The next two questions assume the
following. Suppose a random sample of
size n is drawn from some population. The
population has mean and variance
2 .
Consider the sample mean, defined as
n
i
i
X
n
X
1
(7) Var( (^) X ) =
a. zero.
b. one.
c.
2 .
d.
2 / (n-1).
e.
2 / n.
f.
2 / n
2 .
(8) E( X ) =
a. zero.
b. one.
c. .
d. / (n-1).
e. / n.
f. / n
2 .
(9) An estimator (^) ˆ of an unknown
population parameter is said to be
unbiased if
a.
lim
MSE
n
b.
ˆ E.
c. 0
ˆ E .
d.
lim E
n
e. ^ ^0
lim Prob
n
, for all > 0.
(10) An estimator ˆ of an unknown
population parameter is said to be
asymptotically unbiased if
a.
ˆ E.
b. 0
ˆ E .
c.
lim
E
n
d.
lim E
n
e. ^ ^0
lim Prob
n
, for all > 0.
(11) The principle for finding an estimator
that uses the joint density function of the
sample is called
a. the method of maximum likelihood.
b. the method of moments.
c. the scientific method.
d. Newton’s method.
(12) A 90 percent confidence interval is
necessarily
a. narrower than a 95 percent confidence
interval.
b. wider than a 95 percent confidence
interval.
c. identical to a 95 percent confidence
interval.
d. answer cannot be determined from the
information given.
Drake University, Spring 2008 Page 4 of 7
(1) [Least-squares calculation: 15 pts] Suppose we have three observations on xi and yi as
shown in the graph below.
a. Compute
2
, the least-squares estimate of the slope of the line y = 1
2 x.
b. Compute
1
, the least-squares estimate of the y-intercept of the same line.
c. Compute the three fitted values
y ˆ
i of this least-squares estimated regression line.
d. Compute the three residuals
i of this estimated least-squares regression line.
e. Sketch the least-squares estimated line in the graph above.
Drake University, Spring 2008 Page 5 of 7
(2) [Moments: 12 pts] Suppose X 1 and X 2 are random variables with the following moments.
E(X
1 ) = 7 Var(X 1 ) = 2 Cov(X 1
,X
2
E(X 2 ) = 2 Var(X 2 ) = 8
Now let Y = X 1 + 2X 2. Compute the following and circle your final answers.
a. Compute E(Y).
b. Compute Var(Y).
c. Compute SD(Y).
d. Compute Corr(X 1 ,X 2 ).
Drake University, Spring 2008 Page 7 of 7
(4) [Inference for arbitrary distribution, large sample: 18 pts] Suppose we wish to analyze the
distribution of the number of cars per household in a population. Let denote the unknown
true population mean number of cars per household. Observations X i have been collected on
300 households, with the following summary values. Here, (^) X is the sample mean.
420 2601
300
1
2
300
1
i
i
i
i
X X X
a. [3 pts] Is the population distribution discrete or continuous? Justify your answer.
b. [3 pts] Compute an unbiased estimate of .
c. [3 pts] Compute the standard error of your estimate of .
d. [3 pts] Compute a 95% asymptotic confidence interval for .
e. [6 pts] Test the null hypothesis that = 1 against the one-sided alternative hypothesis
that > 1, at 5% significance using an asymptotic test. Give the value of the test
statistic, the critical point from a table, and your conclusion (whether you can reject null
hypothesis).
Value of test statistic = ________________. Critical point(s) = __________________.
Reject null hypothesis? ______________________.
[end of exam]
