ECON 837 - Econometrics
Midterm exam
Feb. 24th, 2016
Provide detailed calculations and justications to get full credit. Clearly introduce your
notations if needed, and clearly state the regularity assumptions you are using to derive
your results. Partial credit may be given.
Exercise 1 [7 points]:
Consider the following partitioned classical linear regression model:
y = X1 β1 + X2 β2 + u ,
where y is a (n, 1) vector of observations on the dependent variable, and X1 and X2
are (n, k1 ) and (n, k2 ) observation matrices on the regressors. We also introduce X the
(n, k1 + k2 ) matrix, X = (X1 X2 ).
1. Show that if we omit the variables included in X2 , and estimate β1 by running a
regression of y on X1 only, then β̂1 is generally biased with the bias:
E(β̂1 |X) − β1 = P12 β2 , where P12 = (X1′ X1 )−1 X1′ X2 .
2. Interpret the elements of matrix P12 . Under what conditions β̂1 will be unbiased?
3. A researcher is estimating the demand equation for furniture using cross-sectional
data. As regressors she uses an intercept term, the relative price of furniture, and
omits the relevant income variable.
(a) Find an expression for the bias of the OLS estimate of the price variable in
such a regression; discuss its sign.
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(b) What other variables should she have considered, and how could their omission have aected her estimate of the price eect?
Exercise 2 [4 points]:
An economist wishes to estimate from aggregate time series the model:
q = α0 + α1 y + α2 p1 + α3 p2 + α4 n + u
(∗)
where
• q is the volume of food consumption,
• y is the real disposable income,
• p1 is an index of the price of food,
• p2 is an index of all other prices,
• n is the population.
All variables are in logarithm. The economist knows that the correlation between p1 and
p2 is 0.95, and between y and n is 0.93, and so he decides that the equation suers from
multicollinearity. On asking his colleagues for advice, he gets the following suggestions:
(a) Colleague A suggests dropping all variables with t-statistics less than 2;
(b) Colleague B recommends that he should reduce the amount he is asking of the
data by imposing the restrictions α2 + α3 = 0 and 1 − α1 − α4 = 0 which are
suggested by economic theory.
1. Is the economist correct in being sure that (∗) will necessarily suer from multicollinearity?
2. Study each suggestion carefully to decide whether or not it may help with the
multicollinearity issue.
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3. Explain what you would do and why: adopt one of the above suggestions or
something else?
Exercise 3 [6 points]:
Let X1 , X2 , ..., Xn be a sequence of iid random variables with mean E(Xi ) = µ and
variance V ar(Xi ) = σ 2 .
1. Design a (asymptotic) test to test the null hypothesis, H0 : µ = µ0 vs H1 : µ ̸= µ0
at level 0.95. To get full credit you need to: (i) dene your test statistic; (ii)
derive its asymptotic distribution; (iii) and clearly dene your decision rule.
2. Show that your test is consistent against the following xed alternative,
H1 : µ = µ1 where µ1 ̸= µ0 .
3. Derive the asymptotic power of your test under the following sequence of local
√
alternatives, H1,n : µ = µ0 + δ/ n where δ is a non-zero (xed) real number.
Exercise 4 [13 points]:
Consider the model,
where
{
(
ui =
u1,i
u2,i
y1,i = βy2,i + u1,i
y2,i = αxi + u2,i
)
([
∼ iid N
0
0
] [
,
σ11 σ12
σ12 σ22
])
,
y1,i and y2,i are the (univariate) dependent variables, and xi is an exogenous (univariate)
random variable. The sample contains n observations of (y1,i , y2,i , xi ).
1. Derive the OLS estimator of β .
(a) Study the consistency of this estimator.
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(b) If it is not consistent, can you impose some restrictions on the parameters
of the model in order to recover consistency? Interpret and discuss your
ndings.
2. Derive the OLS estimator of α.
(a) Study the consistency of this estimator. If it is not consistent, can you
impose some restrictions on the parameters of the model in order to recover
consistency?
(b) Interpret and discuss your ndings.
3. (a) Derive the IV estimator of β using x as an instrument and study its consistency.
(b) Which estimator of β should be used? Interpret and discuss your ndings.
4. Consider an updated version of the above model where α is replaced by
√
αn = α/ n, and everything else remains the same, that is
{
y1,i = βy2,i + u1,i
y2,i = αn xi + u2,i
(a) How does it aect your study of the OLS estimator of α?
(b) How does it aect your study of the IV estimator of β ?
(c) Take another look at 4(a) and 4(b) when αn = α/nk and,
i. k < 1/2.
ii. k > 1/2.
(d) BONUS: Do you have any idea/intuition/interpretation as to why one would
ever consider such a framework where the value of parameter α is tied to the
sample size? [no computation required].
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