midterm 324

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Econ 210 Midterm Exam
Prof. Sebastien Gay
Winter 2004
This is a closed book exam. You have two hours for this exam.
The exam is out of 120 points, i.e you do not need to do everything to get
100 (DO NOT PANIC!).
No calculator is allowed. You have to solve all the problems.
Please, use a pen to write your answers.
You must always explain your answers.
Good luck!
1. Nuts and Bolts Questions (25 points: 5 points for each answer)
(a) Write down the Gauss-Markov theorem. (give all assumptions).
Write down the Law of Iterated Expectations.
(b) What happens to the OLS intercept and slope estimates when all
observations on the independent variable are identical?
(c) TRUE or FALSE: Consider the following simple regression model:
Yi = α + βXi + ui . All the OLS assumptions are satisfied. Denote
by αˆ and βˆ the OLS estimators of α and β respectively.
Then Y¯ = αˆ+ βˆX¯ ? Explain your answer.
(d) TRUE or FALSE: The slope coefficient from a regression of Yi+ c on
Xi + c is the same as the slope coefficient from a regression of Yi on
Xi.
(e) Two independent RVs X1 and X2 are both normally distributed
N(µ,σ2). We know that σ2 = 50 but do not know the average
µ. The following estimator for µ is proposed:
µˆ =
1
2
X1 +
1
2
X2
Is this an unbiased estimator for m ? What is the variance of this
estimator ?
2. (30 points: 5 points each) Consider the following joint density function:
fX,Y (x, y) =
δxy−δ−1/2 for 0 < x < 2 and 1 < y <∞
0 otherwise
where 1 < δ < 2
1. (a) Show that fX,Y (x, y) is a proper density function.
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(b) Compute the marginal density of Y, fY (y).
(c) What is the variance of Y?
(d) Derive a formula for E(X|Y ).
For parts (e) and (f), assume that δ > 2.
(e) Based on a sample of N i.i.d observations from fY (y), write down the
log likelihood function and derive the maximum likelihood estimator
δˆ
MLE
of δ.
(f) Derive the asymptotic distribution of the estimator δˆ
MLE
you found
in part e).
3 (33 points) Suppose that the relationship between the variables yi and
xi is of the following form:
yNon-standardi = α˜+ β˜xi + γ
2x2i + θi non-standard model
We suppose in all the questions that xi are non-stochastic.
Suppose that θi˜N(0,σ
2) and that θi is such that E[θi] = 0.
However an sloppy econometrician uses the regression of yi on xi with an
intercept. He thinks that the true model is the standard-linear model,
which means that he thinks that yi = y
Standard
i = α+ βxi + εi , with E[εi] = 0.
He computes the OLS estimators αˆ, βˆ.
(a) (4 points) Give the formulas for αˆ, βˆ the OLS estimators of α and β in the
standard-linear model. If the standard-linear model is the true one, are
the OLS estimators αˆ, βˆ unbiased estimators of α and β resp.? No proof
is needed, just answer yes or no and state what it means for αˆ, βˆ.
(b) (6 points) What is the estimated (forecasted) value yˆhi this econometrician
would give using his OLS estimators if given only xi for an individual i.
Just use αˆ and βˆ you don’t need to plug the formulas you found in a).
(c) (7 points) If the econometrician’s model (the standard-linear model) was
true (yi = y
Standard
i ) what would be the error yi−yˆhi in terms of xi,α,β,εi,αˆ
and βˆ ? What would be its expectation?
(d) (5 points) Find the true systematic error yi − yˆhi , which is the differ-
ence between the true model of yi (using the non-standard model yi =
yNon-standardi ) and what the econometrician did forecast given the xi (your
result in question b) yˆhi ) in terms of xi, α˜, β˜, γ, εi, αˆ and βˆ.
(e) (5 points) What is the expectation of this error in terms of xi,α˜,β˜,γ, εi,E[αˆ]
and E[βˆ]? (You don’t need to compute E[αˆ] and E[βˆ]).
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(f) (6 points) Suppose α˜ = E[αˆ], β˜ = E[βˆ], 0 < γ and 0 < xi. What is the
sign of E[yi − yˆhi ]? Compare your result to your answer to question c).
4 (32 points) Consider the following regression model:
Yi = β1 + ui, i = 1, 2, ..., n
where E(ui) = 0, var(ui) = σ
2x2i , where the xi are non stochastic for all
i = 1, 2, ..., n.
1. (a) (5 points) Derive the OLS estimator βˆ
ols
1 for the true parameter β1.
(b) (5 points) Show that the variance of βˆ
ols
1 is given by:
V ar(βˆ
ols
1 ) =
σ2
n
nX
i=1
x2i .
(c) (4 points) Consider now the regression: Y ∗i = β1x
∗
i +u
∗
i , i = 1, 2, ..., n
with
Y ∗i =
Yi
xi
, x∗i =
1
xi
and u∗i =
ui
xi
Show that E(u∗i ) = 0, var(u
∗
i ) = σ
2.
(d) (4 points) Derive the OLS estimator for β˜
ols
1 for the model in question
c).
(e) (4 points) Show that the variance of β˜
ols
1 in this model is given by:
V ar(β˜
ols
1 ) =
σ2Pn
i=1
1
x2
i
(f) (5 points) Show that V ar( βˆ
ols
1 ) > V ar( β˜
ols
1 )
Hint: you can use the fact that (
Pn
i=1
1
x2i
)(
Pn
i=1 x
2
i ) > n
2.
(g) (5 points) Which of these estimators would you prefer? Why?
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