課程名稱︰統計學與計量經濟學暨實習上
課程性質︰必修
課程教師︰張勝凱
開課學院：社會科學院
開課系所︰經濟系
考試日期（年月日）︰2014/01/08
考試時限（分鐘）：150分鐘
試題 :
Econ 2013 Final Exam January 8, 2014
You have 150 minutes. Please write clearly and show all your works. Answer
without any explanation will not earn any points. Good luck!!
Problem 1.(20 points (5,5,5,5))
True or False, explain your answer briefly,
1. If y_i = x'_iβ + ε_i, x_i∈R and E[ε_i|x_i] = 0, then ε_i is
independent of x_i.
2. If y_i = x'_iβ + ε_i, x_i∈R and E[ε_i|x_i] = 0, then E[(x_i)^2ε_i] = 0.
3. If y_i = x'_iβ + ε_i, x_i∈R and E[x_iε_i] = 0, then E[(x_i)^2ε_i] = 0.
4. Since we assume zero conditional mean in the linear regression model, the
assumption of homoskedasticity is redundant.
Sol:
Since we have not learned multiple regression yet, let's use simple regression
┌ 1 ┐' ┌β_0┐
model where x' = │ │ , β = │ │ , Y_i = β_0 + β_1X_i + ε_i.
└X_i┘ └β_1┘
1. False.
E[ε_i|x_i] = 0, E[ε_i] = E[E[ε_i|x_i]] = 0,
E[ε_i|x_i] = E[ε_i], ε_i and x_i are mean independent.
But this does not imply ε_i and x_i are independent.
2. True.
E[(x_i)^2ε_i] = E[E[(x_i)^2ε_i|x_i]] = E[(x_i)^2˙E[ε_i|x_i]]
= E[(x_i)^2˙0] = E[0] = 0.
3. False.
E[ε_ix_i] = 0, E[ε_ix_i] = E[E[ε_ix_i|x_i]] = E[x_i˙E[ε_i|x_i]] = 0,
E[(x_i)^2ε_i] = E[E[(x_i)^2ε_i|x_i]] = E[(x_i)^2˙E[ε_i|x_i]]
If we don's know whether E[ε_i|x_i] equals to zero or not, we cannot make
sure E[(x_i)^2ε_i] equals to zero or not.
4. False.
Zero conditional mean: E[ε_i|x_i] = 0,
homoskedasticity: Var(ε_i|x_i) = σ^2 ∈ C,
If we only assume E[ε_i|x_i) = 0,
then Var(ε_i|x_i) = E[(ε_i)^2|x_i] - (E[ε_i|x_i])^2 = E[(ε_i)^2|x_i]
= (σ_i)^2, (σ_i)^2 ≠ (σ_j)^2 for all i ≠ j.
→ Does not contain homoskedasticity.
Problem 2. (20 points (7,7,6))
For the joint pmf of random variables X and Y in the table below:
┌───┬───┬───┐
│ │ x = 1│ x = 2│
├───┼───┼───┤
│ y = 0│ 0.1 │ 0.2 │
│ │ │ │
│ y = 1│ 0.1 │ 0.1 │
│ │ │ │
│ y = 2│ 0.3 │ 0.2 │
└───┴───┴───┘
1. Find the conditional experctation function E[Y|X = 1) and E[Y|X = 2].
2. Are X and Y stochastically independent?
3. For the best linear predicion E*[Y|X] = α + βX, find α and β.
(Hint: α = E[Y] - E[X] and β = Cov(X,Y)/Var(X))
Sol:
1. E[Y|X = 1] = 1.4, E[Y|X = 2] = 1.
2. Pr(y = 0, x = 1) = 0.1 ≠ Pr(y = 0)˙Pr(x = 1) = 0.3˙0.5 = 0.15
∴X and Y are not stochastically independent .
3. E*[Y|X] = α + βX
E[Y] = 1.2, E[X] = 1.5, Cov(X,Y) = E[XY] - E[X]E[Y] = 1.7 - 1.8 = -0.1,
Var(β) = E[X^2] - (E[X])^2 = 2.5 - 2.25 = 0.25;
β = -(0.1/0.25) = -0.4, α = E[Y] - βE[X] = 1.8
Problem 3. (20 points (5,5,5,5))
Consider a linear model to explain monthly tea consumption:
tea = β_0 + β_1 ink + β_2 price + β_3 edu + u
u = e ×inc ×edu.
where e is a random variable with E(e) = 0 and Var(e) = (σ_e)^2, and e is
independent to inc. price and edu. Suppose the sample you have for tea
consumtion (tea), income (inc), tea price (price) and years of schooling (edu)
is the random sample, and there is a variation is all explanatory variables in
_ _ _ _
your sample. Let β_0, β_1, β_2 and β_3 be an OLS estimators of above
equation.
1. Find E[u|inc, price, edu] and Var(u|inc, price, edu).
_ _ _ _
2. Are β_0, β_1, β_2 and β_3 unbiased estimators for β_0, β_1, β_2 and
β_3? Why of why not.
3. Is the homoskedasticity assumption hold for the above linear model? If yes,
please provide the explanations; if not, write the transformed equation that
has a homoskedastic error term.
_ _ _ _
4. Are β_0, β_1, β_2 and β_3 best linear unbiased estimators for β_0,
β_1, β_2 andβ_3? Why or why not.
Problem 4. (15 points (5,5,5))
Consider the multiple regression model with three independent variables, undet
the classical linear model assumption MLR. 1-MLR. 6:
y = β_0 + β_1x_1 + β_2x_2 + β_3x_3 + u,
You would like to test the null hypothesis H_0: 2β_1 + β_3 = 2
_ _
1. Let β_1 and β_3 denotes the OLS estimators of β_1 and β_3. Find
_ _ _ _
Var(2β_1 + β_3) in terms of the varuance of β_1 and β_3 and the covariance
_ _
between them. What is the standard error of 2β_1 + β_3?
2. Write the t statistics for testing H_0: 2β_1 + β_3 = 2.
_ _ _
3. Define θ_1 = 2β_1 + β_3 and θ_1 = 2β_1 + β_3. Write a regression
equation involving β_0, θ_1, β_2 and β_3 that allows you to directly obtain
_
θ_1 and its standard error.
Problem 5. (30 points (7,7,8,8))
The following wage regression was estimated using 706 observatipns to study the
tradeoff between time spent sleeping and working and to look at other factors
affecting sleep:
_
sleep = 3638.25 - 0.148totwrk - 11.13edu + 2.20age + u SSR = 123455057,
(112.28) (0.017) (5.88) (1.45)
where sleep and totwrk (total work) are in minutes per week and edu and age are
in years.
1. Is either edu or age individually significant at the 5% level against a
two-sided alternative?
2. Dropping edu and age from the equation gives
_
sleep = 3586.38 - 0.151 totwrk + u SSR = 124858119,
(38.91) (0.017)
Are edu and age jointly significant in the original equation at the 5% level?
3. Comments on your finding in 1 and 2.
4. Does including edu and age in the model greatly affect the estimated
tradeoff between sleeping and working? Explain.