Name: Section: Wednesday F5
Nonlinear Econometrics for Finance
Final Exam – 2021
Instructions
Please read these instructions carefully.
1. The maximum score is 550.
2. You have 2 hours to finish the exam.
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(at the beginning) and to scan and upload your work (at the end). Blackboard will not
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documents you are submitting.
6. Please read the questions carefully before answering.
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You cannot discuss the content of the exam with other students during or after the
exam until grades are posted. This includes students from other sections or previous
years. Doing so would be a violation of the Academic Ethics Policy.
The exam has 3 parts:
1. True/False questions (50 points, 10 each)
2. Multiple Choice questions (100 points, 20 each)
3. Long problems (400 points, 80 each)
For full credit on the long problems, please provide detailed explanations, with derivations.
Good luck!
1
TRUE/FALSE (10 points each)
Please circle the correct answer. There will not be partial credits for these questions.
1. The CCAPM implies that assets whose pay-offs co-vary more negatively with the marginal
utility of aggregate consumption should yield relatively lower expected excess returns T F
2. The p-value of a test is the probability of rejecting the null hypothesis if the null hypothesis
is true T F
3. The following Matlab command calculates the p-value of a test statistic (called “test”)
which is distributed as a Chi-Squared random variable with 30 degrees of freedom: 2(1-
chi2cdf(test,30)) T F
4. GMM estimators always have a larger variance than ML estimators for the same parameters
T F
5. Let rt, xt and zt be random variables. If E(rt|xt) = 0 for all t, then E(rt|xt, zt) = 0. T
F
2
MULTIPLE CHOICE (20 points each)
Each multiple choice question is worth 20 points. There will be no partial credits for these questions.
Question 1
Consider three independent random variables x1, x2 and x3 with the same expected value μ and
variances 1/8, 1 and 4, respectively. Consider, also, two estimators θ 1 and θ 2 defined as
θ 1 =
1
2
x3 +
1
2
x2
and
θ 2 = 2×1 x2.
Choose the best answer:
A) Both θ 1 and θ 2 are unbiased for μ and they are equally efficient
B) Both θ 1 and θ 2 are unbiased for μ but θ 2 is more efficient
C) Both θ 1 and θ 2 are biased for μ
D) Both θ 1 and θ 2 are unbiased for μ but θ 3 = x2 is preferable to both
Question 2
Consider a population with mean μ and variance σ2 <∞. After collecting an IID sample of T ob-
servations, you estimate the mean of the population with an estimator α T such that E(α T ) = 0.5μ
and V(α T ) = 1T σ
2. Which statement is correct
A) The estimator α T is not consistent but 2α T is
B) The estimator α T is consistent because its variance goes to zero
C) We cannot determine whether the estimator α T is consistent with this information
D) The estimator 2α T is unbiased but inconsistent
3
Question 3
Consider the CCAPM model and the following equation for the price of an asset in equilibriu
m:
pt = Ct (mt+1, xt+1) +
1
1 +Rf
Et(xt+1),
where pt is the price of the asset at time t; xt+1 is the payoff of the asset at time t+ 1; Rf indicates
the return on the risk-free asset; mt+1 is the stochastic discount factor; and Et and Ct denote the
conditional expectation and the conditional covariance given time-t information, respectively.
Which of the following is true
A) The asset sells at a discount and the covariance term in the equation above is negative
B) The asset sells at a discount and the covariance term in the equation above is negative if the
asset gives a high payoff when consumption is low
C) The asset sells at a discount and the covariance term in the equation above is negative if the
asset gives a high payoff when consumption is high
D) The asset sells at a discount and the covariance term in the equation above is positive if the
asset gives a high payoff when consumption is high
Question 4
We want to estimate the probability that a market index will go up in the next week. As a first
attempt, we collect a large sample of weekly data on the index r0, r1, r2, …, rT and we record xt = 1
if rt rt 1 > 0 and xt = 0 otherwise. Assume that it is reasonable to think of this sample as an
IID sample. Using our sample x1, x2, …., xT , we estimate the probability that the index goes up
using a sample proportion:
p =
1
T
T∑
t=1
xt.
We know that the variance of p is V(p ) = p(1 p)/T . A reasonable estimator for this variance is
V (p ) =
p (1 p )
T
.
Choose the best answer:
A) V (p ) is consistent and unbiased for V(p )
B) V (p ) is consistent but not unbiased for V(p )
C) V (p ) is not consistent (since it goes to zero) but is unbiased for V(p )
D) V (p ) is neither consistent (since it goes to zero) nor unbiased for V(p )
4
Question 5
Suppose you have a sample of T observations from a normally distributed random variable X ～
N(μ, σ2). You are interested in estimating the mean of the population μ but, instead of using the
sample mean X = 1T
∑T
t=1Xt as your estimator, you decide to use the sample median Xmed. After
consulting a statistics book, you discover that
Xmed ～ N
(
μ, 1.5707
σ2
T
)
.
Choose the best asnwer:
A) The median Xmed is not a consistent estimator of μ
B) The median Xmed is as efficient as the mean X because its variance also goes to zero
C) The median Xmed can only be used to estimate the population median and should not be used
to estimate μ
D) None of the above
5
LONGER PROBLEMS
Question 1 (80 points)
Consider the following linear regression model:
Ret,i = αi + βiR
e
t,m + εt,i,
where Ret,i is the excess return on an asset i, R
e
t,m is the excess return on the market and εt,i is a
standard shock with the following properties E[εt,i] = 0 and V[εt,i] = σ2ε , for all assets i.
Notice that the model implies the CAPM if αi = 0 for all assets i. In fact:
E(Ret,i) = αi + βiE(Ret,m).
1. (15 points.) What is the economic interpretation of αi
2. (20 points.) The least-squares estimator of αi is α i =
1
T
∑T
t=1R
e
t,i β i 1T
∑T
t=1R
e
t,m. Using
the Weak Law of Large Numbers and Slutsky’s theorem, show that
√
T (α i αi) d→ N
(
0,
(
1 +
μ2Rem
σ2Rem
)
σ2ε
)
, (1)
where μRem is the expected excess return on the market and σ
2
Rem
is the variance of the excess
market return, using the fact that
√
T (β i βi) d→ N
(
0,
σ2ε
σ2Rem
)
.
6
3. (15 points.) Use the asymptotic distribution in the previous point to test the hypothesis
αi = 0. (Be as precise as possible.)
4. (10 points.) Consider, now, N assets, rather than one. Write the vector of N intercepts as
α =