Name: Section: Wednesday F7
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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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!
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TRUE/FALSE (10 points each)
Please circle the correct answer. There will not be partial credits for these questions.
1. Estimates obtained by MLE are consistent but can be biased. T F
2. Assume returns rt follow a certain GARCH(1, 1) process. Then, the conditional variance ht
is not correlated with ht−2. T F
3. The CCAPM implies that assets whose pay-offs co-vary more positively with aggregate con-
sumption should yield relatively lower expected excess returns. T F
4. If Et−1(rt) = 0 for all t, then E(rt) = 0. T F
5. If E(rt) = 0, then Et−1(rt) = 0 for all t. T F
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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 the CCAPM model and the following equation for the price of an asset in equilibrium:
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 premium and the covariance term in the equation above is positive
B) The asset sells at a premium and the covariance term in the equation above is positive if the
asset gives a high payoff when consumption is low
C) The asset sells at a premium and the covariance term in the equation above is positive if the
asset gives a high payoff when consumption is high
D) The asset sells at a premium and the covariance term in the equation above is negative if the
asset gives a high payoff when consumption is low
Question 2
Consider two lotteries:
Lottery 1: you lose $100 a month if you have a job; you gain $150 a month if you lose your job.
Lottery 2: you gain $100 a month if you have a job; you lose $150 a month if you lose your job.
According to the CCAPM, the price of which lottery ticket is higher?
A) Lottery 1
B) Lottery 2
C) They are the same
D) We could say something if we knew the covariance between the lottery’s payoffs and market
returns
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Question 3
Consider two estimators for the parameter µ, θ̂1 and θ̂2, such that
E(θ̂1) = µ− 15 with V(θ̂1) = 1
and
E(θ̂2) = µ, with V(θ̂2) = 4.
The covariance between the two estimators (i.e., C(θ̂1, θ̂2)) is equal to -2.
By taking a weighted average of θ̂1 and θ̂2, we can define a new estimator θ̂3 = wθ̂1 + (1 − w)θ̂2,
which is, for a certain choice of w, .... (choose the best answer)
A) ... biased with a zero variance, but could be made unbiased
B) ... biased with a zero variance, but could not be made unbiased
C) ... unbiased, but with a non-zero variance
D) ... unbiased, but with a zero variance
Question 4
Consider a population with mean µ and variance σ2 < 0. We collect an IID sample x1, ..., xT and
estimate the mean of the population µ using an estimator θ̂T with the following properties:
E(θ̂T ) = µ and V(θ̂T ) =
2σ2
T
.
Which of the following is the best answer?
A) The estimator θ̂T is the sample mean multiplied by
√
2
B) The estimator is consistent for µ, like the sample mean, and equally efficient since V(θˆT ) goes
to zero.
C) There is not enough information to say whether the estimator is consistent for µ
D) The estimator is consistent for µ and is not the sample mean
Question 5
If X ∼ N(µ, σ2), then Y = eX is log-normal. Which of the following is correct?
A) E(e2X) = e2µ+2σ2
B) E(e2X) = eµ+
σ2
2
C) E(e2X) = e2µ
D) There is not enough information to compute this expectation.
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LONGER PROBLEMS
Question 1 (80 points)
Consider the CCAPM with the following utility function:
u(ct) =
c1−γt
1− γ .
We estimate the model parameters θ = (β, γ) using the Generalized Method of Moments estimator.
1. (30 points.) Consider the following hypothesis:
H0 : γ = 2 and β = 0.95,
HA : γ 6= 2 or β 6= 0.95.
Derive step-by-step an asymptotically Chi-squared test statistic of multiple restrictions and
specify the number of degrees of freedom. (Hint: Start by re-writing the null hypothesis in
terms of a matrix R and a vector r; then use the asymptotic distribution of θ̂ = (β̂, γ̂) to
derive the asymptotic distribution of the test statistic.)
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2. (30 points.) Write the test statistic for the hypothesis:
H0 : β = γ − 1.05,
HA : β 6= γ − 1.05.
Derive step-by-step an asymptotically Normal test statistic of single restrictions. (Hint: follow
a similar strategy as for the previous question.)
3. (20 points.) Suppose that you estimate the parameters θ = (β, γ) using Maximum Likelihood,
instead of GMM. In a large sample, do you expect your estimates to be numerically very
different? Why or why not? In a large sample, do you expect the GMM estimates to be more
precise than the MLE estimates? Why or why not? Explain precisely.
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Question 2 (80 points)
Consider the following GARCH(1,1) model for the S&P500 daily returns:
rt = εt,
εt =
√
htut with ut ∼ N(0, 1),
ht = α0 + α1ht−1 + α2ε2t−1.
The data consists of daily continuously-compounded returns recorded between January 3, 1980 and
December 29, 2006.
Estimate Std. error t-ratio p-value
Intercept 1.01e− 06 1.28e− 07 7.904 ?
ε2t−1 0.068760 ? 46.459 ?
ht−1 0.923680 0.002852 323.87 ?
1. (10 points.) What is the value of the missing standard error?
2. (20 points.) Test formally the hypothesis that the conditional variance (ht) depends on
squared shocks to returns, i.e., ε2t−1.
3. (20 points.) Test formally the hypothesis that the impact of past conditional variance (ht−1)
on conditional variance (ht) is equal to 0.9, i.e. H
0 : α1 = 0.9.
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4. (10 points.) Which of the missing p-values is the largest? Please explain.
5. (5 points.) The last return in the sample is −0.0045 and the value of hT associated with the
last observation in the sample is .0000314. Write the one-day ahead forecast of the variance
for time T + 1 (i.e., the one-day ahead out-of-sample forecast).
6. (10 points.) Use your result from Point 4 to find the one-day ahead 1% Value at Risk on a
million dollar investment in the S&P500 index.
7. (5 points.) Is the variance process ht stationary?
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Question 3 (80 points)
Assume the lifetime (in months) of batteries is modeled as an exponential random variable with
parameter λ. The probability density function of the exponential random variable is
pλ(x) = λe
−λx,
with x ∈ [0,∞). We have 5 batteries whose lifetimes are 5, 7, 9, 6 and 3 months, respectively.
1. (20 points.) What is the standardized log-likelihood of the model?
2. (10 points.) Compute in theory and numerically the MLE estimator.
3. (10 points.) Is the MLE estimator of λ unbiased? Is it consistent?
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4. (20 points.) What is the asymptotic variance of the estimator?
5. (20 points.) Prove the same result as in the previous point by using the facts that (1) an
n sum of exponential random variables with parameter λ (i.e., exp(λ)) is a Gamma random
variable with parameters n and λ (i.e., Gamma(n,λ)) and (2) the inverse of a Gamma random
variable is an inverse-Gamma random variable.
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Question 4 (80 points)
In Maximum Likelihood estimation, the score is
scoret =
∂ log p(xt|xt−1, ..., θ0)
∂θ
.
We know that Et−1[scoret] = 0.
1. (30 points.) Suppose we run the regression:
scoret = a+ bxt−1 + εt.
Using the properties of the score, what is b in the regression above? (Be as precise as possible
and provide derivations.)
2. (30 points.) Using the properties of the score, what are the auto-covariances of the score, i.e.,
C[scoret, score>t+j ] for j ≥ 1? (Be as precise as possible and provide derivations.)
3. (20 points.) When the data is dependent, you should always do HAC estimation in MLE (like
in GMM). Is this statement correct given the answer to Point 2. Why or why not?
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Question 5 (80 points)
Consider the following regression model:
yi = α+ x
β
i + εi,
with E[εi|xi] = 0.
1. (30 points.) Write two moment conditions which can be used to estimate α and β by GMM.
2. (30 points.) What is the 2× 2 matrix Γ0 for this model? (Be as precise as possible.)
3. (20 points.) What is the 2× 2 matrix Φ0 for this model? (Be as precise as possible.)
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