Name: Section: Wednesday F5
Nonlinear Econometrics for Finance
Final Exam - 2021
Instructions
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1. The maximum score is 550.
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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. 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
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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 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 θ̂1 and θ̂2 defined as
θ̂1 =
1
2
x3 +
1
2
x2
and
θ̂2 = 2x1 − x2.
Choose the best answer:
A) Both θ̂1 and θ̂2 are unbiased for µ and they are equally efficient
B) Both θ̂1 and θ̂2 are unbiased for µ but θ̂2 is more efficient
C) Both θ̂1 and θ̂2 are biased for µ
D) Both θ̂1 and θ̂2 are unbiased for µ but θ̂3 = 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 α̂T such that E(α̂T ) = 0.5µ
and V(α̂T ) = 1T σ
2. Which statement is correct?
A) The estimator α̂T is not consistent but 2α̂T is
B) The estimator α̂T is consistent because its variance goes to zero
C) We cannot determine whether the estimator α̂T is consistent with this information
D) The estimator 2α̂T is unbiased but inconsistent
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Question 3
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 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:
p̂ =
1
T
T∑
t=1
xt.
We know that the variance of p̂ is V(p̂) = p(1− p)/T . A reasonable estimator for this variance is
V̂(p̂) =
p̂(1− p̂)
T
.
Choose the best answer:
A) V̂(p̂) is consistent and unbiased for V(p̂)
B) V̂(p̂) is consistent but not unbiased for V(p̂)
C) V̂(p̂) is not consistent (since it goes to zero) but is unbiased for V(p̂)
D) V̂(p̂) is neither consistent (since it goes to zero) nor unbiased for V(p̂)
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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
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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 α̂i =
1
T
∑T
t=1R
e
t,i − β̂i 1T
∑T
t=1R
e
t,m. Using
the Weak Law of Large Numbers and Slutsky’s theorem, show that
√
T (α̂i − 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 (β̂i − βi) d→ N
(
0,
σ2ε
σ2Rem
)
.
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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 =

α1
α2
...
αN
 .
Also, denote by Σε the variance/covariance matrix of the vector ε, where
εt =

εt,1
εt,2
...
εt,N
 .
We have
√
T (α̂− α) d→ N
(
0,
(
1 +
µ2Rem
σ2Rem
)
Σε
)
. (2)
Discuss why Eq. (2) follows logically from Eq. (1).
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5. (20 points.) Use Eq. (2) to construct a one-sided test of the hypothesis that the vector α is
equal to zero.
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Question 2 (80 points)
In order to understand whether positive shocks to returns (induced by favorable aggregate news)
have a different impact on variance than negative shocks to returns (induced by negative aggregate
news), we run an asymmetric GARCH(1,1) model. We also allow for GARCH-in-mean effects:
rt = λht + εt,
εt =
√
htut with ut ∼ N(0, 1),
ht = α0 + α1ht−1 + α2ε2t−1 + α3ε
2
t−11{εt−1<0}.
The conditional mean output:
Estimate Std. error t-ratio p-value
ht 3.3827 1.1670 2.898 0.0037
The conditional variance output:
Estimate Std. error t-ratio p-value
Intercept 1.74e− 06 1.38e− 07 12.66 0.000
ε2t−1 0.024722 0.004225 5.851 0.000
ht−1 0.912771 0.003613 252.61 0.000
ε2t−11{εt−1<0} 0.091709 0.004998 18.347 0.000
1. (10 points.) Test formally the hypothesis that α3 = 0 and interpret your results economically.
2. (10 points.) Test formally the hypothesis that λ is equal to 4.
3. (10 points.) Suppose that λ = 4 is a classical measurement of investors’ risk aversion. How
would you interpret economically the conditional mean of the model, i.e., Et−1(rt) = λht?
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4. (10 points.) If λ = 4, what is the conditional distribution of rt given ht = .0000314?
5. (20 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 5 to find the one-day ahead 1% Value at Risk on a
million dollar investment in the S&P500 index.
7. (10 points.) If ht−2 increases by a specific amount δ, what is the impact on ht?
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Question 3 (80 points)
A bank wants to predict the probability that a borrower will repay a loan. The bank has data on
previous borrowers that can be used to estimate a model for the probability of repayment. Let
xi = (x1i, x2i, ..., xKi) be a 1 ×K vector containing some demographic characteristics of borrower
i, such as income, current debt, employment status, gender, race, location, etc. The bank knows
for each borrower i in the sample whether she/he repaid her/his loan (yi = 1) or not (yi = 0). The
repayment probability is modeled according to a logistic regression model in which the probability
that the borrower is repaid is
P (yi = 1|xi;β) = p(xi;β) = e
xiβ
1 + exiβ
, (3)
where β = (β1, β2, ..., βK)
> is a K × 1 vector of parameters to estimate. Suppose the bank has a
database with an IID sample of n borrowers, so the data are
yi, xi for i = 1, ..., n. (4)
Given this information, please answer the following questions.
1. (25 points.) Write the likelihood of the model.
2. (25 points.) Show that the standardized log-likelihood can be written as
1
n
logL({x}, β) = 1
T
(
n∑
i=1
yixiβ − log
(
n∑
i=1
[
1 + exiβ
]))
.
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3. (30 points.) Show that the ML estimates β̂n can be obtained by solving the following system
of equations:
1
n
n∑
i=1
(yi − p(xi;β))xki = 0 for k = 1, ...,K.
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Question 4 (80 points)
Consider the following model for the short-term interest rate:
drt = α0dt+ γ0dBt, (5)
where drt is the continuous-time change in the interest rate, Bt is a Brownian motion and (α0, γ0)
are two parameters to estimate. We wish to estimate this model using GMM and, in order to do
so, we discretize it in the following way:
rt+∆t − rt = α0∆t + γ0εt+∆t ,
where the variable εt+∆t
d∼ N(0,∆t) is iid. Note that “discretizing” the model simply means going
from “continuous-time” (dt) to “discrete-time” (∆t) which is more consistent with the way in which
data is sampled.
1. (25 points.) Using the discretized model, compute Et [rt+∆t − rt] and Vt [rt+∆t − rt].
2. (30 points.) Use the results in Point 1. to derive two moment conditions to be used for the
identification of the model parameters. What are Γ0 and Φ0 for this model?
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3. (25 points.) Suppose that your model contains another parameter, γ1:
drt = α0dt+ γ0r
γ1
t dBt.
Modify the previous discretized model to obtain a new discretized model (with three param-
eters, i.e., α0, γ0 and γ1) and derive 3 moment conditions to estimate the three parameters.
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Question 5 (80 points)
Consider the following regression model:
yi = α+ e
βxi + 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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