ECON 432 FINAL EXAM
YOUR NAME:
Instruction
• Write legibly your answers on blank papers (or the printed exam).
• Total points are 100. Use your time wisely.
• Examination Rules from Department Policy will be strictly followed.
• The following results may be useful:
Pr(Z ≤ −2.326) = 0.01, Pr(Z ≤ −1.96) = 0.025,
Pr(Z ≤ −1.645) = 0.05. Pr(Z ≤ −1.282) = 0.10,
where Z is a standard normal random variable.
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1. (10 pts) Suppose the simple return R of a stock follows uniform dis-
tribution with support [−0.3, 0.2] and we invest $1,000 on this stock.
The density of R takes the following form:
f(x) =
{
2, for x ∈ [−0.3, 0.2]
0, otherwise.
1.1. [2 pts] Denote by L the profit of this investment. Find the distribution
of L as well as its mean and variance.
1.2 [2 pts] What is the probability that the profit is less than $100?
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1.3 [3 pts] Find the value at risk of α = 0.05, i.e., V aR0.05 of this invest-
ment.
1.4 [3 pts] Find the expected shortfall of α = 0.05, i.e., ES0.05 of this
investment. [Hint: for every α ∈ (0, 1),
ESα = α
−1
∫ V aRα
−∞
xf(x)dx = α−1
∫ α
0
V aRudu
where f(x) denotes the probability density function of the profit.]
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2. (15 pts) Let {Yt}t be a time series generated by
Yt = c+ et + θ2et−2, where {et}t ∼ IID(0, σ2),
where c and θ2 are finite real numbers. Let γ(j) and ρ(j) denote the
auto-covariance function and the auto-correlation function of {Yt}t, re-
spectively.
2.1. [2 pts] Find the mean and variance of Yt.
2.2. [4 pts] Find the auto-covariance function γ(j) for j ≥ 0.
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2.3. [2 pts] Find the auto-correlation function ρ(j) for j ≥ 0.
2.4. [5 pts] Suppose that a covariance-stationary time series has the fol-
lowing auto-correlation function
ρj =
{
3
5
, if j = 1
0, if j > 1
.
Shall we use MA(1) to model this process? Why?
2.5. [2 pts] Do we need to impose the restriction |θ2| < 1 make {Yt}t
covariance stationary? Justify your answer according to the definition
of covariance stationarity.
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Figure 1: Daily CC Return
3. (32 pts) Motivated by the empirical facts in Figure 1, the following
GARCH(1,1) model has been introduced:
rt = σtet, et ∼ iid N(0, 1)
σ2t = ω + α1r
2
t−1 + β1σ
2
t−1
where ω > 0, α1 ≥ 0, β1 ≥ 0 and α1 + β1 < 1. This model successfully
generates the stylized facts on financial returns.
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3.1. [2 pts] Figure 1 presents the daily cc returns, the sample ACF, the fi-
nite sample density/histogram and the Q-Q plot based on apple shares.
Summarize the empirical stylized facts from Figure 1.
3.2. [2 pts] Show that σ2t is the conditional variance of rt given Ft−1 =
{rt−1, σ2t−1, rt−2, σ2t−2, . . . }.
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3.3. [4 pts] Define vt = r
2
t − σ2t . Show that {vt}t is a martingale difference
sequence.
3.4. [4 pts] Show that r2t can be written as an ARMA(1,1) process:
r2t − µ = φ(r2t−1 − µ) + vt + θvt−1,
where µ, φ and θ depend on ω, α1 and β1.
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3.5. [4 pts] Show that
E [r4t ]
(E [r2t ])
2 ≥ 3
and discuss why this result explains the distribution of financial stock
returns better than i.i.d. normal model.
3.6. [4 pts] Find the variance of rt. What is the difference between the
variance of rt and σ
2
t ?
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Using the data, we estimate the GARCH(1,1) model. The estimation
results are summarized in the following table.
Table 1: Estimation of GARCH(1, 1)
Estimate Std. Error
ω 0.000008 0.0000004
α1 0.10 0.008
β1 0.88 0.009
3.7. [4 pts] Find the plug-in estimator of α1+β1. Discuss which stylized fact
of financial return data is well captured by the value of this estimator.
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3.8. [4 pts] For parameters α1 and β1, compute their 90% (asymptotic)
confidence intervals.
3.9. [4 pts] Suppose that r2T = 0.04 and σ
2
T = 0.10. Find the 90% confidence
interval of rT+1 and V aR0.10 with W0 = 1, 000.
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4. (15 pts) The Quantile Kurtosis (QuKurt) of a random variable X is
defined as
QuKurt(X) =
F−1X (1− p1)− F−1X (p1)
F−1X (1− p2)− F−1X (p2)
where FX(·) denotes the CDF of X and 0 < p1 < p2 < 1/2. The
QuKurt is a measure of the tail thickness of X. In this problem we
consider p1 = 0.025 and p2 = 0.25.
4.1. [5 pts] Let qtvα denote the α quantile of tv(0, 1), i.e., the standard
student-t distribution with degree of freedom v. Using R, we obtain
the following
qt(0.025, 2) = -4.303 qt(0.25, 2) = -0.816
qt(0.025, 3) = -3.182 qt(0.25, 3) = -0.765
.
Find the QuKurt for t2(0, 1) and t3(0, 1). Which one (t2(0, 1) or t3(0, 1))
has larger QuKurt? [Hint: the pdf of tv(0, 1) is symmetric around zero.]
4.2. [2 pts] The tail thickness of a random variable may also be measured
using the Kurtosis. What is(are) advantage(s) of QuKurt compared
with Kurtosis in terms of measuring the tail thickness?
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To estimate the QuKurt of a stock, a sample of 74 monthly cc re-
turns was taken and the sample Q̂uKurt was 3.481. To get a confi-
dence interval for the true QuKurt, 1,000 resamples were taken. Let
Q̂uKurtb,boot be the sample QuKurt of the bth resample. The 1,000 val-
ues of Q̂uKurtb,boot were sorted and the table below contains selected
values of Q̂uKurtb,boot ranked from smallest to largest (so rank 1 is the
smallest and so forth).
Rank Value of Q̂uKurtb,boot
25 2.400
50 2.509
250 2.996
750 3.748
950 4.461
975 4.703
.
From R, we also obtain that qnorm(0.75) - qnorm(0.25) = 1.349.
4.3. [2 pts] Find the bootstrap standard error of Q̂uKurt.
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4.4. [2 pts] Find the 90% and 95% confidence intervals using the boot-
strap standard error in part (a) and the asymptotic normal dis-
tribution of Q̂uKurt.
4.5. [4 pts] Find the 90% and 95% equal-tail bootstrap confidence
interval for the true QuKurt.
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5. (18 pts) Figure 2 below shows US quarterly GDP time series data.
Figure 2: US GDP
5.1. [2 pts] Do the US GDP data look like realizations from a covariance
stationary stochastic process? Why or why not?
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5.2. [4 pts] Many empirical researchers used to model US GDP time series
Yt using the following random walk model:
Yt = c+ Yt−1 + εt, εt ∼ iid N(0, 1). (1)
Suppose that c = 5.1 and the US GDP in the second quarter of 2018 is
180. What is the best prediction of the US GDP in the third quarter
of 2018 based on the model in (1)?
5.3. [4 pts] Suppose that Y0 ∼ N(0, 1) which is independent of εt for any
t. Find E[Yt] and V ar (Yt). Is Yt covariance stationary?
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5.4. [4 pts] Show that Yt+j = jc+ Yt +
∑j
s=1 εt+s, for any j ≥ 1.
5.5. [4 pts] Find the correlation coefficient between Yt and Yt+j. Compare
it with the auto-correlation function of the AR(1) process {Xt} from
Xt = c+ φXt−1 + εt, εt ∼ iid N(0, 1)
where |φ| < 1, and explain the main differences.
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6. (10 pts) Indicate whether the following statements are true or false
(circle one). Briefly discuss why it is so.
6.1. [2 pts] If t ∼ mds(0, σ2 ), then Cov(t, t−j) = 0 for any j = 1, 2, . . . .
True False
Why?
6.2. [2 pts] Let Y1, Y2 and Y3 be i.i.d random variables with mean µ and
variance σ2, and let µˆ1 = Y1 and µˆ2 = (Y1 +Y2 +Y3)/3 are two different
point estimators for µ. From the MSE criteria, we prefer µˆ2 to µˆ1.
True False
Why?
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6.3. [2 pts] If rt is continuously compounded (cc) monthly return at month
t for t = 1, 2, . . . , 12, then the annualized cc return is rA = 12rt.
True False
Why?
6.4. [2 pts] Let RGS,t and RAIG,t be 1-month simple returns for Goldman
Sachs Group (GS) and American International Group (AIG). If we
construct a portfolio using the share α ∈ [0, 1] for GS, the portfolio
simple return is Rp,t = αRGS,t + (1− α)RAIG,t.
True False
Why?
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6.5. [2 pts] Let θˆ1 and θˆ2 are two different point estimators for θ. If
V ar(θˆ1) < V ar(θˆ2), then confidence interval based on θˆ1 is more accu-
rate (shorter) so we always prefer to use θˆ1.
True False
Why?
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