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ECON 432 FINAL EXAM

创建日期：2024-04-27 17:47:37

所属分类：经济Economics

标签： ECON 432

FINAL EXAM

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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.
1
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?
2
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.]
3
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.
4
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.
5
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.
6
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, . . . }.
7
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.
8
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 ?
9
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.
10
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.
11
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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