MATLAB代写-ECMT6006

ECMT6006 2019S1 Mid-Semester Exam
18 April 2019
Instructions: This is a closed book exam. Please answer all questions. The
total mark is 100 and the breakdown is shown in square brackets. The duration
of the exam is 1 hour and 10 minutes, including 10 minutes reading time and 1
Problem 1. [30 pionts] Let Pt be the price of a stock at time t, and assume the
stock pays no dividend. Let Rt+1 be the single-period gross return from time t
to t+ 1.
(i) Given the information set Ft at time t, show that a (point) forecast of the
price Pt+1 can be derived from a (point) forecast of Rt+1. 
(ii) Given the information set Ft at time t, show that the conditional variance
of the price Pt+1 can be derived from the conditional variance of Rt+1. 
(iii) What is the log return from time t to t + 1?  Why is it also called the
continuously compounded return? 
(iv) What is the relationship between the log return and the simple net return?
 Why is it often convenient to use log returns? 
(v) Explain at least three limitations of using the normal distribution to model
the gross returns Rt. 
Problem 2. [35 points] Answer the following questions on the test for return
predictability using historical data.
(i) First, you simply run an OLS regression of your returns on a constant
term
rt = β + εt,
for t = 1, 2, . . . , T . What is the OLS estimate for β?  Write down the
test statistic that you would use to test the null hypothesis H0 : β = 0. 
How would you make the decision for this test?  What is the implication
if your null hypothesis is rejected? 
(ii) Next, you add the lagged value of return as another regressor,
rt = β0 + β1rt−1 + εt,
1
for t = 2, 3, . . . , T . What is the relationship between the true parameter
β1 in this regression and the first-order autocorrelation of {rt}?  What
is the relationship between the return autocorrelation and the return pre-
dictability? 
(iii) The table below presents the results of joint tests for autocorrelation in
the daily Nasdaq returns and 3-month T-bill interest rate returns. “LB”
denotes Ljung-Box test and “Robust” denotes the robust test.
Table 1: Joint tests for serial correlation of log returns
L = 5 L = 10 L = 20
95% CV 11.07 18.31 31.41
Test Stat LB Robust LB Robust LB Robust
S&P 500 11.80 6.31 20.54 16.98 31.26 30.52
T-Bill 207.42 32.58 222.23 38.31 376.69 87.69
(a) Describe how the robust test is conducted. 
(b) What is the difference between a LB test and a robust test? 
(c) Interpret the results in the table. What is your conclusion on the
predictability in these two return series? 
Problem 3. [35 points] Consider the following AR(1)-ARCH(1) model for
index stock returns,
rt = φ0 + φ1rt−1 + εt, εt = σtνt
σ2t = ω + αε
2
t−1, νt|Ft−1 ∼ F (0, 1)
where ω > 0, α ≥ 0, σt > 0, Ft−1 denotes the information set up to time t− 1,
and F (0, 1) denotes some distribution with mean 0 and variance 1.
(i) Show that {εt} is a white noise process, and {ε2t} is an AR(1) process. 
(ii) What does “ARCH” stand for?  What is the key difference between this
model and a simple AR(1) model without ARCH specification? 
(iii) What empirical evidence shown in the index stock returns constrains the
use of a simple ARMA model and motivates the ARCH specification?
(iv) ε2t is sometimes used as a proxy for σ
2
t . What is the relationship between
the processes {ε2t} and {σ2t }? 
(v) What is the optimal one-step ahead two-standard-deviation interval fore-
cast for the return using this model?  Explain how you obtain the
feasible version of this forecast. 
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