ECMT6006 2019S1 Mid-Semester Exam
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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 hour writing time. Please do NOT write your name but only your SID on the answer booklet(s).
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 rt at time t, show that a (point) forecast of the price Pt＋1 can be derived from a (point) forecast of Rt＋1 . [5]
(ii) Given the information set rt at time t, show that the conditional variance of the price Pt＋1 can be derived from the conditional variance of Rt＋1 . [5]
(iii) What is the log return from time t to t + 1? [2] Why is it also called the continuously compounded return? [4]
(iv) What is the relationship between the log return and the simple net return? [4] Why is it often convenient to use log returns? [4]
(v) Explain at least three limitations of using the normal distribution to model the gross returns Rt . [6]
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? [3] Write down the test statistic that you would use to test the null hypothesis H0 : β = 0. [4] How would you make the decision for this test? [3] What is the implication if your null hypothesis is rejected? [3]
(ii) Next, you add the lagged value of return as another regressor, rt = β0 + β1 rt − 1 + εt , for t = 2, 3, . . . , T . What is the relationship between the true parameter β 1 in this regression and the first-order autocorrelation of {rt }? [5] What is the relationship between the return autocorrelation and the return pre- dictability? [2]
(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 |
Answer the following.
(a) Describe how the robust test is conducted. [5]
(b) What is the difference between a LB test and a robust test? [5]
(c) Interpret the results in the table. What is your conclusion on the predictability in these two return series? [5]
Problem 3. [35 points] Consider the following AR(1)-ARCH(1) model for index stock returns,
rt = φ0 + φ1 rt − 1 + εt , εt = σt νt
σt(2) = ω + αεt(2)− 1 , νt |rt − 1 ~ F (0, 1)
where ω > 0, α > 0, σt > 0, rt − 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 {εt(2)} is an AR(1) process. [10]
(ii) What does “ARCH” stand for? [2] What is the key difference between this model and a simple AR(1) model without ARCH specification? [3]
(iii) What empirical evidence shown in the index stock returns constrains the use of a simple ARMA model and motivates the ARCH specification? Please explain. [5]
(iv) εt(2) is sometimes used as a proxy for σt(2) . What is the relationship between the processes {εt(2)} and {σt(2)}? [5]
(v) What is the optimal one-step ahead two-standard-deviation interval fore- cast for the return using this model? [6] Explain how you obtain the feasible version of this forecast. [4]
2023-04-06