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MATLAB代写-ECMT6006

时间：2021-04-13

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 Ft 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 Ft 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 + 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}? [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 + 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. [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) ε2t is sometimes used as a proxy for σ

2

t . What is the relationship between

the processes {ε2t} and {σ2t }? [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]

2

学霸联盟

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 Ft 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 Ft 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 + 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}? [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 + 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. [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) ε2t is sometimes used as a proxy for σ

2

t . What is the relationship between

the processes {ε2t} and {σ2t }? [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]

2

学霸联盟