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STAT8003 2023-stat代写-Assignment 2

时间：2023-10-16

STAT8003 2023-24 Sem1 Assignment 2

Due: Monday Oct 23 2023 11:59pm, Please upload to Moodle

You can write your answers by hand (then take a photo)

Or type in Word (then convert to pdf).

1. Consider an ARMA(2,2) model:

= 0.6 − 0.4−1 + 0.5−2 + − 0.9−1 + 0.8−2, where ~(0,

2)

a. Is it stationary? Show your workings.

b. Find out the MA representation of this ARMA(2,2) model.

c. Is it invertible? Show your workings.

d. Find out the AR representation of this ARMA(2,2) model.

2. Let Yt be the daily closing value of a trading pair: At – Bt. In other words, this trading

pair consists of longing 1 share of A and shorting 1 share of B at time t. It is known

that Yt is a stationary MA(1) process specified as follows:

= + + 0.1−1, = 1, 2, …

where µ = 5 and the noise term εt is i.i.d. N(0, 102).

a. Express Y5 in terms of Y1, Y2, Y3, Y4, and ε5. Assume that ε0 = 0.

b. The observed Yt’s for t = 1, 2, 3, 4 are listed in the following table:

t 1 2 3 4

yt 7 18 2 -1

Given the data listed, compute E[Y5|Y1 = y1, …, Y4 = y4, ε0 = 0] and

var[Y5|Y1 = y1, …, Y4 = y4, ε0 = 0].

2

c. Suppose you hold the trading pair at the end of day 4 and you plan to unwind

it (i.e., close out your positions on A and B) at the end of day 5.

i. How many shares of A and B should you buy or sell at the end of day 5?

ii. What are the expected profit and the standard deviation of the profit

generated by this trading strategy?

d. Since the trading strategy in part c. takes a position for only 1 day, it is a

strategy with fixed investment horizon. However, if the position at the end of

day 5 is still profitable for 1 more day, one can always decide not to unwind.

Calculate the probability that you should not unwind your position at the end

of day 5.

3. The sample autocorrelation function ̂() and sample partial autocorrelation

function ̂() for time series {Zt} with a sample size of 150 are given as follows:

k 1 2 3 4 5 6 7 8

̂() 0.8 0.632 0.505 0.42 0.397 0.332 0.282 0.234

̂() ? ? ? 0.045 0.133 -0.1 0.027 -0.011

We assume that {Zt} is stationary.

a. Calculate the three question marks in the table above.

b. Construct approximate 95% confidence intervals for the sample

autocorrelation function ̂() and sample partial autocorrelation function

̂() in the table above.

c. Based on part b., suggest an appropriate AR model for {Zt} and write down its

equation.

d. Estimate the parameters of your model in part c. by means of the method of

moments (i.e., using sample statistics such as sample mean, sample variance,

sample ACF, sample PACF, etc. to estimate model parameters). We have

calculated that the sample mean and sample variance for {Zt} were 3.5 and 0.75

respectively.

3

4. Given the following AR(2) model:

Zt = 0.3 Zt-1 - 0.5 Zt-2 + at, where {at} ~ WN(0, σa2)

a. Show that the roots of {Zt} lie outside the unit circle and hence, {Zt} is

stationary.

b. Find the ACF (auto-correlation function) of {Zt}.

5. Given the following ARMA(1,1) model:

11 tttt eeYY , where {et} is WN(0, σe

2)

a. Show that the variance of Yt is:

2

2

2

0

1

21

e

b. Hence, find the ACF (autocorrelation function) 1,for kYtk .

c. Suppose we estimated the following sample ACFs: r1 = 0.2, r2 = 0.1. Using

these sample ACFs, recover the parameters and . Keep only the stationary

and invertible solution.

d. Suppose the sample variance s2 for Yt is calculated as 0.86, use parts a. and c.

to find an estimate for σe2.

Due: Monday Oct 23 2023 11:59pm, Please upload to Moodle

You can write your answers by hand (then take a photo)

Or type in Word (then convert to pdf).

1. Consider an ARMA(2,2) model:

= 0.6 − 0.4−1 + 0.5−2 + − 0.9−1 + 0.8−2, where ~(0,

2)

a. Is it stationary? Show your workings.

b. Find out the MA representation of this ARMA(2,2) model.

c. Is it invertible? Show your workings.

d. Find out the AR representation of this ARMA(2,2) model.

2. Let Yt be the daily closing value of a trading pair: At – Bt. In other words, this trading

pair consists of longing 1 share of A and shorting 1 share of B at time t. It is known

that Yt is a stationary MA(1) process specified as follows:

= + + 0.1−1, = 1, 2, …

where µ = 5 and the noise term εt is i.i.d. N(0, 102).

a. Express Y5 in terms of Y1, Y2, Y3, Y4, and ε5. Assume that ε0 = 0.

b. The observed Yt’s for t = 1, 2, 3, 4 are listed in the following table:

t 1 2 3 4

yt 7 18 2 -1

Given the data listed, compute E[Y5|Y1 = y1, …, Y4 = y4, ε0 = 0] and

var[Y5|Y1 = y1, …, Y4 = y4, ε0 = 0].

2

c. Suppose you hold the trading pair at the end of day 4 and you plan to unwind

it (i.e., close out your positions on A and B) at the end of day 5.

i. How many shares of A and B should you buy or sell at the end of day 5?

ii. What are the expected profit and the standard deviation of the profit

generated by this trading strategy?

d. Since the trading strategy in part c. takes a position for only 1 day, it is a

strategy with fixed investment horizon. However, if the position at the end of

day 5 is still profitable for 1 more day, one can always decide not to unwind.

Calculate the probability that you should not unwind your position at the end

of day 5.

3. The sample autocorrelation function ̂() and sample partial autocorrelation

function ̂() for time series {Zt} with a sample size of 150 are given as follows:

k 1 2 3 4 5 6 7 8

̂() 0.8 0.632 0.505 0.42 0.397 0.332 0.282 0.234

̂() ? ? ? 0.045 0.133 -0.1 0.027 -0.011

We assume that {Zt} is stationary.

a. Calculate the three question marks in the table above.

b. Construct approximate 95% confidence intervals for the sample

autocorrelation function ̂() and sample partial autocorrelation function

̂() in the table above.

c. Based on part b., suggest an appropriate AR model for {Zt} and write down its

equation.

d. Estimate the parameters of your model in part c. by means of the method of

moments (i.e., using sample statistics such as sample mean, sample variance,

sample ACF, sample PACF, etc. to estimate model parameters). We have

calculated that the sample mean and sample variance for {Zt} were 3.5 and 0.75

respectively.

3

4. Given the following AR(2) model:

Zt = 0.3 Zt-1 - 0.5 Zt-2 + at, where {at} ~ WN(0, σa2)

a. Show that the roots of {Zt} lie outside the unit circle and hence, {Zt} is

stationary.

b. Find the ACF (auto-correlation function) of {Zt}.

5. Given the following ARMA(1,1) model:

11 tttt eeYY , where {et} is WN(0, σe

2)

a. Show that the variance of Yt is:

2

2

2

0

1

21

e

b. Hence, find the ACF (autocorrelation function) 1,for kYtk .

c. Suppose we estimated the following sample ACFs: r1 = 0.2, r2 = 0.1. Using

these sample ACFs, recover the parameters and . Keep only the stationary

and invertible solution.

d. Suppose the sample variance s2 for Yt is calculated as 0.86, use parts a. and c.

to find an estimate for σe2.