xuebaunion@vip.163.com

3551 Trousdale Rkwy, University Park, Los Angeles, CA

留学生论文指导和课程辅导

无忧GPA：https://www.essaygpa.com

工作时间：全年无休-早上8点到凌晨3点

微信客服：xiaoxionga100

微信客服：ITCS521

手写代写-STAT0010

时间：2021-02-21

STAT0010 In-Course Assessment for 2019/2020 Page 1

STAT0010

In-Course Assessment 2019/2020

• This is a closed-book examination.

• The time allowed will be 1 hour 10 minutes. You should answer as many of the paper questions

as you can.

• The numbers in square brackets indicate the relative weight attached to each part question.

Marks will not only be given for the final (numerical) answer but also for the accuracy and

clarity of your answer. As such, make sure to write down your workings, e.g. formulae,

calculations, reasoning.

• This assessment counts for 20% towards your final STAT0010 mark.

• Non-submission of in-course assessment may mean that your overall examination mark is

recorded as ’non-complete’, i.e. you might not obtain a pass for the course.

• Your solutions should be your own work. Any plagiarism will normally result in zero marks

for all students involved, and may also mean that your overall examination mark is recorded as

non-complete. Guidelines as to what constitutes plagiarism may be found in the Departmental

Student Handbooks.

• Your work will be returned to you for feedback during one of the lectures following the exami-

nation, and you will receive a provisional grade – grades are provisional until confirmed by the

Statistics Examiners’ Meeting in June 2020.

• You will have the opportunity during one of the lectures following the examination to study

the comments written on your marked work, which must be returned to the lecturer before you

leave - your work may be required for inspection by the External Examiner.

TURN OVER

STAT0010 In-Course Assessment for 2019/2020 Page 2

• Unless otherwise indicated, in all questions {t} denotes a sequence of uncorrelated zero-mean

random variables with constant variance σ2, i.e. {t} ∼WN(0, σ2), where WN(0, σ2) denotes

white noise.

• For a process {Yt}, we define∇Yt = Yt − Yt−1 and ∇sYy = Yt − Yt−s.

QUESTION 1 [20]

(a) Give an analytical expression for each of the following models using the backshift operator (for

instance, model AR(1) should be written as (1− φB)Xt = t)

(i) ARIMA(2,1,3).

(ii) SARMA(1, 1)× (1, 0)6.

(iii) SARIMA(2, 0, 1)× (1, 1, 2)3.

[10]

(b) We are given the following models:

(i)

Yt = −0.6Yt−2 + t − 0.4t−2

(ii)

(Yt − 0.2) = 1.2(Yt−1 − 0.2) + 0.2(Yt−2 − 0.2) + t − 0.5t−1.

For each of i), ii), classify each of them as an ARIMA(p, d, q) process (that is, determine p, d

and q in each case). Furthermore, determine whether each of the models in i), ii), is stationary

or not, and whether it is invertible or not. Explain your answers. [6]

(c) Find the values of c, if any such values exist, so that the AR(3) model

Yt = Yt−1 + cYt−2 − cYt−3 + t

is stationary. For the values of c such that the AR(3) model is not stationary, suggest a trans-

formation that will provide a stationary model. [4]

QUESTION 2 [20]

(a) For a strictly stationary process Yt, t ≥ 1, indicate for each statement below if it is true or false.

(i) For t, s ≥ 1, then the bivariate vectors (Ys, Yt) and (Yt, Ys) have the same distribution.

(ii) For t, s ≥ 1, then cov(Yt, Ys) = cov(Ys, Yt).

(iii) For t, s ≥ 1, then E(Y1Y1+s) = E(YtYt+s).

(iv) For t, s, h ≥ 1, then the bivariate vectors (Yt, Ys+h) and (Y1, Yh) have the same distribu-

tion.

CONTINUED

STAT0010 In-Course Assessment for 2019/2020 Page 3

Explain your reasoning. [10]

(b) Consider the stationary time series

Yt =

1

2

Yt−1 + t + θ1t−1 + θ12t−12.

(i) Derive and write down a recursive equation connecting γ(k) and γ(k − 1).

(ii) Use the equation you obtained in (i) to calculate ρ(1) and ρ(2).

[10]

QUESTION 3 [20]

Consider the following ARMA(2, 1) model:

Yt =

1

4

Yt−2 + t +

1

3

t−1,

with the variance of the white noise being σ2 = 1.

(a) Rewrite the model as a MA(∞) one. [10]

(b) Use you result in b) to calculate γ(0). [5]

(c) Find the partial autocorrelation function for lags 2 and 3. [5]

QUESTION 4 [20]

Let (Yt)t∈Z be a weakly stationary process. Let

Xt = (1− 0.4B)Yt

and

Zt = (1− 2.5B)Yt.

(a) Show that Xt and Zt have the same autocorrelation function. [10]

(b) Assume now that Yt is given by

Yt = −1.9Yt−1 − 0.88Yt−2 + t + 0.2t−1 + 0.7t−2.

Which ARMA models do the Yt and Xt processes correspond to? Is Yt stationary? What about

invertible? [10]

QUESTION 5 [20]

TURN OVER

STAT0010 In-Course Assessment for 2019/2020 Page 4

(a) A time series of monthly data provided the sample autocorrelations and partial autocorrelations

shown in the following plots:

Suggest a suitable model for this time series. Explain how each of the above plots supports the

choice of model you have made. [10]

(b) Consider a SARMA(0, 0) × (1, 0)10 model. State the conditions on the model parameters

so that the model is stationary. Looking at the analytical model equation, explain briefly for

which values of lag k will the autocovariance function γ(k) of this model be equal to zero.

Furthermore, find γ(20). [10]

END OF PAPER

学霸联盟

STAT0010

In-Course Assessment 2019/2020

• This is a closed-book examination.

• The time allowed will be 1 hour 10 minutes. You should answer as many of the paper questions

as you can.

• The numbers in square brackets indicate the relative weight attached to each part question.

Marks will not only be given for the final (numerical) answer but also for the accuracy and

clarity of your answer. As such, make sure to write down your workings, e.g. formulae,

calculations, reasoning.

• This assessment counts for 20% towards your final STAT0010 mark.

• Non-submission of in-course assessment may mean that your overall examination mark is

recorded as ’non-complete’, i.e. you might not obtain a pass for the course.

• Your solutions should be your own work. Any plagiarism will normally result in zero marks

for all students involved, and may also mean that your overall examination mark is recorded as

non-complete. Guidelines as to what constitutes plagiarism may be found in the Departmental

Student Handbooks.

• Your work will be returned to you for feedback during one of the lectures following the exami-

nation, and you will receive a provisional grade – grades are provisional until confirmed by the

Statistics Examiners’ Meeting in June 2020.

• You will have the opportunity during one of the lectures following the examination to study

the comments written on your marked work, which must be returned to the lecturer before you

leave - your work may be required for inspection by the External Examiner.

TURN OVER

STAT0010 In-Course Assessment for 2019/2020 Page 2

• Unless otherwise indicated, in all questions {t} denotes a sequence of uncorrelated zero-mean

random variables with constant variance σ2, i.e. {t} ∼WN(0, σ2), where WN(0, σ2) denotes

white noise.

• For a process {Yt}, we define∇Yt = Yt − Yt−1 and ∇sYy = Yt − Yt−s.

QUESTION 1 [20]

(a) Give an analytical expression for each of the following models using the backshift operator (for

instance, model AR(1) should be written as (1− φB)Xt = t)

(i) ARIMA(2,1,3).

(ii) SARMA(1, 1)× (1, 0)6.

(iii) SARIMA(2, 0, 1)× (1, 1, 2)3.

[10]

(b) We are given the following models:

(i)

Yt = −0.6Yt−2 + t − 0.4t−2

(ii)

(Yt − 0.2) = 1.2(Yt−1 − 0.2) + 0.2(Yt−2 − 0.2) + t − 0.5t−1.

For each of i), ii), classify each of them as an ARIMA(p, d, q) process (that is, determine p, d

and q in each case). Furthermore, determine whether each of the models in i), ii), is stationary

or not, and whether it is invertible or not. Explain your answers. [6]

(c) Find the values of c, if any such values exist, so that the AR(3) model

Yt = Yt−1 + cYt−2 − cYt−3 + t

is stationary. For the values of c such that the AR(3) model is not stationary, suggest a trans-

formation that will provide a stationary model. [4]

QUESTION 2 [20]

(a) For a strictly stationary process Yt, t ≥ 1, indicate for each statement below if it is true or false.

(i) For t, s ≥ 1, then the bivariate vectors (Ys, Yt) and (Yt, Ys) have the same distribution.

(ii) For t, s ≥ 1, then cov(Yt, Ys) = cov(Ys, Yt).

(iii) For t, s ≥ 1, then E(Y1Y1+s) = E(YtYt+s).

(iv) For t, s, h ≥ 1, then the bivariate vectors (Yt, Ys+h) and (Y1, Yh) have the same distribu-

tion.

CONTINUED

STAT0010 In-Course Assessment for 2019/2020 Page 3

Explain your reasoning. [10]

(b) Consider the stationary time series

Yt =

1

2

Yt−1 + t + θ1t−1 + θ12t−12.

(i) Derive and write down a recursive equation connecting γ(k) and γ(k − 1).

(ii) Use the equation you obtained in (i) to calculate ρ(1) and ρ(2).

[10]

QUESTION 3 [20]

Consider the following ARMA(2, 1) model:

Yt =

1

4

Yt−2 + t +

1

3

t−1,

with the variance of the white noise being σ2 = 1.

(a) Rewrite the model as a MA(∞) one. [10]

(b) Use you result in b) to calculate γ(0). [5]

(c) Find the partial autocorrelation function for lags 2 and 3. [5]

QUESTION 4 [20]

Let (Yt)t∈Z be a weakly stationary process. Let

Xt = (1− 0.4B)Yt

and

Zt = (1− 2.5B)Yt.

(a) Show that Xt and Zt have the same autocorrelation function. [10]

(b) Assume now that Yt is given by

Yt = −1.9Yt−1 − 0.88Yt−2 + t + 0.2t−1 + 0.7t−2.

Which ARMA models do the Yt and Xt processes correspond to? Is Yt stationary? What about

invertible? [10]

QUESTION 5 [20]

TURN OVER

STAT0010 In-Course Assessment for 2019/2020 Page 4

(a) A time series of monthly data provided the sample autocorrelations and partial autocorrelations

shown in the following plots:

Suggest a suitable model for this time series. Explain how each of the above plots supports the

choice of model you have made. [10]

(b) Consider a SARMA(0, 0) × (1, 0)10 model. State the conditions on the model parameters

so that the model is stationary. Looking at the analytical model equation, explain briefly for

which values of lag k will the autocovariance function γ(k) of this model be equal to zero.

Furthermore, find γ(20). [10]

END OF PAPER

学霸联盟