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.
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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]
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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