MAS8382 Summative Practical Report
Summative Practical Report
Submit your solutions to all questions via NESS by
11:45pm on Monday 18th January 2021.
For any questions where you use R, include your R code and any relevant plots and R output.
Marks for each question are indicated. These marks should be used as a rough guide to the relative
weight of individual questions. The final marks for the assignment may be slightly different from
those indicated.
You should submit your work as a single electronic file in PDF or Word format. If you would prefer
not to type your solutions to Questions 3 and 5, you can scan or photograph handwritten answers.
The resulting file(s) can then be embedded in, or appended to, the rest of the report.
Week 2
You should work on the three questions in this section during the Practical for Week 2 and, if
necessary, finish it during your independent study time.
Chapter 3: ARIMA models
1. Consider the general ARIMA(p, d, q) process.
(a) In the solutions to the formative practical report, Question 3(a) gives a function which
simulates from an ARMA(p, q) process. Study this function and then use it to give a new
function which simulates from an ARIMA(p, d, q) process. This function should have d,
the parameter vectors, and the length of the time series as arguments, and it should
return the simulated time series. If you can, allow your function to handle any d. If not,
write a function which handles the d = 0, d = 1 and d = 2 cases and returns with an
error if d > 2.1
(b) Simulate three realisations of an ARIMA(1, 1, 2) process of length 1000 with µ = 0,
φ1 = 0.8, θ1 = 0.8, θ2 = −0.5 and σ
2 = 0.5. Produce two plots: one with each of the
three realisations superimposed and another with the corresponding three series of first
differences superimposed. Comment on your plots.
[7 marks]
2. For an AR(p) model:
(a) Write a function which estimates the model parameters via the Yule-Walker equations.
(b) Test this function by simulating various AR processes and ensuring the original param-
eters are (approximately) recovered. Report the results of your tests. To simulate the
AR processes, you may like to use the function from Question 3(a) in the solutions to
the formative practical report.
[10 marks]
1This second option will be marked for reduced credit.
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MAS8382 Summative Practical Report
3. An ARIMA(2,1,0) process is given by
Xt −Xt−1 = φ1(Xt−1 −Xt−2) + φ2(Xt−2 −Xt−3) + εt
where {εt} is a white noise process with εt ∼ N(0, σ
2). Suppose that φ1 = 0.7, φ2 = −0.1 and
σ2 = 0.09. Suppose further that data are observed up to time n with xn = 17.2, xn−1 = 16.5
and xn−2 = 15.1.
(a) Calculate the forecasts and forecast error variances for times n+ 1, n+ 2 and n+ 3.
(b) Give a 95% prediction interval for the forecasts at these times.
[10 marks]
Week 3
You should work on the three questions in this section during the Practical for Week 3 and, if
necessary, finish it during your independent study time.
Chapter 5: Dynamic linear models
4. Consider a univariate (i.e. scalar Yt) DLM with time-invariant Ft = F, Gt = G, Wt = W
and Vt = V = σ
2
v .
(a) Write an R function which simulates from a DLM of this form. This function should
have F, G, W and V , the length of the time series, and the mean m0 and variance C0
of ϑ0 as arguments, and it should return the simulated time series.
(b) Simulate a long realisation of a linear growth model with m0 = (0, 0)
T , C0 = 10I2,
Wt = diag(0.5, 0.1) and V = 50. For the first 100 samples, generate two separate plots:
one of the simulated data and one of the series of second differences. Using the whole time
series of second differences, report the sample estimates of the mean and autocovariance
function for k = 0, 1, . . . , 5.
[8 marks]
5. Let ϑt = (ϑt1, ϑt2)
T . The linear growth model has observation equation:
Yt = Fϑt + vt where F = (1, 0) and vt ∼ N(0, σ
2
v)
and system equation:
ϑt = Gϑt−1 +wt where G =
(
1 1
0 1
)
and wt =
(
wt1
wt2
)
∼ N2(0,W)
in which
W =
(
σ2w,1 0
0 σ2w,2
)
.
Suppose that we have observed data up to time n. As usual, we denote the conditional
expectation of ϑn as mn = (mn1, mn2)
T and the conditional variance as
Cn =
(
Cn11 Cn12
Cn12 Cn22
)
.
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MAS8382 Summative Practical Report
(a) Derive the mean fn(1) and variance Qn(1) of the one-step ahead forecast of Yt in terms
of the quantities in mn and Cn.
(b) Derive the mean fn(k) of the k-step ahead forecast of Yt in terms of the quantities in mn
and comment on your result.
[10 marks]
Chapter 6: Additional time series models
6. Consider the VARMA(p, q) process.
(a) In the solutions to the formative practical report, Question 3(a) gives a function which
simulates from an ARMA(p, q) process. Study this function and then modify it to give
a new function which simulates from a VARMA(p, q) process. This function should have
the parameters, the length of the time series, and a set of initial values as arguments,
and it should return the simulated time series. Tricky: make this function check if the
entered parameters are such that the process is stationary and/or invertible.
(b) Simulate a long realisation of the process for a VARMA(2, 1) model with µ = (0, 0)T ,
φ1 =
(
0.65 0.1
0.15 0.65
)
, φ2 =
(
0.3 −0.1
0.0 0.3
)
, θ1 =
(
0.6 0.2
0.15 0.5
)
, Σ =
(
1 0.4
0.4 1.2
)
.
Report the sample mean vector, sample variance matrix and sample covariance matrix
between Xt and Xt−1. Hint: in R, to compute the sample covariance matrix between
the n× p data matrix X and the n×m data matrix Y, you can use cov(X, Y). This will
return a p×m matrix whose (i, j)-th entry is the sample covariance between X[,i] and
Y[,j] – this will not be symmetric!
[10 marks]
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