Module Code: MATH380201
1. (a) Consider the time series model
Xt = µ(t) + s(t) + εt.
Assuming the standard notation used in this module, what do each of the terms
Xt, µ(t), s(t) and εt represent?
In a plot of Xt against t, what features would you look for to determine whether
the terms µ(t) and s(t) are required?
Explain why µ(t) and s(t) are functions of t, whilst t is a subscript in Xt and εt.
(b) Quarterly sales of videos in the Leeds “Disney” store are shown in figure 1. Below
is the code and output for an analysis of these data in R, with the sales data stored
in the time series object X.
Explain what is being done at points (i)–(iv) in the R code. Explain what is the
difference between (v) and (vi) in the R code. Explain, giving reasons, which of
(v) and (vi) is preferable. Write out the model with estimated parameters in full.
(The relevant points in the R code are denoted #### (i) #### etc.)
Given that the sales for the four quarters of 2018 were 721, 935, 649, and 1071,
use model-based forecasting to predict sales for the first quarter of 2019. (A point
forecast is sufficient; you do not need to calculate a prediction interval.)
Suggest one change to the fitted model which would improve the analysis. (You
can assume that the choice of stochastic process at (v) in the R code is the correct
one for these data.)
Time
Sa
le
s
2010 2012 2014 2016 2018
10
00
15
00
20
00
Figure 1: Quarterly video sales in Leeds “Disney” store.
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Module Code: MATH380201
> tt = 1:32
> trend.lm = lm(sales ~ tt) #### (i) ####
> summary(trend.lm)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2107.220 57.997 36.33 < 2e-16 ***
tt -43.500 3.067 -14.18 7.72e-15 ***
> trend = ts(fitted(trend.lm), start=start(sales), freq=frequency(sales))
> X = sales - trend #### (ii) ####
> q1 = as.numeric((1:32 %% 4) == 1)
> q2 = as.numeric((1:32 %% 4) == 2)
> q3 = as.numeric((1:32 %% 4) == 3)
> q4 = as.numeric((1:32 %% 4) == 0)
> season.lm = lm(resid(trend.lm) ~ 0 + q1 + q2 + q3 + q4) #### (iii) ####
> summary(season.lm)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
q1 -38.41 43.27 -0.888 0.38232
q2 18.80 43.27 0.435 0.66719
q3 -134.78 43.27 -3.115 0.00422 **
q4 154.38 43.27 3.568 0.00132 **
> season = ts(fitted(season.lm), start=start(sales), freq=frequency(sales))
> Y = X - season #### (iv) ####
> ar(Y, aic=FALSE, order.max=1) #### (v) ####
Coefficients:
1
0.5704
Order selected 1 sigma^2 estimated as 9431
> ar(Y, aic=FALSE, order.max=2) #### (vi) ####
Coefficients:
1 2
0.5574 0.0105
Order selected 2 sigma^2 estimated as 9437
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Module Code: MATH380201
2. Let {Xt} be a moving average process of order q (usually written as MA(q)) defined on
t ∈ Z as
Xt = εt + β1εt−1 + · · ·+ βqεt−q, (1)
where {εt} is a white noise process with variance 1.
(a) Show that for any MA(1) process with β1 6= 1 there exists another MA(1) pro-
cess with the same autocorrelation function, and find the lag 1 moving average
coefficient (β′1 say) of this process.
(b) For an MA(2) process, equation (1) becomes
Xt = εt + β1εt−1 + β2εt−2. (2)
i. Define the backshift operator B, and write equation (2) in terms of a polyno-
mial function β(B), giving a clear definition of this function.
ii. Hence show that equation (2) can be written as an infinite order autoregressive
process under certain conditions on β(B), clearly stating these conditions.
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Module Code: MATH380201
3. Let {Xt} be an autoregressive process of order one, usually written as AR(1).
(a) Write down an equation defining Xt in terms of an autoregression coefficient α
and a white noise process {εt} with variance σ2ε .
Explain what the phrase “{εt} is a white noise process with variance σ2ε” means.
(b) Derive expressions for the variance γ0 and the autocorrelation function ρk, k =
0, 1, . . . of the {Xt} in terms of σ2ε and α.
Use these expressions to suggest an estimate of α in terms of the sample autocor-
relations {ρˆk}.
(c) Suppose that only every second value of Xt is observed, resulting in a time series
Yt = X2t, t = 1, 2, . . ..
Show that {Yt} forms an AR(1) process. Find its autoregression coefficient, say
α′, and the variance of the underlying white noise process, in terms of α and σ2ε .
(d) Given a time series data set X1, . . . , X256 with sample mean x = 9.23 and sample
autocorrelations ρˆ1 = −0.6, ρˆ2 = 0.36, ρˆ3 = −0.22, ρˆ4 = 0.13, ρˆ5 = −0.08,
estimate the autoregression coefficients α and α′ of {Xt} and {Yt}.
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Module Code: MATH380201
4. (a) Given data X1, . . . , Xn, let Xn(l) be the l-step ahead forecast of Xn+l based on a
(possibly infinite) MA model for {Xt}. Show that minimising
E{[Xn+l −Xn(l)]2}
leads to a forecast of the form
Xn(l) =
∞∑
j=l
βjεn+l−j,
where {εt} is a white noise process, stating how the βj might be found.
(b) i. Define difference operators ∇, and write ∇ in terms of the backshift operator
B.
ii. Define autoregressive and moving average process of order (p, q) (usually
written as ARMA(p, q)) and autoregressive integrated moving average pro-
cess of order (p, d, q) (usually written as ARIMA(p, d, q)) and show how an
ARIMA(p, d, q) process can be written as an ARMA(p′, q′) process, giving the
AR and MA orders of this process.
iii. Hourly data were gathered on the bacterial infection of a sample of food under
controlled conditions. These data are denoted by Xt, t = 1, . . . , 478.
Summary statistics for {Xt} and {Yt}, where Yt = ∇Xt, are presented below.
Here, ρˆk and αˆkk are respectively the sample autocorrelation and sample partial
autocorrelation coefficients at lag k.
Identify a suitable model for this time series and estimate the parameters of
Xt : x¯ = 443.7, sx = 341.3.
k 1 2 3 4 5 6 7 8 9 10
ρˆk 0.995 0.991 0.986 0.981 0.976 0.970 0.965 0.959 0.952 0.946
αˆkk 0.995 0.005 −0.022 −0.018 −0.024 −0.031 −0.013 −0.042 −0.027 −0.007
Yt : y¯ = 1.830, sy = 17.17.
k 1 2 3 4 5 6 7 8 9 10
ρˆk −0.019 0.027 0.076 0.058 −0.022 −0.032 0.045 −0.047 −0.014 0.007
αˆkk −0.019 0.026 0.077 0.060 −0.024 −0.043 0.036 −0.043 −0.010 0.007
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