STATS 726
STATISTICS
Time Series
SEMESTER 2, 2020
1 Consider the following processes defined for t ∈ Z:
(I) : Xt = 0.9Xt−1 + εt
(II) : Xt = −0.9Xt−1 + εt
(III) : Xt = 0.9Xt−4 + εt
(IV) : Xt = 0.9Xt−1 − 0.9εt−1 + εt
(V) : Xt = 0.9Xt−1 − 0.81Xt−2 + εt
(VI) : Xt = 0.6Xt−1 + 0.6εt−1 + εt
where εt ∼ IIDN(0, σ2
) and σ
2 > 0.
Figure 1 shows the estimated autocorrelation (left panel) and partial autocorrelation (right panel) plots from a sample of size 200 observations drawn from the processes (I)–(VI) in random order.
Find the correct autocorrelation and partial autocorrelation plots for each process. Briefly state which feature(s) of the autocorrelation and partial autocorrelation function are used to identify the process.
Note: Please refer to appendix for Figure 1 (page 7).
[Total: 18 marks]
2 Figure 2 shows the time plot of total monthly expenditure on cafes, restaurants and takeaway food services in Australia (in billions of dollars) for the period April 1982–September 2017.
Note: Please refer to appendix for Figures 2 and 3 (pages 8–9).
a Describe the time series components that you can observe in Figure 2. [2 marks]
b One of your friends suggested to use a log transformation of the original series before building a model. Based on Figure 2, why do you think he/she has suggested this to you? [3 marks]
c Figure 3 shows the time plot, autocorrelation and partial autocorrelation plots of
❼ the log-transformed original series
❼ the non-seasonally differenced log-transformed series
❼ the seasonally differenced log-transformed series
❼ both non-seasonally and seasonally differenced log-transformed series.
Using Figure 3, suggest an appropriate seasonal ARIMA model for the log-transformed data. Give reasons for your selection. [8 marks]
d Your friend recommended to fit models with different values for p, q, P and Q of the ARIMA(p, d, q)(P, D, Q) model, where d and D are as chosen in part 2c. Among these fitted models, he/she suggested to select the model with largest AICc which is defined as
where L() is the Gaussian likelihood evaluated at ˆα and n is the length of the time series.
Do you agree or disagree with your friend’s recommendation? Give reasons for your selection. [3 marks]
e Assume that the parameter estimates of the model suggested in part 2c are given. Explain how you could use this model to forecast the total expenditure for the next month (i.e., October 2017). [6 marks]
[Total: 22 marks]
3 Let {Xt}t∈N+ be defined as
Xt = Yt + εt
,
where
Yt = Yt−1 + ηt
with the initial condition Y0 = 0. {εt}t∈N+ and {ηt}t∈N+ are independent white noise series with variance > 0 and > 0, respectively. Define Zt = ∇Xt = (1−B)Xt
.
a Determine the orders p, d and q of the ARIMA(p, d, q) model of {Yt}. [1 mark]
b Find an expression for Zt
in terms of ηt
, εt and εt−1. [2 marks]
c Show that the autocovariance function of {Zt} is given by
[9 marks]
d Determine the orders p, d and q of the ARIMA(p, d, q) model of {Zt}. [1 mark]
e Using parts 3a and 3d, determine the orders p, d and q of the ARIMA(p, d, q) model of {Xt}. [2 marks]
[Total: 15 marks]
4 Let {Xt}t∈Z be the stationary AR(1) process defined as
Xt = φXt−1 + εt
,
where |φ| < 1, εt ∼ WN(0, σ2) and σ2 > 0.
Suppose we construct {Yt}t∈Z by using every second element of {Xt} (i.e. Yt = X2t). For example, Y−1 = X−2, Y0 = X0, Y1 = X2 and so on.
a Show that
i
Yt = φ
2Yt−1 + ηt
,
where ηt = φε2t−1 + ε2t
. [3 marks]
ii {ηt} is a white noise process. [6 marks]
iii {ηt} is uncorrelated with {Yt−1, Yt−2, . . . }. [7 marks]
b Determine the orders p, d and q of the ARIMA(p, d, q) model of {Yt}. [1 mark]
c Write down an expression for the autocorrelation function of {Yt}. Denote it by ρ(h), where h is the time lag. [2 marks]
d Let be the best linear predictor of Y5 based on Y3, Y2 and Y1:
= α1Y3 + α2Y2 + α3Y1,
where α1, α2 and α3 are real-valued coefficients.
i Applying the expression for estimating the coefficients of the best linear predictor (refer slides 66–67 in the “Introduction handout”), show that
[4 marks]
ii Suppose the following three choices are given for α1, α2 and α3:
Among these three choices, which one would you select for (α1, α2, α3) to minimize the mean squared error? Justify your selection. [3 marks]
iii Based on your choice for part 4dii, find an expression for E[Y5 − ]2
in terms of φ and γ(0) (i.e., the variance of {Yt}). [3 marks]
[Total: 29 marks]
5 Assume that the observations x1, x2, . . . , xn are generated from the model
Xt = αXt−1 − (1 − α)Xt−2 + εt
,
where εt ∼ IIDN(0, 1) and α is a real-valued constant.
a Write down the conditional likelihood function of the parameter α based on x1, x2, . . . , xn, where n > 2. [3 marks]
b Using part 3a, show that the estimate of α is given by
[5 marks]
c Suppose x1 = 3, x2 = 2, x3 = 0 and x4 = 1. Calculate based on x1, x2, x3 and x4. State the numerical result to 4 decimal places. [3 marks]
d Based on x1, x2, x3 and x4 given in part 5c, calculate 2-step-ahead forecast. State the numerical result to 4 decimal places. [5 marks]
[Total: 16 marks]