STATS 726
STATISTICS
Time Series
SEMESTER 2, 2019
1 State whether the following statements are TRUE or FALSE. In each case, you MUST provide a short explanation for your selection.
a (9 marks) Let for t ∈ N+, where and {εt}t∈N+ ∼ WN(0, σ2). Define Yt = (1 − B6
)Xt
for all t ≥ 7.
Statement: {Yt}t≥7 is weakly stationary.
Hint: For all α, β ∈ R, we have:
cos(α + β) = cos(α)cos(β) − sin(α)sin(β)
cos(α − β) = cos(α)cos(β) + sin(α)sin(β).
b (7 marks) Let {Xt}t∈Z and {Yt}t∈Z be two weakly stationary time series. Define Zt = Xt + Yt
for all t ∈ Z.
Statement: {Zt}t∈Z is weakly stationary.
c (7 marks) For t ≥ 1, Xt = a + bt + εt
, where a, b ∈ R\{0} are constants and {εt}t∈N+ ∼ WN(0, 1). Define Yt = (1 − B)Xt
for all t ≥ 2.
Statement: {Yt}t≥2 is a non-invertible MA(1) process. [23 marks]
2 a (3 marks) Explain briefly what is meant by a causal process. Use your own words since simply copying from the notes will earn no marks.
b Consider the weakly stationary ARMA(1, 2) process defined as
Xt + 0.5Xt−1 = εt − 0.5εt−1 + 0.25εt−2, t = 0, ±1, ±2, . . . ,
where {εt}t∈Z ∼ WN(0, σ2). Show that:
i (4 marks) {Xt}t∈Z is causal.
ii (8 marks) For all t ∈ Z, Xt can be expressed as
Hint: You may find the homogeneous difference equation and the initial conditions defined for an ARMA(p, q) process to be useful in the deriva-tions.
iii (12 marks) The autocovariance function of {Xt}t∈Z is given by
Hint: Use the result from part (ii) or the difference equation defined for the autocovariance function of an ARMA(p, q) process. [27 marks]
3 Suppose that {Xt}t∈Z is defined as
Xt = Acos(ωt) + Bsin(ωt),
where A, B are uncorrelated random variables with zero mean and unit variance, and ω is a fixed frequency in the interval (0, π).
It can be shown that the autocovariance function (ACVF) of {Xt}t∈Z is given by
γX(h) = cos(ωh)
for h ≥ 0.
a (18 marks) Show that the best linear predictor of X3 given X2 and X1 is
˜X3 = 2cos(ω)X2 − X1
and the corresponding mean squared error is zero.
Hint: For all α, β ∈ R, we have:
cos(α + β) = cos(α)cos(β) − sin(α)sin(β)
cos(α − β) = cos(α)cos(β) + sin(α)sin(β)
cos2(α) + sin2
(α) = 1.
For a, b, c, d ∈ R such that ad − bc ≠ 0, the following identity holds:
b (5 marks) Let X = (X1, X2, . . . , Xn)T denote a vector of random variables having mean µ ∈ R
n×1 and variance-covariance matrix Γ ∈ R
n×n
. For an arbitrary vector w ∈ R
n×1
, show that
Var(wT X) = wT Γw.
c (5 marks) Using the result from part (b), show that the ACVF of {Xt}t∈Z is non-negative definite.
Hint: A real-valued function f(·) defined for the integers is said to be non-negative definite if
for all positive integers n and vectors w = (w1, w2, . . . , wn)
> with real-valued entries. [28 marks]
4 Let
φ(B) = (1 − φ1B − · · · − φpB
p
),
θ(B) = (1 + θ1B + · · · + θqB
q
),
Φ(B
s
) = (1 − Φ1B
s − · · · − ΦPB
sP ),
Θ(B
s
) = (1 + Θ1B + · · · + ΘQB
sQ).
A seasonal ARIMA(p, d, q)(P, D, Q)s model is defined as
φ(B)Φ(B
s
)(1 − B)
d
(1 − B
s
)
DXt = c + θ(B)Θ(B
s
)εt
, Eq.(1)
where c = µ(1 − φ1 − · · · − φp)(1 − Φ1 − · · · − ΦP ), µ ∈ R is the mean of Yt = (1 − B)
d
(1 − Bs
)
DXt and {εt}t∈Z ∼ WN(0, σ2
).
Figure 1 shows the time plot of the quarterly visitor nights (in millions) spent by international tourists to Australia for the period 1999–2015.
a (2 marks) Describe the time series components that you can observe in Fig-ure 1.
b Figure 2 shows the time plot, autocorrelation (ACF) and partial autocorrela-tion (PACF) plots of
• the original series
• the non-seasonally differenced time series
• the seasonally differenced time series
• both non-seasonally and seasonally differenced time series.
(8 marks) Using Figure 2, suggest an appropriate seasonal ARIMA model for the given data. Provide reasons for your selection.
c (3 marks) Do you consider it appropriate to include a constant term, c (refer Eq.(1)), in the model suggested in part (b)? Explain your answer.
d (4 marks) Discuss briefly a few residual diagnostics that can be performed to check the adequacy of the model selected.
e (5 marks) Assume that the parameter estimates of the model suggested are given. Explain how you could use this model to forecast the number of visitor nights that will be spent by international tourists to Australia for the next quarter (i.e., Quarter 1 of 2016). [22 marks]