STATS 726
STATISTICS
Time Series
SEMESTER 2, 2018
1 a Explain the following terms in the context of univariate time series. Use your own words since simply copying from the notes will earn no marks.
i (2 marks) Weakly stationary time series.
ii (2 marks) Autocorrelation function (ACF) at lag-h.
iii (2 marks) Partial autocorrelation function (PACF) at lag-h.
b Figures 1–2 (see pages 6–7) give time plots and ACF plots corresponding to five different time series, respectively.
i (5 marks) Briefly explain the time series components that you can observe in each time plot.
ii (4 marks) Match each time plot on page 6 with one of the ACF plots shown in page 7.
For example, time plot (4) and ACF plot (E) is a match.
c Let {Xt} be a random walk process with a drift defined as
Xt = δ + Xt−1 + εt
, t ≥ 1,
where δ is a real-valued constant, X0 = 0 and {εt} ∼ WN(0, σ2).
i (4 marks) Show that {Xt} is not weakly stationary.
ii (3 marks) Let ρX(r, s) = Corr(Xr, Xs) be the autocorrelation function of {Xt} at time points r and s where r, s ∈ {1, 2, . . . }. Prove that
iii (4 marks) Suggest a transformation of the time series {Xt} such that the resulting time series (say {Yt}) is weakly stationary. Show that {Yt} is weakly stationary. [26 marks]
2 Let {Xt}t∈Z be an invertible MA(1) process with a non-zero mean µ, defined by
Xt − µ = θεt−1 + εt
, {εt}t∈Z ∼ WN(0, σ2
). (I)
a (2 marks) State the autocovariance function of {Xt} at lag-h for all h ∈ Z.
b (4 marks) For n ≥ 2, write the expression of the best linear predictor of Xn based on Xn−1, Xn−2, . . . , X1 in terms of µ, θ and {Xn−1, Xn−2, . . . , X1}.
Note: You are not required to simplify the matrix inversion.
c Let X1, X2, X4 and X5 be four random variables from the MA(1) process de-fined in Eq. (I).
i Find the best linear estimator of the missing value X3 in terms of
(A) (4 marks) X1 and X2.
(B) (3 marks) X4 and X5.
(C) (6 marks) X1, X2, X4 and X5.
Hint: Let A ∈ R
r×r
, B ∈ R
r×s
, C ∈ R
s×r and D ∈ R
s×s
. If A, D and D − CA−1B are non-singular, then
ii (9 marks) Compute the mean squared error for each of the estimators in (A)–(C). [28 marks]
3 Let {Zt}t∈Z be random variables satisfying
Zt = φ1Zt−1 + φ2Zt−2 + εt
, (II)
where φ1, φ2 ∈ R and {εt}t∈Z ∼ IID N (0, σ2
).
a (5 marks) Assuming that
Zt = 0 for all t ≤ 0 and
Zt = 0 for all t > n,
write the identity in Eq. (II) for t = 1, 2, . . . , n + 2.
Then re-write the n + 2 equations in the following equivalent matrix form.
y = Xφ + ε,
where y = (Z1, Z2, . . . , Zn, 0, 0)> , φ = (φ1, φ2)
> and (·)
> denotes transposi-tion.
b (6 marks) Assuming that the entries of matrix X are fixed, use the result in part (a) to show that the ordinary least squares (OLS) estimator of vector φ is:
(III)
where ˆρ(h) denotes the sample autocorrelation function of {Zt} at lag-h.
Note: You do not need to derive the first-order conditions of OLS estimation.
c (3 marks) Using the properties of the OLS estimator, show that the estimate of the variance-covariance matrix for φˆ in Eq. (III) is given by
where ˆγ(0) is the sample autocovariance function of {Zt} at lag-0 and ˆσ
2
is the unbiased estimate of σ
2
.
d From a time series of length n = 144, we find that ˆγ(0) = 27.418, ˆρ(1) = −0.620, ρˆ(2) = 0.897 and ˆσ
2 = 5.257.
i (3 marks) Compute the OLS estimate of (φ1, φ2).
ii (6 marks) Construct the 95% confidence intervals for φ1 and φ2. Com-ment whether or not these coefficients are statistically different from zero. Justify your answer.
Note: The 95% central probability region of N (0, 1) is from −1.96 to 1.96. [23 marks]
4 Figure 3 (see page 8) shows the time plot of quarterly European retail trade index from 1996 to 2011 covering wholesale and retail trade, and repair of motor vehicles and motorcycles.
a (1 mark) Describe the time series components that you can observe in Figure 3.
b (2 marks) Is this time series weakly stationary? Briefly justify your answer.
c (3 marks) Figure 4 (see page 9) shows the time plot, ACF and PACF plots of
• original time series
• non-seasonally differenced time series
• seasonally differenced time series
• both non-seasonally and seasonally differenced time series.
Using this figure, find an appropriate differencing which yields a stationary time series. Justify your selection.
d (6 marks) By examining the ACF and PACF plots corresponding to your selection in part (c), suggest a seasonal ARIMA model which is appropriate for this data set. Give reasons for your selection.
e (2 marks) Based on your choice of differencing in part (c) refer either Table 1 or Table 2 or Table 3 (see pages 10–12) and select a model which is best according to the AICc (corrected AIC) value.
f (3 marks) Discuss briefly few residual diagnostics that can be performed to check the adequacy of the model selected in part (e).
g (6 marks) Write the difference equation for the model you have selected in part (e) in terms of the estimated coefficients given in Table 1, 2 or 3. Write the equations which are needed to compute the forecasts for the next 2 quarters. [23 marks]