Description
STAT 4005 Time Series Assignment 1
Let at ∼ W N(0, σ2
)
1. Does the quadratic trend Tt = α + βt2 pass through the moving average
filter (a−1, a0, a1) = ( 1
3
,
1
3
,
1
3
)?
2. Suppose Zt = 8+ 4t+ 2Xt, where Xt is a zero-mean stationary series with
autocovariance function γk.
(a) Find the mean and the autocovariance function of Zt.
(b) Is Zt stationary? Why?
(c) Find the mean and the autocovariance function of ∆Zt = (1 − B)Zt.
(d) Is ∆Zt stationary? Why?
3. Suppose that Zt = (at + at−1 + at−3)/3
(a) Show that Zt is weakly stationary.
(b) Find ρk, k = 0, 1, 2, 3, …
(c) Find Var
1
5
P5
t=1 Zt
.
4. Consider the time series {Zt} satisfying
Zt = 0.2Zt−1 + at.
(a) Assuming that {Zt} is stationary, find the mean E(Zt).
(b) Assuming that {Zt} is stationary and Cov(Zs, at) = 0 for s < t, find the variance Var(Zt). (Hints: take variance on both sides.) (c) Find Cov(Zt, Zt−k) for k = 1, 2, 3, … (Hints: multiply Zt−k on both sides and take expectation.) 5. Consider the time series {Zt} satisfying Z1 = a1; Zt = 0.2Zt−1 + at for t > 1.
(a) By mathematical induction, show that Zt =
Pt−1
k=0 0.2
kat−k.
(b) Find the mean E(Zt) and the variance Var(Zt).
(c) Find Cov(Zt, Zt−k) for t > k and k ≥ 0.
6. Consider the data set monthly milk.csv in the class website that contains the monthly milk production from 1962 to 1975. Using R, decompose
the series into three components.
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STAT 4005 Time Series Assignment 2
Let at ∼ W N(0, σ2
) if not specified otherwise.
1. Let Xt be a stationary time series with mean α and autocovariances γk =
0.8
k and X¯ =
P10
t=6 Xt/5
(a) Find E(X¯).
(b) Find Var(X¯).
2. Consider the process
Zt = at + at−1 + 0.25at−2 , σ
2
a = 20.
(a) Identify the order of the ARIMA model for the process.
(b) Is {Zt} stationary?
(c) Is {Zt} invertible?
(d) Find the ACVF γ(k) and ACF ρ(k) of {Zt} for k = 0, 1, 2, 3,….
(e) Find the values of πk, k = 0, 1, 2, 3,… if the process is written as
at =
P∞
i=0 πtZt−i
.
3. Consider the AR(2) process
Zt = 0.5Zt−1 − 0.06Zt−2 + at ,
where ats are independently and identically distributed as N(0, 1).
(a) Find the roots of the AR characteristic equation.
(b) Is the process Zt stationary and causal? Why?
(c) Find the autocovariances γ(0), γ(1) and γ(2).
4. Find ACVF γ(k), k=0,1,2,3,…. of the process
Zt = 0.7Zt−4 + at .
5. Find the AR and MA representation of the process
Zt = 0.6Zt−1 + at + 0.2at−1 , at ∼ W N(0, 4).
6. Identify the following as specific ARIMA models:
a) Zt = 1.5Zt−1 − 0.5Zt−2 + at − 0.3at−1 + 0.6at−2.
b) Zt = 3Zt−1 − 3Zt−2 + Zt−3 + at + 0.1at−1.
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7. Consider the ARMA(2,1) model
Zt = 0.6Zt−1 − 0.09Zt−2 + at − 0.2at−1 , at ∼ W N(0, 1).
a) Find the AR representation of {Zt}.
b) Find the ACF ρ(k) of {Zt} for k ∈ Z.
8. Show that for |φ| > 1,
Zt =
at
φ2
−
1 −
1
φ2
X∞
k=1
at+k
φk
.
is a white noise process with Var(Zt) = σ
2
φ2 .
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STAT 4005 Time Series Assignment 3
Let Zt ∼ W N(0, σ2
) be white noise. Given a data set
Y = (1.33, −0.56, −1.31, −0.37, 0.05, 0.46, 2.00, −0.19, −0.25, 1.07,
−0.17, 1.14, 0.63, −0.75, 0.15, 0.71, 0.45, −0.14, 0.57, 1.43).
1) Draw a time series plot, ACF and PACF plot for the data.
2) Find the moment estimates of θ, σ
2
for fitting an MA(1) model Yt =
Zt + θZt−1 to the data.
3) Find the least squares estimates of φ1, φ2, σ2
for fitting an AR(2) model
Yt − φ1Yt−1 − φ2Yt−2 = Zt to the data. Find a 95% confidence interval for
each of φ1 and φ2.
4) Find the Yule-Walker estimates of φ1, φ2 for fitting an AR(2) model Yt −
φ1Yt−1 − φ2Yt−2 = Zt to the data.
5) Find the conditional least squares estimates of φ, θ, σ
2
for fitting an
ARMA(1,1) model Yt − φYt−1 = Zt + θZt−1 to the data.
6) Find the maximum likelihood estimates of φ, θ, σ
2
for fitting an ARMA(1,1)
model Yt − φYt−1 = Zt + θZt−1 to the data. What is the maximized value
of the log-likelihood?
7) Among AR(p) with p = 1, 2, 3, 4, 5, which model is the best in terms of
FPE?
8) Among MA(q) with q = 1, 2, 3, 4, 5, which model is the best in terms of
AICC?
9) Fit an MA(1) model to the data. Find the residuals. Find the Ljung-Box
test statistic Q(10). State H0 and H1 of the Ljung-Box test, and then
draw the conclusion.
10) Which model will you select to describe the data?
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STAT 4005 Time Series Assignment 4
1. Given a data set Y1, . . . , Y20.
Y = (1.33, −0.56, −1.31, −0.37, 0.05, 0.46, 2.00, −0.19, −0.25, 1.07,
−0.17, 1.14, 0.63, −0.75, 0.15, 0.71, 0.45, −0.14, 0.57, 1.43).
(a) Fit an MA(2) model to {Yt}, find the k-step ahead forecast and
the 95% prediction intervals for k = 1, 2, 3, . . ..
(b) With the MA(2) model fitted in (a), find the partial autocorrelations φ11, φ22 and φ33 using the first principle.
(c) Fit an AR(1) model to {Yt}, find the k-step ahead forecast and
the 95% prediction intervals for k = 1, 2, 3, . . ..
(d) With the AR(1) model fitted in (c), find Cov(e20(k), e20(l)), where
k 6= l are positive integers.
(e) Fit an ARMA(1,1) model to {Yt}, find the 1st and 2nd-step ahead
forecast and the 95% prediction intervals.
(f) Fit an ARIMA(1,1,0) model to {Yt}, find the 1st and 2nd-step
ahead forecast and the 95% prediction intervals.
Note: You could use the R function arima() for model fitting.
2. Consider the GARCH(1,1) model
Xt = σtt
, t
iid∼ N(0, 1),
σ
2
t = α0 + α1X
2
t−1 + β1σ
2
t−1
,
where α0, α1, β1 ≥ 0 and α1 + β1 < 1.
(a) Express Xt+1 and Xt+2 in terms of Xt
, σt
, t+1 and t+2.
(b) Given observed values of σ1, X1, X2 and X3, express the likelihood
function L(α0, α1, β1) in terms of σ1, X1, X2 and X3.
3. Prove the following result.
Theorem 1 If Xt is a GARCH(p, q) process, then X2
t
is an ARMA(m, p)
process with noise νt = σ
2
t
(
2
t − 1), where m = max(p, q).
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