FINAL EXAM GR5263/GU4263 Section 002 Fall 2021
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Name: UNI:
1. (10pts) Suppose we have data (X1, . . . , X5) = (1.5, 1,−0.5, 1, 2).
(a) (5 pts) If Xt follows an AR(3) model with φ1 = 0.2, φ2 = 0.3, φ3 = 0.1. Find the best linear
predictor of X6 in terms of the data.
(b) (5 pts) If Xt follows an MA(3) model with θ1 = 0.2, θ2 = 0.3, θ3 = 0.1. Find the best linear
predictor of X10 in terms of the data.
2. (15 pts) Bob fits ARMA(p,q) models to a data set of size 1000 with mean 0 for different p and q. The
AIC’s for the model fitting with p = 2, 3 and q = 2, 3 are given in the following tables.
(a) (5 pts) What is the AIC selection of (p,q) based on the table?
(b) (5 pts) What is the selection of (p,q) if we use the BIC criteria instead? (log 1000 ≈ 7)
(c) (5 pts) Bobs observes a spike at every 12 lags in the sample acf and pacf plots of the data. He
wants to try a SARIMA model. Could you write down the equation of Xt as a linear combination
of its own past values and a white noise process Wt for Xt being a ARIMA(1, 1, 1) × (1, 1, 1)12
model? You are free to use φ’s, Φ’s, θ’s, Θ’s for the coefficients and the backshift operator B.
p
q
2 3
2 113 111
3 110 109
Table 1: AIC
3. (15 pts) Consider the linear process given by
Xt =
∞∑
j=0
(
(aB)j + (aB)j+1
)
wt,
where a is a real constant such that |a| < 1; B is the backshift operator; and wt is white noise with
variance σ2w.
(a) (5 pts) Identify Xt as an ARMA(p,q) process. Namely, find (φ1, . . . , φp) and (θ1, . . . , θq) such
that (1− φ1B − · · · − φpBp)Xt = (1 + θ1B + · · ·+ θqBq)wt.
(b) (5 pts) Is Xt causal and/or invertible. Justify your answer.
(c) (5 pts) Find γ(0) and γ(1) of the process. You answer should only involve a and σ2w.
4. (10 pts) A model for a monthly time series is
Yt = β0 + β1t+ β2 cos
(
pit
6
)
+ β3 sin
(
pit
6
)
+ Zt,
where β0, β1, β2 and β3 are non-zero constants and Zt is a causal AR(1) process
Zt − φZt−1 = wt, wt ∼WN(0, σ2w).
Define
Xt = ∇∇12Yt = (1−B)(1−B12)Yt,
where B is the backshift operator.
FINAL EXAM GR5263/GU4263 Section 002 Page 2 of 2
(a) (5 pts) Is Yt weakly stationary? Justify your answer.
(b) (5 pts) Is Xt weakly stationary? Is Xt invertible with respect to wt? Justify your answer. Hint:
consider Xt − φXt−1.
5. (20 pts) Suppose an AR(2) process has autocovariance γ(0) = 1.5, γ(1) = 0.5, φ22 = 0.25
Yt = φ1Yt−1 + φ2Yt−2 +Wt, {Wt} ∼WN(0, σ2).
(a) (10 pts) Find φ1, φ2 and σ
2
w.
(b) ( 5 pts) Find φ33 the pacf at lag 3.
(c) (5 pts) Find the first 3 coefficients in the MA(∞) representation of Yt, namely find ψ0, ψ1, ψ2
where Yt =
∑∞
j=0 ψjWt−j .
6. (30 pts) Consider the GARCH(1,1) model
at = εtσt, εt
i.i.d.∼ N(0, 1)
σ2t = 0.05 + 0.25a
2
t−1 + 0.25σ
2
t−1
Suppose it is known that Ea4t is a finite constant for all t. Useful fact: the fourth moment of a standard
normal random variable is 3.
(a) (5 pts) Given at−1 = −2 and σ2t−1 = 4, what is the distribution of at?
(b) (5 pts) Calculate Cov(at, σ
2
t ).
(c) (10 pts) Calculate Ea4t .
(d) (5 pts) Identify a2t as an ARMA process. Write down explicitly the ARMA equation and the
white noise variance.
(e) (5 pts) Suppose the model is fitted to a set of observations {at, t = 1, . . . , 100}. Suppose the
estimated volatility is given as σˆ2t , t = 1, . . . , 100. How to conduct diagnosis of the appropriateness
of the model on the observations? Give your recommendation.