AMA540 Time series (2022/23) Assignment 2
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AMA540 Time series (2022/23)
Assignment 2
Notes:
(a) Submit your assignment to Blackboard on or before 6:30 pm, 29th Mar. 2023 (b) Heavy deduction of marks will be applied to late submission.
(c) Check your submitted file carefully. We shall mark your submitted file. A wrong file will get no marks. An incomplete file will get incomplete marks.
1. Suppose that {Yt } is an AR(1) process with −1 1. Yt = Yt−1 + et , et NID(0,2 )
(a) Find the autocovariance function for Wt = Yt = Yt − Yt −1 in terms of and 2 .
(b) Show that var(W ) =
22
t 1 + .
2. For the ARMA(1,2) model Yt = 0.7Yt −1 + et + 0.6et−1 + 0.5et−2 , et NID(0,2 ) ,find
(a) (k) in terms of (k − 1) for k >2.
(b) (2) in terms of (1),2 ,& (0). Here (0) = var(Yt ).
(c) (1) in terms of 2 , & (0) .
3. You are given the ARMA process:
X = 0.5X − 0.04X + 0.25Z + Z
Determine the coefficients bi in the infinite series representation of Xt by Zt . Namely, find bi if
Xt = bi Zt−i
i =0
4. The MA(2) process is defined by Xt = 1at−1 +2at−2 + at , at ~ IID(0, )2 . Consider two MA(2) processes, one with 1 = 2 = 1 / 6 and another with
1 = 1, &2 = 6.
(a) Show that these processes have the same autocorrelation function.
(b) How do the roots of the polynomial in the backshift operator B compare?
(c) Determine which of the two processes are invertible. Find the first three partial autocorrelation coefficients of the invertible process.
5. From a time series Yt of length 100, we have computed r1 = 0.8, r2 = 0.5, r3 = 0.4, y = 2 , and a sample variance of 5. Let us assume that an AR(2) model with a constant term is appropriate, namely, (Yt − ) = 1 (Yt−1 − ) + 2 (Yt−2 − ) + et , et NID(0,2 ) . Find estimates for 1 ,2 , , & 2 .
2023-03-21