a) Consider the following moving average process, MA(2):
a) Consider the following moving average process, MA(2):
Yt = ut + α1ut-1 + α2ut-2
where ut is a white noise process, with E(ut)=0, var(ut)=σ2 and cov(ut,us)=0 
.
Derive the mean, E(Yt), the variance, var(Yt), and the covariances cov( Yt,Yt+1 ) and cov(Yt,Yt+2 ), of this process.
b) Give a definition of a (covariance) stationary time series process. Is the MA(2) process (covariance) stationary?
Homework Answers
Is the following MA(2) process covariance-stationary?
Is the following MA(2) process covariance-stationary? Yt = (1 + 2.4L + 0.8L2) εt, with E ( εt εT) = $$ \left\{\begin{array}{c} 1 \text { for } t=\tau \\ 0 \text { otherwise } \end{array}\right. $$ If so, calculate its auto-covariances.
Let {et} denote a white noise process from a normal distribution with E[et] = 0, Var(et)...
Let {et} denote a white noise process from a normal distribution with E[et] = 0, Var(et) = σe2 and Cov(et, es) = 0 for t ≠ s. Define a new time series {Yt} by Yt = et + 0.6 et -- 1 – 0.4 et – 2 + 0.2 et – 3. 1. Find E(Yt ) and Var(Yt ). 2. Find Cov(Yt , Yt – k) for k = 1, 2, ...
Determine which of the following processes are stationary and invertible. In the following we alw...
Determine which of the following processes are stationary and
invertible. In the following we always assume that ut is white
noise with mean zero and variance σ2, i.e. ut ∼ WN(0, σ2)
Problem 2 (20 marks). Determine which of the following processes are stationary and invertible. In the tollowing we always assume that ut is white noise with mean zero and variance σ2, i.e. ut ~ WN(0, σ2). 5 marks 5 marks 5 marks 5 marks 10
Problem 2 (20...
Recall that a time series {εt} is called a white noise process if i. E[εt] =...
Recall that a time series {εt} is called a white noise process if i. E[εt] = 0 t ; ii. Cov(εs, εt) = 0 s ≠ t ; iii. Var(εt) = σ2 < ∞ Construct the autocorrelation function f(h), h=0,-+1,-+2,… for the white noise process.
2. Consider a following time series process Yt = 1.5Yt−1 −0.5Yt−2 +εt a) Rewrite this process...
2. Consider a following time series process Yt = 1.5Yt−1 −0.5Yt−2 +εt a) Rewrite this process in lag polynomial form. b) Is this process invertible? Is this process covariance stationary? c) Difference this process once and show that ΔYt = Yt −Yt−1 is covariance stationary.
Problem 2. Let X be a random variable with mean 0 and variance σ2. Define the...
Problem 2. Let X be a random variable with mean 0 and variance σ2. Define the process Yt-(-1) Compute the mean and covariance function of the process {Yt). Is this process stationary?
2. (a) Consider the following process: where {Z) is a white noise process with unit variance. [1 ...
2. (a) Consider the following process: where {Z) is a white noise process with unit variance. [1 mark] ii. Find the infinite moving average representation of X,i.e., find the scquence [6 marks] i. Explain why the process is stationary. (6) such that Xt = Σ b,2-j. iii. Calculate the mean and the autocovariance "Yo, γι and 72 of the process. 7 marks iv. Given 40 = 0.1 and Xo = 1.8, find the 2-step ahead forecast of the time series...
True or false? You do not have to provide explanations. (a) Any moving average (MA) process is co...
True or false? You do not have to provide explanations. (a) Any moving average (MA) process is covariance stationary. (b) Any autoregressive (AR) process is invertible. (c) The autocorrelation function of an MA process decays gradually while the partial autocorrelation function exhibits a sharp cut-off. (d) Suppose yt is a general linear process. The optimal 2-step-ahead prediction error follows MA(2) process. (e) Any autoregressive moving average (ARMA) process is invertible because any moving average (MA) process is invertible. (f) The...
2. Let (et) be a zero mean white noise process with variance 1. Suppose that the...
2. Let (et) be a zero mean white noise process with variance 1. Suppose that the observed process is h ft + Xt where β is an unknown constant, and Xt-et- Explain why {X.) is stationary. Find its mean function μχ and autocorrelation function p for lk0,1,.. a. b. Show that {Yt3 is not stationary. C. Explain why w. = ▽h = h-K-1 is stationary. d. Calculate Var(Yt) Vt and Var(W) Vt . (Recall: Var(X+c)-Var(X) when c is a constant.)...
1. Consider the following autoregressive process 2+ = 4.0 + 0.8 2t-1 + Ut, where E...
1. Consider the following autoregressive process 2+ = 4.0 + 0.8 2t-1 + Ut, where E (u+12+-1, Zt-2, ....) = 0 and Var (ut|2t-1, 2-2, ...) = 0.3. The unconditional E (Zt) and unconditional variance Var (zt) are: (a) E (2+) = 11.1111, Var (zł) = 0.8333 (b) E (2+) = 11.1111, Var (zt) = 1.5 (c) E (zt) = 20, Var (zt) = 0.8333 (d) E (2+) = 4, Var (zł) = 0.8333 (e) E (Zt) = 4,Var (z+)...
Determine which of the following processes are stationary and
invertible. In the following we always assume that ut is white
noise with mean zero and variance σ2, i.e. ut ∼ WN(0, σ2)
Problem 2 (20 marks). Determine which of the following processes are stationary and invertible. In the tollowing we always assume that ut is white noise with mean zero and variance σ2, i.e. ut ~ WN(0, σ2). 5 marks 5 marks 5 marks 5 marks 10
Problem 2 (20...
Problem 2. Let X be a random variable with mean 0 and variance σ2. Define the process Yt-(-1) Compute the mean and covariance function of the process {Yt). Is this process stationary?
2. (a) Consider the following process: where {Z) is a white noise process with unit variance. [1 mark] ii. Find the infinite moving average representation of X,i.e., find the scquence [6 marks] i. Explain why the process is stationary. (6) such that Xt = Σ b,2-j. iii. Calculate the mean and the autocovariance "Yo, γι and 72 of the process. 7 marks iv. Given 40 = 0.1 and Xo = 1.8, find the 2-step ahead forecast of the time series...
2. Let (et) be a zero mean white noise process with variance 1. Suppose that the observed process is h ft + Xt where β is an unknown constant, and Xt-et- Explain why {X.) is stationary. Find its mean function μχ and autocorrelation function p for lk0,1,.. a. b. Show that {Yt3 is not stationary. C. Explain why w. = ▽h = h-K-1 is stationary. d. Calculate Var(Yt) Vt and Var(W) Vt . (Recall: Var(X+c)-Var(X) when c is a constant.)...
1. Consider the following autoregressive process 2+ = 4.0 + 0.8 2t-1 + Ut, where E (u+12+-1, Zt-2, ....) = 0 and Var (ut|2t-1, 2-2, ...) = 0.3. The unconditional E (Zt) and unconditional variance Var (zt) are: (a) E (2+) = 11.1111, Var (zł) = 0.8333 (b) E (2+) = 11.1111, Var (zt) = 1.5 (c) E (zt) = 20, Var (zt) = 0.8333 (d) E (2+) = 4, Var (zł) = 0.8333 (e) E (Zt) = 4,Var (z+)...
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