Find the spectral density functions of the following MA process:
Xt = Zt + 0.5Zt−1 − 0.3Zt−2
Find the spectral density functions of the following MA process: Xt = Zt + 0.5Zt−1 −...
Find the spectral density of Xt =−0.7Xt−1+Zt −0.3Zt−1+0.7Zt−2, {Zt}~IID(0,2).
Consider the following AR(2) model: Xt – Xt–1 + + X4-2 = Zt, Z4 ~ WN(0,1). (a) Show that X+ is causal. (b) Find the first four coefficients (VO, ..., 43) of the MA(0) representation of Xt. (c) Find the pacf at lag 3, 233, of the AR(2) model.
Suppose Zt = 2 + Xt -2Xt-1+Xt-2, where {Xt} is zero-mean stationary series with autocovariance function. Calculate the autocovariance of Zt
Problem 3 Consider a random walk on the integers. Suppose we start from 0, and at each step, we either go left or right with probability 1/2, ie, Xo--0, and Xt+1 Xt+Zt, where Zt-1 with probability 1/2, and Zt1 with probability 1/2. What is the probability distribution of XT? What is E(X) and Var(XT)? Problem 3 Consider a random walk on the integers. Suppose we start from 0, and at each step, we either go left or right with probability...
Consider the process where B is a backwards shift operator so that BXt-Xt-i and the {Zt) are assumed to be independent random errors. (a) [2 marks] Identify what kind of nonseasonal ARIMA(p,d,q) process this is; that is give the parameters (p,d,q) and give the abbreviated name for this particular process. (b) [3 marks] (i) Is this particular process stationary? Explain. (ii) Is this process invertible? Why? Consider the process where B is a backwards shift operator so that BXt-Xt-i and...
B. Consider the GARCH (1, 1) model Xt-σ.zt, σ -00 + α1XL1 + βισ -1 where Zt are iid N (0, 1) process, ao 0, α120, ai 1 > α1 + β1. Show that 0, A > 0 and 2IV2 t-1't-2 .. B. Consider the GARCH (1, 1) model Xt-σ.zt, σ -00 + α1XL1 + βισ -1 where Zt are iid N (0, 1) process, ao 0, α120, ai 1 > α1 + β1. Show that 0, A > 0...
Use your knowledge of the relationship between spectral density and autocorrelation function in order to answer the following questions. Show your work for full credit. Determine the spectral density of a process with autocorrelation function Rx(t) = 3e-2t a) Determine the spectral density of a process with autocorrelation function Rx(t)-2 sinc(0.51) b) c) Determine the autocorrelation function of a process with spectral density Sx (f) 2 sinc2(f/2) 12 Determine the autocorrelation function ofa process with spectral density Sx(a)-A+ d) Use...
Problem 3 Express growth rates for xt in terms of growth rates of Yt and zt. 1. It = y21 2. It = 4822- 3. It = 4ytzt
PROBLEM 8.3 Explain that the spectral density for an invertible and causal ARMA (p,q) process is continuous on-π, π] and with a minimum value strictly greather than zero. PROBLEM 8.3 Explain that the spectral density for an invertible and causal ARMA (p,q) process is continuous on-π, π] and with a minimum value strictly greather than zero.
1. Consider the following two MA (2) models: ·zt = (1-1.3B + AB*)et with σ-10.0, . x,-(1-1.75B+ .625B2)et with σ-64. (i) Evaluate the autocovariances a,k for lags k 0,1,2 for both models to verify that they have the same acf. [5 marks] (ii) However, only one of these two models is invertible. Which one? 5 marks