Consider the following AR(2) model: Xt – Xt–1 + + X4-2 = Zt, Z4 ~ WN(0,1)....
Consider the ARMA(2,1) model 2+ = 0.624-1 -0.092-2 + at – 0.204-1, a4~WN(0,1) Find the AR representation of {Z}.
2. Consider an ARMA(1,1) process, X4 = 0.5X:-1 +0+ - 0.25a4-1, where az is white noise with zero mean and unit variance. (a) Is the model stationary? Explain your answer briefly. (b) Is the model invertible? Explain your answer briefly. (c) Find the infinite moving-average representation of Xt. Namely, find b; such that X =< 0;&–; j=0 (d) Evaluate the first three lags of the ACF and PACF.
6. (13 marks) where {U, } ~ WN(0,00) is Consider two independent AR(1) series< independent of {K} ~ WN(0,OF). Does their sum Z,-X,-X necessarily follow an AR(1) series? Prove or disprove. (Hint: Compare the causal representation of the sum to that of an AR(1) process) 6. (13 marks) where {U, } ~ WN(0,00) is Consider two independent AR(1) series
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...
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 ARIMA model 8. Consider the ARIMA model X,-4 + Xt-1 + W-0.75W,-1, W, ~ WN(0, σ*) a. Identify p, d, and q. Write the corresponding ARMA (p,q) model. b. Find E VX and VarVX 8. Consider the ARIMA model X,-4 + Xt-1 + W-0.75W,-1, W, ~ WN(0, σ*) a. Identify p, d, and q. Write the corresponding ARMA (p,q) model. b. Find E VX and VarVX
Q1. (10 points total) Consider an ARMA(1,1) model X4 = 0.9X-1 + Z+ +0.527-1, {Z4}~ IID N(0,1). 1. (2 points) (i) Generate n = 200 observations from this ARMA model. (ii) Find the maximum likelihood (ML) estimates of the three parameters o, e, and o2. 2. (8 points) Repeat (i) and (ii) in part 1 nine more times using all different seed numbers. Compare the estimates to their true values. Are the average of 10 estimates for each parameter close...
Consider ARMA(2.1) model X4 - X:-1 +62X-2 = 2+ 2-1. When the process is stationary and causal?
Consider the following AR(1) model: 1. a. Explain why this dynamic model violates TS'3 ZCM assumption made for the unbiasedness of the FDL model estimators. the following random 2. Consider walk model: yeBo yt-1 +ut, t-0,1,..,T a. Show that yt-3βο + yt-3 + ut + ut-1 + ut-2. b. Suppose that 0-0, show that y.-t βο +4 + ut-1 + + u! c. Suppose that that yo -0, and ut for all t are ii.d. with mean 0 and variance...
Consider the following model 1. Consider the following AR(1) model: a. Explain why this dynamic model violates TS'3 ZCM assumption made for the unbiasedness of the FDL model estimators. b. Show that 1 t-2 2. Consider the following random walk model: ytBo yt-1 +ut, t 0,1,...,T Show that ye 3o yt-3 + ut + Ut-1 +t-2 Suppose that yo - 0, show that yt - tPo + ut + ut-1++u, Suppose that that yo -0, and ut for all t...