Consider the ARMA(2,1) model 2+ = 0.624-1 -0.092-2 + at – 0.204-1, a4~WN(0,1) Find the AR...
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.
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
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
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...
QUESTION 3 (a) Consider the ARMA (1, 1) process -Bat-1-where o and θ are model parame- are independent and identically distributed random variables with mean 0 z, oz,-1 ters, and a1, a2, and variance σ (i) Show that the variance of the process is γ,- (ii) Using (i) or otherwise, show that the autocorrelation function (ACF) of the process is: ifk=0. (b) Let Y be an AR(2) process of the special form Y-2Y-2e (i) Find the range of values of...
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.
Consider ARMA(2.1) model X4 - X:-1 +62X-2 = 2+ 2-1. When the process is stationary and causal?
Assume the data Z 7n1 Z1 come from the AR(1) model where øl< 1 and at i.i.d. N(0,1). Give the exact conditional log- likelihood function for φ.
Assume the data Z 7n1 Z1 come from the AR(1) model where øl
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...