Consider y, = у,-1-0.25%-2 + et-1.54-1 + 0.54-2, et ~ WN(0, σ2). a) Determine the order of this A...
b) Consider the I(1) process DeltaYt : m + et, where et ~ WN(0, sigma^2). Take now the first difference of DeltaYt. What, kind of process do you obtain? Is it stationary? Is it invertible? Discuss.
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
2. Consider an ARM A(2,2) model h" ф.xt-1-фг%2 :: at + at-NI 1at-1 + 20.-2, a. Under what condition, the above ARMA(2,2) model is causal/stationary. b. Under what condition, the above ARMA(2,2) model is invertible. State the reason for us to consider an invertible ARMA model. Suppose that xt is causal, i.e. C. Calculate , j = 1,2,3,4,5,6. d. Suppose that x is invertible, i.e. Calculate nj, j-1,2,3,4,5,6. 2. Consider an ARM A(2,2) model h" ф.xt-1-фг%2 :: at + at-NI...
(b) Suppose that {Y, is generated according to Y-10+ et--et-it īet-2,with et ~ N(0, 1) covariance function for (Y). Is (Y) stationary? ustify your answer. (ii) Determine ρ1 and ρ2 (iii) Using (ii) or otherwise, determine oil and ф22
QUESTION 4 Consider the CDF 0 у<0 0.5 F(y) = 0.75 0.90 Osy<2 2sy<3 3sy<5 1 Y>5 Find pſy=5) O A. 0.10 B. 0.15 C. 0.25 D.0.25 QUESTION 5 Consider the PM.F
2. Let [et be a zero mean white noise process with variance 0.25. Suppose that the observed process is k = et + 0.5e-2. a. Explain why {Yt) is stationary. b. Compute yo-V(Y.) c. Compute the autocorrelation pkY, kl-0,1,2,... for Y) d. Let Wt = 3 + 4t + h. i. Find the mean of {W) ii. Is W3 stationary? Why or why not? iii. Let Z Vw, W,- W,_1. Is {Z.1 stationary? Why or why not?
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.
8 Consider the time series model Y1 ~ N(0, 102) and YlYt-1, ,y ~ N øt_i,σ2) for t = 2, , n. Assume priors ρ ~ N(0,T2) and σ2 ~ InvGamma(a, b). (a) Compute the posterior distribution of ρΙσ2Ύ. . . . ,y, . (b) Compute the posterior distribution of σ2le, Yi, . . .
(1 point) Consider the IVP У + 3ty — бу %3D 3, у(0) — 0, У (0) — 0 (a) What is the Laplace transform of the differential equation, after being put into standard form? Y'(s) ) Y(s) = |(b) What is the solution to the differential equation? y(t)
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