1. Define a Stochastic process and briefly discuss the meaning of measurability of a stochastic process.2. Consider the ARMA(1,1) model yt = 0.8yt-1 + et + 0.5et-1 with et ~ WN(0, σe2). Derive the Wold representation of yt.3. Consider the ARMA(2,1) process Φ(L)Xt = Θ(L)et with Φ(L) = 1 − 1.3L + 0.4L2 , Θ(L) = 1 + 0.4L and et ∼ WN(0, σe2). Obtain its Wold representation.4.Consider the ARMA(2,2) process given by Xt =0.4Xt−1+0.45Xt−2+et+et−1+0.25et−2 with et ∼WN(0,σe2).5. Consider the MA(1) process yt = et+1.5et−1 with et ∼ WN(0,σe2).Is the above MA(1) a Wold representation? Why or Why not? If not, obtain a suitable Wold representation.

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MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

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1. Define a Stochastic process and briefly discuss the meaning of measurability of a stochastic process.

2. Consider the ARMA(1,1) model y_t= 0.8y_t-1 + e_t + 0.5e_t-1 with e_t~ WN(0, σ_e²). Derive the Wold representation of y_t.

3. Consider the ARMA(2,1) process Φ(L)X_t = Θ(L)e_t with Φ(L) = 1 − 1.3L + 0.4L² , Θ(L) = 1 + 0.4L and e_t ∼ WN(0, σ_e²). Obtain its Wold representation.

4.Consider the ARMA(2,2) process given by X_t=0.4X_t−1+0.45X_t−2+e_t+e_t−1+0.25e_t−2with e_t ∼WN(0,σ_e²).

5. Consider the MA(1) process y_t = e_t+1.5e_t−1 with et ∼ WN(0,σ_e²).

Is the above MA(1) a Wold representation? Why or Why not? If not, obtain a suitable Wold representation.

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