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# HW8_Solutions - Solutions to HW 8 Exercise 12.4 All pure MA...

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Solutions to HW 8 Exercise 12.4 All pure MA processes, whether univariate or multivariate, are stationary. The model is invertible if all the roots of |() | | |0 BIB Θ =+ Θ = are outside of the unit circle. 1 0 0.6 0.4 1 .6 0.4 () 0 1 0.2 0.4 0.2 1 0.4 B BB B B + B ⎤⎡ Θ= + Θ = + = ⎥⎢ + ⎦⎣ I . 2 11 1. 6 0 . 4 1 .16 0, 5, 1.25 0.2 1 0.4 B B + = + + = =− + . Both roots lie outside the unit circle. Thus the model is invertible. Exercise 12.5 VAR(1) model, 1 tt + t X XZ , where 0.9 0.5 0.1 0.3 Φ= , ˆ h th t + X X , 1 ˆ t + t X X i.e., 1, 1 , 1 ,2 1, 2 ˆ 0.9 0.5 ˆ 0.1 0.3 + + ⎡⎤ = ⎢⎥ ⎣⎦ t t XX X X , or, 1,1 , 1 , 2 1, 2 ˆ 0.9 0.5 , ˆ 0.1 0.3 . + + + X t t X 2 2 ˆ + X X i.e., 2 2,1 ,1 2, 2 ˆ 0.9 0.5 0.76 0.6 ˆ 0.1 0.3 0.12 0.04 t X + + t X == −− t X , or, 2,1 , 2 2,2 , 2 ˆ 0.76 0.6 , ˆ 0.12 0.04 . + + + t X Problem 1 1 10 0 . 7 0 . 7 7 . 7 0 1 0.7 0.7 .7 1 .7 x xx x x Φ = = I x , 22 7 . 7 (1 .7 ) (.7 ) 1 1.4 0, 1/1.4 .714 .7 1 .7 = = = non-stationary.
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