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Consider the following process:
​;​−∞ < t < ∞;​|φ| < 1, |θ| < 1​

where t satisfies the following assumptions:

​) = 0;​∀t

​) = 0;​∀s 6= 0

a) Using the Wold Decomposition Theorem, the process of (3) can be expressed as an MA(∞):
​;​with​− ψ0 = 1​

as (3) has no deterministic components. Derive the expressions for parameters ψi’s in terms of φ and θ.
b) Given the invertibility condition of |θ| < 1, this process can also be represented as an AR(∞):
​;​)​(5)
Derive the expressions for parameters πi’s in terms of φ and θ.

c) Derive γ0 and γ1, the unconditional variance of yt and its first-order autocovariance respectively, in terms of the unknown parameters of the process of (3).

4. Consider the following process:
​;​−∞ < t < ∞​(6)

where c is a constant, and t is normally distributed with the following assumptions:

​) = 0;​∀t

​) = 0;​∀s 6= 0

Assume that the process of (6) is stationary. Suppose we have drawn a sample of T consecutive observations, (y1, y2,··· ,yT), from the process of (6).

Let ˜c, φ˜1, φ˜2, and ˜σ2 denote the conditional maximum likelihood estimators of c, φ1, φ2, and σ2 respectively. Let ˆc, φˆ1, φˆ2, and ˆσ2 denote the least squares (OLS) estimators of c, φ1, φ2, and σ2 respectively. Derive ˜σ2. Demonstrate:

c˜= cˆ φ˜1 = φˆ1 φ˜2 = φˆ2 σ˜2 6= σˆ2
5. Consider the following process:
​;​−∞ < t < ∞;​|θ| < 1​(7)

where t is normally distributed with the following assumptions:

​) = 0;​∀t

​) = 0;​∀s 6= 0

a) Assume |φ| 1. Show that the process of (7) can be expressed as:
​​(9)

​Does the expression of (9) have any practical purpose? Explain your answer.​

Suppose T consecutive observations (y1, y2,··· ,yT) were drawn from the process of (7). Derive the log likelihood function for the exact maximum likelihood estimation of the unknown parameters of (7).​

[Hint: for parts a) and b), write the process of (7) using the lag operator. Notice that if |φ| > 1, |φ−1| < 1. Then, the formula of the geometric sum to infinity can still be used. For part c), use the results obtained in Q3, part c).]