Problem 7.2 This problem shows that (linear) AR(p) time series can lead to (nonlinear) ARCH models when they have random coefficients.

Let {t}t be a strong univariate white noise with t ∼ N(0, 1), and let {φt}t be a p-variate time series independent of {t}t, and such that all the φt are independent of each other, and for each time t, the vector φt = (φt,1, φt,2, . . . , φt,p) is a vector of jointly Gaussian random variables with mean zero and variance/covariance matrix Σ. We study the time series {Yt}t defined by:
Yt = φt,1Yt−1 + φt,2Yt−2 + · · · + φt,pYt−p + t.
1. Determine the conditional distribution of Yt given Y t−1 by integrating out the φ-random variables.
2. Assume that the components φt,1, φt,2, . . ., φt,p of φt are independent, and show that at least in this case, {Yt}t has an ARCH representation.