Consider the first-order bilinear model ???????? = ????????????−1 + ???????? − ????????????−1????????−1, where the ????????

are independent variates with mean 0 and variance ????2

????. Assume the process {????????} is stationary, which involves the condition that ????2 + ????2

????????2 < 1, and assume that {????????}

has a causal stationary representation of the form ???????? = ???????? + ????(????????−1, ????????−2, …).

(a) Verify that ????[????????????????] = ????2

????, and so also that ???????? = ????[????????] satisfies (1 − ????)???????? =

−????????2

????.

(b) Establish that the autocovariances ???????? of {????????} satisfy ???????? = ????????????−1 for ???? > 1, so that the process has the same autocovariance structure as an ARMA(1, 1) process.