Consider the so-called bilinear process yt 12 et-1yt-2 et

Question

Question: Consider the so-called bilinear process Y_t = 1/2 ε_t-1Y_t-2 + ε_t, where ε_t are independent drawings from N(0, σ2). As starting conditions are given ε₀ = 0 and Y_-1= Y₀ = 0

a. Prove that yt is an uncorrelated process. Is it also a white noise process?

b. Prove that Y²_{t is not an uncorrelated process.}

c. Prove that yt cannot be forecasted by linear functions of past observations yt-k (k ≥ 1) but that it can be forecasted by non-linear functions of these past observations.

d. Simulate n = 200 data from this process. Perform a Ljung-Box test and an ARCH test on the resulting time series. What is the relevance of this result for the interpretation of ARCH tests?