Suppose x_t = x_{t 1} + _t where _t = e_t+ e_{t 1}e_{t 2}, 0, and {e_t} is strict white noise.

a) What is the best linear predictor of x_n+1 based on x_n , x_{n 1} , . . . ? Justify your answer.

b) What is the best possible predictor of x_n+1 based on x_n , x_n1 , . . . ? Justify your answer.

c) Compare your answers to a) and b) to decide whether {x_t} is a martingale.

Extra Information for C {xt} is a martingale if E [xn+h | xn , xn1 , . . . ] = xn for all n and for all lead times h > 0. Actually, to establish that {xt} is a martingale, one simply needs to prove the above formula for h = 1 since it can be shown that if it holds for h = 1 it must hold for all h > 0.