Forecasting with estimated parameters: Let x1, x2, . . ., xn be a sample of size n from a causal AR(1) process, xt = xt1

Forecasting with estimated parameters: Let x1, x2, . . ., xn be a sample of size n from a causal AR(1) process, xt = φxt−1 + wt

. Let φˆ be the Yule–Walker estimator of φ.

(a) Show φˆ − φ = Op(n

−1/2

). See Appendix A for the definition of Op(·).

(b) Let x n

n+1 be the one-step-ahead forecast of xn+1 given the data x1, . . ., xn, based on the known parameter, φ, and let xˆ

n n+1 be the one-step-ahead forecast when the parameter is replaced by φˆ. Show x n

n+1

− xˆ

n n+1

= Op(n

−1/2

).