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 φb be the Yule–Walker estimator of φ.

(a) Show φb − φ = Op(n−1/2). See Appendix A for the definition of Op(·).

(b) Let xn n+1 be the one-step-ahead forecast of xn+1 given the data x1, . . . , xn, based on the known parameter, φ, and let xbn n+1 be the one-step-ahead forecast when the parameter is replaced by φb. Show xn n+1−xbn n+1 = Op(n−1/2).

Section 3.7