Suppose x1, . . . , xn are observations from an AR(1) process with = 0. (a) Show the backcasts can be written as

Suppose x1, . . . , xn are observations from an AR(1) process with µ = 0.

(a) Show the backcasts can be written as xn t = φ1−t x1, for t ≤ 1.

(b) In turn, show, for t ≤ 1, the backcasted errors are wbt(φ) = xn t − φxn t−1 = φ1−t

(1 − φ2)x1.

(c) Use the result of

(b) to show P1 t=−∞ wb2 t (φ) = (1 − φ2)x2 1.

(d) Use the result of

(c) to verify the unconditional sum of squares, S(φ), can be written as Pn t=−∞ wb2 t (φ).

(e) Find xt−1 t and rt for 1 ≤ t ≤ n, and show that S(φ) = Xn t=1

(xt − xt−1 t )

2  rt.