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42 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSthe product sequence {q t ² t+1 } is iid normal with mean E(q t ² t+1 ) =0 and variance Var(q t ² t+1 )=E(² 2 t+1)E(qt 2 )=σ² 4 /(1 − ρ 2 ) < ∞. Bythe Lindeberg-Levy central limit theorem, you have √ 1 P T −1T t=1 q t² D t+1 →N (0, σ 4 /(1 − ρ 2 )) as T →∞. For sufficiently large T , the numeratoris a normally distributed random variable and the denominator is aconstant so it follows that√T (ˆρ − ρ) → D N(0, 1 − ρ 2 ). (2.46)You can test hypotheses about ρ by doing the usual t-test.Estimating the Half-Life to ConvergenceIf the sequence {q t } follows the stationary AR(1) process, q t = ρq t−1 +² t ,its unconditional mean is zero, and the expected time, t ∗ , for it toadjust halfway back to 0 following a one-time shock (its half life) canbe calculated as follows. Initialize by setting q 0 =0. Thenq 1 = ² 1and E 1 (q t )=ρ t q 1 = ρ t ² 1 . The half life is that t such that the expectedvalue of q t has reverted to half its initial post-shock size–the t thatsets E 1 (q t )= ² 12.Sowelookforthet ∗ that sets ρ t∗ ² 1 = ² 12t ∗ = − ln(2)ln(ρ) . (2.47)If the process follows higher-order serial correlation, the formulain (2.47) only gives the approximate half life although empirical researcherscontinue to use it anyways. To see how to get the exact halflife, consider the AR(2) process, q t = ρ 1 q t−1 + ρ 2 q t−2 + ² t ,andlet" # " # " #qtρ1 ρy t= ; A =2²t, uq t−1 1 0 t = .0Now rewrite the process in the companion form,y t= Ay t−1 + u t , (2.48)and let e 1 =(1, 0) be a 2 × 1rowselection vector. Now q t =e 1 y t,E 1 (q t )=e 1 A t y 1,whereA 2 = AA, A 3 = AAA, and so forth. The halflife is the value t ∗ such thate 1 A t∗ y 1 = 1 2 e 1y 1= 1 2 ² 1.