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2.4. UNIT ROOTS 41(1 − ρz) = 0 lies outside the unit circle, which in turn is equivalent tosaying that the root is greater than 1. 13The stationary case. To appreciate some of the features of a unit roottime-series, we Þrst review some properties of stationary observations.If 0 ≤ ρ < 1 in (2.43), then {q t } is covariance stationary. It is straightforwardto show that E(q t )=α and Var(q t )=σ² 2 /(1 − ρ 2 ), which areÞnite and time-invariant. By repeated substitution of lagged values ofq t into (2.43), you get the moving-average representation with initialcondition q 0⎛ ⎞Xt−1q t = α(1 − ρ) ⎝ ρ j ⎠ + ρ t Xt−1q 0 + ρ j ² t−j . (2.44)j=0The effect of an ² t−j shock on q t is ρ j . More recent ² t shocks have a ⇐(22)larger effect on q t than those from the more distant past. The effects (eq.2.44)of an ² t shock are transitory because they eventually die out.To estimate ρ, we can simplify the algebra by setting α =0sothat{q t } from (2.43) evolves according toq t+1 = ρq t + ² t+1 ,where 0 ≤ ρ < 1. The OLS estimator is ˆρ = ρ+[( P T −1t=1 q t ² t+1 )/( P T −1t=1 q 2 t )].Multiplying both sides by √ T and rearranging gives√T (ˆρ − ρ) =1 √TP T −11Tj=0t=1 q t ² t+1. (2.45)P T −1t=1 q 2 tThe reason that you multiply by √ T is because that is the correctnormalizing factor to get both the numerator and the denominator onthe right side of (2.45) to remain well behaved as T →∞. By the lawof large numbers, plim 1 P T −1T t=1 qt 2 =Var(q t )=σ² 2 /(1 − ρ 2 ), so for thatsufficiently large T , the denominator can be treated like σ² 2 /(1 − ρ 2 )iidwhich is constant. Since ² t ∼ N(0, σ² 2 )andq t ∼ N(0, σ² 2 /(1 − ρ 2 )),13 Most economic time-series are better characterized with positive values of ρ,but the requirement for stationarity is actually |ρ| < 1. We assume 0 ≤ ρ ≤ 1 tokeep the presentation concrete.