International macroe.. - Free

2.4. UNIT ROOTS 45² 0 = 0, which is the unconditional mean of ² t . Now substitute (2.51)into (2.50). Use the fact that ξ t−1 = q t−1 − γ 0 − γ 1 (t − 1) and subtractq t−1 from both sides to get∆q t =[(1− ρ)γ 0 + ργ 1 ]+(1− ρ)γ 1 t +(ρ − 1)q t−1 + ² t . (2.53)(2.53) says you should run the regression∆q t = α 0 + α 1 t + βq t−1 + ² t , (2.54)where α 0 =(1− ρ)γ 0 + ργ 1 , α 1 =(1− ρ)γ 1 ,andβ = ρ − 1. The nullhypothesis, ρ = 1, can be tested by doing the joint test of the restrictionβ = α 1 = 0. To test if the deviation from a constant is stationary, do ajoint test of the restriction β = α 1 = α 0 = 0. If the random walk withdrift is a reasonable null hypothesis, evidence of trending behavior willprobably be evident upon visual inspection. If this is the case, includinga trend in the test equation would make sense.In most empirical studies, researchers do the Dickey—Fuller test ofthe hypothesis β = 0 instead of the joint tests recommended by Bhargava.Nevertheless, the Bhargava formulation is useful for decidingwhether to include a trend or just a constant. To complicate mattersfurther, the asymptotic distribution of ρ and τ depend on whether aconstant or a trend is included in the test equation so a different setof critical values need to be computed for each speciÞcation of the testequation. Tables of critical values can be found in textbooks on timeserieseconometrics, such as Davidson and MacKinnon [35] or Hamilton[66].Parametric Adjustments for Higher-Ordered Serial CorrelationYou will need to make additional adjustments if ξ t in (2.51) exhibitshigher-order serially correlation. The augmented Dickey—Fuller test isa procedure that employs a parametric correction for such time dependence.To illustrate, suppose that ξ t follows the AR(2) processξ t = ρ 1 ξ t−1 + ρ 2 ξ t−2 + ² t , (2.55)where ² tiid∼ N(0, σ 2 ² ). Then by (2.50), ξ t−1 = q t−1 − γ 0 − γ 1 (t − 1), ⇐(24)