2 Unit Root Asymptotic theories In this section, we develop tools to handle the asymptotics of unit root process 2.1 Random walks and wiener Process Consider a random walk Yt=Yt-1+Et where Yo =0 and Et is i.i.d. with mean zero and Var(et)=02<oo By repeated substitution we have +Et=Yt-2+Et-1+Et S=」 Before we can study the behavior of estimators based on random walks, we must understand in more detail the behavior of the random walk process itsel Thus, consider the random walk Yt, we can write Rescaling, we have /o=T12∑en t=1 (It is important to note here should be read as Var(T-12 E+Et)=ET-I Et)21 Tg==o2 According to the Lindeberg-Levy CLT, we have T-12/o-→N(0,1) More generally, we can construct a variable Yr(r) from the partial sum of Et Trl2 Unit Root Asymptotic Theories In this section, we develop tools to handle the asymptotics of unit root process. 2.1 Random Walks and Wiener Process Consider a random walk, Yt = Yt−1 + εt , where Y0 = 0 and εt is i.i.d. with mean zero and V ar(εt) = σ 2 < ∞. By repeated substitution we have Yt = Yt−1 + εt = Yt−2 + εt−1 + εt = Y0 + X t s=1 εs = X t s=1 εs. Before we can study the behavior of estimators based on random walks, we must understand in more detail the behavior of the random walk process itself. Thus, consider the random walk {Yt}, we can write YT = X T t=1 εt . Rescaling, we have T −1/2YT /σ = T −1/2X T t=1 εt/σ. (It is important to note here σ 2 should be read as V ar(T −1/2 PT t=1 εt) = E[T −1 ( Pεt) 2 ] = T·σ 2 T = σ 2 .) According to the Lindeberg-L´evy CLT, we have T −1/2YT /σ L−→ N(0, 1). More generally, we can construct a variable YT (r) from the partial sum of εt YT (r) = [T r] X∗ t=1 εt , 3