Ch. 21 Univariate Unit Root process 1 Introduction Consider OLS estimation of a AR(1)process, Yt= pYt-1+ut where ut w ii d (0, 0), and Yo=0. The OLS estimator of p is given by and we also have (1) t=1 When the true value of p is less than l in absolute value, then y(so does y ?) is a covariance-stationary process. Applying LLN for a covariance process(see 9.19 of Ch. 4)we have 21) T·a4 1-p2 /T=a2/(1-p2 t=1 Since Yt-lut is a martingale difference sequence with variance E(Y-1)2 and t=1 Applying CLT for a martingale difference sequence to the second term in the righthand side of (1)we have Y-1ut)→N(0 t=1Ch. 21 Univariate Unit Root Process 1 Introduction Consider OLS estimation of a AR(1) process, Yt = ρYt−1 + ut , where ut ∼ i.i.d.(0,σ2 ), and Y0 = 0. The OLS estimator of ρ is given by ρˆT = PT t=1 Yt−1Yt PT t=1 Y 2 t−1 = X T t=1 Y 2 t−1 !−1 X T t=1 Yt−1Yt ! and we also have (ˆρT − ρ) = X T t=1 Y 2 t−1 !−1 X T t=1 Yt−1ut ! . (1) When the true value of ρ is less than 1 in absolute value, then Yt (so does Y 2 t ?) is a covariance-stationary process. Applying LLN for a covariance process (see 9.19 of Ch. 4) we have ( X T t=1 Y 2 t−1 )/T p −→ E[(X T t=1 Y 2 t−1 )/T] =  T · σ 2 1 − ρ 2  /T = σ 2 /(1 − ρ 2 ). (2) Since Yt−1ut is a martingale difference sequence with variance E(Yt−1ut) 2 = σ 2 σ 2 1 − ρ 2 and 1 T X T t=1  σ 2 σ 2 1 − ρ 2  → σ 2 σ 2 1 − ρ 2 . Applying CLT for a martingale difference sequence to the second term in the righthand side of (1) we have 1 √ T ( X T t=1 Yt−1ut) L−→ N(0,σ2 σ 2 1 − ρ 2 ). (3) 1