Substituting(2)and(3)to(1)we have N(0 N(0,1 (6)is not valid for the case when p= 1, however. To see this, recall that the variance of Yt when p= l is to then the Lln as in(2) would not be valid since if we apply clt, then it would incur that ∑2mE(∑y2/=a2一∞ t=1 t=1 Similar reason would show that the CLT would not apply for v(> In stead, T-(Et Yt-1ut)converges. )To obtain the limiting distribution, as we shall prove in the following, for(PT-p)in the unit root case, it turn out that we have to multiply (er -p) by T rather than by VT 1 T t=1 t=1 Thus, the unit root coefficient converge at a faster rate(T)than a coefficient for stationary regression( which converges at VT)Substituting (2) and (3) to (1) we have √ T(ˆρT − ρ) = [(X T t=1 Y 2 t−1 )/T] −1 · √ T[(X T t=1 Yt−1ut)/T] (4) L−→  σ 2 1 − ρ 2 −1 N(0,σ2 σ 2 1 − ρ 2 ) (5) ≡ N(0, 1 − ρ 2 ). (6) (6) is not valid for the case when ρ = 1, however. To see this, recall that the variance of Yt when ρ = 1 is tσ2 , then the LLN as in (2) would not be valid since if we apply CLT, then it would incur that ( X T t=1 Y 2 t−1 )/T p −→ E[(X T t=1 Y 2 t−1 )/T] = σ 2 PT t=1 t T → ∞. (7) Similar reason would show that the CLT would not apply for √ 1 T ( PT t=1 Yt−1ut). ( In stead, T −1 ( PT t=1 Yt−1ut) converges.) To obtain the limiting distribution, as we shall prove in the following, for (ˆρT − ρ) in the unit root case, it turn out that we have to multiply (ˆρT − ρ) by T rather than by √ T: T(ˆρT − ρ) = " ( X T t=1 Y 2 t−1 )/T2 #−1 " T −1 ( X T t=1 Yt−1ut) # . (8) Thus, the unit root coefficient converge at a faster rate (T) than a coefficient for stationary regression ( which converges at √ T). 2