3 Regression with a Unit Root 3.1 Dickey-Fuller Test, Yt is AR(1) process Consider the following simple AR(1)process with a unit root Yt= BYt-1+ut B=1 where Yo =0 and ut is i.i.d. with mean zero and variance o We consider the three least square regression Yt= BYt-1+ut, and Yt=a+ BYt-1+ot+it, nd(a, B, 8)are the conventional least-squares regression coef- ficients. Dickey and Fuller(1979) were concerned with the limiting distribution of the regression in(13),(14), and(15)(B, (&, B), and(a, B, 8)) under the null hypothesis that the data are generated by(11) and(12 We first provide the following asymptotic results of the sample moments which are useful to derive the asymptotics of the Ols estimator Let ut be a i.i.d. sequence with mean zero and variance aand yt ut for t=l (16) with yo=0. Then (a)T-i →σW ∑Y21→→2J0u(r)]2dr3 Regression with a Unit Root 3.1 Dickey-Fuller Test, Yt is AR(1) process Consider the following simple AR(1) process with a unit root, Yt = βYt−1 + ut , (11) β = 1 (12) where Y0 = 0 and ut is i.i.d. with mean zero and variance σ 2 . We consider the three least square regression Yt = βY˘ t−1 + ˘ut , (13) Yt = ˆα + βYˆ t−1 + ˆut , (14) and Yt = ˜α + βY˜ t−1 + ˜δt + ˜ut , (15) where β, ˘ (ˆα, βˆ), and (˜α, β, ˜ ˜δ) are the conventional least-squares regression coef- ficients. Dickey and Fuller (1979) were concerned with the limiting distribution of the regression in (13), (14), and (15) (β, ˘ (ˆα, βˆ), and (˜α, β, ˜ ˜δ)) under the null hypothesis that the data are generated by (11) and (12). We first provide the following asymptotic results of the sample moments which are useful to derive the asymptotics of the OLS estimator. Lemma: Let ut be a i.i.d. sequence with mean zero and variance σ 2 and yt = u1 + u2 + ... + ut for t = 1, 2,...,T, (16) with y0 = 0. Then (a) T − 1 2 P T t=1 ut L−→ σW(1), (b) T −2 P T t=1 Y 2 t−1 L−→ σ 2 R 1 0 [W(r)]2dr, 9