heading Case 2 in table B7 3.1.3 Constant Term and Time Trend Included in the Regression True Process is a random walk with or without drift We finally in the section consider the case that a constant term and a linear trend are added in the regression model, but true process is a random walk with a drift However, the true value of this drift turns out not to matter for the asymptotic distributions of OLS unit root coefficient estimator and t-ratio test statistics in Theorem 3 Let the data yt be generated by(11)and(12); then as T-00, for the regression model (15) T(6-1)→ 1/2{[(1)-2/W()duw(1)+6/W()d-12/brW(r)dh-1 Jow()2dr-40o w(r)dr)2+12o w(r)dr fo rw(r)dr-120o rw(r)dr and (r-1) →T 0 wherea=5][0 1 0] 1, St=Y-at 0 t=1 st denote the OLS estimate of the disturbance variance s=∑(Y and Q=[0 105w(r)dr J [W(r)2dr Jrw(r)dr /2 rW(r)dr 1/8
heading Case 2 in table B.7. 3.1.3 Constant Term and Time Trend Included in the Regression; True Process Is a Random Walk With or Without Drift We finally in the section consider the case that a constant term and a linear trend are added in the regression model, but true process is a random walk with a drift. However, the true value of this drift turns out not to matter for the asymptotic distributions of OLS unit root coefficient estimator and t-ratio test statistics in this case. Theorem 3: Let the data Yt be generated by (11) and (12); then as T → ∞, for the regression model (15), T(β˜ − 1) ⇒ 1/2{[W(1) − 2 R 1 0 W(r)dr][W(1) + 6 R 1 0 W(r)dr − 12 R 1 0 rW(r)dr] − 1} R 1 0 [W(r)]2dr − 4[R 1 0 W(r)dr] 2 + 12 R 1 0 W(r)dr R 1 0 rW(r)dr − 12[R 1 0 rW(r)dr] 2 and t˜= (β˜ T − 1) σ˜β˜ T ⇒ T(β˜ − 1) ÷ p Q, where ˜σ 2 β˜ T = s 2 T 0 1 0 T P T t=1 ξt−1 P T t=1 t P T t=1 ξt−1 P T t=1 ξ 2 t−1 P T t=1 tξt−1 P T t=1 t P T t=1 tξt−1 P T t=1 t 2 −1 0 1 0 , ξt = Yt−αt, s 2 T denote the OLS estimate of the disturbance variance: s 2 T = X T t=1 (Yt − α˜ − β˜ T Yt−1 − ˜δt) 2 /(T − 3), and Q ≡ 0 1 0 1 R W(r)dr 1/2 R W(r)dr R [W(r)]2dr R rW(r)dr 1/2 R rW(r)dr 1/3 −1 0 1 0 . 16
Proof: (a). Let the data generating process be and the regression model be Yt=a+B Note that the regression model of (27)can be equivalently rewritten as Y=(1-B)a+B(Y-1-a(t-1))+(6+a)t +B*Et-1+8*t+ut where a*=(1-B)a, B*=6,8*=8+Ba, and St-1=Yt-1-a(t-1).Moreover under the null hypothesis that B=l and 8=0 t=Y0+u1+a2+….+ut that is, Et is the random walk as described in Lemma 1. Under the maintained hypothesis, a= Co, B= l, and 8=0, which in(28) means that a*=0, B*=1 and 8*=Co. The deviation of the OLS estimate from these true values is given t t=1 5?-1∑ SEt-1u or in shorthand as C=af From Lemma 1, the order of probability of the individual terms in(29) is as O2(2) O2(T2)O2(T2)O2(T O 6*
Proof: (a). Let the data generating process be Yt = α + Yt−1 + ut , and the regression model be Yt = α + βYt−1 + δt + ut . (27) Note that the regression model of (27) can be equivalently rewritten as Yt = (1 − β)α + β(Yt−1 − α(t − 1)) + (δ + βα)t + ut , ≡ α ∗ + β ∗ ξt−1 + δ ∗ t + ut , (28) where α ∗ = (1 − β)α, β ∗ = β, δ ∗ = δ + βα, and ξt−1 = Yt−1 − α(t − 1). Moreover, under the null hypothesis that β = 1 and δ = 0, ξt = Y0 + u1 + u2 + ... + ut ; that is, ξt is the random walk as described in Lemma 1. Under the maintained hypothesis, α = α0, β = 1, and δ = 0, which in (28) means that α ∗ = 0, β ∗ = 1 and δ ∗ = α0. The deviation of the OLS estimate from these true values is given by α˜ ∗ β˜ − 1 ˜δ ∗ − α0 = T P T t=1 ξt−1 P T t=1 t P T t=1 ξt−1 P T t=1 ξ 2 t−1 P T t=1 tξt−1 P T t=1 t P T t=1 tξt−1 P T t=1 t 2 −1 P T t=1 ut P T t=1 ξt−1ut P T t=1 tut , (29) or in shorthand as C = A−1 f. From Lemma 1, the order of probability of the individual terms in (29) is as follows, α˜ ∗ β˜ − 1 ˜δ ∗ − α0 = Op(T) Op(T 3 2 ) Op(T 2 ) Op(T 3 2 ) Op(T 2 ) Op(T 5 2 ) Op(T 2 ) Op(T 5 2 ) Op(T 3 ) −1 Op(T 1 2 ) Op(T 1 ) Op(T 3 2 ) . 17