Chapter 21_Univariate Unit Root Process.pdf

Ch. 21 Univariate Unit Root Process
1 Introduction
Consider OLS estimation of a AR(1) process,
Yt = ρYt 1 + ut,
−
where u .(0, σ2), and Y = 0. The OLS estimator of ρ is given by
t ∼ 0
1
T T − T
Yt 1Yt 2
t=1 −
ρˆT = T = Yt 1 Yt 1Yt
Y 2 −−
P t=1 t 1 t=1 ! t=1 !
− X X
and we also have P
1
T − T
2
(ˆρT ρ) = Yt 1 Yt 1ut . (1)
−−−
t=1 ! t=1 !
X X
2
When the true value of ρ is less than 1 in absolute value, then Yt (so does Yt
?) is a covariance-stationary process. Applying LLN for a covariance process (see
of Ch. 4) we have
T T 2
2 p 2 T σ 2 2
( Yt 1)/T E[( Yt 1)/T ] = · /T = σ/(1 ρ). (2)
−−→− 1 ρ2 −
t=1 t=1
X X  −
Since Yt 1ut is a martingale difference sequence with variance
−
2
2 2 σ
E(Yt 1ut) = σ
− 1 ρ2
−
and
1 T σ2 σ2
σ2 σ2 .
T 1 ρ2 → 1 ρ2
t=1
X  − −
Applying CLT for a martingale difference sequence to the second term in the
righthand side of (1) we have
T 2
1 L 2 σ
( Yt 1ut) N(0, σ). (3)
√−−→ 1 ρ2
T t=1
X −
1
Substituting (2) and (3) to (1) we have
T T
√ 2 1 √
T (ˆρT ρ) = [( Yt 1)/T ]− T [( Yt 1ut)/T ] (4)
−−· −
t=1 t=1
X 1 X
σ2 −σ2
L N(0, σ2 ) (5)
−→ 1 ρ2 1 ρ2
 − −
N(0, 1 ρ2). (6)
≡−
(6) is not valid for the case when ρ= 1, however. To see this, recall that the
2
variance of Yt when ρ= 1 is tσ, then the LLN as in (2) would not be valid since
if we apply CLT, then it would incur that
T T T
2 p 2 2 t=1 t
( Yt 1)/T E[( Yt 1)/T ] = σ. (7)
−−→− T →∞
t=1 t=1
X X P
1 T
Similar reason would show that the CLT would not apply for ( Yt 1ut).
√T t=1 −
1 T
( In stead, T −( Yt 1ut) converges.) To obtain the limiting distribution, as
t=1 − P
we shall prove in the following, for (ˆρρ) in the unit root case, it turn out that
P T −
we have to multiply (ˆρρ) by T rather than by √T :
T −
1
T − T
2 2 1
T (ˆρT ρ) = ( Yt 1)/T T −( Yt 1ut) . (8)
−−−
" t=1 # " t=1 #
X X
Thus, the unit root coefficient conver