Chapter 3 Univariate linear stochastic models-further topics

when φ = 1 there is an immediate problem with this ˆ result, for it seems to imply that T (φT − φ ) has a zero variance, which is not very helpful for hypothesis testing! To obtain a non-degenerate asymptotic
3.1.2 Testing for a unit root: an introduction
The zero mean AR(1) process with normal innovations
t = 1, 2,L, T (3.1) xt = φ xt −1 + at 2 where at ~ NID(0, σ ) and x0 = 0 . The OLS esti-
Functional Central Limit Theorem. Three implications of the theorem are
W (1) ~ N (0,1)
σ ⋅W (r ) ~ N (0, σ 2 r )
[W (r )]2 r ~ x12
A further implication is that, if f (⋅) is a continuous functional on [0,1], then through the continuous mapping theorem,
f ( RT (r )) ⇒ f (W (r ))
We are now in a position to derive the asymptotic distribution of the denominator of (3.2) (see Phillips, 1987a, theorem 3.1)
RT ( r ) = x[ rT ] (r ) σ T = xt −1 σ T
In effect, the interval [0,1] is divided into T + 1 parts at r = 0,1 T , 2 T , K,1 and RT ( r ) is constant at values of r but with jumps at successive integers. As T → ∞ , RT ( r ) becomes increasingly ‘dense’ on [0,1]. In the limit, RT ( r ) weakly converges to standard Brownian motion (or the Weiner process), W (r ) , denoted
Chapter 3 Univariate linear stochastic models: further topics
3.1 Determining the order of integration of a time series
2.1.1. Distinguishing between different values of d.
2
xT (σ T ) is N (0,1) , so its square is x12 , and T −1 2 σ 2 . Thus T ∑ t =1 at converges in probability to
a T ∑ t =1 xt −1at ~ (1 2)σ 2 ( X − 1)
−1 T
T
ˆ T (φT − 1) ⇒
(1 2)([W (1)]2 − 1)
∫
1
(3.3)
0
[W (r )]2 dr
The denominator of (3.3) must be positive. Since [W (1)]2 is x12 and the probability that a x12 variable is less than ˆ unity is 0.68, the probability that T (φT − 1) is negative approaches 0.68 as T becomes large. Thus, in twothirds of samples generated by a random walk, the estiˆ mate φT will be less than unity. Furthermore, when 2 [W (1)] is large so will be the denominator of (3.3), ˆ implying that the limiting distribution of T (φT − 1) will be skewed to the left, with negative values twice as likely as positive values. Note that, from Phillips (1987a)
T
Which has to be scaled by T −2 in order to converge to a finite value T −2 2 2 2 E[T xt −1 ] = (σ 2)(1 − 1 T ) → σ 2 as T → ∞ t =1 Hence T
∑
ˆ T (φT − 1) =
If xt is I (0) , which we will denote xt ~ I (0) even though such a notation has been used previously to denote the distributional characteristics of a series, then, assuming for convenience that it has zero mean: (i) the variance of xt is finite and does not depend on t , (ii) the innovation at has only a temporary effect on the value of xt ,
xt2 = ( xt −1 + at ) 2 = xt2−1 + 2 xt −1at + at2 xt −1at = (1 2)( xt2 − xt2−1 − at2 )
implying that Thus, the numerator of the ratio can be written as
xt −1at = (1 2)( x − x ) − (1 2)∑ t =1 at2 ∑ t =1
(iii) the expected length of times between crossings of x = 0 is finite, i.e., xt fluctuates around its mean of zero, (iv) the autocorrlations, ρ k , decrease steadily in magnitude for large enough k , so that their sum is finite.
If xt ~ I (1) with x0 = 0 then: (i) the variance of xt goes to infinity as t goes to infinity, (ii) an innovation at has a permanent effect on the value of xt because xt is the sum of all previous innovations: see, e.g., equation (2.10), (iii) the expected time between crossings of x0 = 0 is infinite, (iv) the autocorrelations ρ k → 1 for all k as t goes to infinity.
T
−1
T
−2∑Leabharlann ∑t =1 t −1 t T 2 t =1 t −1
x a x
(3.2)
has an asymptotic distribution that is a ratio of a (sca2 led) x1 variable to a non-standard distribution. Let us consider the limiting distribution of the standardised variable xt σ T and map the increasing interval from 0 to T into the fixed interval [0,1] so that results will be invariant to the actual value of T . We thus define the random step function RT ( r ) as follows. Denote [ rT ] as the integer part of rT , where r ∈ [0,1] and define
Where X ~ x12 .
E ( xt2 ) = σ 2t , it follows that the expectation Since
of the denominator of the ratio is
E[∑
T
2 t =1 t −1
x ]=σ
2
(t − 1) = σ 2 (T − 1) T 2 ∑ t =1
mate of φ is given by
T
ˆ = ∑ t =1 φT T
xt −1 xt x
∑
2 t =1 t −1