Lecture 16 Unit Root Tests
RS ? EC2 - Lecture 16
Lecture 16 Unit Root Tests
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Autoregressive Unit Root
? A shock is usually used to describe an unexpected change in a variable or in the value of the error terms at a particular time period. ? When we have a stationary system, effect of a shock will die out gradually. But, when we have a non-stationary system, effect of a shock is permanent. ? We have two types of non-stationarity. In an AR(1) model we have: - Unit root: | 1 | = 1: homogeneous non-stationarity - Explosive root: | 1 | > 1: explosive non-stationarity ? In the last case, a shock to the system become more influential as time goes on. It can never be seen in real life. We will not consider them.
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RS ? EC2 - Lecture 16
Autoregressive Unit Root
? Consider the AR(p) process:
(L) yt t
where (L) 1 1L1 L22 .... p Lp
As we discussed before, if one of the rj's equals 1, (1)=0, or
1 2 .... p 1
? We say yt has a unit root. In this case, yt is non-stationary.
Example: AR(1): yt 1 yt 1 t Unit root: 1 = 1.
H0 (yt non-stationarity): 1 = 1
(or, 1 ? 1 = 0)
H1 (yt stationarity): 1 < 1
(or, 1 ? 1 < 0)
? A t-test seems natural to test H0. But, the ergodic theorem and MDS CLT do not apply: the t-statistic does not have the usual distributions.
Autoregressive Unit Root
? Now, let's reparameterize the AR(1) process. Subtract yt-1 from yt:
yt yt yt1 (1 1) yt1 t 0 yt1 t
? Unit root test:
H0: 0 = 1 ? 1 = 0 H1: 0 < 0.
? Natural test for H0: t-test . We call this test the Dickey-Fuller (DF) test. But, what is its distribution?
? Back to the general, AR(p) process: (L) yt t We rewrite the process using the Dickey-Fuller reparameterization::
yt 0 yt1 1yt1 2yt2 .... p1yt( p1) t ? Both AR(p) formulations are equivalent.
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RS ? EC2 - Lecture 16
Autoregressive Unit Root ? Testing
? AR(p) lag (L): (L) 1 1L1 L22 .... p Lp
? DF reparameterization: (1 L ) 0 1 ( L L2 ) 2 ( L2 L3 ) .... p 1 ( L p 1 L p )
? Both parameterizations should be equal. Then, (1)=-0. unit root hypothesis can be stated as H0: 0=0.
Note: The model is stationary if 0< 0 natural H1: 0 < 0.
? Under H0: 0=0, the model is AR(p-1) stationary in yt. Then, if yt has a (single) unit root, then yt is a stationary AR process.
? We have Augmented
a linear regression framework. Dickey-Fuller (ADF) test.
A
t-test
for
H0
is
the
Autoregressive Unit Root ? Testing: DF
? The Dickey-Fuller (DF) test is a special case of the ADF: No lags are included in the regression. It is easier to derive. We gain intuition from its derivation.
? From our previous example, we have:
y t 1y t 1 t 0 y t 1 t
? If 0 = 0, system has a unit root:: H0 :0 0 H1 :0 0 (| 0 | 0)
? We can test H0 with a t-test:
t 1
^ 1 SE ^
? There is another associated test with H0, the -test:. (T-1)(^ -1).
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RS ? EC2 - Lecture 16
Review: Stochastic Calculus
? Kolmogorov Continuity Theorem ? If for all T > 0, there exist a, b, > 0 such that: ? E(|X(t1, ) ? X(t2, )|a) |t1 ? t2|(1 + b) ? Then X(t, ) can be considered as a continuous stochastic process. ? Brownian motion is a continuous stochastic process. ? Brownian motion (Wiener process): X(t, ) is almost surely continuous, has independent normal distributed (N(0,t-s)) increments and X(t=0, ) =0 ("a continuous random walk").
Review: Stochastic Calculus ? Wiener process
? Let the variable z(t) be almost surely continuous, with z(t=0)=0. ? Define (,v) as a normal distribution with mean and variance v. ? The change in a small interval of time t is z
? Definition: The variable z(t) follows a Wiener process if
? z(0) = 0
? z = t ,
where N(0,1)
? It has continuous paths.
? The values of z for any 2 different (non-overlapping) periods of time are independent.
Notation: W(t), W(t, ), B(t).
Example:
WT (r)
1 T
(1 2
3 ... [Tr] );
r [0,1]
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RS ? EC2 - Lecture 16
Review: Stochastic Process: Wiener process
? What is the distribution of the change in z over the next 2 time units? The change over the next 2 units equals the sum of: - The change over the next 1 unit (distributed as N(0,1)) plus - The change over the following time unit --also distributed as N(0,1). - The changes are independent. - The sum of 2 normal distributions is also normally distributed. Thus, the change over 2 time units is distributed as N(0,2).
? Properties of Wiener processes:
? Mean of z is 0
? Variance of z is t
? Standard deviation of z is t
? Let N=T/t, then
z (T
)
z(0)
n
i 1
i
t
9
Review: Stochastic Calculus ? Wiener process
Example:
WT (r)
1 T
(1 2
3 ... [Tr] )
1 T
S[Tr ] ;
r [0,1]
? If T is large, WT(.) is a good approximation to W(r); r [0,1], defined: W(r) = limT WT(r) E[W(r) ] =0 Var[W(r) ] =t
? Check Billingsley (1986) for the details behind the proof that WT(r) converges as a function to a continuous function W(r).
? In a nutshell, we need - t satisfying some assumptions (stationarity, E[|t|q < for q>2, etc.) - a FCLT (Functional CLT). - a Continuous Mapping Theorem. (Similar to Slutzky's theorem). 10
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RS ? EC2 - Lecture 16
Review: Stochastic Calculus ? Wiener process
? Functional CLT (Donsker's FCLT) If t satisfies some assumptions, then
WT(r) DW(r), where W(r) is a standard Brownian motion for r [0, 1].
Note: That is, sample statistics, like WT(r), do not converge to constants, but to functions of Brownian motions.
? A CLT is a limit for one term of a sequence of partial sums {Sk}, Donsker's FCLT is a limit for the entire sequence {Sk} instead of one term.
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Review: Stochastic Calculus ? Wiener process
? Example: yt = yt-1 + t (Case 1). Get distribution of (X'X/T2)-1 for yt.
T
T t1
T
T2 (yt1)2 T2 [ ti y0]2 T2 [St1 y0]2
t1
t1 i1
t1
T
T2 [(St1)2 2y0St1 y02]
t1
2
T t1
St1 T
2T 1
2y0T1/2
T t1
St1 T
T 1
T 1y02
2
T t1
t /T (t1)T
1 T
S[Tr] 2dr 2y0T 1/2
T t1
t /T (t1)T
1 T
S[Tr] drT 1y02
1
1
2 XT (r)2dr 2y0T1/2 XT (r) drT1y02
0
0
1
d 2 W(r)2dr, T .
0
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RS ? EC2 - Lecture 16
Review: Stochastic Calculus ? Ito's Theorem
? The integral w.r.t a Brownian motion, given by Ito's theorem (integral):
f(t,) dB = f(tk,) Bk
where tk* [tk,tk + 1) as tk + 1 ? tk 0.
As we increase the partitions of [0,T], the sum p to the integral.
? But, this is a probability statement: We can find a sample path where the sum can be arbitrarily far from the integral for arbitrarily large partitions (small intervals of integration).
? You may recall that for a Rienman integral, the choice of tk* (at the start or at the end of the partition) is not important. But, for Ito's integral, it is important (at the start of the partition).
? Ito's Theorem result: B(t,) dB(t,) = B2(t,)/2 ? t/2.
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Autoregressive Unit Root ? Testing: Intuition
? We continue with yt = yt-1 + t (Case 1). Using OLS, we estimate :
T
T
T
yt yt 1
( yt 1 yt 1 ) yt 1
yt 1yt 1
^
t 1 T
t 1
T
1
t 1 T
y2 t 1
t 1
y2 t 1
t 1
y2 t 1
t 1
? This implies:
T
T
yt 1yt 1
( yt 1 / T )(t / T )
T (^ 1) T
t 1 T
y2 t 1
t 1
t 1
1 T
T
( yt 1 /
t 1
T )2
? From the way we defined WT(.), we can see that yt/sqrt(T) converges to a Brownian motion. Under H0, yt is a sum of white noise errors.
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RS ? EC2 - Lecture 16
Autoregressive Unit Root ? Testing: Intuition
? Intuition for distribution under H0: - Think of yt as a sum of white noise errors. - Think of t as dW(t).
Then, using Billingley (1986), we guess that T( ^-1) converges to
1
W (t)dW (t)
T (^ 1) d 0 1
W (t)2dt
0
? We think of t as dW(t). Then, k=0 to t k, which corresponds to 0to(t/T) dW(s)=W( s/T) (for W(0)=0). Using Ito's integral, we have
T
(^
1)
d
1 2
W (1)2 1
1
W (t)2dt
0
Autoregressive Unit Root ? Testing: Intuition
?
T (^ 1)
d
1 2
W
1
(1) 2
1
W (t)2dt
0
Note: W(1) is a N(0,1). Then, W(1)2 is just a 2(1) RV.
? Contrary to the stable model the denominator of the expression for the OLS estimator ?i.e., (1/T)txt2-- does not converge to a constant a.s., but to a RV strongly correlated with the numerator.
? Then, the asymptotic distribution is not normal. It turns out that the limiting distribution of the OLS estimator is highly skewed, with a long tail to the left.
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