Breaks 1
14.384 Time Series Analysis, Fall 2007
Professor Anna Mikusheva
Paul Schrimpf, scribe
October 30, 2007 revised November 3, 2009
Lecture 19
The goal of this lecture is to analyze and test for another type of non-stationarity- break in parameters of a process. Empirical Processes Theory will be our working horse. Estimation of break date (once break is detected) is usually done by OLS, and is not discussed here.
Suppose y t
= β 0 t x t − 1
+ ² t
, where
β t
=
(
β t ≤ t
0
β + γ t > t
0
, x t and E [ ² t
| I t
] = 0 are iid variables. We want to test whether there is a break.
Break at a known date.
Assume that the date of the break t y t
= β 0 t x t − 1
+ γ 0 I t>t
0 x t − 1
² t
F-test (which in this context is called Chow test).
0 is known. Then we can run a regression and test for a break is a test of hypothesis H
0
: γ = 0. We could just do an
F
T t
0
( SSR
1 ,T
( SSR
−
1 ,t
0
( SSR
1 ,t
0
+ SSR
+ SSR t
0
+1 ,T t
0
+1 ,T
)) /k
) / ( T − 2 k ) where SSR t,s is the sum of squared residuals from OLS using the sample from time t to time s . Under the null and assuming that t
0 we have F is known.
T
⇒ χ 2 k
/k
= [
, where k
δT ] (there are increasing number of observations on both sides of the break), is the number of restrictions (the dimension of γ ). This test is valid when t
0
Break at an unknown date.
When t
0 t
0 is not known, a test as above (testing for a break at a single specific date) is not powerful. Note that is a nuisance parameter, which is identified only under the alternative, but not under the null hypothesis.
One test-statistic is the Quandt statistic that aims at a maximum value of a set of F-statistics for different break times:
Q = sup
[ δT ] ≤ t
0
≤ [(1 − δ ) T ]
F
T t
0
(1)
The statistics seemed to be attractive and was known for a long time, but the asymptotic of it was unknown until Andrews’ (1993) paper. Other test statistics are the Mann-Wald (average F-statistic):
M W =
1
T − 2 r t
0
= r
F
T
( t
0
T
) , r = [ δT ] (2)
Cite as: Anna Mikusheva, course materials for 14.384 Time Series Analysis, Fall 2007. MIT OpenCourseWare ( http://ocw.mit.edu
),
Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

Break at an unknown date.
2 and the Andrews-Ploberger (1994) (geometric average):
"
AP = ln
T
1
− 2 r t
0
= r exp
µ
1
2
F
T
( t
T
0
)
¶#
(3)
To make the statistics above usable, we need to answer an important question: what are their asymptotic distributions? For this we’ll use empirical process theory (Functional CLT). We make the following assumptions:
1. Uniform Law of Large Numbers: the following convergence hold uniformly in τ such that 0 < δ < τ <
1 − δ < 1:
Ψ
T
( τ ) =
1
T
[ X ] x t − 1 x 0 t − 1 t =1
The above is a generalization of a law of large numbers 1
T
→ τ Σ xx
P
T t =1 x t − 1 x 0 t − 1
→ Σ xx
.
2. Functional central limit theorem (for which we need some conditions would be that x and ² are independent and x t additional assumptions, e.g. sufficient are iid with finite fourth moments):
√
1
T
[ X ] x t − 1
² t t =1
= ξ
T
( δ ) ⇒ σ Σ 1 / 2 xx
W k
( · ) where W k
( · ) is k-dimensional Brownian motion, and σ 2 is the variance of ² t
.
To derive the limiting distribution of F statistics, we must look at the behavior of SSR . Let β
OLS estimate from the sample from t = 1 , .., r .
SSR
1 ,r
=
X
( y t
−
ˆ 0 x t − 1
)
2 t =1
Then, since ( β − β ) 0
= ² 2 t
− 2( ˆ − β ) 0
= (
P r t =1 t =1 x t − 1 x 0 t − 1
) − 1 (
P r t =1 x t − 1
² t
+ ( β − x t − 1
² t
), we have
β ) 0 x t − 1 x 0 t − 1
( − β )
SSR
1 ,r
= t =1
² 2 t
− ( t =1 x t − 1
² t
) 0 ( t =1 x t − 1 x 0 t − 1
) − 1 ( t =1 x t − 1
² t
)
Combining this, we have, for [ τ T ] = r :
SSR
1 ,r
− t =1
²
2 t
=( √
1
T x t − 1
² t
)
0
(
1
T t =1
= ξ
T
( τ )
0
Ψ ( τ )
− 1
ξ
T
( τ ) t =1
σ 2
⇒
τ
W k
( τ ) 0 W k
( τ ) x t − 1 x
0 t − 1
)
− 1
( √
1
T
Now, looking at the numerator of F , we have t =1 x t − 1
² t
)
⇒
( SSR
1 ,T
σ k
2
µ
−
−
W k
( SSR
(1) 0 W k
1 , [ T τ ]
(1) +
+ SSR
[ T τ ] ,T
)) /k
W k
( τ ) 0
τ
W k
( τ )
+
( W k
(1) − W
K
( τ )) 0 ( W k
(1)
1 − τ
− W k
( τ ))
¶
⇒
σ 2 k
( W k
( τ ) − τ W k
(1)) 0 ( W k
( τ ) − τ W k
(1))
τ (1 − τ )
Cite as: Anna Mikusheva, course materials for 14.384 Time Series Analysis, Fall 2007. MIT OpenCourseWare ( http://ocw.mit.edu
),
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Recursive Estimation 3
Under the null (of no break)
SSR
1 , [ T τ ]
+ SSR
T − 2 k
[ T τ ] ,T
)
→ p
σ
2
Let us introduce a process B k
( τ ) = W k
( τ ) − τ W k
(1) called a Brownian bridge. It is a linear transformation of a Brownian motion that is required to be 0 at t = 0 and t = 1. Thus we have convergence of processes
(as a weak convergence of processes)
F
T
( τ ) ⇒
B k
( τ ) 0 B k
( τ ) kτ (1 − τ )
Now we can get asymptotic distributions of test statistics as they are continuous functionals of F
T
():
Q
M W
AP
⇒
⇒
⇒
δ ≤ τ ≤ 1 − δ
1 − 2 δ log sup
1
1
Z
1
−
δ
B
1 − δ
2 k
τ
δ
( τ
(1
Z
B
δ
) 0 B
− k
( k
τ
τ
(
)
)
τ
0
)
B exp k
( τ kτ (1 − τ )
1 − δ
½
) dτ
B k
( τ ) 0 kτ (1
B
− k
τ
(
)
τ )
¾ dτ
!
Conclusions and Remarks:
• A nice feature of the last statements is that the limiting distributions do not depend on nuisance parameters. We can simulate these distributions to find critical values. The test for H
0
H a
: no break vs.
: there is one break can be performed by calculating the Q statistic in sample and comparing it to the simulated critical values. We will reject large values of Q . An alternative approach is bootstrapping the Q statistic.
• Critical values for Q test are bigger than for Chow test(known break date).
• If errors are auto-correlated one needs to correct for that (use HAC version of Chow test).
• Uniform Law of Large Numbers Assumption implicitly assumes stationarity of x t
.
More than one break
• Theoretically it is possible to get asymptotic distributions for statistics testing for two breaks.
• Practically: you have to simulate critical values, which in a case of even just 2 breaks is heavy computational burden. And complexity increases with dimensionality rapidly.
• Ideologically: If you allow for multiple breaks in a relatively small sample probably you should model the situation differently. Alternatives: describe a process for breaks to occur (deterministic or stochastic), or introduce continuously changing coefficients.
Recursive Estimation
Another idea for how to test for breaks is to look whether the estimation of β on different horizons is stable
(so called recursive estimation). Let:
β r/T ) =( t =1 x t − 1 x
0 t − 1
)
− 1
( t =1 x t − 1 y t
)
Cite as: Anna Mikusheva, course materials for 14.384 Time Series Analysis, Fall 2007. MIT OpenCourseWare ( http://ocw.mit.edu
),
Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

Unit Root tests with a Break 4
One can calculate estimates of β β
˜ changes a lot, it is a sign of a break. Formally, we need to formalize what does it mean “changes a lot”, that is , we have to find the asymptotic distribution. Suppose H
0
: no breaks is true and β is the true coefficient. Then,
√
T ( β
˜(
τ ) − β ) = Ψ
T
( τ ) − 1 ξ
T
( ) ⇒
σ Σ
− 1 / 2 xx
W k
( τ )
τ
The problem is the nuisance parameters, by estimating β by β
β and Σ xx
. We can eliminate them from the asymptotic distribution
˜(1) and looking at the associated t -statistic: t t
( τ ) = σ − 1
²
(
1
T
X x t − 1 x 0 t − 1
) 1 / 2
√
T ( β
˜
( τ ) − β (1)) ⇒
W k
( τ )
τ
− W k
(1) and as above, you can use test statistic sup | t
T
( τ ) | and critical values simulated from sup
τ
{
W k
τ
( τ )
− W k
(1) }
There are many ways to test for breaks, and the way to derive the limiting distribution of test statistics is
.
by using the FCLT.
Unit Root tests with a Break
Consider a unit root process with possible break in trend. The model is walk and d t
= c + γ 1 t>t
0 y ∗ t
= y t
+ d t where is a trend. How do we then test for a unit root? Two observations: y t is a random
1. The distribution of Dickey-Fuller statistics is very sensitive to the trend specification. If the trend in misspecified, for example if y ∗ t regressed on const , y ∗ t − 1
(the break is ignored), then you are likely to accept a unit root, even if there is no one. The intuition is that both a break and a unit root lead to permanent changes in y t
.
2. If the trend is correctly specified say D τ t
= (1 , 1 t> [ τ T ]
) is for a known break date at time t
0
= [ τ T ], then we can get the asymptotics of Dickey-Fuller statistics under unit root assumption. For example,
T ( ρ − 1) ⇒
R
1
0
1
P d τ
W
( P d τ
( s ) dW ( s
W ( s )) 2 ds
)
, where P
N
= I − N ( N 0 N ) − 1 N is a linear projection to a space perpendicular to
In practice, you would simulate critical values imposing the correct break date.
N , and d τ = (1 , 1 t>τ
) 0 .
Perron (1989) found that if you allow for breaks during the Great Depression (1929) and during the oil shocks (1973), then you reject the null of unit root in most macro series. Christiano(1992) and Zivot and
Andrews(1992) objected to this by arguing that it is not fair to treat the break dates as known. If you test for unit roots without assuming the break dates are known, then you have to change a statistic. One suggestion would be t min
DF
= min
τ ∈ [ δ
0
,δ
1
] t ( τ ) , where t ( τ ) is Dickey-Fuller t-statistic for a known break date at t
0
= [ τ T ]. That surely, does not change the value of statistic from what Perron obtained, but suggest different critical values. As a result, one cannot reject the null of unit root in most series if assume “unknown break date”.
Spurious Regression
Let x t and y t be two independent random walks, y t
= y t − 1
+ z t x t
= x t − 1
+ u t
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),
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Spurious Regression 5 where z t and u t are iid and independent of each other. Suppose we run OLS of y t on x t
.
y t
= βx t
+ e t
The true value of β is 0. However, if you simulate this model and test the hypothesis H
0
: β = 0 using OLS t-statistic you will likely faulty reject it in a large portion of cases. That can be seen from the following lines.
β T
1
2
T
1
2
P
P y t x t x 2 t
Let assume that the functional central limit theorem w orks:
µ ¶
ξ
T
( τ ) = √
1
T x
[ τ T ] y
[ τ T ]
⇒ W ( τ ) =
µ
W
1
( τ )
W
2
( τ )
¶ where W ( τ ) is a 2-dimensional Brownian motion. Since integrals are continuous functionals, we have:
1
T
X
ξ
T
( t/T ) ξ
T
( t/T ) 0 ⇒
Z
W ( τ ) W ( τ ) 0 dτ
Thus,
β T
1
2
T
1
2
P
P y t x t x 2 t
⇒
R
R
W
1
W
W 2
2
2 dt dt
That is, β estimate of variance of error term diverges: even in v ery large samples. Next, we can notice that the
σ 2 =
1
T
SSR =
1
T t =1
( y t
−
1
T b 2 b t
⇒
) 2
µZ
= T (
T
1
2
W 2
1 dt − t =1
(
R y 2 t
− 2 β
T
1
2 t =1
1
W
2 dt ) 2
¶
W 2
2 dt y t x t
+ β 2
1
T 2 t =1 x 2 t
)
As a result, t = x 2 t
=
√
T q
1
T
σ 2 r
1
T 2
X x 2 t
We can see that | t | → p ∞ , that is, in large samples you will faulty reject the null in almost all samples. The important point here is that with non-stationary regressors, we get behavior of OLS estimates.
Cite as: Anna Mikusheva, course materials for 14.384 Time Series Analysis, Fall 2007. MIT OpenCourseWare ( http://ocw.mit.edu
),
Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

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