Change-Point Methods for Multiple Structural Breaks
and Regime Switching Models
Birte Muhsal, joint work with Claudia Kirch | March 26th, 2012
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WORKSHOP ON RECENT ADVANCES IN CHANGEPOINT ANALYSIS AT THE UNIVERSITY OF WARWICK
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KIT – University of the State of Baden-Wuerttemberg and
National Laboratory of the Helmholtz Association
www.kit.edu
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Multiple Change-Point Problem
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We want to construct:
an asymptotic test with level α
H0 : no change
H1 : at least one structural change
a consistent estimator of the number of change-points
consistent estimators of the location of change-points.
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
2/21
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Multiple Change-Point Problem
0
100
200
300
400
500
0
100
200
300
400
500
We want to construct:
an asymptotic test with level α
H0 : no change
H1 : at least one structural change
a consistent estimator of the number of change-points
consistent estimators of the location of change-points.
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
2/21
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4
4
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Multiple Change-Point Problem
0
100
200
300
400
500
0
100
200
300
400
500
We want to construct:
an asymptotic test with level α
H0 : no change
H1 : at least one structural change
a consistent estimator of the number of change-points
consistent estimators of the location of change-points.
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
2/21
Outline
1
Introduction
2
Multiple Change-Point Location Model
Classical Multiple Change Point Model
Regime Switching Model
3
Test and Estimation Procedure
4
Simulation Study
5
Conclusion and References
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
3/21
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6
8
10
Classical Multiple Change-Point Model
0
100
Xi =
q +1
X
200
300
dj I {kj −1 < i ≤ kj } + εi ,
400
500
i = 1, ..., n,
j =1
with random errors ε1 , ..., εn and unknown
change points k1 , . . . , kq with 0 = k0 < k1 ≤ . . . ≤ kq ≤ kq +1 = n,
kj = [ϑj n], j = 1, . . . , q, and 0 < ϑ1 ≤ . . . ϑq ≤ 1
number of changes q ∈ N
expectations d1 , . . . , dq +1 with di 6= di +1 for i = 1, . . . , q.
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
4/21
Regime Switching Model
(n)
Xi
(n)
= dQ (n) + εi , i = 1, ..., n,
(n )
i
(n)
with random errors ε1 , ..., εn ,
expectations d1 , ..., dK ∈ R with di 6= dj , for i , j = 1, . . . , K
a non-observable {1, ..., K }-valued stationary process
(n)
{Qi
: i ∈ N}.
(n )
Key feature of {Qi
: i ∈ N}: long duration times.
Differences to the classical change-point model:
both number qn and locations k1 , ..., kqn of structural breaks are
random
the unbounded number of changes qn .
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
5/21
Regime Switching Model
(n)
Xi
(n)
= dQ (n) + εi , i = 1, ..., n,
(n )
i
(n)
with random errors ε1 , ..., εn ,
expectations d1 , ..., dK ∈ R with di 6= dj , for i , j = 1, . . . , K
a non-observable {1, ..., K }-valued stationary process
(n)
{Qi
: i ∈ N}.
(n )
Key feature of {Qi
: i ∈ N}: long duration times.
Differences to the classical change-point model:
both number qn and locations k1 , ..., kqn of structural breaks are
random
the unbounded number of changes qn .
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
5/21
Assumptions on errors
Let the errors ε1 , ..., εn be a strictly stationary sequence with
(1)
E ε1 = 0 ,
2
2
0 < σ = E ε1 < ∞,
X
E |ε1 |
2+ν
< ∞ for some ν > 0,
|γ(h)| < ∞, where γ(h) = cov(ε0 , εh ),
h≥0
2
2
and long run variance τ = σ + 2
X
γ(h) < ∞.
h>0
(2) Invariance principle:
It exists a Wiener process {W (k ), 1 ≤ k ≤ n} such that
max
G≤k ≤n−G
k +G
X
1
1
√
εi − (W (k + G) − W (k )) = op (log(n/G))− 2 .
2G τ i =k +1
1
(3) Hájek-Rényi-type moment condition:
j
X γ
E
εk ≤ C |j − i + 1|ϕ for some γ ≥ 1, ϕ > 1 and some constant C > 0.
k =i
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
6/21
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MOSUM (Moving Sum) Statistic
MOSUM statistic (Hušková and Slabý (2001))
Tn (G) =
max
Tk ,n (G)
G ≤k ≤n −G
k
X
1
Tk ,n (G) = √
2G τ
Xi −
i =k −G+1
with
kX
+G
i =k +1
Xi ,
where G = G(n) is the bandwidth fulfilling
2
n 2+ν log n
G
Introduction
→ 0,
n
G
→∞.
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
7/21
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8
MOSUM (Moving Sum) Statistic
k=G
MOSUM statistic (Hušková and Slabý (2001))
Tn (G) =
max
Tk ,n (G)
G ≤k ≤n −G
k
X
1
Tk ,n (G) = √
2G τ
Xi −
i =k −G+1
with
kX
+G
i =k +1
Xi ,
where G = G(n) is the bandwidth fulfilling
2
n 2+ν log n
G
Introduction
→ 0,
n
G
→∞.
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
7/21
2
4
6
8
MOSUM (Moving Sum) Statistic
k−G+1
k=G
k+G
MOSUM statistic (Hušková and Slabý (2001))
Tn (G) =
max
Tk ,n (G)
G ≤k ≤n −G
k
X
1
Tk ,n (G) = √
2G τ
Xi −
i =k −G+1
with
kX
+G
i =k +1
Xi ,
where G = G(n) is the bandwidth fulfilling
2
n 2+ν log n
G
Introduction
→ 0,
n
G
→∞.
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
7/21
2
4
6
8
MOSUM (Moving Sum) Statistic
k−G+1
k−G+1
k=G
k=G+1
k+G
k+G
MOSUM statistic (Hušková and Slabý (2001))
Tn (G) =
max
Tk ,n (G)
G ≤k ≤n −G
k
X
1
Tk ,n (G) = √
2G τ
Xi −
i =k −G+1
with
kX
+G
i =k +1
Xi ,
where G = G(n) is the bandwidth fulfilling
2
n 2+ν log n
G
Introduction
→ 0,
n
G
→∞.
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
7/21
2
4
6
8
MOSUM (Moving Sum) Statistic
k−G+1
k−G+1
k=G
k=G+1
k+G
k+G
-
k−G+1
k=250
k+G
MOSUM statistic (Hušková and Slabý (2001))
Tn (G) =
max
Tk ,n (G)
G ≤k ≤n −G
k
X
1
Tk ,n (G) = √
2G τ
Xi −
i =k −G+1
with
kX
+G
i =k +1
Xi ,
where G = G(n) is the bandwidth fulfilling
2
n 2+ν log n
G
Introduction
→ 0,
n
G
→∞.
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
7/21
2
4
6
8
MOSUM (Moving Sum) Statistic
k−G+1
k−G+1
k=G
k=G+1
k+G
k+G
-
k−G+1
k=250
-
k+G
k−G+1
k=n−G
k+G
MOSUM statistic (Hušková and Slabý (2001))
Tn (G) =
max
Tk ,n (G)
G ≤k ≤n −G
k
X
1
Tk ,n (G) = √
2G τ
Xi −
i =k −G+1
with
kX
+G
i =k +1
Xi ,
where G = G(n) is the bandwidth fulfilling
2
n 2+ν log n
G
Introduction
→ 0,
n
G
→∞.
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
7/21
Asymptotic Distribution
Theorem (Hušková and Slabý (2001), Kirch and M. (2012))
Let the assumptions on errors (1)-(2) and bandwidth G hold.
Then, under H0 ,
α(n/G)
T k ,n ( G )
max
G≤k ≤n−G
τ
D
− β(n/G) −
→ Γ,
where Γ has a Gumbel extreme value distribution, i.e.
P (Γ ≤ x ) = exp(−2exp(−x )),
and α(x ) =
√
2 log x , β(x ) = 2 log(x ) +
critical value: Dn (G; α) =
1
2
log log x −
1
2
log π.
β(n/G) − log log √11−α
α(n/G)
αn → 0, but not too fast.
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
8/21
Asymptotic Distribution
Theorem (Hušková and Slabý (2001), Kirch and M. (2012))
Let the assumptions on errors (1)-(2) and bandwidth G hold.
Then, under H0 ,
α(n/G)
T k ,n ( G )
max
G≤k ≤n−G
τ
D
− β(n/G) −
→ Γ,
where Γ has a Gumbel extreme value distribution, i.e.
P (Γ ≤ x ) = exp(−2exp(−x )),
and α(x ) =
√
2 log x , β(x ) = 2 log(x ) +
critical value: Dn (G; α) =
1
2
log log x −
1
2
log π.
β(n/G) − log log √11−α
α(n/G)
αn → 0, but not too fast.
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
8/21
Assumptions on errors
Let the errors ε1 , ..., εn be a strictly stationary sequence with
(1)
E ε1 = 0 ,
2
2
0 < σ = E ε1 < ∞,
X
E |ε1 |
2+ν
< ∞ for some ν > 0,
|γ(h)| < ∞, where γ(h) = cov(ε0 , εh ),
h≥0
2
2
and long run variance τ = σ + 2
X
γ(h) < ∞.
h>0
(2) Invariance principle:
It exists a Wiener process {W (k ), 1 ≤ k ≤ n} such that
max
G≤k ≤n−G
k +G
X
1
1
√
εi − (W (k + G) − W (k )) = op (log(n/G))− 2 .
2G τ i =k +1
1
(3) Hájek-Rényi-type moment condition:
j
X γ
E
εk ≤ C |j − i + 1|ϕ for some γ ≥ 1, ϕ > 1 and some constant C > 0.
k =i
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
9/21
2 4 6 8 10 12
Change-Point Estimators (Antoch et al. (2000))
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200
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400
500
0
5
10
0
Test statistic:
Tn (G) :=
max
G≤k ≤n−G
Tk ,n (G)/τ
— critical value Dn (G; αn )
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
10/21
2 4 6 8 10 12
Change-Point Estimators (Antoch et al. (2000))
100
200
300
400
500
0
100
200
300
400
500
0
5
10
0
Test statistic:
Tn (G) :=
max
G≤k ≤n−G
Tk ,n (G)/τ
— critical value Dn (G; αn )
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
10/21
2 4 6 8 10 12
Change-Point Estimators (Antoch et al. (2000))
100
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300
400
500
0
100
200
300
400
500
0
5
10
0
All pairs of indices vj , wj are chosen such that
Tk ,n (G)/τ
≥ Dn (G; αn ) k = vj , ..., wj ,
Tk ,n (G)/τ
< Dn (G; αn ) k = vj − 1, wj + 1
wj − vj
> εG .
The number of change-points q can be estimated by q̂n , the number
of pairs (vj , wj ).
The estimator of change-point kj is defined as
k̂j := arg max Tk ,n (G)/τ .
vj ≤k ≤wj
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
10/21
Consistency of q̂n
Classical model:
Theorem (Kirch and M. (2012))
Let the assumptions on errors (1)-(2), bandwidth G and level {αn }
hold. Furthermore assume
lim sup d0 (n)/G = C > 2
with
n→∞
d0 (n) := min |kj +1 − kj |.
0≤j ≤qn
Then, under H1 ,
P (q̂n = q ) −→ 1 as n → ∞.
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
11/21
Consistency of q̂n
Regime switching model:
Theorem (Kirch and M. (2012))
Let the assumptions on errors (1)-(2), bandwidth G and level {αn }
hold. Furthermore assume
lim sup d0 (n)/G = C > 2
with
n→∞
d0 (n) := min |kj +1 − kj |.
0≤j ≤qn
Then, under H1 ,
P (q̂n = q ) −→ 1 as n → ∞.
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
11/21
Consistency of q̂n
Regime switching model:
Theorem (Kirch and M. (2012))
Let the assumptions on errors (1)-(2), bandwidth G and level {αn }
hold. Furthermore assume
lim P (d0 (n) > 2G) = 1
n→∞
with
d0 (n) := min |kj +1 − kj |.
0≤j ≤qn
Then, under H1 ,
P (q̂n = q n ) −→ 1 as n → ∞.
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
11/21
Assumptions on errors
Let the errors ε1 , ..., εn be a strictly stationary sequence with
(1)
E ε1 = 0 ,
2
2
0 < σ = E ε1 < ∞,
X
E |ε1 |
2+ν
< ∞ for some ν > 0,
|γ(h)| < ∞, where γ(h) = cov(ε0 , εh ),
h≥0
2
2
and long run variance τ = σ + 2
X
γ(h) < ∞.
h>0
(2) Invariance principle:
It exists a Wiener process {W (k ), 1 ≤ k ≤ n} such that
max
G≤k ≤n−G
k +G
X
1
1
√
εi − (W (k + G) − W (k )) = op (log(n/G))− 2 .
2G τ i =k +1
1
(3) Hájek-Rényi-type moment condition:
j
X γ
E
εk ≤ C |j − i + 1|ϕ for some γ ≥ 1, ϕ > 1 and some constant C > 0.
k =i
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
12/21
Consistency of Change-Point Estimators
Classical model:
Theorem (Kirch and M. (2012))
Let the assumptions on errors (1)-(3), bandwidth G and level {αn }
hold. Furthermore assume
lim sup d0 (n)/G = C > 2
with
n→∞
d0 (n) := min |kj +1 − kj |.
0≤j ≤qn
γ
and let P (qn > γn ) → 0, where {γn } satisfies γn log(G)G− 2 → 0.
With q n := min(qn , q̂n ) we have, under H1 ,
2
max |k̂j − kj | = Op (1) .Op γnγ
1≤j ≤q n
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
13/21
Consistency of Change-Point Estimators
Regime switching model:
Theorem (Kirch and M. (2012))
Let the assumptions on errors (1)-(3), bandwidth G and level {αn }
hold. Furthermore assume
lim sup d0 (n)/G = C > 2
with
n→∞
d0 (n) := min |kj +1 − kj |.
0≤j ≤qn
γ
and let P (qn > γn ) → 0, where {γn } satisfies γn log(G)G− 2 → 0.
With q n := min(qn , q̂n ) we have, under H1 ,
2
max |k̂j − kj | = Op (1) .Op γnγ
1≤j ≤q n
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
13/21
Consistency of Change-Point Estimators
Regime switching model:
Theorem (Kirch and M. (2012))
Let the assumptions on errors (1)-(3), bandwidth G and level {αn }
hold. Furthermore assume
lim P (d0 (n) > 2G) = 1
n→∞
with
d0 (n) := min |kj +1 − kj |.
0≤j ≤qn
γ
and let P (qn > γn ) → 0, where {γn } satisfies γn log(G)G− 2 → 0.
With q n := min(qn , q̂n ) we have, under H1 ,
2
max |k̂j − kj | = Op γnγ .
1≤j ≤q n
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
13/21
2 4 6 8 10 12
Variance Estimators (i.i.d. case)
10
0
100
200
300
400
500
0
5
Under H1 the standard variance estimator overestimates the
variance.
0
100
200
σ̂n2
300
400
n
1X
=
(Xi − X n )2
n
500
i =1
Solution: Variance estimator depends on time point k .
σ̂k2,n
Introduction
1
:=
2G
k
X
(Xi − X k −G+1,k )2 +
i =k −G +1
Multiple Change-Point Location Model
kX
+G
!
(Xi − X k +1,k +G )2 .
i =k +1
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
14/21
0
2
4
6
8
Performance of σ̂k2,n
100
200
300
400
500
100
200
300
400
500
100
200
300
400
500
3
4
0
0
1
2
G=50
3
4
0
0
1
2
G=75
0
— σ2
- - - σ̂n2 =
n
1X
(Xi − X n )2
n
i =1
—
Introduction
σ̂k2,n
1
=
2G
k
X
2
(Xi − X k −G+1,k ) +
i =k −G+1
Multiple Change-Point Location Model
kX
+G
!
2
(Xi − X k +1,k +G )
i =k +1
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
15/21
Asymptotic Results with τ̂k ,n
Lemma (Kirch and M. (2012))
If the long run variance estimator τ̂k2,n fulfills
1
max |τ̂k ,n − τ | = op (log(n/G))− 2 under H0
G ≤k ≤n −G
and
max
G ≤k ≤n −G
τ̂k ,n = Op (1) under H1
all of the above results remain true.
Example: σ̂k2,n in the case of i.i.d. random variables.
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
16/21
Simulation Study
Questions:
How good is the general performance of the MOSUM procedure?
How does the choice of the variance estimator influence the
performance of the MOSUM procedure?
How does the bandwidth selection influence the performance?
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
17/21
Simulation Study
ε1 , . . . , εn i.i.d. with ε1 ∼ N (0, 1)
300
400
500
10
6
0
100
200
300
400
500
4
1
400
500
100
200
300
400
500
100
200
300
400
500
100
200
300
400
500
0
100
200
300
400
500
100
200
300
400
500
100
200
300
400
500
100
200
300
400
500
4
1
200
300
400
500
0
100
200
300
400
500
100
200
300
400
500
100
200
300
400
500
4
4
8
8
0
G= 50
G= 50
0
0
Introduction
G= 75
0
Multiple Change-Point Location Model
0
0
0
5
G= 75
5
4
10
0
0
0
−1
0
100
0
0
−1
4
G= 50
0
G= 35
G= 35
0
500
4
3
G= 35
400
8
300
8
200
300
0
0
−1
100
200
G= 25
G= 25
0
100
4
3
G= 25
0
8
200
8
100
2
6
2
6
2
0
2
σ2
10
σk2,n
10
σn2
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
G= 75
0
Simulation Study
Conclusion and References
18/21
Simulation Study
ε1 , . . . , εn i.i.d. with ε1 ∼ N (0, 1)
100
200
300
400
500
4
0
100
200
300
400
500
0
100
200
300
400
500
100
200
300
400
500
100
200
300
400
500
100
200
300
400
500
100
200
300
400
500
4
3
8
0
0
4
0
0
4
8
σ2
8
σk2,n
8
σn2
1
4
G= 25
G= 25
−1
0
0
G= 25
100
200
300
400
500
0
100
200
300
400
500
0
1
4
G= 35
4
3
8
8
0
G= 35
100
200
300
400
500
100
200
300
400
500
100
200
300
400
500
0
100
200
300
400
500
100
200
300
400
500
100
200
300
400
500
G= 50
G= 50
0
0
Introduction
5
G= 75
0
Multiple Change-Point Location Model
0
0
0
5
G= 75
0
10
4
10
0
0
−1
0
1
4
G= 50
0
4
3
8
0
8
−1
0
0
G= 35
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
G= 75
0
Simulation Study
Conclusion and References
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Conclusion
We have theoretically justified the use of the MOSUM procedure for
both the classical model as well as the regime switching model by
analysing the consistency of the change point estimators.
In the simulation study the procedure gives good estimates for the
change points (as long as the bandwidth G is appropriate) and is
additionally easy to implement.
Future research:
MOSUM procedure for change detection in more general models, i.e.
autoregressive or ARMA models.
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
20/21
References
Antoch, J. and Hušková, M. and Jarušková, D. (2000). Change point
detection. 5th ERS IASC Summer School.
Hušková, M. and Slabý, A. (2001). Permutation tests for multiple
changes. Kybernetica, 37, 605 – 622.
Kirch, C. and Muhsal, B. (2012). Change-point methods for multiple
structural breaks and regime switching models. In Preparation.
Thank you for your attention!
Introduction
Multiple Change-Point Location Model
Test and Estimation Procedure
Birte Muhsal – Change-Point Methods for Multiple Structural Breaks and RSM
Simulation Study
Conclusion and References
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