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Structural Break Detection in Time Series Modelsrdavis/lectures/Ottawa_05.pdf · 2005. 7. 8. ·...
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Structural Break Detection Structural Break Detection in Time Series Modelsin Time Series Models
Richard A. Davis
Thomas Lee
Gabriel Rodriguez-Yam
Colorado State University(http://www.stat.colostate.edu/~rdavis/lectures)
This research supported in part by an IBM faculty award.
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IntroductionSetupExamples
ARGARCHStochastic volatility State space model
Model Selection Using Minimum Description Length (MDL)General principlesApplication to AR models with breaks
Optimization using Genetic AlgorithmsBasicsImplementation for structural break estimation
Simulation results
Applications
Simulation results for GARCH and SSM
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Introduction
The Premise (in a general framework):
Base model: Pθ family or probability models for a stationary time series.
Observations: y1, . . . , yn
Segmented model: there exist an integer m ≥ 0 and locations
τ0 = 1 < τ1 < . . . < τm-1 < τm = n + 1
such that
where {Xt,j} is a stationary time series with probability distrand θj≠ θj+1.Objective: estimate
m = number of segmentsτj = location of jth break point θj = parameter vector in jth epoch
, if , 1, jj-jtt tXY τ<≤τ=
jPθ
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Examples
1. Piecewise AR model:
where τ0 = 1 < τ1 < . . . < τm-1 < τm = n + 1, and {εt} is IID(0,1).
Goal: Estimate
m = number of segmentsτj = location of jth break point γj = level in jth epochpj = order of AR process in jth epoch
= AR coefficients in jth epochσj = scale in jth epoch
, if , 111 jj-tjptjptjjt tYYYjj
τ<≤τεσ+φ++φ+γ= −− L
),,( 1 jjpj φφ K
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Piecewise AR models (cont)
Structural breaks: Kitagawa and Akaike (1978)
• fitting locally stationary autoregressive models using AIC• computations facilitated by the use of the Householder transformation
Davis, Huang, and Yao (1995)
• likelihood ratio test for testing a change in the parameters and/or order of an AR process.
Kitagawa, Takanami, and Matsumoto (2001)
• signal extraction in seismology-estimate the arrival time of a seismic signal.
Ombao, Raz, von Sachs, and Malow (2001)
• orthogonal complex-valued transforms that are localized in time and frequency- smooth localized complex exponential (SLEX) transform.• applications to EEG time series and speech data.
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Motivation for using piecewise AR models:
Piecewise AR is a special case of a piecewise stationary process (see Adak 1998),
where , j = 1, . . . , m is a sequence of stationary processes. It is
argued in Ombao et al. (2001), that if {Yt,n} is a locally stationary
process (in the sense of Dahlhaus), then there exists a piecewise
stationary process with
that approximates {Yt,n} (in average mean square).
Roughly speaking: {Yt,n} is a locally stationary process if it has a time-varying spectrum that is approximately |A(t/n,ω)|2 , where A(u,ω) is a continuous function in u.
,)/(~1
),[, 1∑=
ττ −=
m
j
jtnt ntIYY
jj
}{ jtY
, as ,0/ with ∞→→∞→ nnmm nn
}~{ ,ntY
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Examples (cont)
2. Segmented GARCH model:
where τ0 = 1 < τ1 < . . . < τm-1 < τm = n + 1, and {εt} is IID(0,1).
3. Segmented stochastic volatility model:
4. Segmented state-space model (SVM a special case):
, if ,
,
122
1122
112
jj-qtjqtjptjptjjt
ttt
tYY
Y
jjjjτ<≤τσβ++σβ+α++α+ω=σ
εσ=
−−−− LL
. if ,loglog log
,
122
112
jj-tjptjptjjt
ttt
t
Y
jjτ<≤την+σφ++σφ+γ=σ
εσ=
−− L
. if , specified is )|(),...,,,...,|(
111
111
jj-tjptjptjjt
ttttt
typyyyp
jjτ<≤τησ+αφ++αφ+γ=α
α=αα
−−
−
L
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Model Selection Using Minimum Description Length
Basics of MDL:Choose the model which maximizes the compression of the data or, equivalently, select the model that minimizes the code length of the data (i.e., amount of memory required to encode the data).
M = class of operating models for y = (y1, . . . , yn)
LF (y) = = code length of y relative to F ∈ MTypically, this term can be decomposed into two pieces (two-part code),
where
= code length of the fitted model for F
= code length of the residuals based on the fitted model
,ˆ|ˆ( ˆ()( )eL|y)LyL FFF +=
|y)L F(
)|eL Fˆ(
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Model Selection Using Minimum Description Length (cont)
Applied to the segmented AR model:
First term : Let nj = τj – τj-1 and denote the length of the jth segment and the parameter vector of the jth AR process, respectively. Then
, if , 111 jj-tjptjptjjt tYYYjj
τ<≤τεσ+φ++φ+γ= −− L
|y)L F(
)|ˆ()|ˆ(),,(),,( )|ˆ()|ˆ(),,(),,(ˆ(
111
111
yLyLppLnnLL(m)yLyLppLLL(m)|y)L
mmm
mmm
ψ++ψ+++=ψ++ψ++ττ+=
LKK
LKKF
),,, , ( 1 jjpjjj jσφφγ=ψ K
Encoding:integer I : log2 I bits (if I unbounded)
log2 IU bits (if I bounded by IU)
MLE : ½ log2N bits (where N = number of observations used to compute ; Rissanen (1989))
θθ
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Second term : Using Shannon’s classical results on information theory, Rissanen demonstrates that the code length of can be approximated by the negative of the log-likelihood of the fitted model, i.e., by
)eL F|ˆ(
So,∑∑
==
++++=
m
jj
jm
jj n
ppnmm|y)L
12
1222 log
22
logloglogˆ(F
e
)1)ˆ2((log2
ˆ|ˆ(1
22 +σπ≈ ∑
=
m
jj
jn)eL F
For fixed values of m, (τ1,p1),. . . , (τm,pm), we define the MDL as
2)ˆ2(log
2log
22
logloglog
)),(,),,(,(
1 1
222
1222
11
nnn
ppnmm
ppmMDLm
j
m
jj
jj
jm
jj
mm
∑ ∑∑= ==
+σπ++
+++=
ττ K
The strategy is to find the best segmentation that minimizes MDL(m,τ1,p1,…, τm,pm). To speed things up, we use Y-W estimates of AR parameters.
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Optimization Using Genetic Algorithms
Basics of GA:Class of optimization algorithms that mimic natural evolution.
• Start with an initial set of chromosomes, or population, of possible solutions to the optimization problem.
• Parent chromosomes are randomly selected (proportional to the rank of their objective function values), and produce offspring using crossover or mutation operations.
• After a sufficient number of offspring are produced to form a second generation, the process then restarts to produce a thirdgeneration.
• Based on Darwin’s theory of natural selection, the process should produce future generations that give a smaller (or larger)objective function.
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Map the break points with a chromosome c via
where
For example,
c = (2, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, 3, -1, -1, -1, -1,-1)t: 1 6 11 15
would correspond to a process as follows:
AR(2), t=1:5; AR(0), t=6:10; AR(0), t=11:14; AR(3), t=15:20
Application to Structural Breaks—(cont)
Genetic Algorithm: Chromosome consists of n genes, each taking the value of −1 (no break) or p (order of AR process). Use natural selection to find a near optimal solution.
),,,( )),(,),(,( n111 δδ=⎯→←ττ KK cppm mm
⎩⎨⎧
τ=−
=δ− . isorder AR and at timepoint break if ,
,at point break no if ,1
1 jjjt ptp
t
),,,( )),(,),(,( n111 δδ=⎯→←ττ KK cppm mm
⎩⎨⎧
τ=−
=δ− . isorder AR and at timepoint break if ,
,at point break no if ,1
1 jjjt ptp
t
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Simulation Examples-based on Ombao et al. (2001) test cases
1. Piecewise stationary with dyadic structure: Consider a time series following the model,
where {εt} ~ IID N(0,1).⎪⎩
⎪⎨
⎧
≤≤ε+−<≤ε+−
<≤ε+=
−−
−−
−
,1024769 if ,81.32.1 ,769513 if ,81.69.1
,5131 if ,9.
21
21
1
tYYtYY
tYY
ttt
ttt
tt
t
Time
1 200 400 600 800 1000
-10
-50
510
⎪⎩
⎪⎨
⎧
≤≤ε+−<≤ε+−
<≤ε+=
−−
−−
−
,1024769 if ,81.32.1 ,769513 if ,81.69.1
,5131 if ,9.
21
21
1
tYYtYY
tYY
ttt
ttt
tt
t
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1. Piecewise stat (cont)
GA results: 3 pieces breaks at τ1=513; τ2=769. Total run time 16.31 secs
Time0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
0.4
0.5
Time0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
0.4
0.5
True Model Fitted Model
Fitted model: φ1 φ2 σ2
1- 512: .857 .9945513- 768: 1.68 -0.801 1.1134769-1024: 1.36 -0.801 1.1300
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1 200 400 600 800 1000
-6-4
-20
24
Simulation Examples (cont)
2. Slowly varying AR(2) model:
where and {εt} ~ IID N(0,1).
10241 if 81. 21 ≤≤ε+−= −− tYYaY ttttt
)],1024/cos(5.01[8. tat π−=
0 200 400 600 800 1000
time
0.4
0.6
0.8
1.0
1.2
a_t
10241 if 81. 21 ≤≤ε+−= −− tYYaY ttttt
)],1024/cos(5.01[8. tat π−=
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
0.4
0.5
2. Slowly varying AR(2) (cont)
GA results: 3 pieces, breaks at τ1=293, τ2=615. Total run time 27.45 secs
True Model Fitted Model
Fitted model: φ1 φ2 σ2
1- 292: .365 -0.753 1.149293- 614: .821 -0.790 1.176615-1024: 1.084 -0.760 0.960
Time0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
0.4
0.5
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2. Slowly varying AR(2) (cont)
True Model Average Model
In the graph below right, we average the spectogram over the GA fitted models generated from each of the 200 simulated realizations.
Time0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
0.4
0.5
Time
Freq
uenc
y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
0.4
0.5
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Data: Yt = number of monthly deaths and serious injuries in UK, Jan `75 – Dec `84, (t = 1,…, 120)Remark: Seat belt legislation introduced in Feb `83 (t = 99).
Example: Monthly Deaths & Serious Injuries, UK
Year
Cou
nts
1976 1978 1980 1982 1984
1200
1400
1600
1800
2000
2200
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Data: Yt = number of monthly deaths and serious injuries in UK, Jan `75 – Dec `84, (t = 1,…, 120)Remark: Seat belt legislation introduced in Feb `83 (t = 99).
Example: Monthly Deaths & Serious Injuries, UK
Year
Diff
eren
ced
Cou
nts
1976 1978 1980 1982 1984
-600
-400
-200
020
0
Piece 1: (t=1,…, 98) IID; Piece 2: (t=99,…108) IID; Piece 3: t=109,…,120 AR(1)Results from GA: 3 pieces; time = 4.4secs
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Examples
Speech signal: GREASY
Time
1 1000 2000 3000 4000 5000
-150
0-5
000
500
1000
1500
G R EA S Y
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Time
1 1000 2000 3000 4000 5000
-150
0-5
000
500
1000
1500
G R EA S Y
Time0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
0.4
0.5
Speech signal: GREASYn = 5762 observationsm = 15 break pointsRun time = 18.02 secs
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Application to GARCH (cont)
Garch(1,1) model:
⎩⎨⎧
<≤σ++<≤σ++
=σ−−
−−
.1000501 if ,6.1.4. ,5011 if ,5.1.4.
21
21
21
212
tYtY
tt
ttt
. if ,
IID(0,1)~}{ ,
12
121
2jj-tjtjjt
tttt
tY
Y
τ<≤τσβ+α+ω=σ
εεσ=
−−
AG%
GA%
# of CPs
24.019.21
≥ 2
0
0.40.4
72.080.4
AG = Andreou and Ghysels (2002)
⎩⎨⎧
<≤σ++<≤σ++
=σ−−
−−
.1000501 if ,6.1.4. ,5011 if ,5.1.4.
21
21
21
212
tYtY
tt
ttt
Time1 200 400 600 800 1000
-4-2
02
CP estimate = 506
. if ,
IID(0,1)~}{ ,
12
121
2jj-tjtjjt
tttt
tY
Y
τ<≤τσβ+α+ω=σ
εεσ=
−−
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time
y
1 100 200 300 400 500
05
1015
Count Data ExampleModel: Yt | αt ∼ Pois(exp{β + αt }), αt = φαt-1+ εt , {εt}~IID N(0, σ2)
Breaking PointM
DL
1 100 200 300 400 500
1002
1004
1006
1008
1010
1012
1014
True model:
Yt | αt ~ Pois(exp{.7 + αt }), αt = .5αt-1+ εt , {εt}~IID N(0, .3), t < 250
Yt | αt ~ Pois(exp{.7 + αt }), αt = -.5αt-1+ εt , {εt}~IID N(0, .3), t > 250.
GA estimate 251, time 267secs
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time
y
1 500 1000 1500 2000 2500 3000
-2-1
01
23
Model: Yt | αt ∼ N(0,exp{αt}), αt = γ + φ αt-1+ εt , {εt}~IID N(0, σ2)
SV Process Example
True model:
Yt | αt ~ N(0, exp{αt}), αt = -.05 + .975αt-1+ εt , {εt}~IID N(0, .05), t ≤ 750
Yt | αt ~ N(0, exp{αt }), αt = -.25 +.900αt-1+ εt , {εt}~IID N(0, .25), t > 750.
GA estimate 754, time 1053 secs
Breaking Point
MD
L
1 500 1000 1500 2000 2500 3000
1295
1300
1305
1310
1315
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time
y
1 100 200 300 400 500
-0.5
0.0
0.5
Model: Yt | αt ∼ N(0,exp{αt}), αt = γ + φ αt-1+ εt , {εt}~IID N(0, σ2)
SV Process Example
True model:
Yt | αt ~ N(0, exp{αt}), αt = -.175 + .977αt-1+ εt , {εt}~IID N(0, .1810), t ≤ 250
Yt | αt ~ N(0, exp{αt }), αt = -.010 +.996αt-1+ εt , {εt}~IID N(0, .0089), t > 250.
GA estimate 251, time 269s
Breaking PointM
DL
1 100 200 300 400 500
-530
-525
-520
-515
-510
-505
-500
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SV Process Example-(cont)
time
y
1 100 200 300 400 500
-0.5
0.0
0.5
Fitted model based on no structural break:
Yt | αt ∼ N(0, exp{αt}), αt = -.0645 + .9889αt-1+ εt , {εt}~IID N(0, .0935)
True model:
Yt | αt ~ N(0, exp{at}), αt = -.175 + .977αt-1+ et , {εt}~IID N(0, .1810), t ≤ 250
Yt | αt ∼ N(0, exp{αt }), αt = -.010 +.996αt-1+ εt , {εt}~IID N(0, .0089), t > 250.
time
y
1 100 200 300 400 500
-0.5
0.0
0.5
1.0 simulated seriesoriginal series
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32Ottawa 7/05
SV Process Example-(cont)Fitted model based on no structural break:
Yt | αt ∼ N(0, exp{αt}), αt = -.0645 + .9889αt-1+ εt , {εt}~IID N(0, .0935)
time
y
1 100 200 300 400 500
-0.5
0.0
0.5
1.0 simulated series
Breaking Point
MD
L
1 100 200 300 400 500-4
78-4
76-4
74-4
72-4
70-4
68-4
66
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33Ottawa 7/05
Summary Remarks
1. MDL appears to be a good criterion for detecting structural breaks.
2. Optimization using a genetic algorithm is well suited to find a near optimal value of MDL.
3. This procedure extends easily to multivariate problems.
4. While estimating structural breaks for nonlinear time series models is more challenging, this paradigm of using MDL together GA holds promise for break detection in parameter-driven models and other nonlinear models.