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Structural Break Detection in Time Series Models

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1Prague 11/05Structural Break Detection in Time Series ModelsStructural Break Detection in Time Series ModelsRichard A. DavisThomas LeeGabriel Rodriguez-YamColorado State University(http://www.stat.colostate.edu/~rdavis/lectures)This research supported in part by an IBM faculty award.2Prague 11/05Illustrative Exampletime 0 100 200 300 400-6 -4 -2 0 2 4 6 How many segments do you see?τ1 = 51 τ2 = 151 τ3 = 2513Prague 11/05Illustrative Exampletime 0 100 200 300 400-6 -4 -2 0 2 4 6 τ1 = 51 τ2 = 157 τ3 = 259Auto-PARM=Auto-Piecewise AutoRegressive Modeling 4 pieces, 2.58 seconds.4Prague 11/05Data: 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, UKYearCounts1976 1978 1980 1982 19841200 1400 1600 1800 2000 22005Prague 11/05Data: xt= number of monthly deaths and serious injuries in UK, differenced at lag 12; Jan `75 – Dec `84, (t = 13,…, 120)Remark: Seat belt legislation introduced in Feb `83 (t = 99).Example -- Monthly Deaths & Serious Injuries, UK (cont)YearDifferenced Counts1976 1978 1980 1982 1984-600 -400 -200 0 200Traditional regression analysis:⎩⎨⎧≤<≤≤=++=.12098 if 1, ,981 if 0, ,)(ttf(t)WtbfaYtt⎩⎨⎧≤≤=+=−=− .otherwise 0, ,11099 if ,1 )(12tg(t)NtbgYYXttttModel: b=−373.4, {Nt}~AR(13).⎩⎨⎧≤<≤≤=++=.12098 if 1, ,981 if 0, ,)(ttf(t)WtbfaYtt⎩⎨⎧≤≤=+=−=− .otherwise 0, ,11099 if ,1 )(12tg(t)NtbgYYXtttt6Prague 11/05¾ IntroductionyExamples AR GARCH Stochastic volatility  State space models¾ Model selection using Minimum Description Length (MDL)y General principlesy Application to AR models with breaks¾ Optimization using a Genetic Algorithmy Basicsy Implementation for structural break estimation¾ Simulation results¾ Applications¾ Simulation results for GARCH and SSM7Prague 11/05Introduction The Premise (in a general framework): Base model:Pθfamily or probability models for a stationary time series.Observations:y1, . . . , ynSegmented model: there exist an integer m ≥ 0 and locationsτ0= 1 < τ1< . . . < τm-1< τm= n + 1such thatwhere {Xt,j} is a stationary time series with probability distrand θj≠ θj+1.Objective:estimate m = number of segmentsτj= location of jthbreak point θj= parameter vector in jthepoch, if ,1, jj-jtttXYτ<≤τ=jPθ8Prague 11/05Examples 1. Piecewise AR model:where τ0= 1 < τ1< . . . < τm-1< τm= n + 1, and {εt} is IID(0,1).Goal: Estimatem = number of segmentsτj= location of jthbreak point γj= level in jthepochpj= order of AR process in jthepoch= AR coefficients in jthepochσj= scale in jthepoch, if , 111 jj-tjptjptjjttYYYjjτ<≤τεσ+φ++φ+γ=−−L),,(1jjpjφφ K9Prague 11/05Piecewise AR models (cont) Structural breaks: Kitagawa and Akaike (1978)• fitting locally stationary autoregressive models using AIC• computations facilitated by the use of the Householder transformationDavis, 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.10Prague 11/05Motivation 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 withthat 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−=mjjtntntIYYjj}{jtY , as ,0/ with ∞→→∞→ nnmmnn}~{,ntY11Prague 11/05Examples (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 , ,1221122112jj-qtjqtjptjptjjtttttYYYjjjjτ<≤τσβ++σβ+α++α+ω=σεσ=−−−−LL. if ,loglog log ,122112jj-tjptjptjjtttttYjjτ<≤την+σφ++σφ+γ=σεσ=−−L. if , specified is )|(),...,,,...,|(111111jj-tjptjptjjtttttttypyyypjjτ<≤τησ+αφ++αφ+γ=αα=αα−−−L12Prague 11/05Model Selection Using Minimum Description LengthBasics 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 Fˆ()|eL Fˆˆ(13Prague 11/05Illustration Using a Simple Regression Model (see T. Lee `01)Encoding the data: (x1,y1), . . . , (xn,yn)1. “Naïve” case )()( )()(),,( ),,()"("1111nnnnyLyLxLxLyyLxxLnaiveL+++++=+=LLKK2. Linear model; suppose yi= a0+ a1xi, i = 1, . . . , n. Then)()( )()( ),( ),,()"1("101101aLaLxLxLaaLxxLpLnn++++=+==LK3. Linear model with noise; suppose yi= a0+a1xi+ εi, i = 1, . . . , n, where{εi}~IID N(0,σ2). ThenIf A < L(y1) + . . . + L(yn), then “p=1” encoding scheme dominates the “naïve” scheme.4444444434444444421KL )ˆ,ˆ,ˆ|ˆ,,ˆ()ˆ()ˆ()ˆ( )()()"1("21012101σεε+σ+++++== aaLLaLaLxLxLpLnnA14Prague 11/05Model Selection Using Minimum Description Length (cont)Applied to the segmented AR model:First term :, if , 111 jj-tjptjptjjttYYYjjτ<≤τεσ+φ++φ+γ=−−L|y)L Fˆ(∑∑==++++=ψ++ψ++ττ+=mjjjmjjmmmnppnmmyLyLppLLL(m)|y)L121222111log22logloglog )|ˆ()|ˆ(),,(),,(ˆ( LKKFEncoding:integer I : log2I 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))θˆθˆ15Prague 11/05Second term : Using Shannon’s classical results on information theory, Rissanen demonstrates that the code


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