# Structural Break Detection in Time Series Models

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Structural Break Detection in 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 Prague 11 05 1 Illustrative Example 6 4 2 0 2 4 6 How many segments do you see 0 Prague 11 05 1 51 100 200 2 151 300 time 3 251 400 2 Illustrative Example Auto PARM Auto Piecewise AutoRegressive Modeling 6 4 2 0 2 4 6 4 pieces 2 58 seconds 0 Prague 11 05 1 51 100 200 2 157 300 time 3 259 400 3 Example Monthly Deaths Serious Injuries UK 1600 1200 1400 Counts 1800 2000 2200 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 1976 1978 1980 1982 1984 Year Prague 11 05 4 Example Monthly Deaths Serious Injuries UK cont Data 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 99 200 0 bf tt W Wt YYtt aa bf t 0 if if 11 tt 98 98 0 f t f t 1 if 98 t 120 1 if 98 t 120 XXtt YYtt YYtt 1212 bg tt N Ntt bg 600 400 Differenced Counts 200 Traditional regression analysis 1976 1978 1980 1982 1984 99 tt 110 110 11 ifif 99 g t g t 0 otherwise otherwise 0 Year Model b 373 4 Nt AR 13 Prague 11 05 5 Introduction yExamples AR GARCH Stochastic volatility State space models Model selection using Minimum Description Length MDL y General principles y Application to AR models with breaks Optimization using a Genetic Algorithm y Basics y Implementation for structural break estimation Simulation results Applications Simulation results for GARCH and SSM Prague 11 05 6 Introduction The Premise in a general framework Base model P family or probability models for a stationary time series Observations y1 yn Segmented model there exist an integer m 0 and locations 0 1 1 m 1 m n 1 such that Yt X t j if j 1 t j where Xt j is a stationary time series with probability distr P j and j j 1 Objective estimate m number of segments j location of jth break point j parameter vector in jth epoch Prague 11 05 7 Examples 1 Piecewise AR model Yt j j1Yt 1 L jp jYt p j j t if j 1 t j where 0 1 1 m 1 m n 1 and t is IID 0 1 Goal Estimate m number of segments j location of jth break point j level in jth epoch pj order of AR process in jth epoch j1 K jp j AR coefficients in jth epoch j scale in jth epoch Prague 11 05 8 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 Prague 11 05 applications to EEG time series and speech data 9 Motivation for using piecewise AR models Piecewise AR is a special case of a piecewise stationary process see Adak 1998 m Yt n Yt j I j 1 j t n j 1 where Yt j 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 Yt n with mn with mn n 0 as n that approximates Yt n in average mean square Roughly speaking Yt n is a locally stationary process if it has a timevarying spectrum that is approximately A t n 2 where A u is a continuous function in u Prague 11 05 10 Examples cont 2 Segmented GARCH model Yt t t t2 j j1Yt 21 L jp jYt 2 p j j1 t2 1 L jq j t2 q j if j 1 t j where 0 1 1 m 1 m n 1 and t is IID 0 1 3 Segmented stochastic volatility model Yt t t log t2 j j1 log t2 1 L jp j log t2 p j j t if j 1 t j 4 Segmented state space model SVM a special case p yt t 1 yt 1 y1 p yt t is specified t j j1 t 1 L jp j t p j j t Prague 11 05 if j 1 t j 11 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 M Typically this term can be decomposed into two pieces two part code LF y L F y L e F where L F y code length of the fitted model for F L e F code length of the residuals based on the fitted model Prague 11 05 12 Illustration Using a Simple Regression Model see T Lee 01 Encoding the data x1 y1 xn yn 1 Na ve case L naive L x1 K xn L y1 K yn L x1 L L xn L y1 L L yn 2 Linear model suppose yi a0 a1xi i 1 n Then L p 1 L x1 K xn L a0 a1 L x1 L L xn L a0 L a1 3 Linear model with noise suppose yi a0 a1xi i i 1 n where i IID N 0 2 Then L p 1 L x1 L L xn L a 0 L a 1 L 2 L 1 K n a 0 a 1 2 1444444442444444443 A If A L y1 L yn then p 1 encoding scheme dominates the na ve scheme Prague 11 05 13 Model Selection Using Minimum Description Length cont Applied to the segmented AR model Yt j j1Yt 1 L jp jYt p j j t if j 1 t j First term L F y 1 y L L m y L F y L m L 1 K m L p1 K pm L m m log2 m m log2 n log2 p j j 1 j 1 pj 2 2 log2 n j Encoding integer I log2 I bits if I unbounded log2 IU bits if I bounded by IU MLE log2N bits where N number of observations used to compute Rissanen 1989 Prague 11 …