Detekce změn v autoregresních posloupnostech

Transcription

1 Nové Hrady 2012

2 Outline 1 Introduction 2 3 4

3 Change point problem (retrospective) The data Y 1,..., Y n follow a statistical model, which may change once or several times during the observation period Task: decide whether change or changes occurred locate change(s) estimate model before and after change(s) Many theoretical results, many applications Basic reference Csörgő and Horváth, (1997) Chen and Gupta (2000) Antoch, Hušková, Jarušková (2000)

4 On-line approach (sequential monitoring), Chu et al (1996) The observations are coming one by one, after each new observations arrives we make a decision whether change occurs or not. Y 1,... Y m - historical data that satisfy a model We use them as training data and start monitoring in time m + 1 We construct a detector statistic Q(k, m) based on all observations up to m + k and when it cross a prescribed boundary, we stop the monitoring immediately Usually, we solve the problem via the sequential hypotheses testing

5 Change in an autoregression model Model: Y i = β 1 Y i β p Y i p + ε i, i m = (β 1 + δ 1 )Y i (β p + δ p )Y i p + ε i, i > m variance of ε i can be constant or varying

6 Regression model with random regressors Y i = x T i β + x T i δ I {i > m} + ε i, i = p + 1,..., n x i = Y i 1 = (Y i 1,..., Y i p ) T Y i 1 delayed observations, β = (β 1,..., β p ) T, δ = δ n = (δ 1n,..., δ pn ) T 0 Var ε i = σ 2 unknown H 0 : m = n H 1 : p < m < n p < m = m n n change point

7 Tests based on maximum likelihood principle Testing procedures lead to functionals of the partial sums of residuals (Kulperger, 1985, Horváth, 1993, Bai, 1993) S k = k i=p+1 (Y i x T i β n ), k = p + 1,..., n or weighted residuals S k = k i=p+1 x i (Y i x T i β n ), k = p + 1,..., n, β n - LSE based on Y 1,..., Y n

8 Typical test statistics { T n = max S T k C 1 p k n k C n(c }/ k ) 1 S k σ 2 n, (1) T n (ɛ) = { max S T k C 1 k C n(c }/ k nɛ k n(1 ɛ) ) 1 S k σ 2 n, (2) { S T T n (q) = max k C 1 n p k n q 2 (k/n) S k }/ σ 2 n, (3) σ 2 n is a suitable estimator of σ 2, and q is a positive weight function C k = k n x i x T i, C k = x i x T i i=1 i=k+1

9 Limit behaviour under the null hypothesis Let σ 2 n be an estimator of σ 2 such that σ 2 n σ 2 = o P (1). Then, for ɛ (0, 1/2), under causality, as n, ( Tn (ɛ) ) 1/2 B(t) sup ɛ t 1 ɛ t(1 t) ( Tn (q) ) 1/2 B(t) sup 0<t<1 q(t), where {B(t), t [0, 1]} is a p-dimensional Brownian bridge process. HPS (2007)

10 Let σ 2 n be an estimator of σ 2 such that Then, σ 2 n σ 2 = o P ( (log log n) 1 ) lim P( a(log n)tn 1/2 b(log n) t ) = exp{ 2e t } n where a(y) = (2 log y) 1/2, b p (y) = 2 log y + p 2 log log y log Γ( p ) 2 Davis, Huang and Yao (1995); HPS(2007)

11 The limit distribution in the last theorem belongs to the Gumbel-type extreme value distributions, and the convergence is quite slow. Approximations to the critical values can easily be calculated, but they are reliable only for very large n. These approximations can be improved by using resampling techniques HPS+Kirch, 2008 regression bootstrap - resampling residuals only pair bootstrap - resampling pairs (Y i, Y i 1 )

12 Limit behaviour under alternatives Estimators of the change point m m 1 = argmax p<l<n ( S T l C 1 l C n (C l ) 1 S l ) m 2 = argmax p<l<n ( S T l C 1 n S l ) Then, under causality and regularity conditions, m j m = o P (b n ), j = 1, 2, for any sequence {b n } satisfying b n.

13 Estimators of σ 2 or σ 2 n = 1 n p n i=p+1 ( Y in x T i β n ) 2, σn( 2 m) = 1 { m ( Yi x T i n p i=p+1 ) 2 n ( β m + Yi x T i i= m+1 β 0 m ) } 2 where m is an estimator of the change point m, β m and β 0 m are the least squares estimators of β based on Y 1,..., Y m and Y m+1,..., Y n, respectively

14 Application 700 Px50, PX50, daily returns, Index PX50, left, daily returns - right

15 PX50, , ACF PV50, , PACF Sample Autocorrelation Sample Partial Autocorrelations Lag Lag returns PX50, : ACF - left, PACF - right

16 25 PX50, , detection of change PX50 - daily returns, change in autoregression, change

17 PX50, PX50, , ACF Sample Autocorrelation Sample Autocorrelation Lag Lag Index PX50, ACF: before change - left, after change - right

18 PX50, PX50, , PACF Sample Partial Autocorrelations Sample Partial Autocorrelations Lag Lag Index PX50, PACF: before change - left, after change - right

19 model before change: AR(1) X t = X t 1 + ɛ t errors ɛ t satisfy GARCH(1,1) model ɛ t = Z t σ t σ 2 t = ɛ 2 t σ 2 t 1 model after change: GARCH (1,1): X t = σ t Z t σ 2 t = X 2 t σ 2 t 1

20 Efficient score test statistic model: Y i µ = p j=1 β j(y i j µ) + ε i change in ξ = (µ, β 1,..., β p, σ 2 ) ξ n - MLE of ξ L(ξ) score vector (gradient of the log-likelihood function) I(ξ) - Fisher information matrix Test statistic based on Y 1,..., Y n : under H 0 (no change): max B(t) = n 1 2 I 1 2 ( ξ nt )L( ξ nt ) sup 1 j p+2 0 t 1 B j (t) B j (t) = o p (1) B j (t) independent Brownian bridges (Gombay 2008, under normality, Starinská 2011 for general distribution of errors)

21 Sunspots data

22 1 Wolf numbers, sample ACF 1 Wolf numbers, sample PACF 0.8 Sample Autocorrelation Sample Partial Autocorrelations Lag Lag Sunspots: ACF - left, PACF - right

23 150 Wolf index, Sunspots data, prediction in AR(9)

24 change in the mean: AR(2) AR(9) µ n σ n max B(t) change in the mean in AR(2) (at time 1935) mean before change: , mean after change: no change in AR(9) change in AR(2) parameters: changes in years 1826 and 1920 change in variance - ARMA(2,1) model Berkes at al (2009): 1936 (extremal theorem), 1946 (efficient score test)

25 file:///e /beamer/novehrady/sunspotmotyl.gif Sunspots data file:///e /beamer/novehrady/sunspotmotyl.gif [ :23:07]

26 Change in Vector autoregressive process VAR(p) y t = c + β 1 y t β p y t p + ɛ t, t = 1..., k = c + β 1 y t β p y t p + ɛ t, t = k , T H 0 : k = T, H 1 : k = k 0 < T Dvořák (2011,12) - asymptotic pseudo-likelihood ratio test, based on the strong invariance principle, test based on the Gumbel distribution

27 Varying parameters 0.15 MSCI financial index, MSCI financial index, MSCI sector financial index

28 Random coefficients autoregression Basic RCA(p) model: X t = p (β i + B it )X t i + Y t = (β + B t ) T X t 1 + Y t, t Z i=1 β = (β 1..., β p ) T - parameters (constant) B t = (B 1t,..., B pt ) T - iid (0, Σ) - random coefficients X t 1 = (X t 1,..., X t p ) T Y t - iid (0, σ 2 ) - errors

29 Parameters estimation Assume: EB t = 0, E(B t B T t ) = Σ; EY t = 0, EY t = σ 2 ; E(Y t B t ) = 0 X t = (β + B t ) T X t 1 + Y t = β T X t 1 + u t (4) u t = B T t X t 1 + Y t Eu 2 t F t 1 = X T t 1ΣX t 1 + σ 2 = ω T z t 1 + σ 2 (5) F t = σ{(b s, Y s ), s t} ω = vech Σ z t 1 = Kvec (X t 1 X T t 1) K - duplication matrix θ = (β, ω, σ 2 ) T - parameters of the model

30 Maximum- and quasi-maximum likelihood estimators Nichols and Quinn (1982), Aue et al (2006) two-step procedures non-linear optimization nonstationary RCA(1) models - Berkes at al (2009) unified procedures for stationary and nonstationary RCA(1) models - Aue and Horváth (2010)

31 RCA(p) model with parameters θ = (β, ω, σ 2 ) T Γ conditional log-likelihood function under joint normality of (B t, Y t ) (given X 0, X 1,... X p+1 ) L n (θ) = 1 2 n [ (Xt β T X t 1 ) 2 ] ω T z t 1 + σ 2 + log(ω T z t 1 + σ 2 ) t=1 quasi-maximum likelihood estimators - without normality transformed log-likelihood: l n (θ) = 2L n (θ)/n QMLE: l n ( θ) = inf θ Γ l n(θ)

32 l n (θ) = 1 n = 1 n n [ (Xt β T X t 1 ) 2 ] ω T z t 1 + σ 2 + log(ω T z t 1 + σ 2 ) n g t (θ) t=1 t=1 l n (θ) := l n(θ) = 1 n g t(θ) = ( gt (θ) β n g t(θ) t=1, g t(θ) ω {g t(θ)} is strictly stationary mds, g ) t(θ) T σ 2 Eg 1(θ) = 0, Eg 1(θ)g 1(θ) T = A(θ)

33 ( l n 2 l n (θ) ) (θ) = = 1 θ i θ j n n t=1 Eg 1 (θ) = H(θ) = El n (θ) A(θ 0 ) = A, H(θ 0 ) = H nonsingular, g t (θ) n( θ θ0 ) N (0, H 1 AH 1 ) (Nicholls and Quinn 1982, Aue, Horváth, Steinebach, 2006)

34 Monitoring in RCA(p) Model: X t = (β t + Σ 1/2 t b t ) T X t 1 + σ t y t b t iid(0, I), y t iid(0, 1), mutually independent θ t = (β t, ω t, σ 2 t ) T, ω t = vechσ t X 1,..., X m are stable (training, historical data) θ 1 = = θ m = θ 0 continue in observation and stop as soon as a change occurs

35 Sequential testing problem H 0 : θ i = θ 0, i = 1, 2,..., against the alternative that a change occurs, i.e., H A : there exists K 1, θ i = θ 0, 1 i < m + K, θ i = θ 0 + δ m, m + K i <, δ m 0, θ 0, δ m, K - unknown

36 Stopping time of test procedure: τ(m) = inf{k 1, Q(m, k) c q(m, k)} Q(m, k) is detector statistic q(m, k) is a boundary function c > 0 is a constant such that lim P(τ(m) < H 0) = α m (α is the prescribed probability of false alarm), and lim P(τ(m) < H 1) = 1 m (consistency of the test procedure)

37 detector: where Q(m, k) = ( m+k 1 m i=m+1 g i ( θ m ) ) T Â 1 m ( m+k 1 m i=m+1 g i (θ) is gradient vector θ m is QMLE of θ based on the training sample ) g i ( θ m ) Â m = 1 m m i=1 g i ( θ m )g i ( θ m ) T estimator of A based on the training sample boundary function: q(t) = (1 + t) 2 ( t 1+t )2γ, γ [0, 1/2)

38 Theorem. Let {b t }, {y t } be iid, mutually independent, Eb0i 4 <, i = 1,..., p, Ey0 4 <, and condition on strict stationarity holds. Let θ 0 be an inner point of Γ. Then under H 0, as m P( max Q(m, k)/q(k/m) x) P( sup W(t) 2 /t 2γ < x) 1 k< 0<t<1 where W is d-dimensional standard Wiener process with independent components, d = (p + 1)(1 + p/2).

39 Proof: FCLT: nt {n 1/2 A(θ) 1/2 g i (θ), t [0, 1]} {W(t), t [0, 1]} i=1 a few steps that approximate the asymptotic behaviour of by sup 1 k< sup 1 k< m+k i=m+1 g i ( θ) q(k/m) m+k i=m+1 g i (θ) k m m i=1 g i (θ) q(k/m)

40 monitoring under H A δ m is such that δ m 0, m δ m as m, and K = O(m η ), η = η(γ) < 1/2 Then max Q(m, k)/q(k/m) 1 k< in probability (test has asymptotic power 1).

41 0.018 density of stopping time, LSE density of stopping time, CWLSE Densities of stopping times, γ = 0, (blue), γ = 0.25 (green), γ = 0.49 (red). Left: LSE, right: QMLE. True value K=5, m=500, β = 0.3, δ = 0.5, σ 2 B = 0.25, σ 2 Y = 0.25, 3000 observations

42 0.014 density of stopping time, LSE density of stopping time, CWLS Densities of stopping times, γ = 0, (blue), γ = 0.25 (green), γ = 0.49 (red). Left: LSE, right: QMLE. True value K=50, m=500, β = 0.3, δ = 0.5, σ 2 B = 0.25, σ 2 Y = 0.25, 3000 observations

43 THANK YOU!!!