Simultaneous change-point and factor analysis for high-dimensional time series

Transcription

1 Simultaneous change-point and factor analysis for high-dimensional time series Piotr Fryzlewicz Joint work with Haeran Cho and Matteo Barigozzi (slides courtesy of Haeran) CMStatistics 2017 Department of Statistics, London School of Economics, UK.

2 High-dimensional factor models Popular dimension reduction technique for high-dimensional time series in finance, economics, neuroscience, biology, seismology. A small number of unobserved factors drive the co-movement of a large number of time series: x it = λ i f t }{{} χ it +ɛ it = r λ ij f jt + ɛ it, 1 i n; 1 t T. i=1 Typically assume that the factor loadings and factors are time-invariant. Use principal component analysis (PCA) to estimate the factor space. Successfully applied to business cycle analysis, consumer behaviour analysis, asset pricing and economic monitoring and forecasting. 1

3 Change-points in high-dimensional time series During financial crisis, stronger co-movements are observed fewer factors capture larger shares of total variance change in the number of factors and factor structure in general. x it = log returns of daily closing values of stocks composing S&P100 index (04/01/ /08/2016, n = 88 and T = 4177). Goldman Sachs / /2001 9/2003 7/2005 5/2007 3/2009 1/ / /2014 8/2016 Bank of America / /2001 9/2003 7/2005 5/2007 3/2009 1/ / /2014 8/2016 Vertical broken lines: change-points detected from the common components. 2

4 Change-points in high-dimensional time series Minimum number of eigenvalues required to account for 50 95% (y-axis) of the variance of x t. 3

5 Change-points in the common components Standard factor modelling methods are fooled by structural breaks. χ it = a i g 1t = a i (αg 1,t 1 + u t ) for η χ = 1 t ηχ 1, a i g 1t + b i g 2t = a i (αg 1,t 1 + u t ) + b i g 2t for η χ t ηχ 2, c i g 1t + d i g 2t = c i (αg 1,t 1 + u t ) + d i g 2t for η χ t ηχ 3, c i g 1t + d i g 2t = c i (βg 1,t 1 + u t ) + d i g 2t for η χ t ηχ 4 = T, = time-invariant loadings λ i time-varying factors f t = [ ] a i b i c i d i }{{} r=4 (αg 1,t 1 + u t, 0, 0, 0) for 1 t η χ 1, (αg 1,t 1 + u t, g 2t, 0, 0) for η χ t ηχ 2, (0, 0, αg 1,t 1 + u t, g 2t ) for η χ t ηχ 3, (0, 0, βg 1,t 1 + u t, g 2t ) for η χ t T. instabilities in loadings and factor number are absorbed into instabilities in the second-order structure of the factors with r = 4. 4

6 Literature review Importance of chage-point analysis has increasingly been noted in many applications (e.g., forecasting with factor-augmented models). Most of existing change-point methods for factor models test H 0 : loadings are stable. vs H 1 : there exists a single break in the loadings. Investigation is devoted to the effect of a single break in the loadings or factor number on estimation, with accompanying change-point tests. Breitung & Eickmeier (2011), Chen et al. (2014), Han & Inoue (2014), Corradi & Swanson (2014), Baltagi et al. (2015), Yamamoto & Tanaka (2015). Lasso-type estimators: Cheng et al. (2016) for detecting and locating a single change-point, and Ma & Su (2016) for multiple change-point detection in the loadings. So far, the existing literature overwhelmingly focuses on single change-point detection, excluding change-points in the autocorrelations of factors and those in the idiosyncratic components. 5

7 In this talk (a) Propose a comprehensive methodology for consistent estimation of multiple change-points, in the second-order structure of a high-dimensional time series governed by a factor model. Advocate the marriage of factor analysis and change-point analysis. (b) Operate under the most flexible piecewise-stationarity, embracing all possible structural instabilities under factor modelling. (c) Through the use of wavelets, change-point detection in the second-order structure of a high-dimensional time series change-point detection in the means of highdimensional panel data. (d) Circumvent the difficult problem of accurate estimation of the true factor number by adopting a screening procedure. 6

8 Piecewise stationary factor models x it = χ it + ɛ it = λ i f t + ɛ it, 1 i n; 1 t T. f t = (f 1t,..., f rt ) : r factors driving the common components. λ i = (λ i1,..., λ ir ) : time-invariant loadings. loadings with breaks factors = time-invariant loadings piecewise stationary factors (A1.χ) Change-points in common components: B χ = {η χ 1,..., ηχ Bχ }. (i) For each b = 0,..., B χ, weakly stationary processes f b t E f t f b t 2 0 as t η χ b. (ii) Let χ b it = λ i f b t. There exists τ χ < such that for all b, such that ACV of χ b t ACV of χb+1 t O n at some lag τ χ. (A1.ɛ) Change-points in idiosyncratic components: B ɛ = {η ɛ 1,..., ηɛ Bɛ }. 7

9 (A2) E(χ it ɛ i t ) = 0 for any i, i = 1,..., n and t, t = 1,..., T. (A3) There exists C f < such that E(f t f t ) < C f. (A4) n 1 Λ Λ H as n for positive definite H. Also, max i,j λ ij < λ <. (A5) C ɛ < such that, for any {a i } n i=1 satisfying n i=1 a 2 i = 1, sup I,I {1,...,T } 1 I I n n i=1 i =1 t I t I a i a i E(ɛ it ɛ i t ) < C ɛ. (A6) (min 0 b B χ ηχ b+1 ηχ b min 0 b Bɛ ηɛ b+1 ηɛ b ) ct for some c (0, 1). Under (A2) (A6), Γ x = T 1 T t=1 E(x tx t ): (i) its r largest eigenvalues are diverging linearly in n as n ; (ii) the (r + 1)th largest eigenvalue stays bounded for any n. For identification of χ it and ɛ it, we operate in an asymptotic setting where (A7) n as T while n/t = O(log 2 T ). 8

10 Consistency of PCA Γ x = T 1 T t=1 x tx t : sample covariance matrix. ŵ x,j : normalised eigenvector of Γ x corresponding to its jth largest eigenvalue. PCA estimators of χ it and ɛ it for k as a factor number: χ k t = k j=1 ŵx,jŵ x,j x t and ɛ k t = x it χ k t. When k = r: χ r it χ it = O p ( 1 T 1 n ), as good as under stationary factor models. When k r and fixed: for any i and 1 s < e T, 1 e ( ) ( χ k it χ n it) e s + 1 = O p = O p (log T ). T t=s Change-points in the second-order structure of χ it (ɛ it ) appear as those in χ k it ( ɛk it ) for any fixed k r. 9

11 Overview of the proposed methodology Stage 1 χ k it, 1 i n Repeat the following for k R x it, 1 i n; 1 t T PCA ɛ k it, 1 i n Wavelet transformation B χ B ɛ Screening over k R Stage 2 B χ (k) B ɛ (k) DCBS algorithm + Stationary Bootstrap algorithm g j ( χ k it ), h j( χ k it, χk i t ) 1 i i n g j ( ɛ k it ), h j( ɛ k it, ɛk i t ) 1 i i n Stage 1: PCA-based estimation of common and idiosyncratic components. Wavelet-based transformation of the estimated common and idiosyncratic components. Stage 2: Double CUSUM Binary Segmentation (DCBS) algorithm (Cho 2016) for multiple changepoint detection from χ k it and ɛk it, separately. 10

12 Stage 1: Wavelet-based transformation Wavelets: analogous to the Fourier exponentials in the spectral representation for stationary processes, in the Locally Stationary model (Nason et al., 2000). Filtering χ it with a vector of wavelets ψ j = (ψ j,0,..., ψ j,lj 1), Wavelet-based transformation of χ t : wavelet coefficients: d j,it = l χ i,t lψ j,l. g j (χ it ) = d j,it within-series structure of χ it, h j (χ it, χ i t ) = d j,it ± d j,i t cross-series structure of (χ it, χ i t ). Change-points in the second-order structure of χ t are detectable as changepoints in the means of g j (χ it ), 1 i n and h j (χ it, χ i t ), 1 i < i n at multiple wavelet scales j (Cho & Fryzlewicz, 2015). In practice, χ it is unknown. Replace it with g j ( χ k it ) and h j( χ k it, χk i t ). 11

13 Input panel data for segmentation For some fixed k r, consider the N = J T n(n + 1)/2-dimensional panel y lt = { gj ( χ k it ), 1 i n, h j ( χ k it, χk i t ), 1 i i n, l = 1,..., N; t = 1,..., T, = g lt + ε lt. (i) g lt are piecewise constant with all its change-points = B χ = {η χ 1,..., ηχ Bχ }. (ii) max 1 l N max 1 s<e T 1 e s+1 e t=s ε lt = O p (log 1+υ T ). y lt serves as an input to panel data segmentation algorithm in Stage 2. The same arguments hold for ɛ k it. 12

14 Stage 2: Panel data segmentation Problem: detect multiple change-points shared across g lt y lt = g lt + ε lt, l = 1,..., N; t = 1,..., T. CUSUM statistics for univariate data: for a given l, Y l b = b(t b) T ( 1 b b t=1 y lt 1 ) T T b t=b+1 y lt, b = 1,..., T 1. 13

15 Binary segmentation input first BS iteration second BS iteration second BS iteration How do we simultaneously handle the N-dimensional CUSUM series from y lt? 14

16 Double CUSUM statistic (Cho 2016) D b (m) = m(2n m) 2N ( 1 m m l=1 Y(l) b 1 2N m ) N l=m+1 Y(l) b for all b = 1,..., T 1 and m = 1,..., N, where Y (1) b... Y (N) b. N = 100, T = 500, CP at t = T/2 with jump size = 0.15 for 2% of coordinates. 15

17 Double CUSUM statistic (Cho 2016) D b (m) = m(2n m) 2N ( 1 m m l=1 Y(l) b 1 2N m ) N l=m+1 Y(l) b for all b = 1,..., T 1 and m = 1,..., N, where Y (1) b... Y (N) b. N = 100, T = 500, CP at t = T/2 with jump size = 0.15 for 2% of coordinates. 16

18 Double CUSUM statistic (Cho 2016) Db(m) = q m(2n m) 2N 1 m (`) `=1 Yb Pm 1 2N m (`) `=m+1 Yb PN (1) (N ) for all b = 1,..., T 1 and m = 1,..., N, where Yb... Yb. N = 100, T = 500, CP at t = T /2 with jump size = 0.03 for 50% of coordinates. 17

19 Double CUSUM statistic (Cho 2016) D b (m) = m(2n m) 2N ( 1 m m l=1 Y(l) b 1 2N m ) N l=m+1 Y(l) b for all b = 1,..., T 1 and m = 1,..., N, where Y (1) b... Y (N) b. N = 100, T = 500, CP at t = T/2 with jump size = 0.03 for 50% of coordinates. 18

20 Double CUSUM Binary Segmentation algorithm = DC statistic + binary segmentation. max 1 m N D b (m) replaces CUSUM series and a change-point estimate is η = arg max b [s,e) max 1 m N D s,b,e (m) for a given interval [s, e]. Majority of the high-dimensional change-point literature concerns the problem of single change-point detection. DCBS algorithm (a) guarantees the consistency in multiple change-point detection in highdimensional settings; (b) admits both serial- and cross-correlations in the data highly relevant for the time series factor model. 19

21 Consistency of the DCBS algorithm At each η χ b, m b : number of series sharing the change-point, denseness, δ b : average jump size at η χ b, N,T = min 1 b B χ mb δb : min. cross-sectional change size. Suppose ( N log 1+υ T ) 1 N,T T 1/4 (high-dimensional efficiency). Then, for a threshold satisfying C N 1 N,T log2+2υ T < π N,T < C N,T T 1/2, P( B χ = B }{{ χ ; η χ }} b ηχ b {{ < c 1ω N,T } total number location accuracy for b = 1,..., B χ ) 1 where ω N,T = N 2 N,T log2+2υ T. Near-optimality in change-point detection: dense and large changes are estimated with bias logarithmic in T, close to the optimal bias O p (1). 20

22 Bootstrap for threshold selection Approximate the quantiles of T s,e under H 0 using a resampling scheme based on the factor structure and stationary bootstrap (Politis & Romano, 2004). Stationary Bootstrap: block bootstrap which generates SB sample X 1,..., X T that is stationary conditional on the observed X 1,..., X T, using blocks with lengths independently drawn from a geometric distribution. Step 1.χ: For each j {1,..., k}, produce the SB sample of { f jt, 1 t T } as {f jt, 1 t T } and compute χk it = k λ j=1 ij f jt. Step 1.ɛ: Produce the SB sample of { ɛ k t = ( ɛk 1t,..., ɛk nt ), 1 t T } as {ɛ k t = (ɛ k 1t,..., ɛk nt ), 1 t T }. Step 2: Generate y lt through transforming χk it or ɛk it using g j and h j. Step 3: Obtain Y l s,b,e and compute the test statistic T s,e Step 4: Repeat Steps 1 3 R times. The critical value π χ s,e (k) or πɛ s,e (k) for the segment [s, e] is selected as the (1 α)-quantile of the R bootstrap test statistics T s,e at a given significance level α (0, 1). 21

23 Screening over factor number candidates Factor number estimation is challenging particularly in the presence of multiple change-points. Existing change-point tests for factor models rely on accurately estimating r. Our theoretical results hold for over-specified k r, while using k < r leads to χ k it that may not contain all Bχ. Screening of change-points detected from χ k it for a range of k. Factor number candidates: R = {r,..., r} Screen B χ (k), a set of estimated change-points from χ k it for all k R, and select k returning B χ (k) with the largest cardinality: k = max{k R : B χ (k) = max k R B χ (k ) }. Change-points in ɛ it are detected from ɛ k it = x it χ k it. 22

24 Overview of the proposed methodology Stage 1 χ k it, 1 i n Repeat the following for k R x it, 1 i n; 1 t T PCA ɛ k it, 1 i n Wavelet transformation B χ B ɛ Screening over k R Stage 2 B χ (k) B ɛ (k) DCBS algorithm + Stationary Bootstrap algorithm g j ( χ k it ), h j( χ k it, χk i t ) 1 i i n g j ( ɛ k it ), h j( ɛ k it, ɛk i t ) 1 i i n Stage 1: PCA-based estimation of common and idiosyncratic components. Wavelet-based transformation of the estimated common and idiosyncratic components. Stage 2: Double CUSUM Binary Segmentation (DCBS) algorithm (Cho 2016) for multiple changepoint detection from χ k it and ɛk it, separately. 23

25 Finite sample performance: single CP scenario For i = 1,..., n = 100 and t = 1,..., T = 200, x it = q=5 j=1 λ ij f jt + θɛ it, where θ = φ q 1 ρ 2 f 1 ρ Hβ 2, f jt = ρ f,j f j,t 1 + u jt, u jt iid N (0, 1), ɛ it = ρ ɛ,i ɛ i,t 1 + v it + β i v i+k,t, v it iid N (0, 1), k H i, k 0 ρ f,j = (j 1) and ρ ɛ,i iid U( 0.5, 0.5) provide autocorrelations. β i { 0.2, 0.2} and H = 5 supply cross-correlations in ɛ t. In each scenario, a single change-point is introduced to loadings, acf of factors, number of factors and the autocorrelations or cross-correlations of idiosyncratic components at η = T/3, with its denseness controlled by ϱ {1, 0.75, 0.5, 0.25} (dense to sparse). φ {1.0, 1.5, 2.0, 2.5} controls signal-to-noise ratio. Compare DC test (first iteration of DCBS algorithm) against MAX, AVG (pointwise maximum and average of CUSUMs) and DC-NFA (No Factor Analysis, DC test applied to wavelet transformation of x it ). 24

26 Finite sample performance: single CP scenario DC max avg DC NFA A single CP in loadings: detection power of DC, MAX, AVG and DC-NFA vs. φ {1.0, 1.5, 2.0, 2.5} for ϱ {1, 0.75, 0.5, 0.25} (left to right), applied to common (top) and idiosyncratic (bottom) components. 25

27 Finite sample performance: single CP scenario DC max avg DC NFA A single CP in acf of factors: detection power of DC, MAX, AVG and DC-NFA vs. φ {1.0, 1.5, 2.0, 2.5}, applied to common (top) and idiosyncratic (bottom) components. 26

28 Finite sample performance: single CP scenario DC max avg DC NFA A single CP in number of factors: detection power of DC, MAX, AVG and DC-NFA vs. φ {1.0, 1.5, 2.0, 2.5} for ϱ {1, 0.75, 0.5, 0.25} (left to right), applied to common (top) and idiosyncratic (bottom) components. 27

29 Finite sample performance: single CP scenario DC max avg DC NFA A single CP in autocorrelations of ɛ it : detection power of DC, MAX, AVG and DC-NFA vs. φ 1 {2.5, 2.0, 1.5, 1} for ϱ {1, 0.75, 0.5, 0.25} (left to right), applied to common (top) and idiosyncratic (bottom) components. 28

30 Finite sample performance: single CP scenario DC max avg DC NFA A single CP in cross-covariance of ɛ it : detection power of DC, MAX, AVG and DC-NFA vs. φ 1 {2.5, 2.0, 1.5, 1} for ϱ {1, 0.75, 0.5, 0.25} (left to right), applied to common (top) and idiosyncratic (bottom) components. 29

31 Covariance matrix of the idiosyncratic components to the left and right of η = T/3. 30

32 Application to exchange rates against British Sterling x it = log returns of the daily exchange rates of 24 currencies against British Sterling (03/05/ /08/2016, n = 24 and T = 2851). EUR /2005 8/ /2007 2/2009 5/2010 8/ /2012 2/2014 5/2015 8/2016 USD /2005 8/ /2007 2/2009 5/2010 8/ /2012 2/2014 5/2015 8/2016 η χ 9 = 15 June 2016 (UK s EU referendum took place on 23 June 2016). 31

33 Minimum number of eigenvalues required to account for 50 95% (y-axis) of the variance of x t. 32

34 Conclusions New theoretical results on consistency of χ k it estimated with k r. Novel two-stage methodology for multiple change-point detection in the factor model with the most flexible change-point structure. Consistent detection of multiple change-points in loadings, acf of factors, factor number and idiosyncratic components, as well as identifying their origins. M. Barigozzi, H. Cho & P. Fryzlewicz (2017) Simultaneous multiple change-point and factor analysis for high-dimensional time series. Under revision. arxiv:

35 References J. Bai & S. Ng (2002) Determining the number of factors in approximate factor models. Econometrica, 70: B. Baltagi, C. Kao & F. Wang (2015) Identification and estimation of a large factor model with structural instability. Working paper. J. Breitung & S. Eickmeier (2011) Testing for structural breaks in dynamic factor models. J. Econometrics, 163: L. Chen, J. Dolado & J. Gonzalo (2014) Detecting big structural breaks in large factor models. J. Econometrics, 180: X. Cheng, Z. Liao & F. Schorfheide (2016) Shrinkage estimation of high-dimensional factor models with structural instabilities. Rev. Econ. Stud. (to appear). H. Cho (2016) Change-point detection in panel data via double CUSUM statistic. Electron. J. Stat., 10: H. Cho & P. Fryzlewicz (2015) Multiple-change-point detection for high dimensional time series via sparsified binary segmentation. JRSSB, 77: V. Corradi & N. R. Swanson (2014) Testing for structural stability of factor augmented forecasting models. J. Econometrics, 182: X. Han & A. Inoue (2014) Tests for parameter instability in dynamic factor models. Economet. Theor., J. Li, V. Todorov, G. Tauchen & H. Lin (2016) Rank Tests at Jump Events. Preprint. S. Ma & L. Su (2016) Estimation of large dimensional factor models with an unknown number of breaks. Preprint. G. P. Nason, R. von Sachs & G. Kroisandt (2000) Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. JRSSB, 62: D. N. Politis & J. P. Romano (1994) The stationary bootstrap. JASA, 89: Y. Yamamoto & S. Tanaka (2015) Testing for factor loading structural change under common breaks.j. Econometrics, 189: