Long Memory through Marginalization

Transcription

1 Long Memory through Marginalization Hidden & Ignored Cross-Section Dependence Guillaume Chevillon ESSEC Business School (CREAR) and CREST joint with Alain Hecq and Sébastien Laurent Maastricht University Université Aix-Marseille & GREQAM 15th Oxmetrics User Conference, Cass Business School, London September 5, 2014 Chevillon Long Memory through Marginalization 05/09/14 1 / 21

2 There is an emerging consensus in empirical finance that realized volatility series typically display long range dependence with a memory parameter (d) around 0.4 (Andersen et al., JASA, 2001). Lieberman and Phillips (2008, ER). Chevillon Long Memory through Marginalization 05/09/14 2 / 21

3 Some Empirical Evidence Exchange rates RV EUR-USD 1.00 ACF-RV EUR-USD ARFIMA(0,0.38,0) RV USD-YEN ACF-RV USD-YEN ARFIMA(0,0.4,0) Daily Realized Volatility, Autocorrelations and fitted ARFIMA(0, d, 0) Chevillon Long Memory through Marginalization 05/09/14 3 / 21

4 Overview Empirical evidence: volatilities are persistent, exhibiting long memory, ACF decay slowly (hyperbolically rather than exponentially) all seem to present the same degree of memory d = 0.4 when analyzed individually (Andersen et al s magic number). This paper shows: large n multivariate (Big Data) framework may generate (identical) long memory in all the univariate (marginalized) representations. beware small (hidden/ignored) cross-section dependences (Manski, Pesaran, Hendry & Pretis...) Tool: From cross-section to time series dependence in a Final Equation Representation in VAR(1) context Potential applications to realized volatility, prices, river volume... (Cox and Townsend, 1947) Chevillon Long Memory through Marginalization 05/09/14 4 / 21

5 Outline Preliminary: Long Memory and Final Equation Representation Our argument Monte Carlo and stylized empirical facts Chevillon Long Memory through Marginalization 05/09/14 5 / 21

6 Long Memory Many commonly used definitions of short memory or weak dependence imply the property that ) var (T 1/2 S T c (0, ) where S T = T t=1 z t. Any process that does not satisfy this property may be said to exhibit long memory (Diebold & Inoue, 2001). For a covariance stationary process, Beran (1994) shows the equivalence between ) var (T 1/2 S T c v T 2d, as T, ρ z (k) c ρ k 2d 1, as k, f z (ω) c f ω 2d, as ω 0, where d ( 1/2, 1/2), d = 0, c v, c ρ, c f > 0, ρ z (k) = Corr [z t, z t+k ] and f z (ω) is spectral density. Fractionally integrated processes exhibit unit roots. (1) Chevillon Long Memory through Marginalization 05/09/14 6 / 21

7 Known Sources of Long Memory Long memory is commonly observed in economics and finance but its origin is unclear. Müller & Watson (2008) show the difficulty in discriminating between the various models of low frequency variation. Known explanations: aggregation across heterogeneous series, frequencies or economic agents (Granger 1980, Lieberman & Phillips, 2008, Abadir & Talmain, 2002); nonlinearity (Davidson & Sibbertsen 2005, Miller & Park, 2010) structural change (Diebold & Inoue, 2001, Perron and Qu, 2006). learning (bounded rationality) in forward looking models of expectations (Chevillon & Mavroeidis, 2013) Network effects (Schennach, 2013). our contribution: memory as an artefact of univariate modelling and marginalizing small & numerous cross section correlations Chevillon Long Memory through Marginalization 05/09/14 7 / 21

8 Final Equation Representation Consider an n-vector x t admitting an invertible VAR (1) representation (I n A n L) x t = ɛ t x t = (I n A n L) 1 ɛ t Where the inverse satisfies (I n A n L) 1 1 = (I n A n L) det (I n A n L) with (In A n L) the adjugate matrix. yielding a priori an ARMA(n, n 1) det (I n A n L) x t = (I n A n L)ɛ t with common AR polynomial for all variables. The spectral density of element {x 1t } is ( ) 2 n I n A n e iω 1j f (ω) = j=1 det (I n A n e iω ) Chevillon Long Memory through Marginalization 05/09/14 8 / 21

9 Example of FER Consider the trivariate VAR(1) x t y t = a b 0 b a b z t 0 b a with FER A (L) x t = B (L) ɛ t with [ A (L) = (1 al) B (L) = 1 x t 1 y t 1 z t 1 + εx t ε y t ε z t ( al + ) ] [ ( 2b L 1 al ) ] 2b L (1 al) 2 bl (1 al) b 2 L 2 bl (1 al) (1 al) 2 (1 al) 2 b 2 L 2 bl (1 al) (1 al) 2 b 2 L 2 hence all univariate elements follow and ARMA(3, 2) which simplifies e.g. if b = 0. In the context of BEKK, see Hecq, Laurent & Palm (2012). Chevillon Long Memory through Marginalization 05/09/14 9 / 21

10 Outline Preliminary Our argument Monte Carlo and stylized empirical facts Chevillon Long Memory through Marginalization 05/09/14 10 / 21

11 Analytic Framework Assume A n circulant matrix A n = a (n) 0 a (n) 1 a (n) n 1 a (n) n a (n) 1. a (n) 1 a (n) n 1 Define the spectral density of A = lim n A n, g A, with Fourier coefficients a k = lim n a (n) k : ( g A (λ) def = lim n a (n) a (n) 0 ) n a (n) k ekλ ; a k = 1 2π g 2π A (λ) e ikλ dλ 0 k=1 Then the eigenvalues of A n are given by λ k = g A ( 2πk n ), 0 k < n Chevillon Long Memory through Marginalization 05/09/14 11 / 21

12 Our result: choose a parametric function g() Consider (a 2π-periodic version of) the low pass the eigenvalues of C n satisfy: ( ) λ k = g (d) 2πk A = n g (d) A (x) = 1 { x <2πd}, d (0, 1/2), { 1 k = 0,..., nd ; 0 k = nd + 1,..., n 1. nd stochastic trends, n nd cointegration relations. Theorem Any element x t of the process x t, admits a spectral density f n,d such that as (d, n) (1/2, ), for ω > 0, f n,d (ω) (d,n) ( 1, ) f 1/2 (ω) = σɛ 2 1 e iω 1 2 ) f 1/2 (ω) is the spectral density of a flicker noise ARFIMA (0, 1 2, 0. Chevillon Long Memory through Marginalization 05/09/14 12 / 21

13 Intuition of Proof 1/2 1 The coefficients a (n,d) k of A n satisfy lim n a (n,d) 0 = d and a (n,d) k = O ( n 1 + (d 1/2) ) so A n 1 2 I n as (d, n) (1/2, ) 2 Matrix FER for the first row (x t ) A n = [ a (n) A n 1 ] + o (1) det (I n A n L) x 1t = det (I n 1 A n 1 L) ε 1t + o p (1) ( and as (d, n) 12, ) det ( I n 1 A n 1 e iω) f x (ω) det (I n A n e iω ) 2 σ 2 ε (2) Chevillon Long Memory through Marginalization 05/09/14 13 / 21

14 Intuition of Proof 2/2 Szegö s theorem: det ( I n 1 A n 1 e iω) det (I n A n e iω ) [ = exp 1 2πd log 2π 0 = (1 e iω) d n exp [ 1 2π ( 1 e iω) dλ ] 2π 0 ( log 1 g A (λ) e iω) ] dλ Remark: NOT: aggregation of elements of ɛt with Beta-distributed weights as in Granger (1980). Chevillon Long Memory through Marginalization 05/09/14 14 / 21

15 Extension to a more general framework Extension to a wider class of continuous densities as (d, p, n) { ( ) u p } g p,d (u) = exp Γ p+1 p 2πd ( 12,, ). Circulant to Toeplitz Matrices a (n) 0 a (n) 1 a (n) n 1 a (n) A n = n 1., T (n) n = a 1 a (n) 1 a (n) n 1 a (n) 0 as n, z C, det (I n A n z) det (I n T n z). a (n) 0 a (n) 1 a (n) n 1 a (n) a (n) a (n) 1 a (n) n 1 a (n) 1 a (n) 0, General VAR: B n = V n T n V 1 (I n B n L) x t = V n (I n T n L) Vn 1 x t = ɛ t where det (I n B n z) = det (I n T n z). Chevillon Long Memory through Marginalization 05/09/14 15 / 21

16 Outline Preliminary Our argument Monte Carlo and stylized empirical facts Chevillon Long Memory through Marginalization 05/09/14 16 / 21

17 Simulation: Toeplitz matrix Monte Carlo, n = 200, T = 1000, d =.45, M = Density ARFIMA(0, d, 0) One draw t 10 Smooth Log Per t I(d) slope t Log Per t fitted slope (smooth) t ACF-One draw 0.4 a k k Chevillon Long Memory through Marginalization 05/09/14 17 / 21

18 Simulation: Toeplitz matrix Monte Carlo, n = 200, T = 1000, d =.30, M = Density ARFIMA(0, d, 0) One draw t Smooth Log Per t I(d) slope t Log Per t fitted slope (smooth) t ACF-One draw 0.3 a k k Chevillon Long Memory through Marginalization 05/09/14 18 / 21

19 Empirics: Stylized facts Data (provided by TickData) consists of transaction prices at the 5-minute sampling frequency for n = 49 large capitalization stocks from the NYSE, AMEX NASDAQ, covering the period from January 4, 1999 to December 31, 2008 (2,489 trading days). The trading session runs from 9:30 EST until 16:00 EST. Several models are estimated on daily log-returns in % (obtained by summing 5-minute log-returns) on rolling windows of 980 observations. Estimator of realized variation: medrv (robust to jumps) We Estimate a VAR(1) for data and simulated series (d = 0.4) and report the mean of (on/off) diagonal elements as a function of n Chevillon Long Memory through Marginalization 05/09/14 19 / 21

20 VAR(1) Estimation 0.7 (Data) Mean Diag VAR(1) 0.4 (Simulated) Mean Diag VAR(1) n 0.20 (Data) Mean absolute Off Diag VAR(1) (Simulated) Mean abssolute Off Diag VAR(1) n Chevillon Long Memory through Marginalization 05/09/14 20 / 21

21 Conclusions Link from cross section dependence to long memory. Setting Large n-var Toeplitz structure small but nonzero cross correlations fractional number of unit roots nd, d (0, 1/2) Implied univariate ARMA with AR polynomial common to all variables Identical Long Memory appears in all variables as (d, n) (1/2, ) Fractional Integration should disappear in multivariate framework as n increases Extensions to Prices, River flow volume (Tersavirta), hidden depence (Hendry & Pretis). Chevillon Long Memory through Marginalization 05/09/14 21 / 21