Functional data analysis (FDA) has become a popular technique in applied statistics. In particular, this methodology has received considerable attention in recent studies in empirical finance. In this talk we discuss selected topics of functional principal components analysis that are motivated by financial data

Benko, Michal

Kneip, Alois

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            SFB 649 Discussion Paper 2005-016  Common Functional Component Modelling  Alois Kneip* Michal Benko**  * Department Statistics and Mathematics, Johannes Gutenberg-Universität Mainz, Germany ** CASE - Center for Applied Statistics and Economics, Humboldt-Universität zu Berlin, Germany  This research was supported by the Deutsche Forschungsgemeinschaft through the SFB 649 "Economic Risk".  http://sfb649.wiwi.hu-berlin.de ISSN 1860-5664  SFB 649, Humboldt-Universität zu Berlin Spandauer Straße 1, D-10178 Berlin SFB      6 4 9              E C O N O M I C     R I S K                 B E R L I N 1 Common functional componentmodellingAlois Kneip and Michal BenkoACKNOWLEDGEMENTFinancial support of Deutsche Forschungsgemeinschaft via SFB 649 “O¨konomischesRisiko”, Humboldt-Universita¨t zu Berlin, is gratefully acknowledged.1.1 IntroductionFunctional data analysis (FDA) has become a popular technique in applied statistics.In particular, this methodology has received considerable attention in recent studiesin empirical finance. In this talk we discuss selected topics of functional principalcomponents analysis that are motivated by financial data.By definition, FDA deals with the analysis of samples of functions. However, in practicethe functions of interest are often not directly observed but are regression curves whichhave to be reconstructed from discrete, noisy data. In Section 2 we present a newmethod for efficient estimation of functional principal components in such situations.It consists in an adaptation of a technique introduced by Kneip and Utikal (2001) forthe case of density functions.Inference for two independent functional samples is considered. Bootstrap tests aredeveloped to test whether principal components coincide and the two samples thuspossess ”common” functional principal components. The procedure possesses an im-portant application in modelling implied volatilites as described in Benko and Ha¨rdle(2004).In this section we will focus on one sample of i.i.d. smooth random functionsx1(t), . . . , xN (t) ∈ L2[0, 1], t ∈ [0, 1]. For v, w ∈ L2[0, 1] let 〈v, w〉 =∫ bav(t)w(t)dt, andlet ‖ · ‖= 〈·, ·〉1/2 denote the usual L2-norm. The Karhunen-Loe`ve decomposition then1 Common functional component modellingprovides a basic tool to describe the distribution of the random functions xi. Withλ1 ≥ λ2 ≥ . . . and γ1, γ2, . . . denoting eigenvalues and corresponding orthonormaleigenfunctions of the covariance operator C of xi we obtainxi = µ+∑j=1βijγj , i = 1, . . . , N, (1.1)where µ = E(xi) is the mean function and βij = 〈xi−µ, γj〉 are (scalar) factor loadingswith E(β2ij) = λj . Structure and dynamics of the random functions can be assessedby analyzing the ”functional principal components” γj as well as the distribution ofthe factor loadings.An important property of (1.1) consists in the known fact that the first L principalcomponents provide a “best basis” for approximating the sample functions in terms ofthe integrated square error.For any choice of L orthonormal basis functions v1, . . . , vLρ(v1, . . . , vL) = E(‖ xi − µ−L∑j=1〈xi − µ, vj〉vj ‖2)is minimized by vj = γj . In many important applications a small number of functionalprincipal components will suffice to approximate the functions xi with a high degreeof accuracy.For a given sample an empirical analog of (1.1) can be constructed by using eigenvaluesλ1,n ≥ λ2,N ≥ . . . and orthonormal eigenfunctions γ1,N , γ2,N , . . . of the empiricalcovariance operator CN , where CN = 1NN∑i=1〈xi− x¯, ξ〉(xi− x¯). If K denotes the numberof nonzero eigenvalues of CN thenxi = x¯+K∑j=1βij,Nγj,N , i = 1, . . . , N, (1.2)where x¯ is the sample mean, and 1N∑i β2ij,N = λj,N . Obviously, λj,N and γj,N estimateλj and γj for j = 1, 2, . . . .However, in practice, the sample functions xi are often not directly observed, but haveto be reconstructed from noisy observations Yij at discrete design points tik:yik = xi(tik) + εik, k = 1, . . . , Ti, (1.3)where εik are independent noise terms with E(εik) = 0, Var(εik) = σ2i .21.1 IntroductionIn this context the standard approach to estimate functional principal components is tofirst estimate individual functions nonparametrically and then to determine eigenfunc-tions of the resulting estimated empirical covariance operator (compare, e.g., Ramsayand Silverman (1997)).We propose an alternative approach which in a first step relies on estimating theelements of the matrixMlk = 〈xl − x¯, xk − x¯〉, l, k = 1, . . . , N. (1.4)Some simple linear algebra shows that all nonzero eigenvalues λ1,N ≥ λ2,N . . . ofCN and l1 ≥ l2 . . . of M are related by λj,N = lj/N . When using additionally thecorresponding orthonormal eigenvectors p1, p2, . . . of M , the empirical scores βjr,N =〈xj − x¯, γr,N 〉 as well as the empirical eigenfunctions γr,N are obtained by βjr,N =√lrpjr andγr,N =(√lr)−1 N∑i=1pir (xi − x¯) =(√lr)−1 N∑i=1pirxi. (1.5)The elements of M are functionals which can be estimated with asympotically negligi-ble bias and a parametric rate of convergence T−1/2i . If the data in (1.3) is generatedfrom a balanced, equidistant design, then it is easily seen that for i 6= j this rate ofconvergence is achieved by the estimator M̂ij = 1TT∑k=1yikyjk.In the case of a random design some adjustment is necessary: Set T def= min{T1, T2, . . . , TN}and define an equidistant grid {tk def= k/(T − 1), k = 0, 1, . . . , T − 1} on [0, 1]. Then,for each i = 1, . . . , N and k = 1, . . . , T find the index of the first and second nearestneighbor of tk:k(i) = argminj=1,...,Ti|tij − tk| and k′(i) = argminj 6=k′(i)|tij − tk|.Finally, construct the estimatorsM̂ij =1TT∑k=1yik(i)yjk(j) for i 6= j and M̂ii = 1TT∑k=1yik(i)yik′(i). (1.6)The aim of using special estimator (1.6) for the diagonal terms is to avoid the ad-ditional bias. Alternatively we can construct a bias corrected estimator using somenonparametric estimation of variance σ2i .The eigenvalues lˆ1 ≥ lˆ2 . . . and eigenvectors p1, p2, . . . of the resulting matrix M̂ thenprovide estimates λˆr,N = lˆr/N and βˆjr =√lˆrpˆjr. Estimates γˆr,N of the empirical31 Common functional component modellingfunctional principal component γr,N can be determined from (1.5) when replacingthe unknown true functions xi by nonparametric estimates xˆi (as, for example, localpolynomial estimates with bandwidths hi):γˆr,N =(√lˆr)−1 N∑i=1pˆirxˆi. (1.7)When considering (1.7), it is important to note that γˆr,N is defined as a weightedaverage of all estimated sample functions. Averaging reduces variance, and efficientestimation of γr,N therefore requires undersmoothing of individual function estimatesxˆi. Indeed, under suitable additional regularity conditions it can be shown that for anoptimal choice of smoothing parameters and twice continuously differentiable xi, weobtain the rate of convergence ‖ γr,N − γˆr,N ‖= OP ((NT )−2/5). Here, T = mini Ti.A natural question is how many functions K should be used when approximatingthe sample functions by a factor model of the form (1.2), when relying on estimatedprincipal components and scores. This question is crucial from statistical and practicalpoint of view, although it is not discussed often in the literature. We propose abootstrap test which successively tests the hypothesis H0 : λK0+1,N = · · · = λN,N = 0for K0 = 1, 2, 3, . . . based on the respective residual average integrated square error:ρˆN (γ1, . . . , γK0) =1N∑Nr=K0+1lˆr.The idea of this procedure is based on the fact that the estimated eigenfunctions thatcorrespond to small estimated eigenvalues (with relative small importance) are ratherdriven by the sampling noise than can be interpreted as a component of the variation ofthe underlying population. In other words we do not want to use eigenfunctions whoserelative importance cannot be significantly distinguished from random componentsgenerated by noise.1.2 Two sample inferenceClearly, in the framework described by (1.1) - (1.3) we are faced with two sources ofvariability of estimated functional principal components. Due to sampling variationγr,N will differ from the true component γr, and due to (1.3) there will exist anadditional estimation error when approximating γr,N by γˆr,N . The results of Dauxois,Pousse and Romain (1982) imply that ‖ γr − γr,N ‖= OP (N−1/2), and the results ofthe proceeding section therefore imply that the difference between γˆr,N and γr,N is ofsmaller order of magnitude if T is sufficiently large compared to N . Inference aboutfunctional principal components under (1) - (3) will then be first order equivalent toan inference based on known functions xi.41.2 Two sample inferenceWe are mainly interest in two sample problems. Thus letx11(t), x12(t), . . . , x1N1(t) and x21(t), x22(t), . . . , x2N2(t) (1.8)denote two independent samples of smooth functions. The problem of interest is totest whether the functional principal components γr in the respective decompositions(1.1) are common (identical) for both groups. In this case only the factor loadings βirmay vary across samples. Then γr = γ1r = γ2r andxpi = µp +∑r=1βpirγr, p = 1, 2. (1.9)This hypothesis has been used in the work of Fengler, Ha¨rdle and Villa (2003) andBenko and Ha¨rdle (2004) in modelling implied volatilites. It can be seen as a functionalgeneralization of the concept of ”common principal components” as introduced byFlury (1988) in multivariate analysis.If the functions xpi were directly observed, then in order to test the hypothesisH0 : γ1r = γ2rfor r = 1, 2, . . . one could rely on the test statisticsDr =‖ γ1r,N − γ2r,N ‖2 .It can be shown that critical values of Dr can be determined by a bootstrap procedure:Under H0 we have Dr =‖ γ1r,N − γ1r − (γ2r,N − γ2r ) ‖2. The distribution of ‖ γ1r,N −γ1r − (γ2r,N − γ2r ) ‖2 can then be approximated by the bootstrap distribution of ‖γ1∗r,N − γ1r,N − (γ2∗r,N − γ2r,N ) ‖, where γ1∗r,N and γˆ2∗r,N are estimates to be obtained fromindependent bootstrap samplesx1∗1 (t), x1∗2 (t), . . . , x1∗N1(t) and x2∗1 (t), x2∗2 (t), . . . , x2∗N2(t). (1.10)In the practically more relevant situation that all curves in both samples have to bereconstructed from noisy observations according to (1.3), Dr must be replaced byDˆr =‖ γˆ1r,N − γˆ2r,N ‖2, where estimates are determined by the procedure describedabove. Bootstrap estimates are then obtained by resampling the observations corre-sponding to the unknown curves xi. The procedure is asymptotically valid if T issufficiently large such that the additional estimation error is asymptotically negligi-ble. Of course, the test should only be performed for components which can be savelydistinguished from noise, and 1 ≤ r ≤ minK10 ,K20 .51 Common functional component modelling6BibliographyBenko, M.and Ha¨rdle, W. (2004). Common Functional IV Surface Analysis StatisticalTools for Finance and Insurance, edited by Cˇ´ızˇek, P., Ha¨rdle, W., Weron,R. to appear by Springer.Dauxois, J., Pousse, A. and Romain, Y. (1982). Asymptotic Theory for the PrincipalComponent Analysis of a Vector Random Function: Some Applications toStatistical Inference, Journal of Multivariate Analysis 12 : 136-154.Fan, J. and Huang, L. (1999 ). Nonparametric Estimation of Quadratic RegressionFunctionals, Bernoulli 5,: 927-949.Hall, P., Kay, J.W. and Titterington, D.M. (1990 ). Asymptotically optimal difference-based estimation of variance in nonparametric regression, Biometrika 77:520:528.Flury, B. (1988). Common Principal Components and Related Models, Wiley, NewYork.Fengler, M., Ha¨rdle, W. and Villa, P. (2003). The Dynamics of Implied Volatilities: Acommon principle components approach, Review of Derivative Research 6 :179-202.Fengler, M., Ha¨rdle, W. and Mammen, E. (2003). Implied Volatitlity String Dynamics,CASE Discussion Paper, http://www.case.hu-berlin.de.Kneip, A. and Utikal, K. (2001). Inference for Density Families Using Functional Prin-cipal Components Analysis, Journal of the American Statistical Association96 : 519-531.Ramsay, J. and Silverman, B. (1997). Functional Data Analysis, Springer, New York. SFB 649 Discussion Paper Series  For a complete list of Discussion Papers published by the SFB 649, please visit http://sfb649.wiwi.hu-berlin.de.  001 "Nonparametric Risk Management with Generalized Hyperbolic  Distributions" by Ying Chen, Wolfgang Härdle and Seok-Oh Jeong, January 2005. 002 "Selecting Comparables for the Valuation of the European Firms" by Ingolf Dittmann and Christian Weiner, February 2005. 003 "Competitive Risk Sharing Contracts with One-sided Commitment" by Dirk Krueger and Harald Uhlig, February 2005. 004 "Value-at-Risk Calculations with Time Varying Copulae" by Enzo Giacomini and Wolfgang Härdle, February 2005. 005 "An Optimal Stopping Problem in a Diffusion-type Model with Delay" by Pavel V. Gapeev and Markus Reiß, February 2005. 006 "Conditional and Dynamic Convex Risk Measures" by Kai Detlefsen and Giacomo Scandolo, February 2005. 007 "Implied Trinomial Trees" by Pavel Čížek and Karel Komorád, February 2005. 008 "Stable Distributions" by Szymon Borak, Wolfgang Härdle and Rafal Weron, February 2005. 009 "Predicting Bankruptcy with Support Vector Machines" by Wolfgang Härdle, Rouslan A. Moro and Dorothea Schäfer, February 2005. 010 "Working with the XQC" by Wolfgang Härdle and Heiko Lehmann, February 2005. 011 "FFT Based Option Pricing" by Szymon Borak, Kai Detlefsen and Wolfgang Härdle, February 2005. 012 "Common Functional Implied Volatility Analysis" by Michal Benko and Wolfgang Härdle, February 2005. 013 "Nonparametric Productivity Analysis" by Wolfgang Härdle and Seok-Oh Jeong, March 2005. 014 "Are Eastern European Countries Catching Up? Time Series Evidence for Czech Republic, Hungary, and Poland" by Ralf Brüggemann and Carsten Trenkler, March 2005. 015 "Robust Estimation of Dimension Reduction Space" by Pavel Čížek and Wolfgang Härdle, March 2005. 016 "Common Functional Component Modelling" by Alois Kneip and Michal Benko, March 2005.     SFB 649, Spandauer Straße 1, D-10178 Berlin http://sfb649.wiwi.hu-berlin.de  This research was supported by the Deutsche Forschungsgemeinschaft through the SFB 649 "Economic Risk". 

Common Functional Component Modelling

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