Estimation of matrix rank: historical overview and more recent developments

Transcription

1 Estimation of matrix rank: historical overview and more recent developments Vladas Pipiras CEMAT, IST and University of North Carolina, Chapel Hill UTL Probability and Statistics Seminar in Lisbon November 7, 2008

2 Outline: 1. General problem. 2. Several examples. 3. Available rank tests. 4. The case of symmetric matrices (joint work with S. Donald and N. Fortuna). 1 of 16

3 1. General problem 2 of 16

4 General framework: M : p q unknown matrix with unknown rank rk{m}. M = M(N) : matrix estimator such that Nvec( M M) d N (0, W ). There is Ŵ such that Ŵ p W. Interested in testing for matrix rank: H 0 : rk{m} = r, H 1 : rk{m} > r, where r = 0,..., min{p, q} 1. The resulting tests are referred to as (matrix) rank tests. 3 of 16

5 Remarks: Matrix rank is estimated, for example, through sequential testing. This will NOT be discussed further. But here is a typical output for a 4 5 matrix: rank r test statistic critical value P -value Focus here is NOT on estimation of a reduced rank matrix, which can be done in a number of ways. 4 of 16

6 2. Several examples 5 of 16

7 Example 1: (Reduced rank regression) Consider a multivariate linear regression model y i }{{} p 1 = }{{} A }{{} x i p q q 1 }{{}, i = 1,..., n, p 1 + ɛ i with response variables y i R p, regressors (predictors) x i R q, and error terms ɛ i R p. Interested in testing for rk{a}. For a standard, least squares estimator  = n i=1 y ix i ( n i=1 x ix i ) 1, under mild assumptions, n vec(  A) d N (0, W ), where W = E(x i x i ) E(ɛ iɛ i ) has the so-called Kronecker product form, and can be estimated consistently through Ŵ = 1 n n i=1 x ix i 1 n n i=1 (y i Âx i)(y i Âx i). Note: non-kronecker product would arise with heteroscedastic error terms. 6 of 16

8 Example 2: (Number of factors in nonparametric relationship) Consider a multivariate nonparametric model y i }{{} p 1 = F ( }{{} x i ) + ɛ i, i = 1,..., n, d 1 where F is an unknown but otherwise smooth function. Interested in estimating the maximum number of linearly independent component functions of F, denoted as One can show that rk{f }. rk{f } = rk{m}, M = Ef(x i )F (x i )F (x i ). Note that M is symmetric (and positive semi-definite). This will be discussed further and asymptotically normal estimators will be given below. 7 of 16

9 Some other examples: Estimation of p, q in ARMA(p, q) models. Rank of a spectral density matrix at a given frequency; related to cointegration. Cointegrating rank in Vector Error-Correction Models. Identification questions in Simultaneous Equations Models. Number of factors in factor analysis. One general view: Model dimension Matrix rank 8 of 16

10 3. Available rank tests and other work 9 of 16

11 Econometrics literature: LDU rank test: Gill and Lewbel (1992), Cragg and Donald (1996). MINCHI2 rank test: Cragg and Donald (1997). CHART rank test: Robin and Smith (2000). SVD rank tests: Kleibergen and Paap (2006), Ratsimalahelo (2002, 2003). Camba-Méndez, Kapetanios and others. Statistics literature: Reduced rank regression: goes back to the Anderson (1951) and has been studied extensively since by many others. Natural connections to canonical correlation analysis and extensions to VAR series (cointegration). Asymptotics of eigenvalues under quite general assumptions: series of papers by Tyler and coauthors in the 1980 s. 10 of 16

12 Example: MINCHI2 rank test: based on the test statistic ξ minchi2 (r) = N min vec( M M) Ŵ 1 vec( M M). rk{m} r One can show that under H 0 : rk{m} = r : ξ minchi2 (r) under H 1 : rk{m} > r : ξ minchi2 (r) p +. d χ 2 ((p r)(q r)), Remark: Here and in most available rank tests, one assumes that the covariance matrix W is nonsingular. Remark: If W = Ψ 1 Σ 1 has a Kronecker product structure, then ξ minchi2 (r) = N is a classical test statistic, where 0 λ λ 2 p are the ordered eigenvalues of Σ M Ψ M. The same happens for the test statistics corresponding to CHART and SVD tests. p r i=1 λ 2 i, 11 of 16

13 4. The case of symmetric matrices 12 of 16

14 General framework: M, M = M(N) : symmetric, p p matrices such that Nvech( M M) d N (0, V ) or, in a weaker form, M = M1 + M 2 where Mu = 0 if and only if M1 u = 0, M M 1 = O p (1/ N), and Nvech( M2 ) d N (0, V ). There is V such that V p V. 2 different cases: ID case: Indefinite M: indefinite or semidefinite M: V is nonsingular, SD case: Semidefinite M: semidefinite M: V is singular under reduced rank of M. 13 of 16

15 1-dimensional case (1 1 matrices): Rank deficiency rk{m} = 0 M = 0. Under rank deficiency (or H 0 ): Nvech( M M) = N M d N (0, V ). SD case: M positive semidefinite M 0 : V = 0. ID case: M indefinite M R : V > 0. Related work: ID case (DFP (2007)): On rank estimation in symmetric matrices: the case of indefinite matrix estimators (S. Donald, N. Fortuna and V. Pipiras), Econometric Theory 23(6), 2007, SD, ID cases (DFP (2008)): On rank estimation in semidefinite matrices (S. Donald, N. Fortuna and V. Pipiras), Working Paper, of 16

16 Earlier example 2: (Number of factors in nonparametric relationship) In the model y i = F (x i ) + ɛ i, i = 1,..., n, interested in estimating rk{f }, the number of linearly independent component functions of F. One can show that rk{f } = rk{m}, where M = Ef(x i )F (x i )F (x i ). Semidefinite estimator (DFP (2008)): M S = 1 n n i=1 f(x i ) F (x i ) F (x i ). Then, nvech( MS M) d N (0, V S ), V S singular. Indefinite estimator (Donald (1997)): MI = 1 n(n 1) i j y iy j K h(x i x j ). Then, MI = M I,1 + M I,2 with M I u = 0 M I,1 u = 0, MI,1 M = O p ((nh q/2 ) 1 ), and nh q/2 vech( M I,2 ) d N (0, V I ), V I nonsingular. Some other examples: ID case: reduced rank regression with unknown symmetric matrix (Robin and Smith (2000)). SD, ID cases: multiple index mean regression model (Donkers and Schafgans (2005)). ID estimator due to DFP (2008). SD case: spectral density matrices in multivariate time series, cointegration (Camba-Méndez and Kapetanios (2005)). 15 of 16

17 ID case (DFP(2007)): adaptations of available rank tests (LDU, MINCHI2, SVD) to symmetric matrices, and a new rank test (EIG) based on eigenvalues. Notes on results in ID case: EIG test: for symmetric (square) matrices can talk about their eigenvalues directly. LDU test: using symmetric pivoting. MINCHI2 and other tests: adjustment for degrees of freedom in limiting χ 2 distribution: (p r)(p r + 1)/2 instead of (p r) 2. Simulation study: SVD appears to be superior over other rank tests. Main point for ID case: many rank tests are available with well developed theory. SD case (DFP(2008)): Available rank tests for singular W : for example, conditions on W in Robin and Smith (2000) are not satisfied; conditions for using generalized inverses of W as in Camba-Méndez and Kapetanios (2008) are also not satisfied. Main point for SD case: no satisfactory rank tests are available. But: understand where difficulties lie and what possible alternatives are. 16 of 16