A Random Matrix Approach to Dynamic Factors in Macroeconomic Data

Snarska, Małgorzata

doi:10.12693/aphyspola.121.b-110

Cited by 5 publications

(10 citation statements)

References 33 publications

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“…(3.24). This formula is quite convenient for those situations when introducing empirical correlation matrices on both sides of the rectangular random matrix as it is the case in spacial-temporal correlations [56,57,58].…”

Section: Discussionmentioning

confidence: 99%

The supersymmetry method for chiral random matrix theory with arbitrary rotation-invariant weights

Kaymak¹,

Kieburg²,

Guhr³

2014

J. Phys. A: Math. Theor.

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Abstract. In the past few years, the supersymmetry method was generalized to real-symmetric, Hermitean, and Hermitean self-dual random matrices drawn from ensembles invariant under the orthogonal, unitary, and unitary symplectic group, respectively. We extend this supersymmetry approach to chiral random matrix theory invariant under the three chiral unitary groups in a unifying way. Thereby we generalize a projection formula providing a direct link and, hence, a 'short cut' between the probability density in ordinary space and the one in superspace. We emphasize that this point was one of the main problems and critiques of the supersymmetry method since only implicit dualities between ordinary and superspace were known before. As examples we apply this approach to the calculation of the supersymmetric analogue of a Lorentzian (Cauchy) ensemble and an ensemble with a quartic potential. Moreover we consider the partially quenched partition function of the three chiral Gaussian ensembles corresponding to four-dimensional continuum QCD. We identify a natural splitting of the chiral Lagrangian in its lowest order into a part of the physical mesons and a part associated to source terms generating the observables, e.g. the level density of the Dirac operator.

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Section: Discussionmentioning

confidence: 99%

The supersymmetry method for chiral random matrix theory with arbitrary rotation-invariant weights

Kaymak¹,

Kieburg²,

Guhr³

2014

J. Phys. A: Math. Theor.

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“…The Haar measure dµ of the coset Herm + (q|p) is explicitly given by [39,42,57] dµ(U ) = (2πi) −p Sdet p−q (U )[dU ] , (C. 15) where [dU ] is again the flat measure, in particular the product of differentials of all independent matrix entries. We emphasise that there is no natural normalisation of the Haar measure on the coset Herm + (q|p) when pq > 0; namely the volume of the supergroup U(1|1) vanishes due to Cauchylike integration theorems [58][59][60].…”

Section: (C6)mentioning

confidence: 99%

“…Several approximations to this situation have been proposed. First, it was assumed that correlations in time steps and among time series factorise, as was considered in economics [15], climate research [8], sociology [16] and telecommunication [9]. In random matrix theory this problem was analytically discussed recently in [17,18] also for the real case and is called doubly-correlated Wishart ensemble.…”

Section: Introductionmentioning

confidence: 99%

Spectral correlation functions of the sum of two independent complex Wishart matrices with unequal covariances

Akemann¹,

Checinski²,

Kieburg³

2016

J. Phys. A: Math. Theor.

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We compute the spectral statistics of the sum H of two independent complex Wishart matrices, each of which is correlated with a different covariance matrix. Random matrix theory enjoys many applications including sums and products of random matrices. Typically ensembles with correlations among the matrix elements are much more difficult to solve. Using a combination of supersymmetry, superbosonisation and bi-orthogonal functions we are able to determine all spectral k-point density correlation functions of H for arbitrary matrix size N . In the half-degenerate case, when one of the covariance matrices is proportional to the identity, the recent results by Kumar for the joint eigenvalue distribution of H serve as our starting point. In this case the ensemble has a bi-orthogonal structure and we explicitly determine its kernel, providing its exact solution for finite N . The kernel follows from computing the expectation value of a single characteristic polynomial. In the general non-degenerate case the generating function for the k-point resolvent is determined from a supersymmetric evaluation of the expectation value of k ratios of characteristic polynomials. Numerical simulations illustrate our findings for the spectral density at finite N and we also give indications how to do the asymptotic large-N analysis.

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“…Here we present brief recipes for the addition and multiplication of large random matrices and introduce some additional FRV transforms related to moments and cumulants. They found an application in description of spectral properties of large covariance matrices in particular in the financial context [52,53]. Recently, they also turned out to be crucial in the cleaning of noisy covariance matrices [16,54] and studying the properties of the eigenvectors of such matrices [55].…”

Section: Free Random Variable Cookbookmentioning

confidence: 99%

Spectra of large time-lagged correlation matrices from random matrix theory

Nowak¹,

Tarnowski²

2017

J. Stat. Mech.

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We analyze the spectral properties of large, time-lagged correlation matrices using the tools of random matrix theory. We compare predictions of the one-dimensional spectra, based on approaches already proposed in the literature. Employing the methods of free random variables and diagrammatic techniques, we solve a general random matrix problem, namely the spectrum of a matrix 1 T XAX † , where X is an N × T Gaussian random matrix and A is any T × T , not necessarily symmetric (Hermitian) matrix. Using this result, we study the spectral features of the large lagged correlation matrices as a function of the depth of the time-lag. We also analyze the properties of left and right eigenvector correlations for the time-lagged matrices. We positively verify our results by the numerical simulations.

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A Random Matrix Approach to Dynamic Factors in Macroeconomic Data

Cited by 5 publications

References 33 publications

The supersymmetry method for chiral random matrix theory with arbitrary rotation-invariant weights

The supersymmetry method for chiral random matrix theory with arbitrary rotation-invariant weights

Spectral correlation functions of the sum of two independent complex Wishart matrices with unequal covariances

Spectra of large time-lagged correlation matrices from random matrix theory

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