Abstract
We apply random matrix theory to derive the spectral density of large sample covariance matrices generated by multivariate VMA(q), VAR(q) and VARMA(q1, q2) processes. In particular, we consider a limit where the number of random variables N and the number of consecutive time measurements T are large but the ratio N/T is fixed. In this regime, the underlying random matrices are asymptotically equivalent to free random variables (FRV). We apply the FRV calculus to calculate the eigenvalue density of the sample covariance for several VARMA-type processes. We explicitly solve the VARMA(1, 1) case and demonstrate perfect agreement between the analytical result and the spectra obtained by Monte Carlo simulations. The proposed method is purely algebraic and can be easily generalized to q1>1 and q2>1.
Highlights
Vector auto–regressive (VAR) models play an important role in contemporary macro–economics, being an example of an approach called the “dynamic stochastic general equilibrium” (DSGE), which is superseding traditional large– scale macro–econometric forecasting methodologies [1]
Expression (38), along with the fundamental Free Random Variables (FRV) formula (14), allow us to write the equation satisfied by the M – transform M ≡ Mc(z) of the Pearson estimator c = (1/T )YYT = (1/T )YA(5)YT (4) of the cross–covariances in the vector auto–regressive moving average (VARMA)(1, 1) process; it happens to be polynomial of order six, and we print it (45) in appendix 5
The FRV calculus is ideally suited for multidimensional time series problems, provided the dimensions of the underlying matrices are large
Summary
Zdzislaw Burda,1, ∗ Andrzej Jarosz,2, † Maciej A. We apply random matrix theory to derive spectral density of large sample covariance matrices generated by multivariate VMA(q), VAR(q) and VARMA(q1, q2) processes. We consider a limit where the number of random variables N and the number of consecutive time measurements T are large but the ratio N/T is fixed. In this regime the underlying random matrices are asymptotically equivalent to Free Random Variables (FRV). We apply the FRV calculus to calculate the eigenvalue density of the sample covariance for several VARMA–type processes. PACS numbers: 89.65.Gh (Economics; econophysics, financial markets, business and management), 02.50.Sk (Multivariate analysis), 02.60.Cb (Numerical simulation; solution of equations), 02.70.Uu (Applications of Monte Carlo methods)
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