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. 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 a 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.
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 largescale macro-econometric forecasting methodologies [1]. The motivation behind them is based on the assertion that more recent values of a variable are more likely to contain useful information about its future movements than the older ones. On the other hand, a standard tool in multivariate time series analysis is vector moving average (VMA) models, which is really a linear regression of the present value of the time series w.r.t. the past values of a white noise. A broader class of stochastic processes used in macro-economy comprises both these kinds together in the form of vector auto-regressive moving average (VARMA) models. These methodologies can capture certain spatial and temporal structures of multidimensional variables which are often neglected in practice; including them not only results in more accurate estimation, but also leads to models which are more interpretable. They are widely used by academia and central banks (cf. the European Central Bank's Smets-Wouters model for the euro zone [2]), as they constitute quite a simple version of the DSGE equations.
VARMA models are constructed from a number of univariate ARMA (Box-Jenkins; see for example [3]) processes, typically coupled with each other. In this paper, we investigate only a significantly simplified circumstance when there is no coupling between the many ARMA components. One may argue that this is too far fetched and will be of no use in describing an economic reality. However, one may also treat it as a “zeroth-order hypothesis,” analogously to the idea of [4,5] in finance, namely that the case with no cross-covariances is considered theoretically, and subsequently compared to some real-world data modeled by a VARMA process; any discrepancy between the two will reflect VARMA(q 1 , q 2 ) stochastic processes, as we will see below, fall within quite a general set-up encountered in many areas of science where a probabilistic nature of multiple degrees of freedom evolving in time is relevant, for example, multivariate time series analysis in finance, applied macro-econometrics and engineering. To describe this framework, consider a situation of N time-dependent random variables which are measured at T consecutive time moments (separated by some time interval δt); let Y ia be the value od the i-th (i = 1, . . . , N ) random number at the a-th time moment (a = 1, . . . , T ); together, they make up a rectangular N × T matrix Y. In what usually would be the first approximation, each Y ia is supposed to be drawn from a Gaussian probability distribution. We will also assume that they have mean values zero, Y ia = 0. These degrees of freedom may in principle display mutual correlations. A set of correlated zero-mean Gaussian numbers is fully characterized by the two-point covariance function, C ia,jb ≡ Y ia Y jb if the underlying stochastic process generating these numbers is stationary. Linear stochastic processes, including VARMA(q 1 , q 2 ), belong to this category. We will restrict our attention to an even narrower class where the crosscorrelations between different variables and the auto-correlations between different time moments are factorized, i.e.,
In this setting, the inter-variable covariances do not change in time (and are described by an N × N cross-covariance matrix C), and also the temporal covariances are identical for all the numbers (and are included in a T × T autocovariance matrix A; both these matrices are symmetric and positive-definite). The Gaussian probability measure with this structure of covariances is known from textbooks,
where the normalization constant N c.G. = (2π) N T /2 (DetC) T /2 (DetA) N/2 , and the integration measure DY ≡ N i=1 T a=1 dY ia , while the letters “c.G.” stand for “correlated Gaussian.”
Now, a standard way to approach correlated Gaussian random numbers is to recall that they can always be decomposed as linear combinations of uncorrelated Gaussian degrees of freedom; indeed, this is achieved through the transformation
where the square roots of the covariance matrices, necessary to facilitate the transition, exist due to the positivedefiniteness of C and A; the new normalization reads N G. = (2π) N T /2 .
An essential problem in multivariate analysis is to determine (estimate) the covariance matrices C and A from given N time series of length T of the realizations of our random variables Y ia . For simplicity, we do not distinguish in notation between random numbers, i.e., the population, and their realizations in actual experiments, i.e., the sample. Since the realized cross-covariance between degrees i and j at the same time a is Y ia Y ja , the simplest method to estimate the today’s cross-covariance c ij is to compute the time average,
This is usually named the “Pearso
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