The Spectrum of Random Matrices

The Frobenius eigenvector of a positive square matrix is obtained by iterating the multiplication of an arbitrary positive vector by the matrix. Bródy (1997) noticed that, when the entries of the matrix are independently and identically distributed, the speed of convergence increases statistically with the dimension of the matrix. As the speed depends on the ratio between the subdominant and the dominant eigenvalues, Bródy's conjecture amounts to stating that this ratio tends to zero when the dimension tends to infinity. The paper provides a simple proof of the result. Some mathematical and economic aspects of the problem are discussed.     

The Second Eigenvalue of the Leontief Matrix

According to Frobenius, a positive matrix possesses a unique positive eigenvector which belongs to a positive eigenvalue. This eigenvalue is of the largest absolute magnitude and the matrix admits no other positive eigenvector. If an arbitrary positive vector is repeatedly premultiplied by such a matrix, then the result tends towards this positive eigenvector. It is the second largest eigenvalue that determines the speed of convergence. The estimate of the second eigenvalue of a purely random flow coefficient matrix shows that its expected absolute magnitude declines monotonically with the size of the matrix. Hence, the larger the system is the faster is the convergence. A prescribed exactness of the eigenvector (of equilibrium prices or quantities) will be reached after a few—perhaps just a couple of—iterations in a large system.     

The Subdominant Eigenvalue of a Large Stochastic Matrix

Using intuition and computer experimentation, Brady conjectured that the ratio of the subdominant eigenvalue to the dominant eigenvalue of a positive random matrix (with identically and independently distributed entries) converges to zero when the number of the sectors tends to infinity. In this paper, we discuss the deterministic case and, among other things, prove the following version of this conjecture: if each entry of the matrix deviates from 1/n by at most θ/n^1+е, then the modulus of the subdominant root is at most θ/n^е where θ and ε are arbitrary positive real parameters.     

Updating an Input–Output Matrix with Sign-preservation: Some Improved Objective Functions and their Solutions

This paper presents an easy-to-use method for updating input–output (IO) matrices with sign-preservation by combining Lagrangian multipliers and penalty functions. Biproportional methods such as the representative RAS are very simple and popular because a target matrix can be obtained simply by iterative computation. However, they cannot reasonably deal with matrices that include negative entries. Although a generalized version, GRAS, can do so, its objective function is questionable. In contrast, some non-biproportional methods such as those that take weighted or unweighted squared differences between the target and original matrix as objective functions can deal with negative entries, but it is difficult to guarantee the signs of entries. In this study, GRAS and some conventional objective functions were improved and their solutions for preserving the signs of entries are presented. Comparisons of applying these objective functions to a simple example show that both the Improved Normalized Squared Differences (INSD) function and the Improved GRAS (IGRAS) function yield a good target matrix and are close to each other; we suggest that INSD or IGRAS be used for updating IO transaction matrices in practice.     

Markovian lifts of positive semidefinite affine Volterra-type processes

We consider stochastic partial differential equations appearing as Markovian lifts of matrix-valued (affine) Volterra-type processes from the point of view of the generalized Feller property (see, e.g., Dörsek and Teichmann in A semigroup point of view on splitting schemes for stochastic (partial) differential equations, 2010. arXiv:1011.2651). We introduce in particular Volterra Wishart processes with fractional kernels and values in the cone of positive semidefinite matrices. They are constructed from matrix products of infinite dimensional Ornstein–Uhlenbeck processes whose state space is the set of matrix-valued measures. Parallel to that we also consider positive definite Volterra pure jump processes, giving rise to multivariate Hawkes-type processes. We apply these affine covariance processes for multivariate (rough) volatility modeling and introduce a (rough) multivariate Volterra Heston-type model.

     

Alternative Input-Output Matrix Updating Formulations

There has been a recent resurgence of interest in biproportional adjustment methods for updating and interpreting change in matrix representations of regional structures, most commonly input-output accounts. Although the biproportional method, commonly called the RAS technique in the input-output literature, has been shown to have a number of theoretically appealing properties, various alternatives do exist. In this paper, we develop and empirically assess a number of alternatives, comparing performance and examining the attributes of these adjustment methods. Two of these are sign-preserving updating methods for use when tables contain both positive and negative entries. One of these is shown to generate less information gain than does a generalized RAS method that Junius & Oosterhaven (2003) formulated to deal with matrices with both positive and negative values. Overall, while the RAS method continues to be commonly used and its choice is often rational, alternative methods can perform as well or better along certain dimensions and in certain contexts.     

News shock,firm dynamics and business cycles: Evidence and theory

The literature of expectation-driven business cycles has overlooked the role played by endogenous entry. This paper documents empirically news shock as a major source of fluctuations in firm dynamics and comovement between firm entry and GDP using structural vector auto-regressions. We then develop a tractable dynamic stochastic general equilibrium model to study the propagation mechanism assuming fixed operating costs for incumbents and decreasing survival rates for entrants. Our quantitative prediction closely matches the positive comovement between firm entries and core macroeconomic indicators upon news shock. These results remain robust at the sectoral level when the baseline model is extended to a two-sector setup.     

The First Two Eigenvalues of Large Random Matrices and Brody's Hypothesis on the Stability of Large Input–Output Systems

Bródy (1997) notices that for large random Leontief matrices, namely non-negative square matrices with all entries i.i.d., the ratio between the subdominant eigenvalue (in modulus) and the dominant eigenvalue declines generically to zero at a speed of the square root of the size of the matrix as the matrix size goes to infinity. Since then, several studies have been published in this journal in attempting to rigorously verify Bródy's conjecture. This short article, drawing upon some theorems obtained in recent years in the literature on empirical spectral distribution of random matrices, offers a short proof of Bródy's conjecture, and discusses briefly some related issues.     

Stochastic specification and the estimation of share equations

The standard stochastic specification for a system of share equations is obtained by assuming that the shares have a joint normal distribution with means depending upon exogenous variables and a constant covariance matrix. This specification ignores the requirement that shares be between zero and unity by giving positive probability to shares outside this range. An alternative stochastic specification involving the Dirichlet distribution, which automatically limits shares to the unit simplex, is suggested. A comparison of results obtained from the two specifications is made using sampling experiments and data from three different empirical studies. The sampling experiments and empirical applications show that the results are generally quite close, thus providing some justification for the continued use of the normal distribution specification in the estimation of share equations.     

RANDOM WALKS ON REGULAR POLYHEDRA AND OTHER DISTANCE–REGULAR GRAPHS

In this paper we consider Markov chains of the following type: the state space is the set of vertices of a connected, regular graph, and for each vertex transitions are to the adjacent vertices, with equal probabilities. When the mean first–passage matrix F of such a Markov chain is symmetric, the expectation and variance of first–entrance times, recurrence times, number of visits to a vertex and the expectation of the number of different vertices visited, can easily be computed from the entries of F. The method is most effective, when the underlying graph is distance–regular; then F is symmetric and the entries of F can easily be obtained from the graph.     

Central limit theorem for linear eigenvalue statistics of the Wigner and the sample covariance random matrices

We consider n × n real symmetric random matrices n ^?1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n ^?1 A ^T A with independent entries of m × n matrix A. Assuming first that the 4th cumulant (excess) κ ₄ of entries of W and A is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as n → ∞, m → ∞, ${m/n\rightarrow c\in[0,\infty)}$ with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class ${\mathbb{C}^5}$ ). This is done by using a simple “interpolation trick”. Then, by using a more elaborated techniques, we prove the CLT in the case of non-zero excess of entries for essentially ${\mathbb{C}^4}$ test function. Here the variance contains additional term proportional to κ ₄. The proofs of all limit theorems follow essentially the same scheme.     

Asymptotic properties of estimators for the volume fractions of jointly stationary random sets

In the present paper, we show how a consistent estimator can be derived for the asymptotic covariance matrix of stationary 0–1-valued vector fields in R ^d , whose supports are jointly stationary random closed sets. As an example, which is of particular interest for statistical applications, we consider jointly stationary random closed sets associated with the Boolean model in R ^d such that the components indicate the frequency of coverage by the single grains of the Boolean model. For this model, a representation formula for the entries of the covariance matrix is obtained.     

Pure and randomized equilibria in the stochastic von Neumann–Gale model

The paper examines the problem of the existence of equilibrium for the stochastic analogue of the von Neumann–Gale model of economic growth. The mathematical framework of the model is a theory of set-valued random dynamical systems defined by positive stochastic operators with certain properties of convexity and homogeneity. Existence theorems for equilibria in such systems may be regarded as generalizations of the Perron–Frobenius theorem on eigenvalues and eigenvectors of positive matrices. The known results of this kind are obtained under rather restrictive assumptions. We show that these assumptions can be substantially relaxed if one allows for randomization. The main result of the paper is an existence theorem for randomized equilibria. Some special cases (models defined by positive matrices) are considered in which the existence of pure equilibria can be established.     

Estimation of Non-Stationary Social Accounting Matrix Coefficients with Supply-Side Information

Given aggregated data, a framework for estimating the entries of a social accounting matrix (SAM), or any large matrix of expenditures, trade or income flows, is developed. Under this framework it is possible to evaluate the contribution of structural and supply-side information, as well as policy variables, within the generalized context of a non-stationary SAM. Inference and diagnostic properties are developed as well. This new estimator can be viewed as a generalized maximum likelihood estimator. Stationary and non-stationary estimates of the US SAM for the years 1987-1994 together with the effects of supply-side variables are analyzed.     

A characterization of the extended serial correspondence

We study the problem of assigning objects to a group of agents. We focus on probabilistic methods that take agents’ ordinal preferences over the objects. Importantly, we allow for indifferences among objects. Katta and Sethuraman (2006) propose the extended serial correspondence to solve this problem. Our main result is a characterization of the extended serial correspondence in welfare terms by means of stochastic dominance efficiency, stochastic dominance no-envy and “limited invariance,” a requirement we adapt from Heo (2014a). We also prove that an assignment matrix is selected by the extended serial correspondence if and only if it satisfies “non-wastefulness” and “ordinal fairness,” which we adapt from Kesten et al. (2011).     

The error components model: Conditions for the existence of the maximum likelihood estimates

The paper provides a stochastic specification of the error components model that ensures a positive maximum likelihood estimate of the error component variance. Also, it is shown that all of the stochastic parameters of the error components are identifiable albeit with certain qualifications. The model analyzed is the simple, two-component model, in which the stochastic variable is decomposed into a random individual effect and an overall error term. However, the results can be easily generalized to include an additional random time-effect variable.     

Usual stochastic ordering of the sample maxima from dependent distribution-free random variables

In this paper, we discuss stochastic comparison of the largest order statistics arising from two sets of dependent distribution-free random variables with respect to multivariate chain majorization, where the dependency structure can be defined by Archimedean copulas. When a distribution-free model with possibly two parameter vectors has its matrix of parameters changing to another matrix of parameters in a certain mathematical sense, we obtain the first sample maxima is larger than the second sample maxima with respect to the usual stochastic order, based on certain conditions. Applications of our results for scale proportional reverse hazards model, exponentiated gamma distribution, Gompertz–Makeham distribution, and location-scale model, are also given. Meanwhile, we provide two numerical examples to illustrate the results established here.     

Bayesian stochastic search for VAR model restrictions

We propose a Bayesian stochastic search approach to selecting restrictions for vector autoregressive (VAR) models. For this purpose, we develop a Markov chain Monte Carlo (MCMC) algorithm that visits high posterior probability restrictions on the elements of both the VAR regression coefficients and the error variance matrix. Numerical simulations show that stochastic search based on this algorithm can be effective at both selecting a satisfactory model and improving forecasting performance. To illustrate the potential of our approach, we apply our stochastic search to VAR modeling of inflation transmission from producer price index (PPI) components to the consumer price index (CPI).     

The Solution of Updating or Regionalizing a Matrix with both Positive and Negative Entries

Normally, when updating or regionalizing input-output matrices with negative entries, the negative numbers are first brought outside the matrix, then the matrix is updated or regionalized, then the negative numbers are added back to the result. This is theoretically, and sometimes also empirically, a rather unsatisfactory procedure. This paper proposes a theoretically sound alternative for the presently used ad hoc procedure. Based on the first-order conditions of a restated information loss problem, we generalize the RAS-procedure using reciprocals of the exponential transformations of the related Lagrange multipliers. The diagonal matrices that update or regionalize a given matrix optimally are the solutions of a fixed-point problem. To derive a numerical solution, the paper presents the GRAS-algorithm, which is illustrated in terms of a simple updating example.     

A stochastic differential game of capitalism

We develop a stochastic differential game of capitalism to analyze the role of uncertainty. In the deterministic game, the firm’s rent is completely taxed away and the firm stops investing completely. In the stochastic game, the government does not tax the firm’s rent completely. The firm posts a positive rate of investment if the firm’s rent exceeds the labor’s income. Although the cooperative solution is indeterminate, cooperation is always Pareto optimal compared to the non-cooperative Markovian Nash equilibrium. For individual rationality, we apply a payoff distribution procedure based on Yeung and Petrosyan (2006) to derive a subgame-consistent solution.