Stochastic Recurrence Equations Research Articles

Branching processes with immigration in a random environment—The Grincevičius–Grey setup

We determine the tail asymptotics of the stationary distribution of a branching process with immigration in a random environment, when the immigration distribution dominates the offspring distribution. The assumptions are the same as in the Grincevičius–Grey theorem for the stochastic recurrence equation.

Dynamic partial correlation models

We introduce a new scalable model for dynamic conditional correlation matrices based on a recursion of dynamic bivariate partial correlation models. By exploiting the model’s recursive structure and the theory of perturbed stochastic recurrence equations, we establish stationarity, ergodicity, and filter invertibility in the multivariate setting using conditions for bivariate slices of the data only. From this, we establish consistency and asymptotic normality of the maximum likelihood estimator for the model’s static parameters. The new model outperforms benchmarks like the t-cDCC and the multivariate t-GAS, both in simulations and in an in-sample and out-of-sample asset pricing application to US stock returns.

Autoregressive conditional proportion: A multiplicative‐error model for (0,1)‐valued time series

We propose a multiplicative autoregressive conditional proportion (ARCP) model for (0,1)‐valued time series, in the spirit of GARCH (generalized autoregressive conditional heteroscedastic) and ACD (autoregressive conditional duration) models. In particular, our underlying process is defined as the product of a (0,1)‐valued independent and identically distributed (i.i.d.) sequence and the inverted conditional mean, which, in turn, depends on past reciprocal observations in such a way that is larger than unity. The probability structure of the model is studied in the context of the stochastic recurrence equation theory, while estimation of the model parameters is performed with the exponential quasi‐maximum likelihood estimator (EQMLE). The consistency and asymptotic normality of the EQMLE are both established under general regularity assumptions. Finally, the usefulness of our proposed model is illustrated with two real datasets.

Tails of bivariate stochastic recurrence equation with triangular matrices

We study bivariate stochastic recurrence equations with triangular matrix coefficients and we characterize the tail behavior of their stationary solutions W=(W1,W2). Recently it has been observed that W1,W2 may exhibit regularly varying tails with different indices, which is in contrast to well-known Kesten-type results. However, only partial results have been derived. Under typical “Kesten–Goldie” and “Grey” conditions, we completely characterize tail behavior of W1,W2. The tail asymptotics we obtain has not been observed in previous settings of stochastic recurrence equations.

Asymptotic independence ex machina: Extreme value theory for the diagonal SRE model

We consider multivariate stationary processes satisfying a stochastic recurrence equation of the form , where are i.i.d. random vectors and are i.i.d. diagonal matrices and (Mt) are i.i.d. random variables. We obtain a full characterization of the vector scaling regular variation properties of , proving that some coordinates Xt, i and Xt, j are asymptotically independent even though all coordinates rely on the same random input (Mt). We prove the asynchrony of extreme clusters among marginals with different tail indices. Our results are applied to some multivariate autoregressive conditional heteroskedastic (BEKK‐ARCH and CCC‐GARCH) processes and to log‐returns. Angular measure inference shows evidences of asymptotic independence among marginals of diagonal SRE with different tail indices.

The dynamics of stochastic mono-molecular reaction systems in stochastic environments

We study the stochastic dynamics of a system of interacting species in a stochastic environment by means of a continuous-time Markov chain with transition rates depending on the state of the environment. Models of gene regulation in systems biology take this form. We characterise the finite-time distribution of the Markov chain, provide conditions for ergodicity, and characterise the stationary distribution (when it exists) as a mixture of Poisson distributions. The mixture measure is uniquely identified as the law of a fixed point of a stochastic recurrence equation. This recursion is crucial for statistical computation of moments and other distributional features.

CHARACTERIZATION OF THE TAIL BEHAVIOR OF A CLASS OF BEKK PROCESSES: A STOCHASTIC RECURRENCE EQUATION APPROACH

We consider conditions for strict stationarity and ergodicity of a class of multivariate BEKK processes $(X_t : t=1,2,\ldots )$ and study the tail behavior of the associated stationary distributions. Specifically, we consider a class of BEKK-ARCH processes where the innovations are assumed to be Gaussian and a finite number of lagged $X_t$ ’s may load into the conditional covariance matrix of $X_t$ . By exploiting that the processes have multivariate stochastic recurrence equation representations, we show the existence of strictly stationary solutions under mild conditions, where only a fractional moment of $X_t$ may be finite. Moreover, we show that each component of the BEKK processes is regularly varying with some tail index. In general, the tail index differs along the components, which contrasts with most of the existing literature on the tail behavior of multivariate GARCH processes. Lastly, in an empirical illustration of our theoretical results, we quantify the model-implied tail index of the daily returns on two cryptocurrencies.

Homogeneous mappings of regularly varying vectors

It is well known that the product of two independent regularly varying random variables with the same tail index is again regularly varying with this index. In this paper, we provide sharp sufficient conditions for the regular variation property of product-type functions of regularly varying random vectors, generalizing and extending the univariate theory in various directions. The main result is then applied to characterize the regular variation property of products of i.i.d. regularly varying quadratic random matrices and of solutions to affine stochastic recurrence equations under nonstandard conditions.

Two classes of dynamic binomial integer-valued ARCH models

This paper introduces two classes of binomial integer-valued ARCH models with dynamic survival probabilities, each of which is controlled by a stochastic recurrence equation. Stationarity and ergodicity of the process are established, and stochastic properties are given. Conditional least squares and conditional maximum likelihood estimators for the parameters of interest are considered, and their large-sample properties are established. The performances of these estimators are compared via simulation studies. Finally, we demonstrate the usefulness of the proposed models by analyzing real datasets.

Truncated moments of perpetuities and a new central limit theorem for GARCH processes without Kesten’s regularity

We consider a class of perpetuities which admit direct characterization of asymptotics of the key truncated moment. The class contains perpetuities without polynomial decay of tail probabilities thus not satisfying Kesten’s theorem. We show how to apply this result in deriving a new weak law of large numbers for solutions to stochastic recurrence equations and a new central limit theorem for GARCH(1,1) processes in the critical case.

Tail indices for mathbf{A}mathbf{X}+mathbf{B} Recursion with Triangular Matrices

We study multivariate stochastic recurrence equations (SREs) with triangular matrices. If coefficient matrices of SREs have strictly positive entries, the classical Kesten result says that the stationary solution is regularly varying and the tail indices are the same in all directions. This framework, however, is too restrictive for applications. In order to widen applicability of SREs, we consider SREs with triangular matrices and we prove that their stationary solutions are regularly varying with component-wise different tail exponents. Several applications to GARCH models are suggested.

A new time-varying model for forecasting long-memory series

In this work we propose a new class of long-memory models with time-varying fractional parameter. In particular, the dynamics of the long-memory coefficient, d, is specified through a stochastic recurrence equation driven by the score of the predictive likelihood, as suggested by Creal et al. (J Appl Econom 28:777–795, 2013) and Harvey (Dynamic models for volatility and heavy tails: with applications to financial and economic time series, Cambridge University Press, Cambridge, 2013). We demonstrate the validity of the proposed model by a Monte Carlo experiment and an application to two real time series.

Exponential-Type GARCH Models With Linear-in-Variance Risk Premium

One of the implications of the intertemporal capital asset pricing model is that the risk premium of the market portfolio is a linear function of its variance. Yet, estimation theory of classical GARCH-in-mean models with linear-in-variance risk premium requires strong assumptions and is incomplete. We show that exponential-type GARCH models such as EGARCH or Log-GARCH are more natural in dealing with linear-in-variance risk premia. For the popular and more difficult case of EGARCH-in-mean, we derive conditions for the existence of a unique stationary and ergodic solution and invertibility following a stochastic recurrence equation approach. We then show consistency and asymptotic normality of the quasi-maximum likelihood estimator under weak moment assumptions. An empirical application estimates the dynamic risk premia of a variety of stock indices using both EGARCH-M and Log-GARCH-M models. Supplementary materials for this article are available online.

Componentwise different tail solutions for bivariate stochastic recurrence equations with application to ${\rm GARCH}(1,1)$ processes

Componentwise different tail solutions for bivariate stochastic recurrence equations with application to ${\rm GARCH}(1,1)$ processes

Specification Tests for Time Series Models with GARCH-Type Conditional Variance

Specification tests are developed for the conditional distribution of a dependent process {Xi} in a family of nonlinear time-series models. The family includes several Generalized AutoRegressive Conditional Heteroscedastic [GARCH] models that are widely used in finance and economics. Such tests are essential for ensuring the validity of density forecasts that are based on the assumed model. The tests are implemented using a bootstrap procedure, because the test statistics are not asymptotically pivotal. A novel method is developed for verifying the validity of the bootstrap tests for a given model; this involves simultaneously embedding the process under the null hypothesis and the bootstrap process in a system of stochastic recurrence equations. The method is illustrated for a well-known GARCH model. Within the specification testing literature, a feature that distinguishes this paper is that the conditional distribution of Xi under the null model, depends on the unobservable past values {. . . , X−2, X−1} extending back to the infinite past. The theory for testing the specification of the conditional distribution of Xi in such parametric models has not been developed in the general setting considered in this paper. The tests performed well in a simulation study, and a data example illustrates the tests.

Integer‐Valued Autoregressive Models With Survival Probability Driven By A Stochastic Recurrence Equation

This paper proposes a new class of integer‐valued autoregressive models with a dynamic survival probability. The peculiarity of this class of models lies in the specification of the survival probability through a stochastic recurrence equation. The proposed models can effectively capture changing dependence over time and enhance both the in‐sample and out‐of‐sample performance of integer‐valued autoregressive models. This point is illustrated through an empirical application to a real‐time series of crime reports. Additionally, this paper discusses the reliability of likelihood‐based inference for the class of models. In particular, this study proves the consistency of the maximum likelihood estimator and a plug‐in estimator for the conditional probability mass function in a misspecified model setting.

Existence and uniqueness of a stationary and ergodic solution to stochastic recurrence equations via Matkowski’s FPT

AbstractWe establish the existence of a unique stationary and ergodic solution for systems of stochastic recurrence equations defined by stochastic self-maps on Polish metric spaces based on the fixed point theorem of Matkowski. The results can be useful in cases where the stochastic Lipschitz coefficients implied by the currently used method either do not exist, or lead to the imposition of unnecessarily strong conditions for the derivation of the solution.

A Stochastic Recurrence Equation Approach to Stationarity and Phi-Mixing of a Class of Nonlinear ARCH Models

This article generalises the results of Saidi and Zakoian (2006) to a considerably larger class of nonlinear ARCH models with discontinuities, leverage effects and robust news impact curves. We propose a new method of proof for the existence of a strictly stationary and phi-mixing solution. Moreover, we show that any path converges to this solution. The proof relies on stochastic recurrence equation theory and builds on the work of Bougerol (1993) and Straumann (2005). The assumptions that we need for this approach are less restrictive than those imposed in Saidi and Zakoian (2006) and typically found in Markov chain theory, as they require very little from the distribution of the underlying process. Furthermore, they can be stated in a general setting for random functions on a separable Banach space as is done in Straumann and Mikosch (2006). Finally, we state sufficient conditions for the existence of moments.

Asymptotics for parametric GARCH-in-Mean models

In this paper we develop an asymptotic theory for the Quasi-Maximum Likelihood Estimator (QMLE) of the parametric GARCH-in-Mean model. The asymptotics is based on a study of the volatility as a process of the model parameters. The proof makes use of stochastic recurrence equations for this random function and uses exponential inequalities to localize the problem. Our results show why the asymptotics for this specification is quite complex although it is a rather standard parametric model. Nevertheless, our theory does not yet treat all standard specifications of the mean function.

A stochastic recurrence equations approach for score driven correlation models

ABSTRACTWe describe stationarity and ergodicity (SE) regions for a recently proposed class of score driven dynamic correlation models. These models have important applications in empirical work. The regions are derived from sufficiency conditions in Bougerol (1993) and take a nonstandard form. We show that the nonstandard shape of the sufficiency regions cannot be avoided by reparameterizing the model or by rescaling the score steps in the transition equation for the correlation parameter. This makes the result markedly different from the volatility case. Observationally equivalent decompositions of the stochastic recurrence equation yield regions with different shapes and sizes. We use these results to establish the consistency and asymptotic normality of the maximum likelihood estimator. We illustrate our results with an analysis of time-varying correlations between U.K. and Greek equity indices. We find that also in empirical applications different decompositions can give rise to different conclusions regarding the stability of the estimated model.