Autocovariance Estimation in Regression with a Discontinuous Signal and m‐Dependent Errors: A Difference‐Based Approach

We discuss a class of difference‐based estimators for the autocovariance in nonparametric regression when the signal is discontinuous and the errors form a stationary m‐dependent process. These estimators circumvent the particularly challenging task of pre‐estimating such an unknown regression function. We provide finite‐sample expressions of their mean squared errors for piecewise constant signals and Gaussian errors. Based on this, we derive biased‐optimized estimates that do not depend on the unknown autocovariance structure. Notably, for positively correlated errors, that part of the variance of our estimators that depend on the signal is minimal as well. Further, we provide sufficient conditions for ‐consistency; this result is extended to piecewise Hölder regression with non‐Gaussian errors. We combine our biased‐optimized autocovariance estimates with a projection‐based approach and derive covariance matrix estimates, a method that is of independent interest. An R package, several simulations and an application to biophysical measurements complement this paper.     

Empirically feasible solutions and explicit dynamics for rational expectation models

Fairly general rational expectation (RE) models are proved to have linear vector autoregressive moving average (VARMA) models as their reduced forms, using Muth's method of undetermined coefficients (MUC). An advantage of using VARMAs is that RE estimation can benefit from well-developed theory of solutions of stochastic difference equations and computer packages for VARMA and state space (SS) models. For example, we report theoretical derivations of explicit dynamics associated with the RE structure for Muth's and Lucas-Sargent-Wallace's models, suggesting generalizations and new solutions. We illustrate an RE structure estimation by using energy prices and Fackler and Krieger's (J. Bus. Econom. Statist. 4 (1986), 71–80) study of five major macroeconomic variables.     

A note on the asymptotic and exact Fisher information matrices of a Markov switching VARMA process

Statistical Methods & Applications - We study the asymptotic and exact Fisher information (FI) matrices of Markov switching vector autoregressive moving average (MS VARMA) models. In a related...     

Calculating the autocovariances and the likelihood for periodic V ARMA models

This paper concerns the autocovariance calculation and likelihood evaluation for periodic vector ARMA models (PV ARMA). Based on a state space representation of PV ARMA models, we derive an algorithm for computing the PV ARMA autocovariances. The proposed method computes the autocovariances for distinct seasons separately, thereby facilitating efficient calculation for models with a large period. As a result, the obtained autocovariance calculation procedure is exploited in a periodic Chandrasekhar-type filter to evaluate the exact likelihood for Gaussian PV ARMA series. Empirical evidence shows the superiority of the periodic Chandrasekhar algorithm for likelihood evaluation over the Kalman-based one.     

R2 Bounds for Predictive Models: What Univariate Properties Tell us About Multivariate Predictability

ABSTRACT

A long-standing puzzle in macroeconomic forecasting has been that a wide variety of multivariate models have struggled to out-predict univariate models consistently. We seek an explanation for this puzzle in terms of population properties. We derive bounds for the predictive R² of the true, but unknown, multivariate model from univariate ARMA parameters alone. These bounds can be quite tight, implying little forecasting gain even if we knew the true multivariate model. We illustrate using CPI inflation data. Supplementary materials for this article are available online.     

Two Canonical VARMA Forms: Scalar Component Models Vis-à-Vis the Echelon Form

In this article we study two methodologies which identify and specify canonical form VARMA models. The two methodologies are: (1) an extension of the scalar component methodology which specifies canonical VARMA models by identifying scalar components through canonical correlations analysis; and (2) the Echelon form methodology, which specifies canonical VARMA models through the estimation of Kronecker indices. We compare the actual forms and the methodologies on three levels. Firstly, we present a theoretical comparison. Secondly, we present a Monte Carlo simulation study that compares the performances of the two methodologies in identifying some pre-specified data generating processes. Lastly, we compare the out-of-sample forecast performance of the two forms when models are fitted to real macroeconomic data.     

A transitional Markov switching autoregressive model

ABSTRACT

This paper is concerned with properties of a transitional Markov switching autoregressive (TMSAR) model, together with its maximum-likelihood estimation and inference. We extend existing MSAR models by allowing dependence of AR parameters on hidden states at time points prior to the current time t. A stationary solution is given and expressions for the theoretical autocovariance function are derived. Two time series are analyzed and the new model outperforms two existing MSAR models in terms of maximized log-likelihood, residual correlations, and one-step-ahead forecasting performance. The new model also gives more regime changes in agreement with real events.     

Continuous processes derived from the solution of generalized Langevin equation: theoretical properties and estimation

ABSTRACT

In this paper we present a class of continuous-time processes arising from the solution of the generalized Langevin equation and show some of its properties. We define the theoretical and empirical codifference as a measure of dependence for stochastic processes. As an alternative dependence measure we also consider the spectral covariance. These dependence measures replace the autocovariance function when it is not well defined. Results for the theoretical codifference and theoretical spectral covariance functions for the mentioned process are presented. The maximum likelihood estimation procedure is proposed to estimate the parameters of the process arising from the classical Langevin equation, i.e. the Ornstein–Uhlenbeck process, and of the so-called Cosine process. We also present a simulation study for particular processes arising from this class showing the generation, and the theoretical and empirical counterpart for both codifference and spectral covariance measures.     

Model fitting and inference under latent equilibrium processes

This paper presents a methodology for model fitting and inference in the context of Bayesian models of the type f(Y | X,θ)f(X|θ)f(θ), where Y is the (set of) observed data, θ is a set of model parameters and X is an unobserved (latent) stationary stochastic process induced by the first order transition model f(X ^(t+1)|X ^(t),θ), where X ^(t) denotes the state of the process at time (or generation) t. The crucial feature of the above type of model is that, given θ, the transition model f(X ^(t+1)|X ^(t),θ) is known but the distribution of the stochastic process in equilibrium, that is f(X|θ), is, except in very special cases, intractable, hence unknown. A further point to note is that the data Y has been assumed to be observed when the underlying process is in equilibrium. In other words, the data is not collected dynamically over time. We refer to such specification as a latent equilibrium process (LEP) model. It is motivated by problems in population genetics (though other applications are discussed), where it is of interest to learn about parameters such as mutation and migration rates and population sizes, given a sample of allele frequencies at one or more loci. In such problems it is natural to assume that the distribution of the observed allele frequencies depends on the true (unobserved) population allele frequencies, whereas the distribution of the true allele frequencies is only indirectly specified through a transition model. As a hierarchical specification, it is natural to fit the LEP within a Bayesian framework. Fitting such models is usually done via Markov chain Monte Carlo (MCMC). However, we demonstrate that, in the case of LEP models, implementation of MCMC is far from straightforward. The main contribution of this paper is to provide a methodology to implement MCMC for LEP models. We demonstrate our approach in population genetics problems with both simulated and real data sets. The resultant model fitting is computationally intensive and thus, we also discuss parallel implementation of the procedure in special cases.     

Some asymptotic properties in INAR(1) processes with Poisson marginals

The first-order integer-valued autoregressive (INAR(1)) process with Poisson marginal distributions is considered. It is shown that the sample autocovariance function of the model is asymptotically normally distributed. We derive asymptotic distribution of Yule-Walker type estimators of parameters. It turns out that our Yule-Walker type estimators are better than the conditional least squares estimators proposed by Klimko and Nelson (1978) and Al-Osh and Alzaid (1987). also, we study the relationship between the model andM/M/∞ queueing system.     

Upper bounds for ruin probabilities under model uncertainty

Abstract

In this paper, we investigate some ruin problems for risk models that contain uncertainties on both claim frequency and claim size distribution. The problems naturally lead to the evaluation of ruin probabilities under the so-called G-expectation framework. We assume that the risk process is described as a class of G-compound Poisson process, a special case of the G-Lévy process. By using the exponential martingale approach, we obtain the upper bounds for the two-sided ruin probability as well as the ruin probability involving investment. Furthermore, we derive the optimal investment strategy under the criterion of minimizing this upper bound. Finally, we conclude that the upper bound in the case with investment is less than or equal to the case without investment.     

Empirical study of robust estimation methods for PAR models with application to the air quality area

Abstract

This paper compares three estimators for periodic autoregressive (PAR) models. The first is the classical periodic Yule-Walker estimator (YWE). The second is a robust version of YWE (RYWE) which uses the robust autocovariance function in the periodic Yule-Walker equations, and the third is the robust least squares estimator (RLSE) based on iterative least squares with robust versions of the original time series. The daily mean particulate matter concentration (PM10) data is used to illustrate the methodologies in a real application, that is, in the Air Quality area.     

Smoothed empirical likelihood for GARCH models with heavy-tailed errors

ABSTRACT

This paper proposes an empirical likelihood (EL) method for estimating the GARCH(p, q) models with heavy-tailed errors. Using the kernel smoothing method, we derive a smoothed EL ratio statistic, which yields a smoothed EL estimator. Moreover, we derive a profile EL for the partial parameters in the presence of nuisance parameters. Simulations and empirical results are conducted to illustrate our proposed method.     

Spectral analysis of Markov switching GARCH models with statistical inference

We derive matrix expressions in closed form for the autocovariance function and the spectral density of Markov switching GARCH models and their powers. For this, we apply the Riesz–Fischer theorem which defines the spectral representation as the Fourier transform of the autocovariance function. Under suitable assumptions, we prove that the sample estimator of the spectral density is consistent and asymptotically normally distributed. Further statistical implications in terms of order identification and parameter estimation are discussed. A simulation study confirms the validity of the asymptotic properties. These methods are also well suited for financial market applications, and in particular for the analysis of time series in the frequency domain, as shown in some proposed real-world examples.     

A method for computing the autocovariance of renewal processes

In this paper we derive formulae for the autocovariance functions of renewal and renewal reward processes. The derivation is based on a Poissonization technique of a renewal process. The formulae are expressed in the form of Laplace transforms. In some cases we may invert the Laplace transforms analytically, but in general we have to invert them numerically.     

Multivariate time series prediction using a hybridization of VARMA models and Bayesian networks

In this paper, a new hybrid model of vector autoregressive moving average (VARMA) models and Bayesian networks is proposed to improve the forecasting performance of multivariate time series. In the proposed model, the VARMA model, which is a popular linear model in time series forecasting, is specified to capture the linear characteristics. Then the errors of the VARMA model are clustered into some trends by K-means algorithm with Krzanowski–Lai cluster validity index determining the number of trends, and a Bayesian network is built to learn the relationship between the data and the trend of its corresponding VARMA error. Finally, the estimated values of the VARMA model are compensated by the probabilities of their corresponding VARMA errors belonging to each trend, which are obtained from the Bayesian network. Compared with VARMA models, the experimental results with a simulation study and two multivariate real-world data sets indicate that the proposed model can effectively improve the prediction performance.     

Dependence Estimation for High‐frequency Sampled Multivariate CARMA Models

The paper considers high‐frequency sampled multivariate continuous‐time autoregressive moving average (MCARMA) models and derives the asymptotic behaviour of the sample autocovariance function to a normal random matrix. Moreover, we obtain the asymptotic behaviour of the cross‐covariances between different components of the model. We will see that the limit distribution of the sample autocovariance function has a similar structure in the continuous‐time and in the discrete‐time model. As a special case, we consider a CARMA (one‐dimensional MCARMA) process. For a CARMA process, we prove Bartlett's formula for the sample autocorrelation function. Bartlett's formula has the same form in both models; only the sums in the discrete‐time model are exchanged by integrals in the continuous‐time model. Finally, we present limit results for multivariate MA processes as well, which are not known in this generality in the multivariate setting yet.     

Analysis of an unreliable MX/G1G2/1/repeated service queue with delayed repair under randomized vacation policy

Abstract

In this article we consider an unreliable M^X/G/1 queue with two types of general heterogeneous service and optional repeated service subject to server’s break down and delayed repair under randomized vacation policy. We assume that customer arrive to the system according to a compound Poisson process. The server provides two types of general heterogeneous service and a customer can choose either type of service before its service start. After the completion of either type of service, the customer has the further option to repeat the same type of service once again. While the server is working with any types of service or repeated service, it may breakdown at any instant. Further the concept of randomized vacation is also introduced. For this model, we first derive the joint distribution of state of the server and queue size by considering both elapsed and remaining time, which is one of the objective of this article. Next, we derive Laplace Stieltjes transform of busy period distribution. Finally, we obtain some important performance measure and reliability indices of this model.     

The analysis of MX/M/1 queue with two-stage vacations policy

Abstract

In this article, we consider a batch arrival M^X/M/1 queue with two-stage vacations policy that comprises of single working vacation and multiple vacations, denoted by M^X/M/1/SWV?+?MV. Using the matrix analytic method, we derive the probability generating function (PGF) of the stationary system size and investigate the stochastic decomposition structure of stationary system size. Further, we obtain the Laplace–Stieltjes transform (LST) of stationary sojourn time of a customer by the first passage time analysis. At last, we illustrate the effects of various parameters on the performance measures numerically and graphically by some numerical examples.     

Some Second-Order Asymptotics for Extreme Value Linear Regression Models

ABSTRACT

In this article we derive finite-sample corrections in matrix notation for likelihood ratio and score statistics in extreme-value linear regression models. We consider three corrected score tests that perform better than the usual score test. We also derive general formulae for second-order biases of maximum likelihood estimates of the linear parameters. Some simulations are performed to compare the likelihood ratio and score statistics with their modified versions and to illustrate the bias correction.