Abstract: The paper develops a discrete state space solution method for a class of nonlinear rational expectations models. The method works by using numerical quadrature rules to approximate the integral operators that arise in stochastic intertemporal models. The method is particularly useful for approximating asset pricing models and has potential applications in other problems as well. An empirical application uses the method to study the relationship between the risk premium and the conditional variability of the equity return under an ARCH endowment process. NONLINEAR DYNAMIC RATIONAL EXPECTATIONS MODELS rarely admit explicit solutions. Techniques like the method of undetermined coefficients or forward- looking expansions, which often work well for linear models, rarely provide explicit solutions for nonlinear models. The lack of explicit solutions compli- cates the tasks of analyzing the dynamic properties of such models and generat- ing simulated realizations for applied policy work and other purposes. This paper develops a discrete state-space approximation method for a specific class of nonlinear rational expectations models. The class of models is distinguished by two features: First, the solution functions for the endogenous variables are functions of at most a finite number of lags of an exogenous stationary state vector. Second, the expectational equations of the model take the form of integral equations, or more precisely, Fredholm equations of the second type. The key component of the method is a technique, based on numerical quadrature, for forming a discrete approximation to a general time series conditional density. More specifically, the technique provides a means for calibrating a Markov chain, with a discrete state space, whose probability distribution closely approximates the distribution of a given time series. The quality of the approximation can be expected to get better as the discrete state space is made sufficiently finer. The term "discrete" is used here in reference to the range space of the random variables and not to the time index; time is always discrete in our analysis. The discretization technique is primarily useful for taking a discrete approxi- mation to the conditional density of the strictly exogenous variables of a model. The specification of this conditional density could be based on a variety of 1Financial support under NSF Grants SES-8520244 and SES-8810357 is acknowledged. We thank the co-editor and referees of earlier drafts for many, many helpful comments that substantially improved the manuscript.

Abstract: Generalized linear models have unified the approach to regression for a wide variety of discrete, continuous, and censored response variables that can be assumed to be independent across experimental units. In applications such as longitudinal studies, genetic studies of families, and survey sampling, observations may be obtained in clusters. Responses from the same cluster cannot be assumed to be independent. With linear models, correlation has been effectively modeled by assuming there are cluster-specific random effects that derive from an underlying mixing distribution. Extensions of generalized linear models to include random effects has, thus far, been hampered by the need for numerical integration to evaluate likelihoods. In this article, we cast the generalized linear random effects model in a Bayesian framework and use a Monte Carlo method, the Gibbs sampler, to overcome the current computational limitations. The resulting algorithm is flexible to easily accommodate changes in the number...

Abstract: The mixed model equations as presented by C. R. Henderson offers the base for a methodology that provides flexibility of fitting models with various fixed and random elements with the possible assumption of correlation among random effects. The advantage of teaching analysis of variance applications from this methodology is presented. Particular emphasis is placed upon the relationship between choice of estimable function and inference space.

Abstract: The utility of the Hammerstein model, which is composed of a static nonlinear element in series with a linear dynamic part, was investigated to represent the dynamics of nonlinear chemical processes. Different methods to identify the parameters of Hammerstein models were tested. The methods were applied to the identification of simulated distillation columns and to an experimental heat exchanger process. The results show that the dynamics of such processes can be better represented by Hammerstein-type models than by linear models.

Abstract: Analyses of linear models based on robust estimates of regression coefficients offers the user an attractive robust alternative to the classical least squares analysis in analyzing linear models. Much of the work done on robust analyses of linear models has concerned their asymptotic properties. To be of practical interest, though, the small sample properties of these analyses need to be ascertained. This article discusses studentization of these robust analyses and surveys past studies of it. With increasing speed of computation, resampling techniques have become feasible solutions to this studentizing problem. Some discussion of these techniques is also offered. To illustrate the discussion a Monte Carlo study of several experiments is included.

Abstract: Multivariate linear models discrimination and allocation frequency analysis of time series time domain analysis linear models for spatial data.

Abstract: The problem of modeling change in a vector time series is studied using a dynamic linear model with measurement matrices that switch according to a time-varying independent random process. We derive filtered estimators for the usual state vectors and also for the state occupancy probabilities of the underlying nonstationary measurement process. A maximum likelihood estimation procedure is given that uses a pseudo-expectation-maximization algorithm in the initial stages and nonlinear optimization. We relate the models to those considered previously in the literature and give an application involving the tracking of multiple targets.

Abstract: A common way to identify the inertial parameters of robots is to use a linear model in relation to the parameters and standard least squares (LS) techniques. The authors present a method to generate exciting identification trajectories in order to minimize the effect of noise and error modeling on the LS solution. Using nonlinear optimization techniques, the condition number of a matrix W obtained from the energy model is minimized and the scaling of its terms is carried out. An example of a 3 degree of freedom robot is presented. >

Abstract: This paper derives several properties unique to nonlinear model hypothesis testing problems involving linear or nonlinear inequality constraints in the null or alternative hypothesis. The paper is organized around a lemma which characterizes the set containing the least favorable parameter value for a nonlinear model inequality constraints hypothesis test. We then present two examples which illustrate several implications of this lemma. We also discuss the impact of these properties on the empirical implementation and interpretation of these test procedures.

Abstract: Even if robust regression estimators have been around for nearly 20 years, they have not found widespread application. One obstacle is the diversity of estimator types and the necessary choices of tuning constants, combined with a lack of guidance for these decisions. While some participants of the IMA summer program have argued that these choices should always be made in view of the specific problem at hand, we propose a procedure which should fit many purposes reasonably well. A second obstacle is the lack of simple procedures for inference, or the reluctance to use the straightforward inference based on asymptotics.

Abstract: Summary This paper gives a review and development of what is currently known about the directionality (irreversibility) of time series models, together with briefer coverage of the still limited statistical methodology. Reversibility is shown to imply stationarity; Weiss's result concerning the reversibility of linear Gaussian processes is stressed, and contrasted to the directional nature of much time series data. Reversed ARMA models are explored, and non-linear examples given; the stationarity and invertibility conditions of ARMA models are shown to be implicitly directional, and a consequence of the future-independent nature of such models. Invertibility is extended to the two-sided futuredependent generalised linear model, and applied to reversible moving average models. The directional and reversible implications of autoregressive roots are covered. Work applying directional-sensitive methods of statistical analysis to reversed data series is mentioned; possible dangers in transforming directional series to Gaussian marginal distributions are noted. The directional nature of most non-linear models is invoked to emphasise the current importance of the area.

Abstract: We find that long-term uncertainty in a linear model of the interest rate term structure can have dramatic effects on variance bounds implied by the expectations theories of the term structure. We bootstrap fractionally integrated models of the term structure of interest rates. The fractional order of integration's bootstrapped standard errors simulate uncertainty surrounding long-term forecasts of interest rates, and we find that it is possible to overstate the significance of variance-bounds violations by at least a factor of three and perhaps by a factor of ten when long-term uncertainty is ignored.

Abstract: In this article, we present the multivariable variogram, which is defined in a way similar to that of the traditional variogram, by the expected value of a distance, squared, in a space withp dimensions. Combined with the linear model of coregionalization, this tool provides a way for finding the elementary variograms that characterize the different spatial scales contained in a set of data withp variables. In the case in which the number of elementary components is less than or equal to the number of variables, it is possible, by means of nonlinear regression of variograms and cross-variograms, to estimate the coregionalization parameters directly in order to obtain the elementary variables themselves, either by cokriging or by direct matrix inversion. This new tool greatly simplifies the procedure proposed by Matheron (1982) and Wackernagel (1985). The search for the elementary variograms is carried out using only one variogram (multivariable), as opposed to thep(p + 1)/2 required by the Matheron approach. Direct estimation of the linear coregionalization model parameters involves the creation of semipositive definite coregionalization matrices of rank 1.

Abstract: In predicting the course of individual realizations of an epidemic it is important to know the magnitude of the variability of such realizations about their mean. In this paper and in the context of the general stochastic epidemic, some methods of obtaining approximate estimates of this variability are investigated; one is a multivariate normal approximation based on an asymptotic Gaussian diffusion process, and another uses an approximating linear stochastic process. The extension of these methods to the more detailed models used to describe the transmission dynamics of HIV infection and AIDS is discussed.

Abstract: 1. Introduction.- 2. Bayesian Statistical Analysis.- 3. Computational Aspects of Bayesian Analysis.- 4. Prediction with Parameter Uncertainty.- 5. The Credibility Problem.- 6. The Hierarchical Bayesian Approach.- 7. The Hierarchical Normal Linear Model.- 8. Examples.- 9. Modifications to the Hierarchical Normal Linear Model.- Appendix. Algorithms, Programs, and Data Sets.- A. The Simplex Method of Function Maximization.- B. Adaptive Gaussian Integration.- C. Gauss-Hermite Integration.- D. Polar Method for Generating Normal Deviates.- E. GAUSS Programs.- 1. Simplex Maximization.- 2. Adaptive Gaussian Integration.- 3. Gauss-Hermite Integration.- 4. Monte Carlo Integration.- 5. Tierney-Kadane Integration.- F. Data Sets.- 1. Data Set 1.- 2. Data Sets 2-4.

Abstract: Shift-share analysis continues to be popular among geographers, regional scientists, and planners despite widespread criticism of the method. In this paper, it is argued that insufficient attention has been paid to model-based approaches to shift—share analysis. It is shown that conventional shift—share and stochastic shift—share yield identical conclusions. Stochastic shift—share is easily extended dynamically and along the lines suggested by Arcelus. Thus, in the stochastic models several of the most persistent criticisms of the technique are addressed by allowing for testing of hypotheses, while preserving the practicality of the conventional accounting approach. It is suggested that stochastic shift—share should be used whenever practical.

Abstract: The authors present a regression approach to the backcalculation of flexible linear models of the HIV infection curve. They note that "because expected AIDS incidence can be expressed as a linear function of unknown parameters regression methods may be used to obtain parameter and covariance estimates for a variety of interesting quantities such as the expected number of people infected in previous time intervals and the projected AIDS incidence in future time intervals. We exploit these ideas to show that estimates based on maximum likelihood are for practical purposes equivalent to approximate estimates based on quasi-likelihood and on Poisson regression. These algorithms are readily implemented on a personal computer." These concepts are illustrated by projecting AIDS incidence in the United States up to 1993. (EXCERPT)

Abstract: A loss function approach is used to define the concepts of explained residual variation and explained risk for general regression models. Explained risk measures the ability of the covariates in a correctly specified model to distinguish differing outcomes. Explained residual variation, which is R 2 for a linear model, estimates the explained risk with a penalty for poorly fitting models. Application of the general definitions to linear regression, logistic regression, and survival analysis is given. The importance of distinguishing the concepts of explained residual variation, explained risk, and goodness of fit is discussed.

Abstract: The empirical and quantitative (linear Arrhenius) model of Davey (1989a) was assessed against a range of published lag phase data, some of which included combined temperature—water activity as influencing factors while others had temperature as the sole influencing factor. For two large (n≧ 27) sets of combined temperature and water activity data, the model explained 74·2 and 86·4% of the variation in results, and for eight wide ranging data sets with temperature as the sole factor, it explained between 80·8 and 99·7% of the variation in results, with an overall mean of 95·7%. These published data included both Gram-positive and Gram-negative bacteria and a spore former. The model has four coefficients which can be obtained by relatively unsophisticated users with linear regression analyses on most desk top calculators or personal computers.

Abstract: The polytomous unidimensional Rasch model with equidistant scoring, also known as the rating scale model, is extended in such a way that the item parameters are linearly decomposed into certain basic parameters. The extended model is denoted as the linear rating scale model (LRSM). A conditional maximum likelihood estimation procedure and a likelihood-ratio test of hypotheses within the framework of the LRSM are presented. Since the LRSM is a generalization of both the dichotomous Rasch model and the rating scale model, the present algorithm is suited for conditional maximum likelihood estimation in these submodels as well. The practicality of the conditional method is demonstrated by means of a dichotomous Rasch example with 100 items, of a rating scale example with 30 items and 5 categories, and in the light of an empirical application to the measurement of treatment effects in a clinical study.

Abstract: This paper addresses the problems of identification of a constrained nonlinear system (CNLS) described by Volterra functional series. A workable and practical method in terms of a penalty function principle and orthogonal expansion was developed, which involves an extension of the classical least squares solution to include a penalty function with arbitrarily defined weights which are adjusted to ensure that the constraints are satisfied. A total of nine basins across a range of climates and catchment areas in China were selected for examination of both daily and hourly rainfall-runoff forecasting. It was found that the method described in this paper can provide a more reasonable and robust response function where hydrologic constraints are required. The nonlinear model yields a better streamflow forecasting than the linear model, particularly in peak flows.

Abstract: Statistical inference for linear models has classically focused on either estimation or hypothesis testing of linear combinations of fixed effects or of variance components for random effects. A third form of inference—prediction of linear combinations of fixed and random effects—has important advantages over conventional estimators in many applications. None of these approaches will result in accurate inference if the data contain strong, unaccounted for local gradients, such as spatial trends in field-plot data. Nearest neighbor methods to adjust for such trends have been widely discussed in recent literature. So far, however, these methods have been developed exclusively for classical estimation and hypothesis testing. In this article a method of obtaining nearest neighbor adjusted (NNA) predictors, along the lines of “best linear unbiased prediction,” or BLUP, is developed. A simulation study comparing “NNABLUP” to conventional NNA methods and to non-NNA alternatives suggests considerable pot...