Typologies of linear dynamic systems and models

Open and closed linear dynamic systems are formulated and considered for stationary and integrated data processes. A typology of linear dynamic systems is developed, extending that available for individual dynamic equations. Then an overlapping typology of models of these systems is examined and methods for analysing econometric models are described. General to simple modelling strategies are briefly considered.     

Parallel tempering for dynamic generalized linear models

ABSTRACT

Markov chain Monte Carlo (MCMC) methods can be used for statistical inference. The methods are time-consuming due to time-vary. To resolve these problems, parallel tempering (PT), as a parallel MCMC method, is tried, for dynamic generalized linear models (DGLMs), as well as the several optimal properties of our proposed method. In PT, two or more samples are drawn at the same time, and samples can exchange information with each other. We also present some simulations of the DGLMs in the case and provide two applications of Poisson-type DGLMs in financial research.     

Automatic structure discovery for varying-coefficient partially linear models

Varying-coefficient partially linear models provide a useful tools for modeling of covariate effects on the response variable in regression. One key question in varying-coefficient partially linear models is the choice of model structure, that is, how to decide which covariates have linear effect and which have non linear effect. In this article, we propose a profile method for identifying the covariates with linear effect or non linear effect. Our proposed method is a penalized regression approach based on group minimax concave penalty. Under suitable conditions, we show that the proposed method can correctly determine which covariates have a linear effect and which do not with high probability. The convergence rate of the linear estimator is established as well as the asymptotical normality. The performance of the proposed method is evaluated through a simulation study which supports our theoretical results.     

GQL estimation in linear dynamic models for panel data

In the econometrics literature, it is standard practice to use the existing instrumental variables as well as generalized method of moments approaches for the estimation of the parameters of a linear dynamic mixed model for panel data. In this paper, we introduce a generalized quasi-likelihood estimation approach that produces estimates with smaller mean squared errors when compared with the aforementioned and other existing approaches.     

Multivariate stochastic volatility with Bayesian dynamic linear models

This paper develops a Bayesian procedure for estimation and forecasting of the volatility of multivariate time series. The foundation of this work is the matrix-variate dynamic linear model, for the volatility of which we adopt a multiplicative stochastic evolution, using Wishart and singular multivariate beta distributions. A diagonal matrix of discount factors is employed in order to discount the variances element by element and therefore allowing a flexible and pragmatic variance modelling approach. Diagnostic tests and sequential model monitoring are discussed in some detail. The proposed estimation theory is applied to a four-dimensional time series, comprising spot prices of aluminium, copper, lead and zinc of the London metal exchange. The empirical findings suggest that the proposed Bayesian procedure can be effectively applied to financial data, overcoming many of the disadvantages of existing volatility models.     

A note on transformed likelihood approach in linear dynamic panel models

The transformed likelihood approach to estimation of fixed effects dynamic panel data models is shown to present very good inferential properties but it is not directly implemented in the most diffused statistical software. The present paper aims at showing how a simple model reformulation can be adopted to describe the problem in terms of classical linear mixed models. The transformed likelihood approach is based on the first differences data transformation, the following results derive from a convenient reformulation in terms of deviations from the first observations. Given the invariance to data transformation, the likelihood functions defined in the two cases coincide. Resulting in a classical random effect linear model form, the proposed approach significantly improves the number of available estimation procedures and provides a straightforward interpretation for the parameters. Moreover, the proposed model specification allows to consider all the estimation improvements typical of the random effects model literature. Simulation studies are conducted in order to study the robustness of the estimation method to mean stationarity violation.     

Admissible linear estimation in singular linear models

The admissibility results of Rao (1976), proved in the context of a nonsingular covariance matrix, are exteneded to the situation where the covariance matrix is singular. Admi.s s Lb Le linear estimators in the Gauss-Markoff model are characterized and admis-sibility of the best linear unbiased estimator is investigated.     

Projectors and linear estimation in general linear models

The paper gives a self-contained account of minimum disper­sion linear unbiased estimation of the expectation vector in a linear model with the dispersion matrix belonging to some, rather arbitrary, set of nonnegative definite matrices. The approach to linear estimation in general linear models recommended here is a direct generalization of some ideas and results presented by Rao (1973, 19 74) for the case of a general Gauss-Markov model

A new insight into the nature of some estimation problems originaly arising in the context of a general Gauss-Markov model as well as the correspondence of results known in the literature to those obtained in the present paper for general linear models are also given. As preliminary results the theory of projectors defined by Rao (1973) is extended.     

Automatic specification of piecewise linear additive models: application to forecasting natural gas demand

When facing any forecasting problem not only is accuracy on the predictions sought. Also, useful information about the underlying physics of the process and about the relevance of the forecasting variables is very much appreciated. In this paper, it is presented an automatic specification procedure for models that are based on additivity assumptions and piecewise linear regression. This procedure allows the analyst to gain insight about the problem by examining the automatically selected model, thus easily checking the validity of the forecast. Monte Carlo simulations have been run to ensure that the model selection procedure behaves correctly under weakly dependent data. Moreover, comparison over other well-known methodologies has been done to evaluate its accuracy performance, both in simulated data and in the context of short-term natural gas demand forecasting. Empirical results show that the accuracy of the proposed model is competitive against more complex methods such as neural networks.     

Collinearity in generalized linear models

This paper defines collinearity for generalized linear models (GLMs), investigates its consequences and proposes diagnostic criteria. The relationship between collinearity in GLMs and standard linear models (SLMs) is explored and bounds which relate the degree of collinearity in these two models are given. Estimation based on ridge methods is discussed.     

Atypical observations in linear models

The problem of ‘atypical data point’ in the estimation of model parameters and its effect on prediction are discussed. A cross-validity procedure is then proposed to accommodate the unusual observations in the estimation and thereby to improve the prediction of future data points. Each atypical point, weighted according to cross-validitory procedure, is used in the estimation of model parameters.     

Selection procedures in linear models

Many authors have studied variable selection in multiple linear regression models. In this paper, we derive some generalized selection procedures for the linear models. An approximation of noncentral F distribution has also been obtained.     

Prediction in linear mixed models

Following estimation of effects from a linear mixed model, it is often useful to form predicted values for certain factor/variate combinations. The process has been well defined for linear models, but the introduction of random effects into the model means that a decision has to be made about the inclusion or exclusion of random model terms from the predictions. This paper discusses the interpretation of predictions formed including or excluding random terms. Four datasets are used to illustrate circumstances where different prediction strategies may be appropriate: in an orthogonal design, an unbalanced nested structure, a model with cubic smoothing spline terms and for kriging after spatial analysis. The examples also show the need for different weighting schemes that recognize nesting and aliasing during prediction, and the necessity of being able to detect inestimable predictions.     

Optimal allocation for linear combinations of linear predictors in generalized linear models

The problem of the allocation of experimental units to experimental groups is studied within the context of generalized linear models. Optimal designs for the estimation of linear combinations of linear predictors are characterized, using concepts from the theory of optimal design. If there is only one linear combination of interest, then the D-optimal allocation is equivalent to the well-known Neyman allocation of subsamples in stratified sampling. However, if the number of linear combinations equals the number of design points, or experimental groups, then the equal replication of all design points is D-optimal. For cases in between, there are no easily accessible general solutions to the problem, although some particular cases are solved, including: i estimation of the n- 1 possible comparisons with a control group in an n-point, one-factor design; and ii estimation of 2 one or two of the four natural parameters of a 2 factorial design. The A-optimal allocations are determined in general.     

Reversible jump methods for generalised linear models and generalised linear mixed models

A reversible jump algorithm for Bayesian model determination among generalised linear models, under relatively diffuse prior distributions for the model parameters, is proposed. Orthogonal projections of the current linear predictor are used so that knowledge from the current model parameters is used to make effective proposals. This idea is generalised to moves of a reversible jump algorithm for model determination among generalised linear mixed models. Therefore, this algorithm exploits the full flexibility available in the reversible jump method. The algorithm is demonstrated via two examples and compared to existing methods.     

Adaptive varying-coefficient linear models

Summary. Varying-coefficient linear models arise from multivariate nonparametric regression, non-linear time series modelling and forecasting, functional data analysis, longitudinal data analysis and others. It has been a common practice to assume that the varying coefficients are functions of a given variable, which is often called an index . To enlarge the modelling capacity substantially, this paper explores a class of varying-coefficient linear models in which the index is unknown and is estimated as a linear combination of regressors and/or other variables. We search for the index such that the derived varying-coefficient model provides the least squares approximation to the underlying unknown multidimensional regression function. The search is implemented through a newly proposed hybrid backfitting algorithm. The core of the algorithm is the alternating iteration between estimating the index through a one-step scheme and estimating coefficient functions through one-dimensional local linear smoothing. The locally significant variables are selected in terms of a combined use of the t -statistic and the Akaike information criterion. We further extend the algorithm for models with two indices. Simulation shows that the methodology proposed has appreciable flexibility to model complex multivariate non-linear structure and is practically feasible with average modern computers. The methods are further illustrated through the Canadian mink–muskrat data in 1925–1994 and the pound–dollar exchange rates in 1974–1983.     

Transformed generalized linear models

The estimation of data transformation is very useful to yield response variables satisfying closely a normal linear model. Generalized linear models enable the fitting of models to a wide range of data types. These models are based on exponential dispersion models. We propose a new class of transformed generalized linear models to extend the Box and Cox models and the generalized linear models. We use the generalized linear model framework to fit these models and discuss maximum likelihood estimation and inference. We give a simple formula to estimate the parameter that index the transformation of the response variable for a subclass of models. We also give a simple formula to estimate the r$r$th moment of the original dependent variable. We explore the possibility of using these models to time series data to extend the generalized autoregressive moving average models discussed by Benjamin et al. [Generalized autoregressive moving average models. J. Amer. Statist. Assoc. 98, 214–223]. The usefulness of these models is illustrated in a simulation study and in applications to three real data sets.     

Parameter estimation in dynamic models

In this paper we study the parameter estimation of a first-order dynamic model for intervention and transfer function analysis, A new parameterization is proposed to avoid the “overshoot” problem in nonlinear estimation in a frequently used parameterization (Box and Tiao, 1975).     

Missing plot technique in linear models

This paper extends the missing plot substitution technique to the case where the missing observations-cause some previously estimable functions to become non-estimable. It is shown that with appropriate modifications, the usual methods of analysis remain valid. We also obtain necessary and sufficient conditions under which the sum of squares due to a hypothesis can be calculated without “re-estimating” the missing observations