Testing non-nested log-linear models with pseudo estimator

As the number of random variables for the categorical data increases, the possible number of log-linear models which can be fitted to the data increases rapidly, so that various model selection methods are developed. However, we often found that some models chosen by different selection criteria do not coincide. In this paper, we propose a comparison method to test the final models which are non-nested. The statistic of Cox (1961, 1962) is applied to log-linear models for testing non-nested models, and the Kullback-Leibler measure of closeness (Pesaran 1987) is explored. In log-linear models, pseudo estimators for the expectation and the variance of Cox's statistic are not only derived but also shown to be consistent estimators.     

Beta seasonal autoregressive moving average models

In this paper, we introduce the class of beta seasonal autoregressive moving average (βSARMA) models for modelling and forecasting time series data that assume values in the standard unit interval. It generalizes the class of beta autoregressive moving average models [Rocha AV and Cribari-Neto F. Beta autoregressive moving average models. Test. 2009;18(3):529–545] by incorporating seasonal dynamics to the model dynamic structure. Besides introducing the new class of models, we develop parameter estimation, hypothesis testing inference, and diagnostic analysis tools. We also discuss out-of-sample forecasting. In particular, we provide closed-form expressions for the conditional score vector and for the conditional Fisher information matrix. We also evaluate the finite sample performances of conditional maximum likelihood estimators and white noise tests using Monte Carlo simulations. An empirical application is presented and discussed.     

Bayesian identification of double seasonal autoregressive time series models

ABSTRACT

Seasonal autoregressive (SAR) models have been modified and extended to model high frequency time series characterized by exhibiting double seasonal patterns. Some researchers have introduced Bayesian inference for double seasonal autoregressive (DSAR) models; however, none has tackled the problem of Bayesian identification of DSAR models. Therefore, in order to fill this gap, we present a Bayesian methodology to identify the order of DSAR models. Assuming the model errors are normally distributed and using three priors, i.e. natural conjugate, g, and Jeffreys’ priors, on the model parameters, we derive the joint posterior mass function of the model order in a closed-form. Accordingly, the posterior mass function can be investigated and the best order of DSAR model is chosen as a value with the highest posterior probability for the time series being analyzed. We evaluate the proposed Bayesian methodology using simulation study, and we then apply it to real-world hourly internet amount of traffic dataset.     

Focused information criterion for locally misspecified vector autoregressive models

This paper investigates the focused information criterion and plug-in average for vector autoregressive models with local-to-zero misspecification. These methods have the advantage of focusing on a quantity of interest rather than aiming at overall model fit. Any (su?ciently regular) function of the parameters can be used as a quantity of interest. We determine the asymptotic properties and elaborate on the role of the locally misspecified parameters. In particular, we show that the inability to consistently estimate locally misspecified parameters translates into suboptimal selection and averaging. We apply this framework to impulse response analysis. A Monte Carlo simulation study supports our claims.     

Adaptive order selection for autoregressive models

Autoregressive model is a popular method for analysing the time dependent data, where selection of order parameter is imperative. Two commonly used selection criteria are the Akaike information criterion (AIC) and the Bayesian information criterion (BIC), which are known to suffer the potential problems regarding overfit and underfit, respectively. To our knowledge, there does not exist a criterion in the literature that can satisfactorily perform under various situations. Therefore, in this paper, we focus on forecasting the future values of an observed time series and propose an adaptive idea to combine the advantages of AIC and BIC but to mitigate their weaknesses based on the concept of generalized degrees of freedom. Instead of applying a fixed criterion to select the order parameter, we propose an approximately unbiased estimator of mean squared prediction errors based on a data perturbation technique for fairly comparing between AIC and BIC. Then use the selected criterion to determine the final order parameter. Some numerical experiments are performed to show the superiority of the proposed method and a real data set of the retail price index of China from 1952 to 2008 is also applied for illustration.     

Identification of non-linear additive autoregressive models

Summary.  We propose a lag selection method for non-linear additive autoregressive models that is based on spline estimation and the Bayes information criterion. The additive structure of the autoregression function is used to overcome the 'curse of dimensionality', whereas the spline estimators effectively take into account such a structure in estimation. A stepwise procedure is suggested to implement the method proposed. A comprehensive Monte Carlo study demonstrates good performance of the method proposed and a substantial computational advantage over existing local-polynomial-based methods. Consistency of the lag selection method based on the Bayes information criterion is established under the assumption that the observations are from a stochastic process that is strictly stationary and strongly mixing, which provides the first theoretical result of this kind for spline smoothing of weakly dependent data.     

Bayesian estimation with integrated nested Laplace approximation for binary logit mixed models

In multilevel models for binary responses, estimation is computationally challenging due to the need to evaluate intractable integrals. In this paper, we investigate the performance of integrated nested Laplace approximation (INLA), a fast deterministic method for Bayesian inference. In particular, we conduct an extensive simulation study to compare the results obtained with INLA to the results obtained with a traditional stochastic method for Bayesian inference (MCMC Gibbs sampling), and with maximum likelihood through adaptive quadrature. Particular attention is devoted to the case of small number of clusters. The specification of the prior distribution for the cluster variance plays a crucial role and it turns out to be more relevant than the choice of the estimation method. The simulations show that INLA has an excellent performance as it achieves good accuracy (similar to MCMC) with reduced computational times (similar to adaptive quadrature).     

Variable selection and estimation for high-dimensional spatial autoregressive models

Spatial regression models are important tools for many scientific disciplines including economics, business, and social science. In this article, we investigate postmodel selection estimators that apply least squares estimation to the model selected by penalized estimation in high-dimensional regression models with spatial autoregressive errors. We show that by separating the model selection and estimation process, the postmodel selection estimator performs at least as well as the simultaneous variable selection and estimation method in terms of the rate of convergence. Moreover, under perfect model selection, the ℓ₂ rate of convergence is the oracle rate of $\sqrt{s/n}$, compared with the convergence rate of $\text{◂√▸}\sqrt{s\log p/n}$ in the general case. Here, n is the sample size and p, s are the model dimension and number of significant covariates, respectively. We further provide the convergence rate of the estimation error in the form of $\sup$ norm, and ideally the rate can reach as fast as $\text{◂√▸}\sqrt{\log s/n}$.     

Separated hypotheses testing for autoregressive models with non-negative residuals

Normal residual is one of the usual assumptions in autoregressive model but sometimes in practice we are faced with non-negative residuals. In this paper, we have derived modified maximum likelihood estimators of parameters of the residuals and autoregressive coefficient. Also asymptotic distribution of modified maximum likelihood estimators in both stationary and non-stationary models are computed. So that, we can derive asymptotic distribution of unit root, Vuong's and Cox's tests statistics in stationary situation. Using simulation, it shows that Akaike information criterion and Vuong's test work to select the optimal autoregressive model with non-negative residuals. Sometimes Vuong's test select two competing models as equivalent models. These models may be suitable or unsuitable equivalent models. So we consider Cox's test to make inference after model selection. Kolmogorov–Smirnov test confirms our results. Also we have computed tracking interval for competing models to choosing between two close competing models when Vuong's test and Cox's test cannot detect the differences.     

Tree-structured generalized autoregressive conditional heteroscedastic models

We propose a new generalized autoregressive conditional heteroscedastic (GARCH) model with tree-structured multiple thresholds for the estimation of volatility in financial time series. The approach relies on the idea of a binary tree where every terminal node parameterizes a (local) GARCH model for a partition cell of the predictor space. The fitting of such trees is constructed within the likelihood framework for non-Gaussian observations: it is very different from the well-known regression tree procedure which is based on residual sums of squares. Our strategy includes the classical GARCH model as a special case and allows us to increase model complexity in a systematic and flexible way. We derive a consistency result and conclude from simulation and real data analysis that the new method has better predictive potential than other approaches.     

Testing for random effects and serial correlation in spatial autoregressive models

This paper constructs and evaluates tests for random effects and serial correlation in spatial autoregressive panel data models. In these models, ignoring the presence of random effects not only produces misleading inference but inconsistent estimation of the regression coefficients. Two different estimation methods are considered: maximum likelihood and instrumental variables. For each estimator, optimal tests are constructed: Lagrange multiplier in the first case; Neyman's C(α)$C(\alpha )$ in the second. In addition, locally size-robust tests, for individual hypotheses under local misspecification of the unconsidered parameter, are constructed. Extensive Monte Carlo evidence is presented.     

A model averaging approach for the ordered probit and nested logit models with applications

This paper considers model averaging for the ordered probit and nested logit models, which are widely used in empirical research. Within the frameworks of these models, we examine a range of model averaging methods, including the jackknife method, which is proved to have an optimal asymptotic property in this paper. We conduct a large-scale simulation study to examine the behaviour of these model averaging estimators in finite samples, and draw comparisons with model selection estimators. Our results show that while neither averaging nor selection is a consistently better strategy, model selection results in the poorest estimates far more frequently than averaging, and more often than not, averaging yields superior estimates. Among the averaging methods considered, the one based on a smoothed version of the Bayesian Information criterion frequently produces the most accurate estimates. In three real data applications, we demonstrate the usefulness of model averaging in mitigating problems associated with the ‘replication crisis’ that commonly arises with model selection.     

Penalized regression models with autoregressive error terms

Penalized regression methods have recently gained enormous attention in statistics and the field of machine learning due to their ability of reducing the prediction error and identifying important variables at the same time. Numerous studies have been conducted for penalized regression, but most of them are limited to the case when the data are independently observed. In this paper, we study a variable selection problem in penalized regression models with autoregressive (AR) error terms. We consider three estimators, adaptive least absolute shrinkage and selection operator, bridge, and smoothly clipped absolute deviation, and propose a computational algorithm that enables us to select a relevant set of variables and also the order of AR error terms simultaneously. In addition, we provide their asymptotic properties such as consistency, selection consistency, and asymptotic normality. The performances of the three estimators are compared with one another using simulated and real examples.     

A nonparametric test of conditional autoregressive heteroscedasticity for threshold autoregressive models

Threshold autoregressive models are widely used in time‐series applications. When building or using such a model, it is important to know whether conditional heteroscedasticity exists. The authors propose a nonparametric test of this hypothesis. They develop the large‐sample theory of a test of nonlinear conditional heteroscedasticity adapted to nonlinear autoregressive models and study its finite‐sample properties through simulations. They also provide percentage points for carrying out this test, which is found to have very good power overall.     

Testing for variance changes in autoregressive models with unknown order

The problem of change point in autoregressive process is studied in this article. We propose a Bayesian information criterion-iterated cumulative sums of squares algorithm to detect the variance changes in an autoregressive series with unknown order. Simulation results and two examples are presented, where it is shown to have good performances when the sample size is relatively small.     

Testing a linear relationship in varying coefficient spatial autoregressive models

In this article, we propose a test to check a linear relationship in varying coefficient spatial autoregressive models, in which a residual-based bootstrap procedure is suggested to approximate the null distribution of the resulting test statistic. We conduct simulation studies to assess the performance of the test, including the validity of the bootstrap approximation to the null distribution of the test statistic and the power of the test. The simulation results demonstrate that the residual-based bootstrap procedure gives very accurate estimate of the null distribution of the test statistic and the test is of satisfactory power. Furthermore, a real example is given to demonstrate the application of the proposed test.     

Transformed symmetric generalized autoregressive moving average models

Cordeiro and Andrade [Transformed generalized linear models. J Stat Plan Inference. 2009;139:2970–2987] incorporated the idea of transforming the response variable to the generalized autoregressive moving average (GARMA) model, introduced by Benjamin et al. [Generalized autoregressive moving average models. J Am Stat Assoc. 2003;98:214–223], thus developing the transformed generalized autoregressive moving average (TGARMA) model. The goal of this article is to develop the TGARMA model for symmetric continuous conditional distributions with a possible nonlinear structure for the mean that enables the fitting of a wide range of models to several time series data types. We derive an iterative process for estimating the parameters of the new model by maximum likelihood and obtain a simple formula to estimate the parameter that defines the transformation of the response variable. Furthermore, we determine the moments of the original dependent variable which generalize previous published results. We illustrate the theory by means of real data sets and evaluate the results developed through simulation studies.     

Detection of outliers in mixed regressive-spatial autoregressive models

ABSTRACT

This article studies the outlier detection problem in mixed regressive-spatial autoregressive model. The formulae for testing outliers and their approximate distributions are derived under the mean-shift model and the variance-weight model, respectively. The simulation studies are conducted for examining the power and size of the test, as well as for the detection of outliers when a simulated data contains several outliers. A real data is analyzed to illustrate the proposed method, and modified models based on mean-shift and variance-weight models in which detected outliers are taken into account are suggested to deal with the outliers and confirm theconclusions.     

A test of separate hypotheses for comparing linear mixed models with non nested fixed effects

As researchers increasingly rely on linear mixed models to characterize longitudinal data, there is a need for improved techniques for selecting among this class of models which requires specification of both fixed and random effects via a mean model and variance-covariance structure. The process is further complicated when fixed and/or random effects are non nested between models. This paper explores the development of a hypothesis test to compare non nested linear mixed models based on extensions of the work begun by Sir David Cox. We assess the robustness of this approach for comparing models containing correlated measures of body fat for predicting longitudinal cardiometabolic risk.     

Joint tests of non-nested models and general error specifications

This paper is concerned with joint tests of non-nested models and simultaneous departures from homoskedasticity, serial independence and normality of the disturbance terms. Locally equivalent alternative models are used to construct joint tests since they provide a convenient way to incorporate more than one type of departure from the classical conditions. The joint tests represent a simple asymptotic solution to the “pre-testing” problem in the context of non-nested linear regression models. Our simulation results indicate that the proposed tests have good finite sample properties.