Regression model selection—a residual likelihood approach

Summary. We obtain the residual information criterion RIC, a selection criterion based on the residual log-likelihood, for regression models including classical regression models, Box–Cox transformation models, weighted regression models and regression models with autoregressive moving average errors. We show that RIC is a consistent criterion, and that simulation studies for each of the four models indicate that RIC provides better model order choices than the Akaike information criterion, corrected Akaike information criterion, final prediction error, C _p and R _adj², except when the sample size is small and the signal-to-noise ratio is weak. In this case, none of the criteria performs well. Monte Carlo results also show that RIC is superior to the consistent Bayesian information criterion BIC when the signal-to-noise ratio is not weak, and it is comparable with BIC when the signal-to-noise ratio is weak and the sample size is large.     

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.     

On Variational Bayes Estimation and Variational Information Criteria for Linear Regression Models

Variational Bayes (VB) estimation is a fast alternative to Markov Chain Monte Carlo for performing approximate Baesian inference. This procedure can be an efficient and effective means of analyzing large datasets. However, VB estimation is often criticised, typically on empirical grounds, for being unable to produce valid statistical inferences. In this article we refute this criticism for one of the simplest models where Bayesian inference is not analytically tractable, that is, the Bayesian linear model (for a particular choice of priors). We prove that under mild regularity conditions, VB based estimators enjoy some desirable frequentist properties such as consistency and can be used to obtain asymptotically valid standard errors. In addition to these results we introduce two VB information criteria: the variational Akaike information criterion and the variational Bayesian information criterion. We show that variational Akaike information criterion is asymptotically equivalent to the frequentist Akaike information criterion and that the variational Bayesian information criterion is first order equivalent to the Bayesian information criterion in linear regression. These results motivate the potential use of the variational information criteria for more complex models. We support our theoretical results with numerical examples.     

Accounting for uncertainty in health economic decision models by using model averaging

Summary.  Health economic decision models are subject to considerable uncertainty, much of which arises from choices between several plausible model structures, e.g. choices of covariates in a regression model. Such structural uncertainty is rarely accounted for formally in decision models but can be addressed by model averaging. We discuss the most common methods of averaging models and the principles underlying them. We apply them to a comparison of two surgical techniques for repairing abdominal aortic aneurysms. In model averaging, competing models are usually either weighted by using an asymptotically consistent model assessment criterion, such as the Bayesian information criterion, or a measure of predictive ability, such as Akaike's information criterion. We argue that the predictive approach is more suitable when modelling the complex underlying processes of interest in health economics, such as individual disease progression and response to treatment.     

Forecasting time series of economic processes by model averaging across data frames of various lengths

This paper presents an extension of mean-squared forecast error (MSFE) model averaging for integrating linear regression models computed on data frames of various lengths. Proposed method is considered to be a preferable alternative to best model selection by various efficiency criteria such as Bayesian information criterion (BIC), Akaike information criterion (AIC), F-statistics and mean-squared error (MSE) as well as to Bayesian model averaging (BMA) and naïve simple forecast average. The method is developed to deal with possibly non-nested models having different number of observations and selects forecast weights by minimizing the unbiased estimator of MSFE. Proposed method also yields forecast confidence intervals with a given significance level what is not possible when applying other model averaging methods. In addition, out-of-sample simulation and empirical testing proves efficiency of such kind of averaging when forecasting economic processes.     

Forecasting Performance of Information Criteria with Many Macro Series

Stock & Watson (1999) consider the relative quality of different univariate forecasting techniques. This paper extends their study on forecasting practice, comparing the forecasting performance of two popular model selection procedures, the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). This paper considers several topics: how AIC and BIC choose lags in autoregressive models on actual series, how models so selected forecast relative to an AR(4) model, the effect of using a maximum lag on model selection, and the forecasting performance of combining AR(4), AIC, and BIC models with an equal weight.     

Generalized Nelson–Siegel term structure model: do the second slope and curvature factors improve the in-sample fit and out-of-sample forecasts?

The dynamic Nelson–Siegel (DNS) model and even the Svensson generalization of the model have trouble in fitting the short maturity yields and fail to grasp the characteristics of the Japanese government bonds yield curve, which is flat at the short end and has multiple inflection points. Therefore, a closely related generalized dynamic Nelson–Siegel (GDNS) model that has two slopes and curvatures is considered and compared empirically to the traditional DNS in terms of in-sample fit as well as out-of-sample forecasts. Furthermore, the GDNS with time-varying volatility component, modeled as standard EGARCH process, is also considered to evaluate its performance in relation to the GDNS. The GDNS model unanimously outperforms the DNS in terms of in-sample fit as well as out-of-sample forecasts. Moreover, the extended model that accounts for time-varying volatility outpace the other models for fitting the yield curve and produce relatively more accurate 6- and 12-month ahead forecasts, while the GDNS model comes with more precise forecasts for very short forecast horizons.     

Model selection for factor analysis: Some new criteria and performance comparisons

This paper derives Akaike information criterion (AIC), corrected AIC, the Bayesian information criterion (BIC) and Hannan and Quinn’s information criterion for approximate factor models assuming a large number of cross-sectional observations and studies the consistency properties of these information criteria. It also reports extensive simulation results comparing the performance of the extant and new procedures for the selection of the number of factors. The simulation results show the di?culty of determining which criterion performs best. In practice, it is advisable to consider several criteria at the same time, especially Hannan and Quinn’s information criterion, Bai and Ng’s IC_p2 and BIC₃, and Onatski’s and Ahn and Horenstein’s eigenvalue-based criteria. The model-selection criteria considered in this paper are also applied to Stock and Watson’s two macroeconomic data sets. The results differ considerably depending on the model-selection criterion in use, but evidence suggesting five factors for the first data and five to seven factors for the second data is obtainable.     

VARIATIONAL BAYESIAN ANALYSIS FOR HIDDEN MARKOV MODELS

The variational approach to Bayesian inference enables simultaneous estimation of model parameters and model complexity. An interesting feature of this approach is that it also leads to an automatic choice of model complexity. Empirical results from the analysis of hidden Markov models with Gaussian observation densities illustrate this. If the variational algorithm is initialized with a large number of hidden states, redundant states are eliminated as the method converges to a solution, thereby leading to a selection of the number of hidden states. In addition, through the use of a variational approximation, the deviance information criterion for Bayesian model selection can be extended to the hidden Markov model framework. Calculation of the deviance information criterion provides a further tool for model selection, which can be used in conjunction with the variational approach.     

Bayesian model selection for join point regression with application to age-adjusted cancer rates

Summary.  The method of Bayesian model selection for join point regression models is developed. Given a set of K +1 join point models M ₀,  M ₁, …,  M _K with 0, 1, …,  K join points respec-tively, the posterior distributions of the parameters and competing models M _k are computed by Markov chain Monte Carlo simulations. The Bayes information criterion BIC is used to select the model M _k with the smallest value of BIC as the best model. Another approach based on the Bayes factor selects the model M _k with the largest posterior probability as the best model when the prior distribution of M _k is discrete uniform. Both methods are applied to analyse the observed US cancer incidence rates for some selected cancer sites. The graphs of the join point models fitted to the data are produced by using the methods proposed and compared with the method of Kim and co-workers that is based on a series of permutation tests. The analyses show that the Bayes factor is sensitive to the prior specification of the variance σ ², and that the model which is selected by BIC fits the data as well as the model that is selected by the permutation test and has the advantage of producing the posterior distribution for the join points. The Bayesian join point model and model selection method that are presented here will be integrated in the National Cancer Institute's join point software ( http://www.srab.cancer.gov/joinpoint/ ) and will be available to the public.     

联合广义线性模型中的变量选择

在联合广义线性模型中,散度参数与均值都被赋予了广义线性模型的结构,本文主要考虑在只有分布的一阶矩和二阶矩指定的条件下,联合广义线性模型中均值部分的变量选择问题。本文采用广义拟似然函数,提出了新的模型选择准则(EAIC);该准则是Akaike信息准则的推广。论文通过模拟研究验证了该准则的效果。     

A High‐dimensional Focused Information Criterion

The focused information criterion for model selection is constructed to select the model that best estimates a particular quantity of interest, the focus, in terms of mean squared error. We extend this focused selection process to the high‐dimensional regression setting with potentially a larger number of parameters than the size of the sample. We distinguish two cases: (i) the case where the considered submodel is of low dimension and (ii) the case where it is of high dimension. In the former case, we obtain an alternative expression of the low‐dimensional focused information criterion that can directly be applied. In the latter case, we use a desparsified estimator that allows us to derive the mean squared error of the focus estimator. We illustrate the performance of the high‐dimensional focused information criterion with a numerical study and a real dataset.     

Focused Information Criterion for Capture–Recapture Models for Closed Populations

Abstract. We propose a criterion for selecting a capture–recapture model for closed populations, which follows the basic idea of the focused information criterion (FIC) of Claeskens and Hjort. The proposed criterion aims at selecting the model which, among the available models, leads to the smallest mean‐squared error (MSE) of the resulting estimator of the population size and is based on an index which, up to a constant term, is equal to the asymptotic MSE of the estimator. Two alternative approaches to estimate this FIC index are proposed. We also deal with multimodel inference; in this case, the population size is estimated by using a weighted average of the estimates coming from different models, with weights chosen so as to minimize the MSE of the resulting estimator. The proposed model selection approach is compared with more common approaches through a series of simulations. It is also illustrated by an application based on a dataset coming from a live‐trapping experiment.     

What is the effective sample size of a spatial point process?

Point process models are a natural approach for modelling data that arise as point events. In the case of Poisson counts, these may be fitted easily as a weighted Poisson regression. Point processes lack the notion of sample size. This is problematic for model selection, because various classical criteria such as the Bayesian information criterion (BIC) are a function of the sample size, n, and are derived in an asymptotic framework where n tends to infinity. In this paper, we develop an asymptotic result for Poisson point process models in which the observed number of point events, m, plays the role that sample size does in the classical regression context. Following from this result, we derive a version of BIC for point process models, and when fitted via penalised likelihood, conditions for the LASSO penalty that ensure consistency in estimation and the oracle property. We discuss challenges extending these results to the wider class of Gibbs models, of which the Poisson point process model is a special case.     

Simple Formula for Calculating Bias‐corrected AIC in Generalized Linear Models

In real‐data analysis, deciding the best subset of variables in regression models is an important problem. Akaike's information criterion (AIC) is often used in order to select variables in many fields. When the sample size is not so large, the AIC has a non‐negligible bias that will detrimentally affect variable selection. The present paper considers a bias correction of AIC for selecting variables in the generalized linear model (GLM). The GLM can express a number of statistical models by changing the distribution and the link function, such as the normal linear regression model, the logistic regression model, and the probit model, which are currently commonly used in a number of applied fields. In the present study, we obtain a simple expression for a bias‐corrected AIC (corrected AIC, or CAIC) in GLMs. Furthermore, we provide an ‘R’ code based on our formula. A numerical study reveals that the CAIC has better performance than the AIC for variable selection.     

A note on the unification of the Akaike information criterion

To measure the distance between a robust function evaluated under the true regression model and under a fitted model, we propose generalized Kullback–Leibler information. Using this generalization we have developed three robust model selection criteria, AICR*, AICCR* and AICCR, that allow the selection of candidate models that not only fit the majority of the data but also take into account non-normally distributed errors. The AICR* and AICCR criteria can unify most existing Akaike information criteria; three examples of such unification are given. Simulation studies are presented to illustrate the relative performance of each criterion.     

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.     

A diagnostic of influential cases based on the information complexity criteria in generalized linear mixed models

ABSTRACT

Modeling diagnostics assess models by means of a variety of criteria. Each criterion typically performs its evaluation upon a specific inferential objective. For instance, the well-known DFBETAS in linear regression models are a modeling diagnostic which is applied to discover the influential cases in fitting a model. To facilitate the evaluation of generalized linear mixed models (GLMM), we develop a diagnostic for detecting influential cases based on the information complexity (ICOMP) criteria for detecting influential cases which substantially affect the model selection criterion ICOMP. In a given model, the diagnostic compares the ICOMP criterion between the full data set and a case-deleted data set. The computational formula of the ICOMP criterion is evaluated using the Fisher information matrix. A simulation study is accomplished and a real data set of cancer cells is analyzed using the logistic linear mixed model for illustrating the effectiveness of the proposed diagnostic in detecting the influential cases.     

Spatio-temporal model for crop yield forecasting

This paper proposes a linear mixed model (LMM) with spatial effects, trend, seasonality and outliers for spatio-temporal time series data. A linear trend, dummy variables for seasonality, a binary method for outliers and a multivariate conditional autoregressive (MCAR) model for spatial effects are adopted. A Bayesian method using Gibbs sampling in Markov Chain Monte Carlo is used for parameter estimation. The proposed model is applied to forecast rice and cassava yields, a spatio-temporal data type, in Thailand. The data have been extracted from the Office of Agricultural Economics, Ministry of Agriculture and Cooperatives of Thailand. The proposed model is compared with our previous model, an LMM with MCAR, and a log transformed LMM with MCAR. We found that the proposed model is the most appropriate, using the mean absolute error criterion. It fits the data very well in both the fitting part and the validation part for both rice and cassava. Therefore, it is recommended to be a primary model for forecasting these types of spatio-temporal time series data.     

Autoregressive model selection based on a prediction perspective

The autoregressive (AR) model is a popular method for fitting and prediction in analyzing time-dependent data, where selecting an accurate model among considered orders is a crucial issue. Two commonly used selection criteria are the Akaike information criterion and the Bayesian information criterion. However, the two criteria are known to suffer potential problems regarding overfit and underfit, respectively. Therefore, using them would perform well in some situations, but poorly in others. In this paper, we propose a new criterion in terms of the prediction perspective based on the concept of generalized degrees of freedom for AR model selection. We derive an approximately unbiased estimator of mean-squared prediction errors based on a data perturbation technique for selecting the order parameter, where the estimation uncertainty involved in a modeling procedure is considered. Some numerical experiments are performed to illustrate the superiority of the proposed method over some commonly used order selection criteria. Finally, the methodology is applied to a real data example to predict the weekly rate of return on the stock price of Taiwan Semiconductor Manufacturing Company and the results indicate that the proposed method is satisfactory.