Methods for the Analysis of Explanatory Linear Regression Models with Missing Data Not at Random

Since the work of Little and Rubin (1987) not substantial advances in the analysisof explanatory regression models for incomplete data with missing not at randomhave been achieved, mainly due to the difficulty of verifying the randomness ofthe unknown data. In practice, the analysis of nonrandom missing data is donewith techniques designed for datasets with random or completely random missingdata, as complete case analysis, mean imputation, regression imputation, maximumlikelihood or multiple imputation. However, the data conditions required to minimizethe bias derived from an incorrect analysis have not been fully determined. In thepresent work, several Monte Carlo simulations have been carried out to establishthe best strategy of analysis for random missing data applicable in datasets withnonrandom missing data. The factors involved in simulations are sample size,percentage of missing data, predictive power of the imputation model and existenceof interaction between predictors. The results show that the smallest bias is obtainedwith maximum likelihood and multiple imputation techniques, although with lowpercentages of missing data, absence of interaction and high predictive power ofthe imputation model (frequent data structures in research on child and adolescentpsychopathology) acceptable results are obtained with the simplest regression imputation.     

A simulation study of estimation methods in the analytic hierarchy process

This paper uses a simulation analysis to investigate the statistical accuracy and rank preservation capability of the AHP estimation methods. The methods under study consist of: the eigenvalue, mean transformation, row geometric mean, column geometric mean, harmonic mean and simple row average. The methods are compared under three distributions for error term—gamma, lognormal and uniform—and under two types of input matrices of various sizes.     

A note on Musgrave asymmetrical trend-cycle filters

In this paper, we derive the implicit forecasts in the asymmetrical trend-cycle averages used in the X-11 seasonal adjustment method. We give an algorithm to calculate them, and we study their statistical properties. We express the forecasts as Stein estimators. We derive expressions for their bias, variance, covariances and prediction mean squared errors. We show that the prediction mean squared errors of the implied predictors are always smaller or equal to those obtained using the least squares predictors. Finally, we derive the prior distributions under which the implied predictors are Bayes estimators.     

Quantile cointegrating regression

Quantile regression has important applications in risk management, portfolio optimization, and asset pricing. The current paper studies estimation, inference and financial applications of quantile regression with cointegrated time series. In addition, a new cointegration model with quantile-varying coefficients is proposed. In the proposed model, the value of cointegrating coefficients may be affected by the shocks and thus may vary over the innovation quantile. The proposed model may be viewed as a stochastic cointegration model which includes the conventional cointegration model as a special case. It also provides a useful complement to cointegration models with (G)ARCH effects. Asymptotic properties of the proposed model and limiting distribution of the cointegrating regression quantiles are derived. In the presence of endogenous regressors, fully-modified quantile regression estimators and augmented quantile cointegrating regression are proposed to remove the second order bias and nuisance parameters. Regression Wald tests are constructed based on the fully modified quantile regression estimators. An empirical application to stock index data highlights the potential of the proposed method.     

基于信息熵的多属性决策方法在商品住宅价格中的研究

在市场经济条件下,房地产价格在房地产经济发展和房地产经济运行中有着重要的功能和作用,因此,对房价变动的预测以及如何能够合理制定房价,显得尤为重要。价格的预测,究其本质,是一种体现在数值上的决策活动,本文结合了多属性综合决策模型与回归分析对房价进行了预测研究。通过选取影响房价的部分宏观因素,建立基于熵的多属性综合决策模型,得到各因素与房价之间的关系,并通过回归分析,对房价进行预测。     

Estimation strategies for the regression coefficient parameter matrix in multivariate multiple regression

We consider improved estimation strategies for the parameter matrix in multivariate multiple regression under a general and natural linear constraint. In the context of two competing models where one model includes all predictors and the other restricts variable coefficients to a candidate linear subspace based on prior information, there is a need of combining two estimation techniques in an optimal way. In this scenario, we suggest some shrinkage estimators for the targeted parameter matrix. Also, we examine the relative performances of the suggested estimators in the direction of the subspace and candidate subspace restricted type estimators. We develop a large sample theory for the estimators including derivation of asymptotic bias and asymptotic distributional risk of the suggested estimators. Furthermore, we conduct Monte Carlo simulation studies to appraise the relative performance of the suggested estimators with the classical estimators. The methods are also applied on a real data set for illustrative purposes.     

Robust Hierarchical Bayes Small Area Estimation for the Nested Error Linear Regression Model

Standard model‐based small area estimates perform poorly in presence of outliers. Sinha & Rao ( 2009 ) developed robust frequentist predictors of small area means. In this article, we present a robust Bayesian method to handle outliers in unit‐level data by extending the nested error regression model. We consider a finite mixture of normal distributions for the unit‐level error to model outliers and produce noninformative Bayes predictors of small area means. Our modelling approach generalises that of Datta & Ghosh ( 1991 ) under the normality assumption. Application of our method to a data set which is suspected to contain an outlier confirms this suspicion, correctly identifies the suspected outlier and produces robust predictors and posterior standard deviations of the small area means. Evaluation of several procedures including the M‐quantile method of Chambers & Tzavidis ( 2006 ) via simulations shows that our proposed method is as good as other procedures in terms of bias, variability and coverage probability of confidence and credible intervals when there are no outliers. In the presence of outliers, while our method and Sinha–Rao method perform similarly, they improve over the other methods. This superior performance of our procedure shows its dual (Bayes and frequentist) dominance, which should make it attractive to all practitioners, Bayesians and frequentists, of small area estimation.     

Variable Selection and Shrinkage: Comparison of Some Approaches

A common strategy within the framework of regression models is the selection of variables with possible predictive value, which are incorporated in the regression model. Two recently proposed methods, Breiman's Garotte (B reiman , 1995) and Tibshirani's Lasso (T ibshirani , 1996) try to combine variable selection and shrinkage. We compare these with pure variable selection and shrinkage procedures. We consider the backward elimination procedure as a typical variable selection procedure and as an example of a shrinkage procedure an approach of V an H ouwelingen and L e C essie (1990). Additionally an extension of van Houwelingens and le Cessies approach proposed by S auerbrei (1999) is considered. The ordinary least squares method is used as a reference.
With the help of a simulation study we compare these approaches with respect to the distribution of the complexity of the selected model, the distribution of the shrinkage factors, selection bias, the bias and variance of the effect estimates and the average prediction error.     

The discrete Kalman filter applied to linear regression models: statistical considerations and an application

In this paper we show how the Kalman filter, which is a recursive estimation procedure, can be applied to the standard linear regression model. The resulting "Kalman estimator" is compared with the classical least-squares estimator.
The applicability and (dis)advantages of the filter are illustrated by means of a case study which consists of two parts. In the first part we apply the filter to a regression model with constant parameters and in the second part the filter is applied to a regression model with time-varying stochastic parameters. The prediction-powers of various "Kalman predictors" are compared with "least-squares predictors" by using T heil 's prediction-error coefficient U.     

Complete subset regressions

This paper proposes a new method for combining forecasts based on complete subset regressions. For a given set of potential predictor variables we combine forecasts from all possible linear regression models that keep the number of predictors fixed. We explore how the choice of model complexity, as measured by the number of included predictor variables, can be used to trade off the bias and variance of the forecast errors, generating a setup akin to the efficient frontier known from modern portfolio theory. In an application to predictability of stock returns, we find that combinations of subset regressions can produce more accurate forecasts than conventional approaches based on equal-weighted forecasts (which fail to account for the dimensionality of the underlying models), combinations of univariate forecasts, or forecasts generated by methods such as bagging, ridge regression or Bayesian Model Averaging.     

Relative error prediction in nonparametric deconvolution regression model

In this paper, we studied an alternative estimator of the regression function when the covariates are observed with error. It is based on the minimization of the relative mean squared error. We obtain expressions for its asymptotic bias and variance together with an asymptotic normality result. Our technique is illustrated on simulation studies. Numerical results suggest that the studied estimator can lead to tangible improvements in prediction over the usual kernel deconvolution regression estimator, particularly in the presence of several outliers in the dataset.     

Retransformation bias: A look at the box-cox transformation to linear balanced mixed ANOVA models

Summary After a Box-Cox transformation to data following a linear balanced mixed ANOVA model, final results may be presented after retransformation to the original scale of measurement. Consequently, estimation of means which may be unbiased in the transformed scale will not be so after retransformation. In this article, the bias introduced together with the corresponding variance is assessed. It is found that whereas bias may not be a serious problem, the variances are inflated for positive transformation parameter the closer it is to zero.     

Predictive regressions under asymmetric loss: Factor augmentation and model selection

This paper discusses the specifics of forecasting using factor-augmented predictive regressions under general loss functions. In line with the literature, we employ principal component analysis to extract factors from the set of predictors. In addition, we also extract information on the volatility of the series to be predicted, since the volatility is forecast-relevant under non-quadratic loss functions. We ensure asymptotic unbiasedness of the forecasts under the relevant loss by estimating the predictive regression through the minimization of the in-sample average loss. Finally, we select the most promising predictors for the series to be forecast by employing an information criterion that is tailored to the relevant loss. Using a large monthly data set for the US economy, we assess the proposed adjustments in a pseudo out-of-sample forecasting exercise for various variables. As expected, the use of estimation under the relevant loss is found to be effective. Using an additional volatility proxy as the predictor and conducting model selection that is tailored to the relevant loss function enhances the forecast performance significantly.     

A count-amount model with endogenous recording of observations

This paper discusses some of the problems which are encountered if an event ? is only recorded if its value satisfies a recording criterion A. It follows that we get an incorrect idea of the frequency of the events and of its true distribution. In order to solve these problems, an econometric model has been constructed by means of which consistent estimation of the true parameters is possible. The model is estimated on consumer purchases, where the number of purchases is assumed to be NEGBIN-distributed and the purchase amounts obey a lognormal distribution. Purchases are only recorded if their value exceeds Dfl. 10. It is shown that ignoring the recording condition will result in biased estimates and invalid predictions. Apart from this, the model is, among others, relevant for insurance problems, marketing surveys and criminological and epidemiological phenomena.     

Quantile regression for dynamic panel data with fixed effects

This paper studies a quantile regression dynamic panel model with fixed effects. Panel data fixed effects estimators are typically biased in the presence of lagged dependent variables as regressors. To reduce the dynamic bias, we suggest the use of the instrumental variables quantile regression method of Chernozhukov and Hansen (2006) along with lagged regressors as instruments. In addition, we describe how to employ the estimated models for prediction. Monte Carlo simulations show evidence that the instrumental variables approach sharply reduces the dynamic bias, and the empirical levels for prediction intervals are very close to nominal levels. Finally, we illustrate the procedures with an application to forecasting output growth rates for 18 OECD countries.     

Targeting predictors in random forest regression

Random forest (RF) regression is an extremely popular tool for analyzing high-dimensional data. Nonetheless, its benefits may be lessened in sparse settings due to weak predictors, and a pre-estimation dimension reduction (targeting) step is required. We show that proper targeting controls the probability of placing splits along strong predictors, thus providing an important complement to RF’s feature sampling. This is supported by simulations using finite representative samples. Moreover, we quantify the immediate gain from targeting in terms of the increased strength of individual trees. Macroeconomic and financial applications show that the bias–variance trade-off implied by targeting, due to increased correlation among trees in the forest, is balanced at a medium degree of targeting, selecting the best 5%–30% of commonly applied predictors. Improvements in the predictive accuracy of targeted RF relative to ordinary RF are considerable, up to 21%, occurring both in recessions and expansions, particularly at long horizons.     

Conditional feature screening for mean and variance functions in models with multiple-index structure

The existing methods for feature screening focus mainly on the mean function of regression models. The variance function, however, plays an important role in statistical theory and application. We thus investigate feature screening for mean and variance functions with multiple-index framework in high dimensional regression models. Notice that some information about predictors can be known in advance from previous investigations and experience, for example, a certain set of predictors is related to the response. Based on the conditional information, together with empirical likelihood, we propose conditional feature screening procedures. Our methods can consistently estimate the sets of active predictors in the mean and variance functions. It is interesting that the proposed screening procedures can avoid estimating the unknown link functions in the mean and variance functions, and moreover, can work well in the case of high correlation among the predictors without iterative algorithm. Therefore, our proposal is of computational simplicity. Furthermore, as a conditional method, our method is robust to the choice of the conditional set. The theoretical results reveal that the proposed procedures have sure screening properties. The attractive finite sample performance of our method is illustrated in simulations and a real data application.     

A general analysis of bias in the estimated standard errors of least squares coefficients

It is well known that when errors in the usual regression model are not independently distributed with equal variances, the application of ordinary least squares leads to calculated variances of the coefficient estimates which are biased and inconsistent. The nature of this bias has been investigated extensively, but the existing literature is limited in two significant ways. First, derivations of exact expressions for the bias have been restricted to special cases and, except for the simplest of these, the expressions derived are almost unmanageably complex. Second, for general error specifications, attention has been focused exclusively on deriving bounds for the bias, which are usually wide and do not allow even the probable direction of any bias to be determined. This paper derives an asymptotic expression for the bias which allows both its sign and approximate magnitude to be described easily in most regression problems. This expression is then used to investigate the bias in the cases of serial correlation of an arbitrary degree, variance components models and approximation of a non-linear relationship with a linear specification.     

Preliminary test Liu estimators based on the conflicting W,LR and LM tests in a regression model with multivariate Student-<Emphasis Type="Italic">t</Emphasis> error

In this paper, the problem of estimation of the regression coefficients in a multiple regression model with multivariate Student-t error is considered under the multicollinearity situation when it is suspected that the regression coefficients may be restricted to a linear manifold. The preliminary test Liu estimators (PTLE) based on the Wald, Likelihood ratio (LR) and Lagrangian multiplier (LM) tests are given. The bias and mean square error (MSE) of the proposed estimators are derived and conditions of superiority of these estimators are provided. In particular, we show that in the neighborhood of the null hypothesis, the PTLE based on the LM test has the best performance followed by the estimators based on LR and W tests, while the situation is reversed when the parameter moves away from the manifold of the restriction. Furthermore, the optimum choice of the level of significance is also discussed.     

Utilization of ambulatory care studies: testing the assumptions

A linear regression procedure is usually used to estimate the effect of a set of predictors on utilization of ambulatory health care. The implicit assumptions embedded in the linear regression model have never been examined. Here, with utilization data of a sample of 48292 patients from the file of the Québec National Health Plan, four implicit hypotheses embedded in the linear regression model are tested: (1) the transition from the state of utilization to the state of no utilization, and vice-versa, depends on the level of the transition rates, (2) the effect of independent variables depends on the transitions being predicted from or to the state of utilization, (3) the transition is time dependent, and (4) the system of transitions from one state to another is not at equilibrium. The analysis shows that the first three hypotheses cannot be rejected. Thus, the use of the familiar linear regression procedure in this study to estimate the effect of a set of factors on utilization would have yielded biased estimates.