MODEL BUILDING AND DATA MINING

This paper defines the phenomenon of data mining in econometrics and discusses various outcomes of and solutions to data mining. Both classical and Bayesian approaches are considered, each with notable advantages and disadvantages, and with the choice of loss function affecting critical values. Illustrative examples include variable addition and exclusion in a standard linear regression model, the choice of lag structure in a dynamic single equation, and specification in a simultaneous equations model.     

A pre-test like estimator dominating the least-squares method

We develop a pre-test type estimator of a deterministic parameter vector β$\mathit{\beta }$ in a linear Gaussian regression model. In contrast to conventional pre-test strategies, that do not dominate the least-squares (LS) method in terms of mean-squared error (MSE), our technique is shown to dominate LS when the effective dimension is greater than or equal to 4. Our estimator is based on a simple and intuitive approach in which we first determine the linear minimum MSE (MMSE) estimate that minimizes the MSE. Since the unknown vector β$\mathit{\beta }$ is deterministic, the MSE, and consequently the MMSE solution, will depend in general on β$\mathit{\beta }$ and therefore cannot be implemented. Instead, we propose applying the linear MMSE strategy with the LS substituted for the true value of β$\mathit{\beta }$ to obtain a new estimate. We then use the current estimate in conjunction with the linear MMSE solution to generate another estimate and continue iterating until convergence. As we show, the limit is a pre-test type method which is zero when the norm of the data is small, and is otherwise a non-linear shrinkage of LS.     

MORE ON THE PRE-TEST ESTIMATOR IN RIDGE REGRESSION

ABSTRACT

The problem of estimation of the regression coefficients in a multiple regression model is considered under a multicollinearity situation when it is suspected that the regression coefficients may be restricted to a subspace. The objective of this paper is to compare the usual preliminary test estimator and the preliminary test ridge regression estimator in the sense of the dispersion matrix of one dominating that of the other. In particular we proved two results giving necessary and sufficient conditions for the superiority of the preliminary test ridge regression estimator over the preliminary test estimator associated with the δ = 0 (or Δ = 0) and δ ≠ 0 (or Δ ≠ 0).     

ON THE USE OF THE STEIN VARIANCE ESTIMATOR IN THE DOUBLE k-CLASS ESTIMATOR IN REGRESSION

This paper investigates the predictive mean squared error performance of a modified double k-class estimator by incorporating the Stein variance estimator. Recent studies show that the performance of the Stein rule estimator can be improved by using the Stein variance estimator. However, as we demonstrate below, this conclusion does not hold in general for all members of the double k-class estimators. On the other hand, an estimator is found to have smaller predictive mean squared error than the Stein variance-Stein rule estimator, over quite large parts of the parameter space.     

Some sampling properties of the two-stage test in a linear regression with a proxy variable

In thia paper, we examine the sampling properties of the two-stage test which consists of a pre-test for disturbance variance followed by a main-test a regression coefficient of an unobservable variable when a proxy variable is used instead of the unobservable variable. It is shown that the pre-test works effectively if the significance level of the pre-test is appropriately chosen.     

Risk performance of a pre-test estimator for normal variance with the Stein-variance estimator under the LINEX loss function

In this paper, we derive the exact formula of the risk function of a pre-test estimator for normal variance with the Stein-variance (PTSV) estimator when the asymmetric LINEX loss function is used. Fixing the critical value of the pre-test to unity which is a suggested critical value in some sense, we examine numerically the risk performance of the PTSV estimator based on the risk function derived. Our numerical results show that although the PTSV estimator does not dominate the usual variance estimator when under-estimation is more severe than over-estimation, the PTSV estimator dominates the usual variance estimator when over-estimation is more severe. It is also shown that the dominance of the PTSV estimator over the original Stein-variance estimator is robust to the extension from the quadratic loss function to the LINEX loss function.     

MSE Performance of a Heterogeneous Pre-Test Ridge Regression Estimator

This article describes how diagnostic procedures were derived for symmetrical nonlinear regression models, continuing the work carried out by Cysneiros and Vanegas (2008 Cysneiros , F. J. A. , Vanegas , L. H. ( 2008 ). Residuals and their statistical properties in symmetrical nonlinear models . Statist. Probab. Lett. 78 : 3269 – 3273 .[Crossref], [Web of Science ®] , [Google Scholar]) and Vanegas and Cysneiros (2010 Vanegas , L. H. , Cysneiros , F. J. A. ( 2010 ). Assesment of diagnostic procedures in symmetrical nonlinear regression models . Computat. Statist. Data Anal. 54 : 1002 – 1016 .[Crossref], [Web of Science ®] , [Google Scholar]), who showed that the parameters estimates in nonlinear models are more robust with heavy-tailed than with normal errors. In this article, we focus on assessing if the robustness of this kind of models is also observed in the inference process (i.e., partial F-test). Symmetrical nonlinear regression models includes all symmetric continuous distributions for errors covering both light- and heavy-tailed distributions such as Student-t, logistic-I and -II, power exponential, generalized Student-t, generalized logistic, and contaminated normal. Firstly, a statistical test is shown to evaluating the assumption that the error terms all have equal variance. The results of simulation studies which describe the behavior of the test for heteroscedasticity proposed in the presence of outliers are then given. To assess the robustness of inference process, we present the results of a simulation study which described the behavior of partial F-test in the presence of outliers. Also, some diagnostic procedures are derived to identify influential observations on the partial F-test. As ilustration, a dataset described in Venables and Ripley (2002 Venables , W. N. , Ripley , B. D. ( 2002 ). Modern Applied with S. , 4th ed. New York : Springer .[Crossref] , [Google Scholar]), is also analyzed.     

The Correlated Bivariate Noncentral F Distribution and Its Application

This article proposes the singly and doubly correlated bivariate noncentral F (BNCF) distributions. The probability density function (pdf) and the cumulative distribution function (cdf) of the distributions are derived for arbitrary values of the parameters. The pdf and cdf of the distributions for different arbitrary values of the parameters are computed, and their graphs are plotted by writing and implementing new R codes. An application of the correlated BNCF distribution is illustrated in the computations of the power function of the pre-test test for the multivariate simple regression model (MSRM).     

On the sensitivity of pre-test estimators to covariance misspecification

In statistical and econometric practice it is not uncommon to find that regression parameter estimates obtained using estimated generalized least squares (EGLS) do not differ much from those obtained through ordinary least squares (OLS), even when the assumption of spherical errors is violated. To investigate if one could ignore non-spherical errors, and legitimately continue with OLS estimation under the non-spherical disturbance setting, Banerjee and Magnus (1999) developed statistics to measure the sensitivity of the OLS estimator to covariance misspecification. Wan et al. (2007) generalized this work by allowing for linear restrictions on the regression parameters. This paper extends the aforementioned studies by exploring the sensitivity of the equality restrictions pre-test estimator to covariance misspecification. We find that the pre-test estimators can be very sensitive to covariance misspecification, and the degree of sensitivity of the pre-test estimator often lies between that of its unrestricted and restricted components. In addition, robustness to non-normality is investigated. It is found that existing results remain valid if elliptically symmetric, instead of normal, errors are assumed.     

Shrinkage estimator of regression model under asymmetric loss

This article investigates the performance of the shrinkage estimator (SE) of the parameters of a simple linear regression model under the LINEX loss criterion. The risk function of the estimator under the asymmetric LINEX loss is derived and analyzed. The moment-generating functions and the first two moments of the estimators are also obtained. The risks of the SE have been compared numerically with that of pre-test and least-square estimators (LSEs) under the LINEX loss criterion. The numerical comparison reveals that under certain conditions the LSE is inadmissible, and the SE is the best among the three estimators.