Estimation of Standard Errors of Empirical Bayes Estimators in Capm-Type Models

This paper is concerned with the problem of estimating the standard errors of the empirical Bayes estimators in linear regression models. The problem of deriving an exact expression for the standard error of this estimator is generally intractable. We suggest a procedure based on Efron’s bootstrap method as a way of estimating the standard error. It is shown, through simulations, that the bootstrap method provides a more accurate estimate of the standard error of the empirical Bayes estimator than the traditional large sample method.     

Linear Regression With Nested Errors Using Probability‐Linked Data

Probabilistic matching of records is widely used to create linked data sets for use in health science, epidemiological, economic, demographic and sociological research. Clearly, this type of matching can lead to linkage errors, which in turn can lead to bias and increased variability when standard statistical estimation techniques are used with the linked data. In this paper we develop unbiased regression parameter estimates to be used when fitting a linear model with nested errors to probabilistically linked data. Since estimation of variance components is typically an important objective when fitting such a model, we also develop appropriate modifications to standard methods of variance components estimation in order to account for linkage error. In particular, we focus on three widely used methods of variance components estimation: analysis of variance, maximum likelihood and restricted maximum likelihood. Simulation results show that our estimators perform reasonably well when compared to standard estimation methods that ignore linkage errors.     

Standard errors for EM estimation

The EM algorithm is a popular method for computing maximum likelihood estimates. One of its drawbacks is that it does not produce standard errors as a by-product. We consider obtaining standard errors by numerical differentiation. Two approaches are considered. The first differentiates the Fisher score vector to yield the Hessian of the log-likelihood. The second differentiates the EM operator and uses an identity that relates its derivative to the Hessian of the log-likelihood. The well-known SEM algorithm uses the second approach. We consider three additional algorithms: one that uses the first approach and two that use the second. We evaluate the complexity and precision of these three and the SEM in algorithm seven examples. The first is a single-parameter example used to give insight. The others are three examples in each of two areas of EM application: Poisson mixture models and the estimation of covariance from incomplete data. The examples show that there are algorithms that are much simpler and more accurate than the SEM algorithm. Hopefully their simplicity will increase the availability of standard error estimates in EM applications. It is shown that, as previously conjectured, a symmetry diagnostic can accurately estimate errors arising from numerical differentiation. Some issues related to the speed of the EM algorithm and algorithms that differentiate the EM operator are identified.     

Using Heteroscedasticity-Consistent Standard Errors for the Linear Regression Model with Correlated Regressors

The use of heteroscedasticity-consistent covariance matrix (HCCM) estimators is very common in practice to draw correct inference for the coefficients of a linear regression model with heteroscedastic errors. However, in addition to the problem of heteroscedasticity, linear regression models may also be plagued with some considerable degree of collinearity among the regressors when two or more regressors are considered. This situation causes many adverse effects on the least squares measures and alternatively, the ordinary ridge regression method is used as a common practice. But in the available literature, the problems of multicollinearity and heteroscedasticity have not been discussed as a combined issue especially, for the inference of the regression coefficients. The present article addresses the inference about the regression coefficients taking both the issues of multicollinearity and heteroscedasticity into account and suggests the use of HCCM estimators for the ridge regression. This article proposes t- and F-tests, based on these HCCM estimators, that perform adequately well in the numerical evaluation of the Monte Carlo simulations.     

The Extent of Gross Errors Eliminated by Robust Multiple Linear Regressions

Robust estimation methods can effectively eliminate the influence of gross errors on parameter estimation. However, the extent of gross errors eliminated (EGEE) by robust estimation methods is far-reaching. This article presents a new approach to determine EGEE by robust estimation method. Taking multiple linear regressions (2–5) as examples, simulation experiments were conducted to compare the EGEE of 14 frequently used robust estimation methods. This article confirms several additional efficient robust estimation methods for dealing with multiple linear regressions, as well as the minimum number of observations needed to eliminate gross errors in certain ranges completely.     

Testing for the Equality of Two Nonparametric Regression Curves with Long Memory Errors

This article discusses the problem of testing the equality of two nonparametric regression functions against two-sided alternatives for uniform design on [0,1] with long memory moving average errors. The standard deviations and the long memory parameters are possibly different for the two errors. The article adapts the partial sum process idea used in the independent observations settings to construct the tests and derives their asymptotic null distributions. The article also shows that these tests are consistent for general alternatives and obtains their limiting distributions under a sequence of local alternatives. Since the limiting null distributions of these tests are unknown, we first conducted a Monte Carlo simulation study to obtain a few selected critical values of the proposed tests. Then based on these critical values, another Monte Carlo simulation is conducted to study the finite sample level and power behavior of these tests at some alternatives. The article also contains a simulation study that assesses the effect of estimating the nonparametric regression function on an estimate of the long memory parameter of the errors. It is observed that the estimate based on direct observations is generally preferable over the one based on the estimated nonparametric residuals.     

Fitting Time Series Models by Minimizing Multistep-ahead Errors: a Frequency Domain Approach

This paper brings together two topics in the estimation of time series forecasting models: the use of the multistep-ahead error sum of squares as a criterion to be minimized and frequency domain methods for carrying out this minimization. The methods are developed for the wide class of time series models having a spectrum which is linear in unknown coefficients. This includes the IMA(1, 1) model for which the common exponentially weigh-ted moving average predictor is optimal, besides more general structural models for series exhibiting trends and seasonality. The method is extended to include the Box–Jenkins `air line' model. The value of the multistep criterion is that it provides protection against using an incorrectly specified model. The value of frequency domain estimation is that the iteratively reweighted least squares scheme for fitting generalized linear models is readily extended to construct the parameter estimates and their standard errors. It also yields insight into the loss of efficiency when the model is correct and the robustness of the criterion against an incorrect model. A simple example is used to illustrate the method, and a real example demonstrates the extension to seasonal models. The discussion considers a diagnostic test statistic for indicating an incorrect model.     

Standard errors for the parameters of graphical Gaussian models

The comparison of an estimated parameter to its standard error, the Wald test, is a well known procedure of classical statistics. Here we discuss its application to graphical Gaussian model selection. First we derive the Fisher information matrix and its inverse about the parameters of any graphical Gaussian model. Both the covariance matrix and its inverse are considered and a comparative analysis of the asymptotic behaviour of their maximum likelihood estimators (m.l.e.s) is carried out. Then we give an example of model selection based on the standard errors. The method is shown to produce almost identical inference to likelihood ratio methods in the example considered.     

Diagnostics for a Linear Model with First-Order Autoregressive Symmetrical Errors

In this article, we focus on some diagnostics for linear regression model with first-order autoregressive and symmetrical errors. The symmetrical class includes both light- and heavy-tailed univariate symmetrical distributions, which offers a more flexible framework for modeling. Maximum likelihood estimates are computed via the Fisher-score method. Score statistic and its adjustment are proposed for testing autocorrelation of the random errors. Local influence diagnostics are also derived for the model under some usual perturbation schemes. The performances of the test statistics are investigated through Monte Carlo simulations. Finally, a real data set is used to illustrate our diagnostic methods.     

Uniform Test for Predictive Regression With AR Errors

Testing predictability is of importance in economics and finance. Based on a predictive regression model with independent and identically distributed errors, some uniform tests have been proposed in the literature without distinguishing whether the predicting variable is stationary or nearly integrated. In this article, we extend the empirical likelihood methods of Zhu, Cai, and Peng with independent errors to the case of an AR error process. Again, the proposed new tests do not need to know whether the predicting variable is stationary or nearly integrated, and whether it has a finite variance or an infinite variance. A simulation study shows the new methodologies perform well in finite sample.     

A Note on parameter and standard error estimation in adaptive robust regression

This paper introduces practical methods of parameter and standard error estimation for adaptive robust regression where errors are assumed to be from a normal/independent family of distributions. In particular, generalized EM algorithms (GEM) are considered for the two cases of t and slash families of distributions. For the t family, a one step method is proposed to estimate the degree of freedom parameter. Use of empirical information is suggested for standard error estimation. It is shown that this choice leads to standard errors that can be obtained as a by-product of the GEM algorithm. The proposed methods, as discussed, can be implemented in most available nonlinear regression programs. Details of implementation in SAS NLIN are given using two specific examples.     

Time Series Properties of the Class of Generalized First-Order Autoregressive Processes with Moving Average Errors

A new class of time series models known as Generalized Autoregressive of order one with first-order moving average errors has been introduced in order to reveal some hidden features of certain time series data. The variance and autocovariance of the process is derived in order to study the behaviour of the process. It is shown that in special cases these new results reduce to the standard ARMA results. Estimation of parameters based on the Whittle procedure is discussed. We illustrate the use of this class of model by using two examples.     

Transformations for Smooth Regression Models with Multiplicative Errors

We consider whether one should transform to estimate nonparametrically a regression curve sampled from data with a constant coefficient of variation, i.e. with multiplicative errors. Kernel-based smoothing methods are used to provide curve estimates from the data both in the original units and after transformation. Comparisons are based on the mean-squared error (MSE) or mean integrated squared error (MISE), calculated in the original units. Even when the data are generated by the simplest multiplicative error model, the asymptotically optimal MSE (or MISE) is surprisingly not always obtained by smoothing transformed data, but in many cases by directly smoothing the original data. Which method is optimal depends on both the regression curve and the distribution of the errors. Data-based procedures which could be useful in choosing between transforming and not transforming a particular data set are discussed. The results are illustrated on simulated and real data.     

Standard errors for the Laspeyres index number with autocorrelated error models

This paper deals with the stochastic approach to Laspeyres price index number with the assumption of serial correlation of orders 1 and 2. The first round of estimation provides the estimates of Laspeyres index numbers in the presence of serial correlation assuming that variance is independent of time. In the second round of estimation, we use the weighted least square approach to derive the standard errors of Laspeyres index number assuming variance is dependent on time. These standard errors are linked to the variability of relative prices and are simple to evaluate. It shows that the larger index numbers are expected to estimate with less degree of precision. The results are illustrated with price data of Pakistan.     

Estimation in Second-Order Models with Errors in the Factor Levels

The impact of errors in the factor levels is examined on the estimation of parameters in second-order response models. Errors can occur in setting the factor levels for response surface and robust parameter design models. These errors can lead to heterogeneity of variances in model errors that make ordinary least squares estimation inappropriate. Weighted least squares and maximum likelihood estimation approaches are developed as viable alternatives where it is assumed the variances and covariances of the errors are known. Performance of these estimation techniques are examined in simulation studies for two examples. Another example is given that applies these results.     

网络调查中的非抽样误差及其预防措施

互联网的迅速发展,给统计调查方法带来了巨大的影响,在互联网的基础之上发展起来的网络调查方法,以其独特的优势,日益受到人们的青睐。本文结合调查误差分析的理论,根据网络自身的特点,分析了网络调查的非抽样误差的来源,并提出了减少误差的方法。     

A Bayesian Analysis of Autoregressive Models with Exogenous Variables and Power-Transformed and Threshold GARCH Errors

Consider a class of autoregressive models with exogenous variables and power transformed and threshold GARCH (ARX-PTTGARCH) errors, which is a natural generalization of the standard and special GARCH model. We propose a Bayesian method to show that combining Gibbs sampler and Metropolis-Hastings algorithm to give a Bayesian analysis can be applied to estimate parameters of ARX-PTTGARCH models with success.     

Local Linear Regression for Non Grid Spatiotemporal Models with Autoregressive Errors

In various environmental studies, spatiotemporal correlated data are involved, so there has been increasing demand of proposing spatiotemporal estimation methods that capture spatiotemporal correlation so as to improve the accuracy of estimation. In this article, we construct estimators for non grid spatiotemporal models with autoregressive errors. It is proved that the estimators are asymptotic normality. Simulation results also show the estimators perform well.     

On Cross-Lagged Panel Models With Serially Correlated Errors

Cross-lagged panel studies are studies in which two or more variables are measured for a large number of subjects at each of several points in time. The variables divide naturally into two sets, and the purpose of the analysis is to estimate and test the cross-effects between the two sets. One approach to this analysis is to treat the cross-effects as parameters in regression equations. This study contributes to this approach by extending the regression model to a multivariate model that captures the correlation among the variables and allows the errors in the model to be correlated over time.     

Ain't Behavin': Forecast Errors and Measurement Errors in Early GNP Estimates

Revisions of the early GNP estimates may contain elements of measurement errors as well as forecast errors. These types of error behave differently but need to satisfy a common set of criteria for well-behavedness. This article tests these criteria for U.S. GNP revisions. The tests are similar to tests of rationality and are based on the generalized method of moments estimator. The flash, 15-day, and 45-day estimates are found to be ill behaved, but the 75-day estimate satisfies the criteria for well-behavedness.