Mean square error tests for restrictions in singular linear models

Weak and strong mean square error tests of restrictions presented in Wallace (1972) are generalized to apply to singular linear models. The singularity necessitates a slight change in the strong m.s.e. criterion and the requirement that the restrictions be estimable, but otherwise the tests are applied in a fashion analogous to the non-singular case. Use of those tests implies that the solution for the linear model parameter vector is contingent on a test result. The risk behavior of these contingent solutions is discussed.     

IV估计的最优工具变量选取方法

IV估计的有限样本性质对工具变量的选取十分敏感,尤其是存在弱工具变量的情形。本文在Donald和Newey(2001)的基础上研究了常用的IV估计———2SLS的最优工具变量选取方法。首先通过对2SLS估计量进行Nagar分解,从理论上推导出估计量的近似MSE表达式;根据这一表达式,提出IV估计的最优工具变量选取准则,并证明选取准则的渐近有效性。模拟结果表明,本文提出的工具变量选取准则能够极大地改善2SLS估计量的有限样本表现。本研究为实证中面临的工具变量选择问题提供了理论依据。     

Randomized response: Estimating mean square errors of linear estimators and finding optimal unbiased strategies

Summary General procedures are described to generate quantitative randomized response (RR) required to estimate the finite population total of a sensitive variable. Permitting sample selection with arbitrary probabilities a formula for the mean square error (MSE) of a linear estimator of total based on RR is noted indicating the simple modification over one that might be based on direct response (DR) if the latter were available. A general formula for an unbiased estimator of the MSE is presented. A simple approximation is proposed in case the RR ratio estimator is employed based on a simple random sample (SRS) taken without replacement (WOR). Among sampling strategies employing unbiased but not necessarily linear estimators based on RR, certain optimal ones are identified under two alternative models analogously to well-known counterparts based on DR, if available. Unlike Warner’s (1965) treatment of categorical RR we consider quantitative RR here.     

Estimation of models with grouped and ungrouped data by means of “2SLS”

This paper deals with a special case of estimation with grouped data, where the dependent variable is only available for groups, whereas the endogenous regressor(s) is available at the individual level. By estimating the first stage using the available individual data, and then estimating the second stage at the aggregate level, it might be possible to gain efficiency relative to the OLS and 2SLS estimators that use only grouped data. We term this the mixed-2SLS estimator (M2SLS). The M2SLS estimator is consistent and asymptotically normal. We also provide a test of efficiency of M2SLS relative to OLS and “2SLS” estimators.     

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.     

Mean squared error of the empirical best linear unbiased predictor in an orthogonal finite discrete spectrum linear regression model

The mean squared error (MSE) of the empirical best linear unbiased predictor in an orthogonal finite discrete spectrum linear regression model is derived and a comparison with the MSE of the best linear unbiased predictor in this model is made. It is shown that under weak conditions these two mean square errors are asymptotically the same.     

A ridge-like method for simultaneous estimation of simultaneous equations

In the context of full information estimation in a linear simultaneous equations model, this paper considers a ridge-like modification of the 3SLS estimator. The proposed method is particularly desirable where the square matrix of the 3SLS normal equationsis singular (or near-singular) leading to non-existence (or poor performance) of the estimator. Furthermore, the type of solution suggested here does seem to result in the existence of the finite sample moments of the estimator even when the degrees of over identification are as low as zero (just identified models). This paper considers only a simple scalar form of the ‘ridge-matrix” with a relatively simple choice of the modifying scalar that preserves the asymptotic properties of the 3SLS estimator. A value of this scalar is derived which minimizes an appropriatequadratic risk criterion. The approximate quadratic risk function is based upon the asymptotic approximation of the relevant moments in the manner of Nagar (1959). A range of risk reducing values of the ‘ridge-scalar” is also given.     

Mean square error estimation in multi-stage sampling

Summary: Suppose for a homogeneous linear unbiased function of the sampled first stage unit (fsu)-values taken as an estimator of a survey population total, the sampling variance is expressed as a homogeneous quadratic function of the fsu-values. When the fsu-values are not ascertainable but unbiased estimators for them are separately available through sampling in later stages and substituted into the estimator, Raj (1968) gave a simple variance estimator formula for this multi-stage estimator of the population total. He requires that the variances of the estimated fsu-values in sampling at later stages and their unbiased estimators are available in certain `simple forms'. For the same set-up Rao (1975) derived an alternative variance estimator when the later stage sampling variances have more ‘complex forms’. Here we pursue with Raj's (1968) simple forms to derive a few alternative variance and mean square error estimators when the condition of homogeneity or unbiasedness in the original estimator of the total is relaxed and the variance of the original estimator is not expressed as a quadratic form.  We illustrate a particular three-stage sampling strategy and present a simulation-based numerical exercise showing the relative efficacies of two alternative variance estimators. Received: 19 February 1999     

Testing for serial correlation,spatial autocorrelation and random effects using panel data

This paper considers a spatial panel data regression model with serial correlation on each spatial unit over time as well as spatial dependence between the spatial units at each point in time. In addition, the model allows for heterogeneity across the spatial units using random effects. The paper then derives several Lagrange multiplier tests for this panel data regression model including a joint test for serial correlation, spatial autocorrelation and random effects. These tests draw upon two strands of earlier work. The first is the LM tests for the spatial error correlation model discussed in Anselin and Bera [1998. Spatial dependence in linear regression models with an introduction to spatial econometrics. In: Ullah, A., Giles, D.E.A. (Eds.), Handbook of Applied Economic Statistics. Marcel Dekker, New York] and in the panel data context by Baltagi et al. [2003. Testing panel data regression models with spatial error correlation. Journal of Econometrics 117, 123–150]. The second is the LM tests for the error component panel data model with serial correlation derived by Baltagi and Li [1995. Testing AR(1) against MA(1) disturbances in an error component model. Journal of Econometrics 68, 133–151]. Hence, the joint LM test derived in this paper encompasses those derived in both strands of earlier works. In fact, in the context of our general model, the earlier LM tests become marginal LM tests that ignore either serial correlation over time or spatial error correlation. The paper then derives conditional LM and LR tests that do not ignore these correlations and contrast them with their marginal LM and LR counterparts. The small sample performance of these tests is investigated using Monte Carlo experiments. As expected, ignoring any correlation when it is significant can lead to misleading inference.     

Asymptotically optimal tests for some growth curve models under non-normal error structure

This paper considers growth curve models consisting of two stages. In the first stage we have a regression model with random parameter. The second stage is given by a linear model for the mean of the random parameter of the first stage. In the present paper we prove asymptotic optimality of tests for linear functions of the mean of the random parameter under non-normal error. Received: September 1999     

A predictability test for a small number of nested models

We introduce quasi-likelihood ratio tests for one sided multivariate hypotheses to evaluate the null that a parsimonious model performs equally well as a small number of models which nest the benchmark. The limiting distributions of the test statistics are non-standard. For critical values we consider: (i) bootstrapping and (ii) simulations assuming normality of the mean square prediction error difference. The proposed tests have good size and power properties compared with existing equal and superior predictive ability tests for multiple model comparison. We apply our tests to study the predictive ability of a Phillips curve type for the US core inflation.     

Tourism forecasting: accuracy of alternative econometric models

This study evaluates the forecasting accuracy of six alternative econometric models in the context of the demand for international tourism in Denmark. These econometric models are special cases of a general autoregressive distributed lag specification. In addition, the forecasting accuracy of two univariate time series models is evaluated for benchmark comparison purposes. The forecasting competition is based on annual data on inbound tourism to Denmark. Individual models are estimated for each of the six major origin countries over the period 1969–93 and forecasting performance is assessed using data for the period 1994–97. Rankings of these forecasting models over different time horizons are established based on mean absolute percentage error and root mean square percentage error.     

Cointegration Testing in Single Error-Correction Equations in the Presence of Linear Time Trends

Banerjee, Dolado and Mestre (J. Time Ser. Anal. 19 (1998) 267−283)introduce an error-correction test for the null hypothesis of no cointegration. The present paper supplements their work. They provide critical values forregressions with and without detrending. Here it is shown that the latter arenot appropriate if the series display linear trends. This does not mean thatdetrending is required. Correct percentiles are suggested for the case thatseries follow linear time trends but tests are based on regressions withoutdetrending. They are readily available from the literature.     

Identifiability of mean‐reverting measurement error with instrumental variable

In the context where one main regressor is measured with error and at least one instrumental variable is available for the correction of measurement error, this paper provides, to the best of our knowledge, a first point‐identification result on the variance of measurement error, the variance of latent variable, and their covariance. We show that the parameters are identified if the regression model is not de facto linear. We illustrate the method in an application to identify mean‐reverting measurement error, a typical issue in reported income where the measurement error of income is negatively correlated with the true income.     

Asymptotic expansion of the minimax value in survey sampling

Summary Suppose that a real numbery _u is associated with each unitu of a populationU and that the functiony:u →y _u onU is known to be an element of the parameter space Θ. The statistician has to select a samples ⊂U ofn units and to employy _u;u ∈s to estimate the arithmetic mean of ally _u,u ∈U. The performance of such a strategy is assessed by its mean square error or, more simply, by the supremum of the mean square error. This supremum cannot be determined exactly for the parameter space of Scott/Smith (1975). We propose, therefore, an asymptotic approximation; this approximation is based on the assumption, that the sample sizen is fixed and that linear estimators have to be used.     

Risk management in life insurance companies: Evidence from Taiwan

The solvency issue of life insurance companies has become more important in recent years as business risks turn increasingly greater. This study examines the relationship among investing risk, underwriting risk, and the capital ratio during the post risk-based capital regulation period of 2004–2009 in Taiwan. In addition to the two-stage least square regression (2SLS), we also adopt the two-stage quantile regression (2SQR) to capture the effects of low capital (or risk) levels and high capital (or risk) levels. 2SLS do not fully explain the capital-risk relation. Contrary to previous evidence reported in the U.S., our findings in 2SQR model indicate that the relationship between capital and underwriting risk is positive, while the relationship between investing risk and capital shows a reverse pattern. Overall, the 2SQR provides stronger evidence than the 2SLS.     

An efficient variant of the product and ratio estimators

Abstract  This article presents a variant of the usual ratio and product methods of estimation, with the intention 10 improve their efficiency. The first order large sample approximations to the bias and the mean square error of the proposed estimator are obtained and compared with those of the well-known methods (simple expansion, ratio, product, difference and linear regression methods). For a special case, the accuracy of the first order approximation (terms up to the order n-¹ ) is examined by including terms upto the order n^-2 . With suitable choice of a design parameter, the proposed estimator turns out to be superior to the three methods mentioned first. The relation to the other two methods is examined; if the design parameter can be chosen near to the optimal value, the proposed method is seen to be approximately as efficient as the linear regression estimator. Finally some extensions are indicated.     

Goodness-of-fit tests for long memory moving average marginal density

This paper addresses the problem of fitting a known density to the marginal error density of a stationary long memory moving average process when its mean is known and unknown. In the case of unknown mean, when mean is estimated by the sample mean, the first order difference between the residual empirical and null distribution functions is known to be asymptotically degenerate at zero, and hence can not be used to fit a distribution up to an unknown mean. In this paper we show that by using a suitable class of estimators of the mean, this first order degeneracy does not occur. We also investigate the large sample behavior of tests based on an integrated square difference between kernel type error density estimators and the expected value of the error density estimator based on errors. The asymptotic null distributions of suitably standardized test statistics are shown to be chi-square with one degree of freedom in both cases of the known and unknown mean. In addition, we discuss the consistency and asymptotic power against local alternatives of the density estimator based test in the case of known mean. A finite sample simulation study of the test based on residual empirical process is also included.     

Acknowledgement

This paper examines the conditions under which it is possible to make comparisons between restricted least squares estimators under a generalized mean square error criterion. Our results demonstrate the rather extreme limitations on such comparisons — in many cases it is not possible to compare two restricted least squares estimators using the generalized mean square error criterion.     

Estimating an economic model of crime using panel data from North Carolina

This paper replicates the Cornwell and Trumbull ( 1994 ) estimation of a crime model using panel data on 90 counties in North Carolina over the period 1981–1987. While the Between and Within estimates are replicated, the fixed effects 2SLS as well as the 2SLS estimates are not. In fact, the fixed effects 2SLS estimates turn out to be insignificant for all important deterrent variables as well as legal opportunity variables. We argue that the usual Hausman test, based on the difference between fixed effects and random effects, may lead to misleading inference when endogenous variables of the conventional simultaneous equation type are among the regressors. We estimate the model using random effects 2SLS and perform a Hausman test based on the difference between fixed effects 2SLS and random effects 2SLS. We cannot reject the consistency of the random effects 2SLS estimator and this estimator yields plausible and significant estimates of the crime model. This result should be tempered by the legitimacy of the chosen instruments. Copyright © 2006 John Wiley & Sons, Ltd.