The asymptotic equivalence between the iterated improved 2sls estimator and the 3sls estimator

In this paper we show that the 3SLS estimator of a system of equations is asymptotically equivalent to an iterative 2SLS estimator applied to each equation, augmented with the residuals from the other equations. This result is a natural extension of Telser (1964).     

Stein-like 2SLS estimator

ABSTRACT

Maasoumi (1978 Maasoumi, E. (1978). A modified Stein-like estimator for the reduced form coefficients of simultaneous equations. Econometrica 46:695–703.[Crossref], [Web of Science ®] , [Google Scholar]) proposed a Stein-like estimator for simultaneous equations and showed that his Stein shrinkage estimator has bounded finite sample risk, unlike the three-stage least square estimator. We revisit his proposal by investigating Stein-like shrinkage in the context of two-stage least square (2SLS) estimation of a structural parameter. Our estimator follows Maasoumi (1978 Maasoumi, E. (1978). A modified Stein-like estimator for the reduced form coefficients of simultaneous equations. Econometrica 46:695–703.[Crossref], [Web of Science ®] , [Google Scholar]) in taking a weighted average of the 2SLS and ordinary least square estimators, with the weight depending inversely on the Hausman (1978 Hausman, J. A. (1978). Specification tests in econometrics. Econometrica 46:1251–1271.[Crossref], [Web of Science ®] , [Google Scholar]) statistic for exogeneity. Using a local-to-exogenous asymptotic theory, we derive the asymptotic distribution of the Stein estimator and calculate its asymptotic risk. We find that if the number of endogenous variables exceeds 2, then the shrinkage estimator has strictly smaller risk than the 2SLS estimator, extending the classic result of James and Stein (1961 James W, ., Stein, C. M. (1961). Estimation with quadratic loss. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability 1:361–380. [Google Scholar]). In a simple simulation experiment, we show that the shrinkage estimator has substantially reduced finite sample median squared error relative to the standard 2SLS estimator.     

The successive raising estimator and its relation with the ridge estimator

The raised estimators are used to reduce collinearity in linear regression models by raising a column in the experimental data matrix which may be nearly linear with the other columns. The raising procedure has two components, namely stretching and rotating, which we can analyze separately. We give the relationship between the raised estimators and the classical ridge estimators. Using a case study, we show how to determine the perturbation parameter for the raised estimators by controlling the amount of precision to be retained in the original data.     

The relative performances of improved ridge estimators and an empirical bayes estimator: some monte carlo results

The relative 'performances of improved ridge estimators and an empirical Bayes estimator are studied by means of Monte Carlo simulations. The empirical Bayes method is seen to perform consistently better in terms of smaller MSE and more accurate empirical coverage than any of the estimators considered here. A bootstrap method is proposed to obtain more reliable estimates of the MSE of ridge esimators. Some theorems on the bootstrap for the ridge estimators are also given and they are used to provide an analytical understanding of the proposed bootstrap procedure. Empirical coverages of the ridge estimators based on the proposed procedure are generally closer to the nominal coverage when compared to their earlier counterparts. In general, except for a few cases, these coverages are still less accurate than the empirical coverages of the empirical Bayes estimator.     

Further results on the two-stage hierarchial information (2shi) estimators in the linear regression models

The paper considers a class of 2SHI estimators for the linear regression models and provides some results regarding the dominance in quadratic loss of this class over the OLS and usual Stein-rule estimators.     

The ARCH(2) model: Pseudo-maximum estimation and asymptotic results under dependent innovations

This paper investigates the pseudo-maximum likelihood (PML) estimation of an ARCH(2) model when the innovations' law belongs to the quadratic exponential family. In addition, the error terms are conditionally independent, but not necessarily dependent. The consistency and asymptotic normality of the PML estimator are obtained by means of martingale techniques.     

Measuring the degree of severity of heteroskedasticity and the choice between the ols estimator and the 2sae

This paper dwells on the choice between the ordinary least squares and the estimated generalized least squares estimators when the presence of heteroskedasticity is suspected. Since the estimated generalized least squares estimator does not dominate the ordinary least squares estimator completely over the whole parameter space, it is of interest to the researcher to know in advance whether the degree of severity of heteroskedasticity is such that OLS estimator outperforms the estimated generalized least squares (or 2SAE). Casting the problem in the non-spherical error mold and exploiting the principle underlying the Bayesian pretest estimator, an intuitive non-mathematical procedure is proposed to serve as an aid to the researcher in deciding when to use either the ordinary least squares (OLS) or the estimated generalized least squares (2SAE) estimators.     

Asymptotic approximations to the gain of the 2shi over stein estimators in linear regression models when the disturbances are small

The paper considers a new family of explicit or fully operational two-stage Stein or hierarchial information (2SHI) estimators for linear regression models, and provides an expression for the difference between the risks of these estimators and the usual Stein-rule estimator when the variance of the disturbance is small. The condition under which the 2SHI estimators have smaller average MSE than the Stein-rule estimator is also given.     

A study of the Gamma-Pareto (IV) distribution and its applications

ABSTRACT

Pareto distributions and their close relatives and generalizations provide very flexible families of heavy-tailed distributions that may be used to model income distributions as well as a wide variety of other social and economic distributions. On the other hand, gamma distribution has a wide application in various social and economic spheres such as survival analysis, to model aggregate insurance claims, and the amount of rainfall accumulated in a reservoir etc. Combining the above two heavy-tailed distributions, using the technique by Alzaatreh et al. (2012 Alzaatreh, A., Famoye, F., Lee, C. (2012). Gamma-Pareto distribution and its applications. J. Modern Appl. Stat. Methods. 11:78–94.[Crossref] , [Google Scholar]), we define a new distribution, namely Gamma-Pareto (IV) distribution, hereafter called as GPD^(IV) distribution. Various properties of the GPD^(IV) are investigated such as limiting behavior, moments, mode, and Shannon entropy. Also some characterizations of the GPD^(IV) distribution are mentioned in this paper. Maximum likelihood method is proposed for estimating the model parameters. For illustrative purposes, real data sets are considered as applications of the GPD^(IV) distribution.     

An efficient and fast algorithm for estimating the frequencies of 2-D superimposed exponential signals in presence of zero-mean multiplicative and additive noise

In this paper, a computationally efficient algorithm is proposed for estimating the parameters of two-dimensional (2-D) superimposed exponential signals in presence of independently and identically distributed (i.i.d.) zero-mean multiplicative and additive noise. It is observed that the estimator is consistent and works quite well in terms of biases and mean squared errors. Moreover, the algorithm is efficient when multiple 2-D frequencies pairs share a same 1-D frequency component and the estimators attain the same convergence rate with the least squares estimator (LSE) in presence of additive noise. Finally, it is observed that the algorithm can be used to estimate the frequencies of the evanescent component of texture accurately.     

Consistency and asymptotic normality of the estimated effective doses in bioassay

In order to estimate the effective dose such as the 0.5 quantile _ED50${\mathrm{ED}}_{50}$ in a bioassay problem various parametric and semiparametric models have been used in the literature. If the true dose–response curve deviates significantly from the model, the estimates will generally be inconsistent. One strategy is to analyze the data making only a minimal assumption on the model, namely, that the dose–response curve is non-decreasing. In the present paper we first define an empirical dose–response curve based on the estimated response probabilities by using the “pool-adjacent-violators” (PAV) algorithm, then estimate effective doses _ED100p${\mathrm{ED}}_{100p}$ for a large range of p by taking inverse of this empirical dose–response curve. The consistency and asymptotic distribution of these estimated effective doses are obtained. The asymptotic results can be extended to the estimated effective doses proposed by Glasbey [1987. Tolerance-distribution-free analyses of quantal dose–response data. Appl. Statist. 36 (3), 251–259] and Schmoyer [1984. Sigmoidally constrained maximum likelihood estimation in quantal bioassay. J. Amer. Statist. Assoc. 79, 448–453] under the additional assumption that the dose–response curve is symmetric or sigmoidal. We give some simulations on constructing confidence intervals using different methods.     

Convergence rate of the kernel regression estimator for associated and truncated data

This paper studies the behaviour of the kernel estimator of the regression function for associated data in the random left truncated model. The uniform strong consistency rate over a real compact set of the estimate is established. The finite sample performance of the estimator is investigated through extensive simulation studies.     

The asymptotic covariance structure of log-linear model estimated parameters for the multiple recapture census

This paper presents a matrix formulation for log-linear model analysis of the incomplete contingency table which arises from multiple recapture census data. Explicit matrix product expressions are given for the asymptotic covariance structure of the maximum likelihood estimators of both the log-linear model parameter vector and the predicted value vector for the observed and missing cells. These results are illustrated for data pertaining to a population of children possessing a common congenital anomaly.     

The empirical Bayes estimators of the mean and variance parameters of the normal distribution with a conjugate normal-inverse-gamma prior by the moment method and the MLE method

Most of the samples in the real world are from the normal distributions with unknown mean and variance, for which it is common to assume a conjugate normal-inverse-gamma prior. We calculate the empirical Bayes estimators of the mean and variance parameters of the normal distribution with a conjugate normal-inverse-gamma prior by the moment method and the Maximum Likelihood Estimation (MLE) method in two theorems. After that, we illustrate the two theorems for the monthly simple returns of the Shanghai Stock Exchange Composite Index.     

On the duality between long-run relations and common trends in the i(1) versus(2)model.an application to aggregate money holdings

Long-run relations and common trends are discussed in terms of the multivariate cointegration model given in the autoregressive and the moving average form. The basic results needed for the analysis of I(1) and 1(2)processes are reviewed and the results applied to Danish monetary data. The test procedures reveal that nominal money stock is essentially I(2). Long-run price homogeneity is supported by the data and imposed on the system. It is found that the bond rate is weakly exogenous for the long-run parameters and therefore act as a driving trend. Using the nonstationarity property of the data, “excess money” is estimated and its effect on the other determinants of the system is investigated. In particular, it is found that “excess money” has no effect on price inflation.     

Pseudo maximum likelihood estimation of the univariate GARCH (2,2) and asymptotic normality under dependent innovations

In this paper, we first consider the pseudo maximum likelihood estimation of the univariate GARCH (2,2) model and derive the underlying estimator. Then, we make use of the technique of martingales to establish the asymptotic normality of the pseudo-maximum likelihood estimator (PMLE) of the univariate GARCH (2,2) model. Contrary to previous approaches encountered in the statistical literature, the pseudo-likelihood function uses the general form of the density laws of the quadratic exponential family.     

Bayesian Inference in Cointegrated I (2) Systems: A Generalization of the Triangular Model

This paper generalizes the cointegrating model of Phillips (1991 Phillips , P. C. B. ( 1991 ). Optimal inference in cointegrated systems . Econometrica 59 : 283 – 306 .[Crossref], [Web of Science ®] , [Google Scholar]) to allow for I (0), I (1) and I (2) processes. The model has a simple form that permits a wider range of I (2) processes than are usually considered, including a more flexible form of polynomial cointegration. Further, the specification relaxes restrictions identified by Phillips (1991 Phillips , P. C. B. ( 1991 ). Optimal inference in cointegrated systems . Econometrica 59 : 283 – 306 .[Crossref], [Web of Science ®] , [Google Scholar]) on the I (1) and I (2) cointegrating vectors and restrictions on how the stochastic trends enter the system. To date there has been little work on Bayesian I (2) analysis and so this paper attempts to address this gap in the literature. A method of Bayesian inference in potentially I (2) processes is presented with application to Australian money demand using a Jeffreys prior and a shrinkage prior.     

Bayesian Inference in Cointegrated I (2) Systems: A Generalization of the Triangular Model

This paper generalizes the cointegrating model of Phillips (1991) to allow for I (0), I (1) and I (2) processes. The model has a simple form that permits a wider range of I (2) processes than are usually considered, including a more flexible form of polynomial cointegration. Further, the specification relaxes restrictions identified by Phillips (1991) on the I (1) and I (2) cointegrating vectors and restrictions on how the stochastic trends enter the system. To date there has been little work on Bayesian I (2) analysis and so this paper attempts to address this gap in the literature. A method of Bayesian inference in potentially I (2) processes is presented with application to Australian money demand using a Jeffreys prior and a shrinkage prior.     

The empirical Bayes estimators of the parameter of the Poisson distribution with a conjugate gamma prior under Stein's loss function

For the hierarchical Poisson and gamma model, we calculate the Bayes posterior estimator of the parameter of the Poisson distribution under Stein's loss function which penalizes gross overestimation and gross underestimation equally and the corresponding Posterior Expected Stein's Loss (PESL). We also obtain the Bayes posterior estimator of the parameter under the squared error loss and the corresponding PESL. Moreover, we obtain the empirical Bayes estimators of the parameter of the Poisson distribution with a conjugate gamma prior by two methods. In numerical simulations, we have illustrated: The two inequalities of the Bayes posterior estimators and the PESLs; the moment estimators and the Maximum Likelihood Estimators (MLEs) are consistent estimators of the hyperparameters; the goodness-of-fit of the model to the simulated data. The numerical results indicate that the MLEs are better than the moment estimators when estimating the hyperparameters. Finally, we exploit the attendance data on 314 high school juniors from two urban high schools to illustrate our theoretical studies.     

煤炭矿区生态环境改善的支付意愿与受偿意愿的差异性分析——以榆林市神木县、府谷县和榆阳区为例

支付意愿（WTP）与受偿意愿（WTA）是条件价值评估法（CVM）评估生态环境影响价值的两个测量尺度。然而在实际应用中,同一种环境物品的WTP与Ⅵ吓A之间却存在着显著的差异。以榆林市神木县、府谷县和榆阳区为例,运用CVM对煤炭矿区居民的WTP与WTA进行了研究,发现WTP与WTA之间的差异较大。通过对WTP与WTA之间差异的经济社会因素进行实证分析,发现WTP与WTA之间的差异受被调查者的职业状况、年龄及所在行政区域的影响较大。在中国煤炭矿区的CVM应用中,应以WTP作为首选,可以将WTA作为WTP的有效补充。