Optimal linear estimators of location and scale parameters using order statistics and related empirical Bayes estimation

Summary

An estimator which is a linear function of the observations and which minimises the expected square error within the class of linear estimators is called an “optimal linear” estimator. Such an estimator may also be regarded as a “linear Bayes” estimator in the spirit of Hartigan (1969). Optimal linear estimators of the unknown mean of a given data distribution have been described by various authors; corresponding “linear empirical Bayes” estimators have also been developed.

The present paper exploits the results of Lloyd (1952) to obtain optimal linear estimators based on order statistics of location or/and scale parameter (s) of a continuous univariate data distribution. Related “linear empirical Bayes” estimators which can be applied in the absence of the exact knowledge of the optimal estimators are also developed. This approach allows one to extend the results to the case of censored samples.     

Credibility ratemaking using collateral information

Credibility ratemaking is a technique used in pricing health care, property and casualty, workers’ compensation, and group life coverages. It has been a part of actuarial practice since the time of Mowbray's (1914) contribution. In earlier work, we showed how many types of credibility models could be expressed as special cases of mixed linear models. This article extends this approach to credibility by formally introducing collateral information through the use of Bayesian methods.

Specifically, we derive credibility estimators and mean square errors for normal hierarchical linear models. We provide intuition for the credibility estimators by establishing the link between these estimators and homogeneous and inhomogeneous estimators that appear in non-Bayesian credibility theory.     

Parametric,semiparametric, and nonparametric estimation of characteristic values within mass assessment and hedonic pricing models

Parametric estimators, such as OLS, attain high efficiency for well-specified models. Nonparametric estimators greatly reduce specification error but at the cost of efficiency. Semiparametric estimators compromise between these dual goals of efficiency and specification error. Semiparametric estimators can assume general forms within classes of functional forms. This paper applies OLS, the kernel nonparametric regression estimator, and the semi-parametric estimator of Powell, Stock, and Stoker (1989) to a data set, which should, based on theory and previous empirical work, yield positive coefficients. The semiparametric estimator, on average, displayed the performance most consistent with prior expectations followed by the nonparametric and parametric estimators. In addition, the paper shows how the semiparametric estimator can provide insights into the form of misspecification and suggest data transformations.     

GMM estimation of the number of latent factors: With application to international stock markets

We propose new generalized method of moments (GMM) estimators for the number of latent factors in linear factor models. The estimators are appropriate for data with a large (small) number of cross-sectional observations and a small (large) number of time series observations. The estimation procedure is simple and robust to the configurations of idiosyncratic errors encountered in practice. In addition, the method can be used to evaluate the validity of observable candidate factors. Monte Carlo experiments show that the proposed estimators have good finite-sample properties. Applying the estimators to international stock markets, we find that international stock returns are explained by one strong global factor. This factor is highly correlated with the Fama–French factors from the U.S. stock market. This result can be interpreted as evidence of market integration. We also find two weak factors closely related to markets in Europe and the Americas, respectively.     

Granularity adjustment for default risk factor model with cohorts

This paper examines granularity adjustments to parameter estimators in a default risk model with cohorts. The model is an extension of the Vasicek model (Vasicek, 1991) and includes a general factor and cohort specific factors. The granularity adjustments derived in the paper concern the mean and/or the variance of observed default frequencies and are easy to implement in practice. For illustration, the method is applied to the S&P corporate ratings. The Granularity Adjusted (GA) estimators are compared to the unadjusted estimators in terms of their asymptotic properties and in finite sample.     

On risk management problems related to a coherence property

Value at Risk has lost the battle against Expected Shortfall on theoretical grounds, the latter satisfying all coherence properties while the former may, on carefully constructed cases, lack the sub-additivity property that is in a sense, the most important property a risk measure ought to satisfy. While the superiority of Expected Shortfall is evident as a theoretical tool, little has been researched on the properties of estimators proposed in the literature. Since those estimators are the real tools for calculating bank capital reserves in practice, the natural question that one may ask is whether a given estimator of Expected Shortfall also satisfies the coherence properties. In this paper, we show that it is possible to have estimators of Expected Shortfall that do not satisfy the sub-additivity condition. This finding should motivate risk managers and quantitative asset managers to investigate further the properties of the estimators of the risk measures they are currently utilizing.     

Presenting hypothesis testing by analogy

Auditing students often have difficulty with hypothesis testing. In this article, a concise approach for presenting hypothesis testing is discussed. Specifically, the author shows that the construction of an hypothesis test for the mean per-unit estimator is analogous to the formulation of such a test for auxiliary estimators. Because these tests are comparable, the approach encompasses a logical progression from the mean per-unit estimator to the auxiliary estimators. This approach enables students to perceive the similarities among these estimators and to better understand and assimilate the subject matter.     

The Geometry of the Maximum Likelihood Estimator of the Zero-Beta Return

This paper explores geometric relations, in mean-variance space, among the sample frontier, the maximum likelihood estimator, and two other estimators of the zerobeta return. It is also demonstrated that a partition of the portfolio space is determined by a family of parabolas; the zeros of each parabola are the maximum likelihood estimators associated with all portfolios on the parabola. This observation is the basis for an additional interpretation of the statistic of the Likelihood Ratio Test of portfolio efficiency without a riskless asset.     

The heteroskedasticity-consistent covariance estimator in accounting

The main purpose of this paper is to compare the White (1980) heteroskedasticity-consistent (HC) covariance matrix estimator with alternative estimators. Many regression packages compute the White (1980) heteroskedasticity-consistent (HC) covariance matrix estimator. The common procedure in Accounting and Finance research to deal with the heteroskedasticity problem is based on this estimator, despite its worse finite-samples properties when compared with other consistent estimators. In this paper we compare several HC covariance matrix estimators based on a sample of 3706 European listed companies from Austria, Finland, France, Germany, Greece, Ireland, Italy, Netherlands, Norway, Portugal, Spain, Sweden and the United Kingdom. We conclude that HC standard errors increase when finite-samples more appropriate estimators are considered and in the most part of countries the Ohlson (1995) model coefficients estimates became statistically insignificant. This can be explained by the high leverage points in the design matrix. To the best of our knowledge it is the first time that these alternative estimators are compared with the one of White (1980) in accounting research.     

Robust Estimation of Beta Coefficients: Evidence from a Small Stock Market

In this paper we demonstrate that robust estimators improve the reliability of estimates of beta coefficients on small, thinly traded stock markets. We outline several different types of robust and bounded influence regression estimators and assess them using a jackknife methodology on data from the Johannesburg Stock Exchange. The empirical evidence confirms the hypothesis that robust estimators are more efficient than least squares estimators and indicates that least squares estimators may over-estimate systematic risk in some cases.     

Improved estimation of the covariance matrix of stock returns with an application to portfolio selection

This paper proposes to estimate the covariance matrix of stock returns by an optimally weighted average of two existing estimators: the sample covariance matrix and single-index covariance matrix. This method is generally known as shrinkage, and it is standard in decision theory and in empirical Bayesian statistics. Our shrinkage estimator can be seen as a way to account for extra-market covariance without having to specify an arbitrary multifactor structure. For NYSE and AMEX stock returns from 1972 to 1995, it can be used to select portfolios with significantly lower out-of-sample variance than a set of existing estimators, including multifactor models.     

Portfolio analysis of intraday covariance matrix in the Greek equity market

The intraday nonparametric estimation of the variance–covariance matrix adds to the literature in portfolio analysis of the Greek equity market. This paper examines the economic value of various realized volatility and covariance estimators under the strategy of volatility timing. I use three types of portfolios: Global Minimum Variance, Capital Market Line and Capital Market Line with only positive weights. The estimators of volatilities and covariances use 5-min high-frequency intraday data. The dataset concerns the FTSE/ATHEX Large Cap index, FTSE/ATHEX Mid Cap index, and the FTSE/ATHEX Small Cap index of the Greek equity market (Athens Stock Exchange). As far as I know, this is the first work of its kind for the Greek equity market. Results concern not only the comparison of various estimators but also the comparison of different types of portfolios, in the strategy of volatility timing. The economic value of the contemporary non-parametric realized volatility estimators is more significant than this when the covariance is estimated by the daily squared returns. Moreover, the economic value (in b.p.s) of each estimator changes with the volatility timing.     

Nonparametric methods with applications to hedonic models

Current real estate statistical valuation involves the estimation of parameters within a posited specification. Suchparametric estimation requires judgment concerning model (1) variables; and (2) functional form. In contrast,nonparametric regression estimation requires attention to (1) but permits greatly reduced attention to (2). Parametric estimators functionally model the parameters and variables affectingE(y¦x) while nonparametric estimators directly modelpdf(y, x) and henceE(y¦x).This article applies the kernel nonparametric regression estimator to two different data sets and specifications. The article shows the nonparametric estimator outperforms the standard parametric estimator (OLS) across variable transformations and across data subsets differing in quality. In addition, the article reviews properties of nonparametric estimators, presents the history of nonparametric estimators in real estate, and discusses a representation of the kernel estimator as a nonparametric grid method.     

信用风险模型的贝叶斯改进研究

基于小样本数据和外部先验信息,本文运用贝叶斯(Bayes)估计量来改进信用风险模型的违约预测力。同时,运用中国上市公司财务数据,分别对贝叶斯估计量和标准Logit估计量进行了模拟估计,并通过统计量AUC值和布莱尔分数(Brier Score)对其预测精度进行比较。结果表明,贝叶斯估计量具有更高的预测精度和稳定性。     

Nonparametric estimation of scalar diffusion models of interest rates using asymmetric kernels

This paper proposes an asymmetric kernel-based method for nonparametric estimation of scalar diffusion models of spot interest rates. We derive the asymptotic theory for the asymmetric kernel estimators of the drift and diffusion functions for general and positive recurrent processes and illustrate the advantages of the Gamma kernel for bias correction and efficiency gains. The finite-sample properties and the practical relevance of the proposed nonparametric estimators for bond and option pricing are evaluated using actual and simulated data for U.S. interest rates.     

Empirical Bayes credibility

Abstract

A credibility estimator is Bayes in the restricted class of linear estimators and may be viewed as a linear approximation to the (unrestricted) Bayes estimator. When the structural parameters occurring in a credibility formula are replaced by consistent estimators based on data from a collective of similar risks,we obtain an empirical credibility estimator, which is a credibility counterpart of empirical Bayes estimators. Empirical credibility estimators are proposed under various model assumptions, and sufficient conditions for asymptotic optimality are established.     

Empirical estimation of tail dependence using copulas: application to Asian markets

This paper introduces non-parametric estimators for upper and lower tail dependence whose confidence intervals are obtained with a bootstrap method. We call these estimators ‘naïve estimators’ as they represent a discretization of Joe's formulae linking copulas to tail dependence. We apply the methodology to an empirical data set composed of three composite indexes for the three Tigers (Thailand, Malaysia and Indonesia). The extremes show a dependence structure which is symmetric for the Thai and Malaysian markets and asymmetric for the Thai and Indonesian markets and for the Malaysian and the Indonesian markets. Using these results we estimate the copula (which belongs to the Student or Archimedean copula families) for each pair of markets by two methods. Finally, we provide risk measurements using the best copula associated with each pair of markets.     

Aggregation of Nonparametric Estimators for Volatility Matrix

An aggregated method of nonparametric estimators based on time-domainand state-domain estimators is proposed and studied. To attenuatethe curse of dimensionality, we propose a factor modeling strategy.We first investigate the asymptotic behavior of nonparametricestimators of the volatility matrix in the time domain and inthe state domain. Asymptotic normality is separately establishedfor nonparametric estimators in the time domain and state domain.These two estimators are asymptotically independent. Hence,they can be combined, through a dynamic weighting scheme, toimprove the efficiency of volatility matrix estimation. Theoptimal dynamic weights are derived, and it is shown that theaggregated estimator uniformly dominates volatility matrix estimatorsusing time-domain or state-domain smoothing alone. A simulationstudy, based on an essentially affine model for the term structure,is conducted, and it demonstrates convincingly that the newlyproposed procedure outperforms both time- and state-domain estimators.Empirical studies further endorse the advantages of our aggregatedmethod.     

Estimation of DSGE models when the data are persistent

Dynamic stochastic general equilibrium (DSGE) models are often solved and estimated under specific assumptions as to whether the exogenous variables are difference or trend stationary. However, even mild departures of the data generating process from these assumptions can severely bias the estimates of the model parameters. This paper proposes new estimators that do not require researchers to take a stand on whether shocks have permanent or transitory effects. These procedures have two key features. First, the same filter is applied to both the data and the model variables. Second, the filtered variables are stationary when evaluated at the true parameter vector. The estimators are approximately normally distributed not only when the shocks are mildly persistent, but also when they have near or exact unit roots. Simulations show that these robust estimators perform well especially when the shocks are highly persistent yet stationary. In such cases, linear detrending and first differencing are shown to yield biased or imprecise estimates.