Testing the capital asset pricing model efficiently under elliptical symmetry: a semiparametric approach

We develop new tests of the capital asset pricing model that take account of and are valid under the assumption that the distribution generating returns is elliptically symmetric; this assumption is necessary and sufficient for the validity of the CAPM. Our test is based on semiparametric efficient estimation procedures for a seemingly unrelated regression model where the multivariate error density is elliptically symmetric, but otherwise unrestricted. The elliptical symmetry assumption allows us to avoid the curse of dimensionality problem that typically arises in multivariate semiparametric estimation procedures, because the multivariate elliptically symmetric density function can be written as a function of a scalar transformation of the observed multivariate data. The elliptically symmetric family includes a number of thick‐tailed distributions and so is potentially relevant in financial applications. Our estimated betas are lower than the OLS estimates, and our parameter estimates are much less consistent with the CAPM restrictions than the corresponding OLS estimates. Copyright © 2002 John Wiley & Sons, Ltd.     

A flexible test of symmetry

Research in consumer behavior, or other surveys, where there is interest in the comparison of response measurement can produce data that is conveniently summarized in an m×m contingency table. In this case, interest may lie in the symmetry of the data in the table or in the homogeneity of the marginal distributions. The purpose of this paper is to examine some of these ideas and propose a test of symmetry which has advantages over the commonly used statistics.     

Estimation methods for male labor supply functions How to take account of nonlinear taxes

Nonlinear taxes create econometric difficulties when estimating labor supply functions. One estimation method that tackles these problems accounts for the complete form of the budget constraint and uses the maximum likelihood method to estimate parameters. Another method linearizes budget constraints and uses instrumental variables techniques. Using Monte Carlo simulations I investigate the small-sample properties of these estimation methods and how they are affected by measurement errors in independent variables. No estimator is uniquely best. Hence, in actual estimation the choice of estimator should depend on the sample size and type of measurement errors in the data. Complementing actual estimates with a Monte Carlo study of the estimator used, given the type of measurement errors that characterize the data, would often help interpreting the estimates. This paper shows how such a study can be performed.     

Econometrically accounting for identities and restrictions in models of interregional migration : A comment

This comment clarifies and expands on an earlier article in this Journal by Greenwood and Hunt which addresses problems of estimating interregional migration models when non-stochastic adding-up constraints are present. The comment focuses on issues of specification, estimation and hypothesis testing within a closed system of migration equations stressing solutions to the degrees of freedom problem and tests for homogeneity and symmetry.     

Asymmetry and nonstationarity for a seasonal time series model

Tests for symmetry and seasonal unit roots are developed for an extended model of Hylleberg et al. (1990. Seasonal integration and cointegration. Journal Econometrics 44, 215–238.) which can represent both partial seasonal unit roots and threshold effects. Methods based on ordinary least squares (OLS) estimation and instrumental variable (IV) estimation are proposed and compared. For adjusting mean functions, ordinary mean adjustment and recursive mean adjustment are both considered. Several tests are constructed from various combination of estimation schemes and mean adjustment schemes. Among the tests, the tests based on IV-estimation are recommended because they have very simple limiting null distributions and have finite sample power properties comparable to those based on the OLSE. The recommended tests are applied to a US unemployment rate data set and find evidences for both nonstationarities associated with zero frequency and threshold effects.     

Sequential estimation in generalized linear models when covariates are subject to errors

Sequential methods have been used for many applications; especially, when fixed sample procedures are not possible and/or when “early stopping” of sampling is beneficial for applications. At the same time, the issue of how to make correct inferences when measurement errors are present has drawn considerable attention from statisticians. In this paper, the problems of sequential estimation of generalized linear models when there are measurement errors in both adaptive and fixed design cases are studied. The proposed sequential procedure is proved to be asymptotically consistent and efficient in the sense of Chow and Robbins [Ann Math Stat 36(2):457–462, 1965] when measurement errors decay gradually as the number of sequentially selected design points increases. This assumption is useful in sequentially designed experiments, and can also be fulfilled in the case when replicate measurements are available. Some numerical studies based on a Rasch model and a logistic regression model are conducted to evaluate the performance of the proposed procedure.     

Problems in the estimation and interpretation of the reliability of survey data

In this paper I discuss several of the difficulties involved in estimating the reliability of survey measurement. Reliability is defined on the basis of classical true-score theory, as the correlational consistency of multiple measures of the same construct, net of true change. This concept is presented within the framework of a theoretical discussion of the sources of error in survey data and the design requirements for separating response variation into components representing such response consistency and measurement errors. Discussion focuses on the potential sources of random and nonrandom errors, including “invalidity” of measurement, the term frequently used to refer to components of method variance. Problems with the estimation of these components are enumerated and discussed with respect to both cross-sectional and panel designs. Empirical examples are given of the estimation of the quantities of interest, which are the basis of a discussion of the interpretational difficulties encountered in reliability estimation. Data are drawn from the ISR's Quality of Life surveys, the National Election Studies and the NORC's General Social Surveys. The general conclusion is that both cross-sectional and panel estimates of measurement reliability are desirable, but for the purposes of isolating the random component of error, panel designs are probably the most advantageous.     

Nonparametric identification and estimation of sample selection models under symmetry

Under a conditional mean restriction Das et al. (2003) considered nonparametric estimation of sample selection models. However, their method can only identify the outcome regression function up to a constant. In this paper we strengthen the conditional mean restriction to a symmetry restriction under which selection biases due to selection on unobservables can be eliminated through proper matching of propensity scores; consequently we are able to identify and obtain consistent estimators for the average treatment effects and the structural regression functions. The results from a simulation study suggest that our estimators perform satisfactorily.     

Testing and imposing Slutsky symmetry in nonparametric demand systems

Maximization of utility implies that consumer demand systems have a Slutsky matrix which is everywhere symmetric. However, previous non- and semi-parametric approaches to the estimation of consumer demand systems do not give estimators that are restricted to satisfy this condition, nor do they offer powerful tests of this restriction. We use nonparametric modeling to test and impose Slutsky symmetry in a system of expenditure share equations over prices and expenditure. In this context, Slutsky symmetry is a set of nonlinear cross-equation restrictions on levels and derivatives of consumer demand equations. The key insight is that due to the differing convergence rates of levels and derivatives and due to the fact that the symmetry restrictions are linear in derivatives, both the test and the symmetry restricted estimator behave asymptotically as if these restrictions were (locally) linear. We establish large and finite sample properties of our methods, and show that our test has advantages over the only other comparable test. All methods we propose are implemented with Canadian micro-data. We find that our nonparametric analysis yields statistically significantly and qualitatively different results from traditional parametric estimators and tests.     

A simple estimator for nonlinear error in variable models

We propose a simple estimator for nonlinear method of moment models with measurement error of the classical type when no additional data, such as validation data or double measurements, are available. We assume that the marginal distributions of the measurement errors are Laplace (double exponential) with zero means and unknown variances and the measurement errors are independent of the latent variables and are independent of each other. Under these assumptions, we derive simple revised moment conditions in terms of the observed variables. They are used to make inference about the model parameters and the variance of the measurement error. The results of this paper show that the distributional assumption on the measurement errors can be used to point identify the parameters of interest. Our estimator is a parametric method of moments estimator that uses the revised moment conditions and hence is simple to compute. Our estimation method is particularly useful in situations where no additional data are available, which is the case in many economic data sets. Simulation study demonstrates good finite sample properties of our proposed estimator. We also examine the performance of the estimator in the case where the error distribution is misspecified.     

Complete Neyman measurement procedure

Traditional estimation theory generally starts from point estimators, and based on them confidence regions with given confidence level are constructed. However, this approach works only in some special cases and, even more severe, it is based on the unrealistic but mathematical necessary assumption of a generally unbounded parameter space.  The procedures derived in this paper, start from a bounded measurement range which contains the potential values of the parameter of interest. For given measurement range and given reliability requirement measurement procedures including a point estimator are developed. The result are complete measurement procedures for distribution parameters. Most precise procedures are derived and called complete Neyman measurement procedures.     

Indirect inference with time series observed with error

We propose the indirect inference estimator as a consistent method to estimate the parameters of a structural model when the observed series are contaminated by measurement error by considering the noise as a structural feature. We show that the indirect inference estimates are asymptotically biased if the error is neglected. When the condition for identification is satisfied, the structural and measurement error parameters can be consistently estimated. The issues of identification and misspecification of measurement error are discussed in detail. We illustrate the reliability of this procedure in the estimation of stochastic volatility models based on realized volatility measures contaminated by microstructure noise.     

Semiparametric and nonparametric estimation of sample selection models under symmetry

This paper considers the semiparametric estimation of binary choice sample selection models under a joint symmetry assumption. Our approaches overcome various drawbacks associated with existing estimators. In particular, our method provides root-n$n$ consistent estimators for both the intercept and slope parameters of the outcome equation in a heteroscedastic framework, without the usual cross equation exclusion restriction or parametric specification for the error distribution and/or the form of heteroscedasticity. Our two-step estimators are shown to be consistent and asymptotically normal. A Monte Carlo simulation study indicates the usefulness of our approaches.     

Robustness tests of the augmented Solow model

This paper demonstrates some techniques for testing the robustness of cross-section and panel data regressions, and applies them to the influential augmented Solow growth model. The paper focuses on robust estimation and analysis of sensitivity to measurement error. In particular, it is shown that estimated technology parameters and convergence rates are highly sensitive to measurement error. © 1998 John Wiley & Sons, Ltd.     

The Wishart Autoregressive process of multivariate stochastic volatility

The Wishart Autoregressive (WAR) process is a dynamic model for time series of multivariate stochastic volatility. The WAR naturally accommodates the positivity and symmetry of volatility matrices and provides closed-form non-linear forecasts. The estimation of the WAR is straighforward, as it relies on standard methods such as the Method of Moments and Maximum Likelihood. For illustration, the WAR is applied to a sequence of intraday realized volatility–covolatility matrices from the Toronto Stock Market (TSX).     

Identification and Estimation of Engel Curves with Endogenous and Unobserved Expenditures

When dealing with the estimation of Engel curves, measurement errors in expenditure data and simultaneity are likely sources of endogeneity. In this paper we study identification of the parameters that characterize an Engel curve in the presence of both. We consider specifications where budget shares are polynomials in the logarithm of total expenditure, which is the case frequently encountered in empirical applications. We propose an estimation procedure which is an extension of that in Lewbel ( 1996 ) and exploits a control function assumption to correct for the endogeneity of the true unobserved total expenditure. Copyright © 2013 John Wiley & Sons, Ltd.     

ECONOMETRIC ISSUES IN MACROECONOMIC MODELS WITH GENERATED REGRESSORS

Abstract. The paper critically reviews the literature on the econometric issues raised by the use of generated regressors (GR) in empirical models. The economic rationale for the use of GR is considered, with examples being drawn from several macroeconomic examples, including New Classical Macroeconomic (NCM) models which postulate monetary ncutrality. Various estimation methods are discussed for models which include 'surprise' or 'unexpected' terms and the strengths and weaknesses of each approach are investigated. Drawing upon the work of McAleer and McKenzie (1991b), situations where the typically inefficient two-step estimation (2SE) method will be efficient are highlighted. Problems of model misspecification and measurement errors are also investigated. An empirical section highlights some of the dangers of using uncorrected 2SE estimation results through a careful consideration of many recent attempts to test the NCM monetary neutrality hypothesis.     

Well-posedness of measurement error models for self-reported data

This paper considers the widely admitted ill-posed inverse problem for measurement error models: estimating the distribution of a latent variable $X^1$ from an observed sample of $X$, a contaminated measurement of $X^1$. We show that the inverse problem is well-posed for self-reporting data under the assumption that the probability of truthful reporting is nonzero, which is supported by empirical evidences. Comparing with ill-posedness, well-posedness generally can be translated into faster rates of convergence for the nonparametric estimators of the latent distribution. Therefore, our optimistic result on well-posedness is of importance in economic applications, and it suggests that researchers should not ignore the point mass at zero in the measurement error distribution when they model measurement errors with self-reported data. We also analyze the implications of our results on the estimation of classical measurement error models. Then by both a Monte Carlo study and an empirical application, we show that failing to account for the nonzero probability of truthful reporting can lead to significant bias on estimation of the latent distribution.     

Zero expenditures and the estimation of engel curves

Over the brief interview periods typical of budget surveys (such as the U.K. Family Expenditure Survey) households' recorded expenditures are liable to be misleading indicators of their underlying consumption. Most starkly, a household may not be observed to purchase commodities that it nevertheless consumes. This paper considers the implications of such measurement errors for the estimation of demand systems, and derives a consistent estimator for the special case of linear Engel curves.     

Retransformation bias: A look at the box-cox transformation to linear balanced mixed ANOVA models

Summary After a Box-Cox transformation to data following a linear balanced mixed ANOVA model, final results may be presented after retransformation to the original scale of measurement. Consequently, estimation of means which may be unbiased in the transformed scale will not be so after retransformation. In this article, the bias introduced together with the corresponding variance is assessed. It is found that whereas bias may not be a serious problem, the variances are inflated for positive transformation parameter the closer it is to zero.