Yoshihiko Nishiyama

The Bootstrap and the Edgeworth Correction for Semiparametric Averaged Derivatives

In a number of semiparametric models, smoothing seems necessary in order to obtain estimates of the parametric component which are asymptotically normal and converge at parametric rate. However, smoothing can inflate the error in the normal approximation, so that refined approximations are of interest, especially in sample sizes that are not enormous. We show that a bootstrap distribution achieves a valid Edgeworth correction in the case of density-weighted averaged derivative estimates of semiparametric index models. Approaches to bias reduction are discussed. We also develop a higher-order expansion to show that the bootstrap achieves a further reduction in size distortion in the case of two-sided testing. The finite-sample performance of the methods is investigated by means of Monte Carlo simulations from a Tobit model. Copyright The Econometric Society 2005.

OLS Estimation and the T Test Revisited in Rank-Size Rule Regression

The rank-size rule and Zipfs law for city sizes have been traditionally examined by means of OLS estimation and the t test. This paper studies the accurate and approximate properties of the OLS estimator and obtains the distribution of the t statistic under the assumption of Zipfs law (i.e., Pareto distribution). Indeed, we show that the t statistic explodes asymptotically even under the null, indicating that a mechanical application of the t test yields a serious type I error. To overcome this problem, critical regions of the t test are constructed to test the Zipfs law. Using these corrected critical regions, we can conclude that our results are in favor of the Zipfs law for many more countries than in the previous researches such as Rosen and Resnick (1980) or Soo (2005). By using the same database as that used in Soo (2005), we demonstrate that the Zipf law is rejected for only one of 24 countries under our test whereas it is rejected for 23 of 24 countries under the usual t test. We also propose a more efficient estimation procedure and provide empirical applications of the theory for some countries.

Nonparametric Estimation Methods of Integrated Multivariate Volatilities

Estimation of integrated multivariate volatilities of an Itô process is an interesting and important issue in finance, for example, in order to evaluate portfolios. New non-parametric estimators have been recently proposed by Malliavin and Mancino (2002) and Hayashi and Yoshida (2005a) as alternative methods to classical realized quadratic covariation. The purpose of this article is to compare these alternative estimators both theoretically and empirically, when high frequency data is available. We found that the Hayashi–Yoshida estimator performs the best among the alternatives in view of the bias and the MSE. The other estimators are shown to have possibly heavy bias mostly toward the origin. We also applied these estimators to Japanese Government Bond futures to obtain the results consistent with our simulation.

Corporate hierarchy, promotion, and firm growth: Japanese internal labor market in transition

Abstract This paper investigates the incentive systems and the hierarchical design of the Japanese firms as integral parts of employment structure. Using the survey cross-section data on job ranks and wages, we analyze the promotion policy and compensations system as the key incentive mechanism in these firms with highly developed internal labor markets. We find that the incentive as well as hierarchical structures of the large Japanese firms are highly sensitive to the longrun growth rates of these firms. This finding is supported by a prediction of a model of internal promotions developed in the paper. We also find that the span of control, incentive effects of promotion, and wage-age profile at each job rank are all increasing in the longrun growth rates of these firms. These findings are jointly consistent with and in support of the hypothesis that the expected gains from the promotion is the key incentive in inducing efforts of the employees.

A PUZZLING PHENOMENON IN SEMIPARAMETRIC ESTIMATION PROBLEMS WITH INFINITE-DIMENSIONAL NUISANCE PARAMETERS

This note considers a puzzling phenomenon that is observed in some semiparametric estimation problems. In some cases, using estimated values of the nuisance parameters provides a more efficient estimator for the parameters of interest than does using the true values. This phenomenon takes place even in cases of semi-nonparametric models in which the nuisance parameters are infinite-dimensional and cannot be estimated at the parametric rate. We examine the structure and present the necessary and sufficient condition for the occurrence of this puzzle. We also provide a simple sufficient condition. It shows that the puzzle occurs when the term accounting for the effect of estimation of nuisance parameters is included in the tangent space. This condition is often satisfied when the estimating equation does not bring any restriction on the form of the nuisance parameters. Our simple sufficient condition can be applied to many important estimators.

Studentization in Edgworth expansions for estimates of semiparametric index models

We establish valid theoretical and empirical Edgeworth expansions for density-weighted averaged derivative estimates of semiparametric index models.

Maximum empirical likelihood estimation of continuous-time models with conditional characteristic functions

For some popular financial continuous-time models, tractable expressions of likelihood functions are unknown. For that reason, the maximum likelihood estimation method is infeasible. Fortunately, closed functional forms of conditional characteristic functions of some of these models are known. We construct an empirical likelihood estimation method using tractable conditional characteristic functions to estimate such a model. This method resolves the problem of covariance matrix singularity in the standard generalized method of moments and fully utilizes information in conditional moment restrictions. It is applicable to many popular financial models such as some diffusion models, jump diffusion models, and stochastic volatility models. Using a Monte Carlo comparison, we show that this method provides superior performance compared to other methods in some situations.

Kernel order selection by minimum bootstrapped MSE for density weighted averages

Density weighted averages are nonparametric quantities expressed by the expectation of a function of random variables with density weight. It is associated with parametric components of some semiparametric models, and we are concerned with an estimator of these quantities. Asymptotic properties of semiparametric estimators have been studied in econometrics since the end of 1980s and it is now widely recognized that they are n-consistent in many cases. Many of them involve estimates of nonparametric functions such as density and regression function but they are biased estimators for the true functions. Because of this, we typically need to use some bias reduction techniques in the nonparametric estimates for n-consistency of the semiparametric estimators. When we use a kernel estimator, a standard way is to take a higher order kernel function. For density estimation, the higher the kernel order is, the less becomes the bias without changing the order of variance in theory. However, it is also known that higher order kernels can inflate the variance which may cause the result that the mean squared error with very high order kernel becomes larger than that with low order kernel in small sample. This paper proposes to select the bandwidth and kernel order simultaneously by minimizing bootstrap mean squared error for a plug-in estimator of density weighted averages. We show that standard bootstrap does not work at all for bias approximation as in density estimation, but smoothed bootstrap is useful in our problem if suitably transformed.

Minimum normal approximation error bandwidth selection for averaged derivatives

Density-weighted averaged derivative estimator gives a computationally convenient consistent and asymptotically normally (CAN) distributed estimate of the parametric component of a semiparametric single index model. This model includes some important parametric models as special cases such as linear regression, Logit/Probit, Tobit and Box-Cox and other transformation models. This estimator involves a nonparametric kernel density estimate and thus it faces the problem of bandwidth selection. A reasonable way of bandwidth selection for point estimation is one minimizing the mean squared error. Alternatively, for the purposes of hypothesis testing and confidence interval estimation, we may like to choose it such that it minimizes the normal approximation error. The purpose of this paper is to propose a new bandwidth suitable for these purposes by minimizing the normal approximation error in the tail of exact distribution of the statistics using higher order asymptotic theory of Edgeworth expansion or bootstrap method.