"Improving Small Sample Properties of the Empirical Likelihood Estimation"

We propose to use a simple modification of the maximum empirical likelihood (MEL) method for estimating structural equation in econometrics. The modified estimator improves both the asymptotic bias and the mean squared error of the MEL estimator in the orders of O(n -1) and O(n -2), respectively, at the same time. It also improves the asymptotic bias of the generalized method of moments (GMM) estimation (or the estimating equation (EE) method) significantly when there are many instruments in the econometric literatures.

"An Optimal Modification of the LIML Estimation for Many Instruments and Persistent Heteroscedasticity"

We consider the estimation of coefficients of a structural equation with many instrumental variables in a simultaneous equation system. It is mathematically equivalent to an estimating equation estimation or a reduced rank regression in the statistical linear models when the number of restrictions or the dimension increases with the sample size. As a semi- parametric method, we propose a class of modifications of the limited information maximum likelihood (LIML) estimator to improve its asymptotic properties as well as the small sample properties for many instruments and persistent heteroscedasticity. We show that an asymptotically optimal modification of the LIML estimator, which is called AOM-LIML, improves the LIML estimator and other estimation methods. We give a set of sufficient conditions for an asymptotic optimality when the number of instruments or the dimension is large with persistent heteroscedasticity including a case of many weak instruments.

"Applications of the Asymptotic Expansion Approach based on Malliavin-Watanabe Calculus in Financial Problems"

This paper reviews the asymptotic expansion approach based on Malliavin-Watanabe Calculus in Mathematical Finance. We give the basic formulation of the asymptotic expansion approach and discuss its power and usefulness to solve important problems arised in nance. As illustrations we use three major problems in nance and give some useful formulae and new results including numerical analyses.

"The Conditional Limited Information Maximum Likelihood Approach to Dynamic Panel Structural Equations"

We propose the conditional limited information maximum likelihood (CLIML) approach for estimating dynamic panel structural equation models. When there are dynamic effects and endogenous variables with individual effects at the same time, the CLIML estimation method for the doubly-filtered data does give not only a consistent estimation, but also it attains the asymptotic efficiency when the number of orthogonal condition is large. Our formulation includes Alvarez and Arellano (2003), Blundell and Bond (2000) and other linear dynamic panel models as special cases.

"Improving the Rank-Adjusted Anderson-Rubin Test with Many Instruments and Persistent Heteroscedasticity"

Anderson and Kunitomo (2007) have developed the likelihood ratio criterion, which is called the Rank-Adjusted Anderson-Rubin (RAAR) test, for testing the coefficients of a structural equation in a system of simultaneous equations in econometrics against the alternative hypothesis that the equation of interest is identified. It is related to the statistic originally proposed by Anderson and Rubin (1949, 1950), and also to the test procedures by Kleibergen (2002) and Moreira (2003). We propose a modified procedure of RAAR test, which is suitable for the cases when there are many instruments and the disturbances have persistent heteroscedasticities.

"A Generalized SSAR Model and Predictive Distribution with an Application to VaR"

The asymmetrical movements between the downward and upward phases of the sample paths of time series have been sometimes observed. By generalizing the SSAR (simultaneous switching autoregressive) models, we introduce a class of nonlinear time series models having the asymmetrical sample paths in the upward and downward phases. We show that the class of generalized SSAR models is useful for estimating the asymmetrical predictive distribution given the present and past information. Applications to the prediction based on the predictive median and the estimation of the VaR (value at risk) in financial risk management are discussed.

"Some Properties of the LIML Estimator in a Dynamic Panel Structural Equation"

We investigate the finite sample and asymptotic properties of several estimation methods (Within-Group, GMM and LIML) for a panel autoregressive structural equation model with random effects when both T and N are large. When we use the forward-filtering to a structural model as Alvarez and Arellano (2003), both the WG and GMM estimators are significantly biased when both T and N go to infinity while T/N is different from zero. The LIML (limited information maximum likelihood) estimator has consistency and the asymptotic normality when T/N converges to a constant as both T and N go to infinity. Its asymptotic distribution has some bias and covariance which depend on the limiting behavior of T/N.

"Separating Information Maximum Likelihood Estimation of Realized Volatility and Covariance with Micro-Market Noise"

For estimating the realized volatility and covariance by using high frequency data, we introduce the Separating Information Maximum Likelihood (SIML) method when there are possibly micro-market noises. The resulting estimator is simple and it has the representation as a specific quadratic form of returns. The SIML estimator has reasonable asymptotic properties; it is consistent and it has the asymptotic normality (or the stable convergence in the general case) when the sample size is large under general conditions including non-Gaussian processes and volatility models. Based on simulations, we find that the SIML estimator has reasonable finite sample properties and thus it would be useful for practice. It is also possible to use the limiting distribution of the SIML estimator for constructing testing procedures and confidence intervals.

"Asymptotic Expansions of the Distributions of Semi-Parametric Estimators in a Linear Simultaneous Equations System"

Asymptotic expansions are made of the distributions of a class of semi-parametric estimators including the Maximum Empirical Likelihood (MEL) method and the Generalized Method of Moments (GMM) for the coefficients of a single structural equation in the linear simultaneous equations system. The expansions in terms of the sample size, when the non-centrality parameters increase proportionally, are carried out to the order of O(n-2). Comparisons of the distributions of the MEL and GMM estimators are also made.

"Empirical Likelihood Estimation of Levy Processes (Revised: March 2005)"

We propose a new parameter estimation procedure for the Levy processes and the class of infinitely divisible distribution. We shall show that the empirical likelihood method gives an easy way to estimate the key parameters of the infinitely divisible distributions including the class of stable distributions as a special case. The maximum empirical likelihood estimator by using the empirical characteristic functions gives the consistency, the asymptotic normality, and the asymptotic efficiency for the key parameters when the number of restrictions on the empirical characteristic functions is large. Test procedures can be also developed. Some extensions to the estimating equations problem with the infinitely divisible distributions are discussed.