Takeaki Kariya

Generalized Least Squares

Preface.1 Preliminaries.1.1 Overview.1.2 Multivariate Normal and Wishart Distributions.1.3 Elliptically Symmetric Distributions.1.4 Group Invariance.1.5 Problems.2 Generalized Least Squares Estimators.2.1 Overview.2.2 General Linear Regression Model.2.3 Generalized Least Squares Estimators.2.4 Finiteness of Moments and Typical GLSEs.2.5 Empirical Example: CO2 Emission Data.2.6 Empirical Example: Bond Price Data.2.7 Problems.3 Nonlinear Versions of the Gauss-Markov Theorem.3.1 Overview.3.2 Generalized Least Squares Predictors.3.3 A Nonlinear Version of the Gauss-Markov Theorem in Prediction.3.4 A Nonlinear Version of the Gauss-Markov Theorem in Estimation.3.5 An Application to GLSEs with Iterated Residuals.3.6 Problems.4 SUR and Heteroscedastic Models.4.1 Overview.4.2 GLSEs with a Simple Covariance Structure.4.3 Upper Bound for the Covariance Matrix of a GLSE.4.4 Upper Bound Problem for the UZE in an SUR Model.4.5 Upper Bound Problems for a GLSE in a Heteroscedastic Model.4.6 Empirical Example: CO2 Emission Data.4.7 Problems.5 Serial Correlation Model.5.1 Overview.5.2 Upper Bound for the Risk Matrix of a GLSE.5.3 Upper Bound Problem for a GLSE in the Anderson Model.5.4 Upper Bound Problem for a GLSE in a Two-equation Heteroscedastic Model.5.5 Empirical Example: Automobile Data.5.6 Problems.6 Normal Approximation.6.1 Overview.6.2 Uniform Bounds for Normal Approximations to the Probability Density Functions.6.3 Uniform Bounds for Normal Approximations to the Cumulative Distribution Functions.6.4 Problems.7 Extension of Gauss-Markov Theorem.7.1 Overview.7.2 An Equivalence Relation on S(n).7.3 A Maximal Extension of the Gauss-Markov Theorem.7.4 Nonlinear Versions of the Gauss-Markov Theorem.7.5 Problems.8 Some Further Extensions.8.1 Overview.8.2 Concentration Inequalities for the Gauss-Markov Estimator.8.3 Efficiency of GLSEs under Elliptical Symmetry.8.4 Degeneracy of the Distributions of GLSEs.8.5 Problems.9 Growth Curve Model and GLSEs.9.1 Overview.9.2 Condition for the Identical Equality between the GME and the OLSE.9.3 GLSEs and Nonlinear Version of the Gauss-Markov Theorem .9.4 Analysis Based on a Canonical Form.9.5 Efficiency of GLSEs.9.6 Problems.A. Appendix.A.1 Asymptotic Equivalence of the Estimators of theta in the AR(1) Error Model and Anderson Model.Bibliography.Index.

Minimax estimators in the normal MANOVA model

This paper considers the problem of estimating the coefficient matrix B: m - p in a normal multivariate regression model under the risk matrix : m - m and obtains classes of minimax estimators for Baranchik type, Strawderman type, Efron-Morris type, and Stein type estimators.

Bounds for the Covariance Matrices of Zellner's Estimator in the SUR Model and the 2SAE in a Heteroscedastic Model

Abstract This article gives a bound for the covariance matrix of Zellners estimator with the unrestricted sample covariance matrix in the two equations seemingly unrelated regression (SUR) model, and a bound for the covariance matrix of the two-stage Aitken estimator (2SAE) in a heteroscedastic model with two distinct variances. Some efficiencies of these estimators relative to the ordinary least squares estimator (OLSE) are also considered.

Pricing Mortgage-Backed Securities (MBS)

This paper presents a pricing formula for MBSs andproposes a specific model for MBS prices thatdescribes the so-called burnout phenomenon ofprepayments due to refinancing. A numerical exampleof the model is demonstrated by Monte Carlosimulation. An estimation procedure is alsodescribed.

A method for approximations to the pdf′s and cdf′s of GLSE′s and its application to the seemingly unrelated regression model

This paper first develops a valid method for approximations to the pdfs and cdfs of GLSE in linear models and, applying this method to the Zellner estimator with an unrestricted sample covariance in the seemingly unrelated regression model, obtains an approximate pdf with a bound of ordern−2 and an approximate covariance matrix with a bound of ordern−3

Testing Gaussianity and Linearity of Japanese Stock Returns

In this article, we first investigate the Gaussianity of Japanese stock return time series (214 daily, 18 weekly) by the Gaussianity test proposed by Kariya, Tsay, Terui and Li (1994) comprehensively and consistently. And it is observed that all the series are not Gaussian when the 6th order moment structures are taken into account. Up to the 4th order moments there are some series which are compatible with the Gaussianity. Secondly, we apply five well-known nonlinearity tests for stationary time series to the data set and examine the specific nonlinearity of the series. Some series strongly exhibit the specific types of nonlinearity. Typically the Nikkei daily index shows the TAR (Threshold Autoregressive) type nonlinearity. Comparing daily return series with weekly series, it is also shown that a central limit effect is working on the weekly stock returns, where daily information is accumulated over a week, in the sense that weekly returns are relatively closer to Gaussian.

Tests for independence of two multivariate regression equations with different design matrices

In this paper, the authors considered various procedures for testing for the independence of two multivariate regression equations with different design matrices. Asymptotic null distributions as well as nonnull distributions under local alternatives of the test statistics associated with the above procedures are also derived.

Note on a Condition for Equality of Sample Variances in a Linear Model

Abstract This article gives a necessary and sufficient condition for s 2(Σ) = s 2(I), where s 2(Σ) is a usual sample variance for σ2 in a linear model y = Xβ + u with cov(u) = σ2Σ. The result is associated with a testing problem for covariance structure.

An Extensive Analysis on the Japanese Markets via S. Taylor's Model

Applying S. Taylors approach (1986), we make an extensive analysis on the Japanese stock market, foreign exchange market and the Japanese Government Bond Futures market. The purpose of this paper is to empirically reveal the structure of the Japanese markets via Taylors model rather than to propose a new model. For this reason, we include a variety of analyzed data particularly for the Japanese stock market and the foreign exchange market because the results can be used in a different manner. The paper consists of three parts. But each part can be read separately.Part 1: Overshooting hypothesis for Japanese stock pricesPart 2: A trend movement in daily/weekly Yen-Dollar exchange ratesPart 3: Price variations of Japanese Government Futures.In the first part, the stock prices are shown to over-respond to new information, which is different from the behaviors of stock prices in other markets. In Part 2, a trend movement is revealed in Yen-Dollar exchange rates. In Part 3, a strategy in the Japanese Government Bond futures markets is shown to perform better than a buy and hold strategy.

New Bond Pricing Models with Applications to Japanese Data

In this paper, the cross-sectional bond pricing model for individual bonds Kariya (1993) proposed by formulating stochastic discount function (term structure) is first applied to Japanese Government bond (JG-bond) data. The model performs very well as it stands. Second, we generalize the cross-sectional model to two types of time-dependent Markov models (TDMs) with the term structure of discount rates of each bond att being dependent on the one att−1, and apply them to the same data to find significantly improved results over those of the cross-sectional model. In fact, almost all the differences between actual prices and model values are less than 0.5 yen in each month over 12 years, implying that the error rate is less than 0.5%. On the basis of our analysis, we propose a TDM as a model for JG-bond trading.