Kimio Morimune

Asymptotic expansions of the distributions of the estimates of coefficients in a simultaneous equation system

Abstract In this paper asymptotic expansions are derived for the density functions of the TSLS and LIML estimates of coefficients in a simultaneous equation system when the sample size increases and the effect of the exogenous variables increases along the sample size. These approximations are used to compare the asymptotic moments of the TSLS and LIML estimates and the concentration of probability around the true value of the estimates.

The numerical values of some key parameters in econometric models

Abstract In the case of two endogenous variables, exogenous predetermined variables, and normally distributed disturbances, the distributions of the Two-Stage Least Squares (TSLS) and Limited Information Maximum Likelihood (LIML) estimators can be compared on the basis of three key parameters: the non-centrality parameter, a standardization of the structural coefficient, and the number of excluded exogenous variables. In this paper the values of these parameters are estimated in eleven structural equations from various actual econometric models. The distribution functions of the normalized TSLS and LIML estimators are given for the first two key parameters set at approximately their trimmed means, and the third at its median.

Test in a Structural Equation

Properties of t ratios associated with the limited information maximum likelihood, two-stage least squares, and ordinary least squares estimators in a structural form estimation are studied. The existence of moments of these t ratios, including the limited information maximum likelihood form, is proved first. Second, Monte Carlo simulations are performed to find out real sizes of the t test and the likelihood ratio test. Third, asymptotic expansions of the distributions of t ratios are derived to find out deviations of real sizes from nominal sizes. The t ratios associated with the limited information maximum likelihood and two-stage least squares estimators are proved asymptotically as powerful as the likelihood ratio test. Copyright 1989 by The Econometric Society.

Improving the Maximum Likelihood Estimate in Linear Functional Relationships for Alternative Parameter Sequences

Abstract We propose an improvement of the maximum likelihood (ML) estimate in linear functional relationships. The improved estimate is a linear combination of the ML and the least squares estimate so as to remove the bias of the former. Approximations to the distribution of the estimate are derived for two alternative parameter sequences: a sequence in which the noncentrality parameter (the spread of the true values) increases while the number of observations stays fixed, and that in which the number of observations increases. The mean squared errors of the improved estimate, in terms of its asymptotic distributions, are obtained and shown to be smaller than those of the ML. Implications to large-scale simultaneous econometric models are also given.

Improving the Limited Information Maximum Likelihood Estimator When the Disturbances are Small

Abstract The limited information maximum likelihood (LIML) estimator is shown to be inadmissible in terms of asymptotic expansions of the distributions of the estimators. The LIML estimator is improved by combining it linearly with the two-stage or the ordinary least squares estimator.

Testing homogeneity of a large data set by bootstrapping

It is not rare to analyze large data sets these days. Large data is usually of census type which is called the micro data in econometrics. The basic method of analysis is to estimate a single regression equation with common coefficients over the whole data. The same applies to other methods of estimation such as the discrete choice models, Tobit models, and so on. Heterogeneity in the data is usually adjusted by the dummy variables which represent socioeconomic differences among individuals in the sample. Including the coefficients of dummy variables, only one equation is estimated for the whole large sample, and it is usually not preferred to divide the whole sample into sub-samples. Data is said to be homogenous in this paper if a single equation is fit to the whole data, and if it explains socioeconomic properties of the data well. We may estimate an equation in each sub-population if the whole population is divided into known sub-populations. Regression coefficients are differentfrom one sub-population to another in this case, and the data is said to be heterogeneous in our paper. The analysis of variance is applied if sub-populations are known, and a sub-sample is collected from each sub-population. The sub-sample test statistics can be correlated with each other since the sub-samples can be overlapped. Critical values of the test statistics are calculated by simulations. An example follows.

Asymptotic Expansions of the Distribution of an Improved Limited Information Maximum Likelihood Estimator

Abstract An improvement of the limited information maximum likelihood (LIML) estimator was proposed by Morimune (1978). In this note two asymptotic expansions of the distribution of the improved estimator are derived, both for large sample size and for large values of the noncentrality parameter. The improved estimator uniformly dominates the LIML estimator in terms of the probability of concentration around the true coefficient for both the parameter sequences. The distributions of the coefficient of income multiplier in the consumption function are also derived.

Decision rules for the choice of structural equations

Abstract In practical econometric analysis we are faced with the problem of how to specify structural equations. The conventional t -test of coefficients is apparently inappropriate. The smallest root, say λ, of a certain determinantal equation provides us with basis for the test of overidentifying restrictions. The preliminary test, based on λ, may give us a possible decision rule for choosing a structural equation from nested alternatives. However, ambiguity remains in specifying the significance level. We propose a decision method called the unbiased decision rule; unbiased in the sense that we attain a correct decision with probability of more than a half. The critical points are found as the medians of non-central F -distributions. The degrees of freedom and the non-centrality parameter of non-central F -distributions are determined by the properties of contending models. We also discuss the implications of the unbiased decision rule in the context of the conventional pre-test.

Distribution-free statistical inference for generalized Lorenz dominance based on grouped data

One income distribution is preferable to another under any increasing and Schur-concave (S-concave) social welfare function (SWF) if and only if the generalized Lorenz (GL) curve of the first distribution lies above that of the second. Thus, testing for GL dominance of one distribution over another is of interest. The paper focuses on inference based on grouped data and makes two contributions: (i) it gives a new formula for the asymptotic variance-covariance matrix of a vector of sample GL curve ordinates, interpreting it as a method-of-moments (MM) estimator, and (ii) it proposes a new test for multivariate inequality restrictions, of which GL dominance is a special case. For the Japanese household income data grouped into deciles, the test accepts the null hypothesis that income distribution in Japan improved from 1979 to 1994.

Unit Root Analyses of the Causality Between Japanese Money and Income

Unit root techniques and cointegration analysis have develop ed considerably in the last ten years. At the same time, the nonstationary test for Granger causality has been developed. We shed some new light on Japanese money supply and income causality by using nonstationary techniques. We specify univariate ARMA models of money, income, GNP deflator and rate of interest, initially by using the Dickey and Fuller (DF) or the augmented DF (ADF) tests. Two diagnostic tests are applied to each selected ARMA regression. One is the residual DF test, and the other is the moving average (MA) unit root test of residuals . After selecting the ARMA model, some causality tests are applied to the error correction model (ECM) of a vector autoregression (VAR) one of which is ordinary least squares (OLS) and another is the maximum likelihood (ML) method. The former requires only the standard F-test on the deleted variables in the ECM. The latter requires the Johansens ML method in estimating cointegration. Causality is found to go from income to money supply but not the other way. Appendices include a simple implementation of the MA unit root test, a pedagogical proof of the Granger causality tests developed by Toda and Phillips (1993) and an interpretation of the test proposed by Toda and Yamamoto (1995). JEL Classification Numbers: C32, E50