"On Likelihood Ratio Tests of Structural Coefficients: Anderson-Rubin (1949) revisited"

We develop the likelihood ratio criterion (LRC) for testing the coefficients of a structural equation in a system of simultaneous equations in econometrics. We relate the likelihood ratio criterion to the AR statistic proposed by Anderson and Rubin (1949, 1950), which has been widely known and used in econometrics over the past several decades. The method originally developed by Anderson and Rubin (1949, 1950) can be modified to the situation when there are many (or weak in some sense) instruments which may have some relevance in recent econometrics. The method of LRC can be extended to the linear functional relationships (or the errors-in-variables) model, the reduced rank regression and the cointegration models.

"On the Asymptotic Optimality of the LIML Estimator with Possibly Many Instruments"

We consider the estimation of the coefficients of a linear structural equation in a simultaneous equation system when there are many instrumental variables. We derive some asymptotic properties of the limited information maximum likelihood (LIML) estimator when the number of instruments is large; some of these results are new and we relate them to results in some recent studies. We have found that the variance of the LIML estimator and its modifications often attain the asymptotic lower bound when the number of instruments is large and the disturbance terms are not necessarily normally distributed, that is, for the micro-econometric models with many instruments.

"A New Light from Old Wisdoms : Alternative Estimation Methods of Simultaneous Equations with Possibly Many Instruments"

We compare four dffierent estimation methods for a coefficient of a linear structural equation with instrumental variables. As the classical methods we consider the limited information maximum likelihood (LIML) estimator and the two-stage least squares (TSLS) estimator, and as the semi-parametric estimation methods we consider the maximum emirical likelihood (MEL) estimator and the generalized method of moments (GMM) (or the estimating equation) estimator. We prove several theorems on the asymptotic optimality of the LIML estimator when the number of instruments is large, which are new as well as old, and we relate them to the results in some recent studies. Tables and figures of the distribution functions of four estimators are given for enough values of the parameters to cover most of interest. We have found that the LIML estimator has good performance when the number of instruments is large, that is, the micro-econometric models with many instruments in the terminology of recent econometric literature.

On the Formation Age of the First Planetary System

Recently, it has been observed the extreme metal-poor stars in the Galactic halo, which must be formed just after Pop III objects. On the other hand, the first gas clouds of mass $\sim 10^6 M_{\odot}$ are supposed to be formed at $z \sim$ 10, 20, and 30 for the $1\sigma$, $2\sigma$ and $3\sigma$, where the density perturbations are assumed of the standard $\Lambda$CDM cosmology. If we could apply this gaussian distribution to the extreme small probability, the gas clouds would be formed at $z \sim$40, 60, and 80 for the $4\sigma$, $6\sigma$, and $8\sigma$. The first gas clouds within our galaxy must be formed around $z\sim 40$. Even if the gas cloud is metal poor, there is a lot of possibility to form the planets around such stars. The first planetary systems could be formed within $\sim 6\times 10^7$ years after the Big Bang in the universe. Even in our galaxies, it could be formed within $\sim 1.7\times 10^8$ years. It is interesting to wait the observations of planets around metal-poor stars. For the panspermia theory, the origin of life could be expected in such systems.Comment: 5 pages,Proceedings IAU Symposium No. 249, 2007, Exoplanets:Y-S. Sun, S. Ferraz-Mello and J.-L, Zhou, eds. (p325

Construction of N = 2 Chiral Supergravity Compatible with the Reality Condition

We construct N = 2 chiral supergravity (SUGRA) which leads to Ashtekar's canonical formulation. The supersymmetry (SUSY) transformation parameters are not constrained at all and auxiliary fields are not required in contrast with the method of the two-form gravity. We also show that our formulation is compatible with the reality condition, and that its real section is reduced to the usual N = 2 SUGRA up to an imaginary boundary term.Comment: 16 pages, late

Quasinormal modes prefer supersymmetry ?

One ambiguity in loop quantum gravity is the appearance of a free parameter which is called Immirzi parameter. Recently Dreyer has argued that this parameter may be fixed by considering the quasinormal mode spectrum of black holes, while at the price of changing the gauge group to SO(3) rather than the original one SU(2). Physically such a replacement is not quite natural or desirable. In this paper we study the relationship between the black hole entropy and the quasi normal mode spectrum in the loop quantization of N=1 supergravity. We find that a single value of the Immirzi parameter agrees with the semiclassical expectations as well. But in this case the lowest supersymmetric representation dominates, fitting well with the result based on statistical consideration. This suggests that, so long as fermions are included in the theory, supersymemtry may be favored for the consistency of the low energy limit of loop quantum gravity.Comment: 3 page

Supersymmetry algebra in N = 1 chiral supergravity

We consider the supersymmetry (SUSY) transformations in the chiral Lagrangian for $N = 1$ supergravity (SUGRA) with the complex tetrad following the method used in the usual $N = 1$ SUGRA, and present the explicit form of the SUSY trasformations in the first-order form. The SUSY transformations are generated by two independent Majorana spinor parameters, which are apparently different from the constrained parameters employed in the method of the 2-form gravity. We also calculate the commutator algebra of the SUSY transformations on-shell.Comment: 10 pages, late

Minimal Off-Shell Version of N = 1 Chiral Supergravity

We construct the minimal off-shell formulation of N = 1 chiral supergravity (SUGRA) introducing a complex antisymmetric tensor field $B_{\mu \nu}$ and a complex axial-vector field $A_{\mu}$ as auxiliary fields. The resulting algebra of the right- and left-handed supersymmetry (SUSY) transformations closes off shell and generates chiral gauge transforamtions and vector gauge transformations in addition to the transformations which appear in the case without auxiliary fields.Comment: 9 pages, late

N = 3 chiral supergravity compatible with the reality condition and higher N chiral Lagrangian density

We obtain N = 3 chiral supergravity (SUGRA) compatible with the reality condition by applying the prescription of constructing the chiral Lagrangian density from the usual SUGRA. The $N = 3$ chiral Lagrangian density in first-order form, which leads to the Ashtekar's canonical formulation, is determined so that it reproduces the second-order Lagrangian density of the usual SUGRA especially by adding appropriate four-fermion contact terms. We show that the four-fermion contact terms added in the first-order chiral Lagrangian density are the non-minimal terms required from the invariance under first-order supersymmetry transformations. We also discuss the case of higher N theories, especially for N = 4 and N = 8.Comment: 20 pages, Latex, some more discussions and new references added, some typos corrected, accepted for publication in Physical Review

Canonical formulation of N = 2 supergravity in terms of the Ashtekar variable

We reconstruct the Ashtekar's canonical formulation of N = 2 supergravity (SUGRA) starting from the N = 2 chiral Lagrangian derived by closely following the method employed in the usual SUGRA. In order to get the full graded algebra of the Gauss, U(1) gauge and right-handed supersymmetry (SUSY) constraints, we extend the internal, global O(2) invariance to local one by introducing a cosmological constant to the chiral Lagrangian. The resultant Lagrangian does not contain any auxiliary fields in contrast with the 2-form SUGRA and the SUSY transformation parameters are not constrained at all. We derive the canonical formulation of the N = 2 theory in such a manner as the relation with the usual SUGRA be explicit at least in classical level, and show that the algebra of the Gauss, U(1) gauge and right-handed SUSY constraints form the graded algebra, G^2SU(2)(Osp(2,2)). Furthermore, we introduce the graded variables associated with the G^2SU(2)(Osp(2,2)) algebra and we rewrite the canonical constraints in a simple form in terms of these variables. We quantize the theory in the graded-connection representation and discuss the solutions of quantum constraints.Comment: 19 pages, Latex, corrected some typos and added a referenc