Summary. We consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined. Using an information theoretic argument we derive a measure pD for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest. In general pD approximately corresponds to the trace of the product of Fisher's information and the posterior covariance, which in normal models is the trace of the ‘hat’ matrix projecting observations onto fitted values. Its properties in exponential families are explored. The posterior mean deviance is suggested as a Bayesian measure of fit or adequacy, and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages. Adding pD to the posterior mean deviance gives a deviance information criterion for comparing models, which is related to other information criteria and has an approximate decision theoretic justification. The procedure is illustrated in some examples, and comparisons are drawn with alternative Bayesian and classical proposals. Throughout it is emphasized that the quantities required are trivial to compute in a Markov chain Monte Carlo analysis.

/pdf/bayesian-measures-of-model-complexity-and-fit-1ouzmtfu7x.pdf

Bayesian measures of model complexity and fit

In this paper, we propose a classical approach to model selection. Using the Kullback-Leibler Information measure, we propose simple and directional likelihood-ratio tests for discriminating and choosing between two competing models whether the models are nonnested, overlapping or nested and whether both, one, or neither is misspecified. As a prerequisite, we fully characterize the asymptotic distribution of the likelihood ratio statistic under the most general conditions.

/pdf/likelihood-ratio-tests-for-model-selection-and-non-nested-5asf4hfg3m.pdf

Likelihood Ratio Tests for Model Selection and Non-Nested Hypotheses

https://w4.stern.nyu.edu/finance/docs/WP/2000/pdf/wpa00041.pdf

Illiquidity and stock returns: cross-section and time-series effects $

New tests are presented on the effects of stock illiquidity on stock return. Over time, expected market illiquidity positively affects ex ante stock excess return (usually called â¬Srisk premiumâ¬?). This complements the positive cross-sectional return-illiquidity relationship. The illiquidity measure here is the average daily ratio of absolute stock return to dollar volume, which is easily obtained from daily stock data for long time series in most stock markets. Illiquidity affects more strongly small firms stocks, suggesting an explanation for the changes â¬Ssmall firm effectâ¬? over time. The impact of market illiquidity on stock excess return suggests the existence of illiquidity premium and helps explain the equity premium puzzle.

Illiquidity and Stock Returns: Cross-Section and Time-Series Effects

This paper develops asymptotic distribution theory for instrumental variable regression when the partial correlation between the instruments and a single included endogenous variable is weak, here modeled as local to zero. Asymptotic representations are provided for various instrumental variable statistics, including the two-stage least squares (TSLS) and limited information maximum- likelihood (LIML) estimators and their t-statistics. The asymptotic distributions are found to provide good approximations to sampling distributions with just 20 observations per instrument. Even in large samples, TSLS can be badly biased, but LIML is, in many cases, approximately median unbiased. The theory suggests concrete quantitative guidelines for applied work. These guidelines help to interpret Angrist and Krueger's (1991) estimates of the returns to education: whereas TSLS estimates with many instruments approach the OLS estimate of 6%, the more reliable LIML and TSLS estimates with fewer instruments fall between 8% and 10%, with a typical confidence interval of (6%, 14%).

/pdf/instrumental-variables-regression-with-weak-instruments-4yftqv6rc0.pdf

Instrumental variables regression with weak instruments

Some decision rules for discriminating among alternative regression models are proposed and mutually compared. They are essentially based on the Akaike Information Criterion as well as the Kullback-Leibler Information Criterion (KLIC) : namely, the distance between a postulated model and the true unknown structure is measured by the KLIC. The proposed criteria combine the parsimony of parameters with the goodness of fit. Their relationships with conventional criteria are discussed in terms of a new concept of unbiasedness .

/pdf/information-criteria-for-discriminating-among-alternative-10pguahm1p.pdf

Information Criteria for Discriminating Among Alternative Regression Models

The exact moments of the least squares estimator for the autoregressive model

The distributions of the Limited Information Maximum Likelihood estimator for the coefficient of one endogenous variable are evaluated numerically. Tables are given for enough values of the parameters to cover all cases of interests. Comparisons are made with the Two-Stage Least Squares estimator.

/pdf/evaluation-of-the-distribution-function-of-the-limited-3o5vir0mfk.pdf

Takamitsu Sawa

Papers

Information Criteria for Discriminating Among Alternative Regression Models

The exact moments of the least squares estimator for the autoregressive model

Evaluation of the Distribution Function of the Limited Information Maximum Likelihood Estimator

Finite-Sample Properties of the k-Class Estimators

Evaluation of the Distribution Function of the Two-Stage Least Squares Estimate