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This article presents a large class of probability densities f(x, θ) for which, with positive probability, the maximum likelihood estimator based on a sample of size 2 is non unique, and the possible values of do not form an interval. Such a density f(x, θ) can even be chosen to be unimodal, and one such example is the Cauchy density with a location parameter. A discrete version of the argument gives examples in which the nonuniqueness of the maximum likelihood estimator persists for samples of arbitrary size.     

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Selection processes that are inherent in litigation complicate policy-oriented research of medical malpractice tort reforms. In regard to their deterrent impact, the range of potential inferences based on analyses of claim frequency is limited because plaintiffs only file a subset of potential claims. In regard to their impact on litigation costs, researchers often analyze data on claim disposition, but it is difficult to determine whether effects attributed to tort reforms reflect changes in litigant behavior or their influence on the selection of claims. In this article, we evaluate these problems and report results of our study of the effects of medical malpractice reforms on claim disposition.     

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ABSTRACT

In the class of stochastic volatility (SV) models, leverage effects are typically specified through the direct correlation between the innovations in both returns and volatility, resulting in the dynamic leverage (DL) model. Recently, two asymmetric SV models based on threshold effects have been proposed in the literature. As such models consider only the sign of the previous return and neglect its magnitude, this paper proposes a dynamic asymmetric leverage (DAL) model that accommodates the direct correlation as well as the sign and magnitude of the threshold effects. A special case of the DAL model with zero direct correlation between the innovations is the asymmetric leverage (AL) model. The dynamic asymmetric leverage models are estimated by the Monte Carlo likelihood (MCL) method. Monte Carlo experiments are presented to examine the finite sample properties of the estimator. For a sample size of T = 2000 with 500 replications, the sample means, standard deviations, and root mean squared errors of the MCL estimators indicate only a small finite sample bias. The empirical estimates for S&;P 500 and TOPIX financial returns, and USD/AUD and YEN/USD exchange rates, indicate that the DAL class, including the DL and AL models, is generally superior to threshold SV models with respect to AIC and BIC, with AL typically providing the best fit to the data.     

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This article develops a new identification procedure to estimate the contemporaneous relation between monetary policy and the stock market within a vector autoregression (VAR) framework. The approach combines high-frequency data from the futures market with the VAR methodology to circumvent exclusion restrictions and achieve identification. Our analysis casts doubt on VAR models imposing a recursive structure between innovations in policy rates and stock returns. We find that a tightening in policy rates has a negative impact on stock prices and that the Federal Reserve (Fed) has responded significantly to movements in the stock market. Estimates are robust to various model specifications.     

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Thanks and congratulations to all of the discussants for covering a wide array of important topics. Since space does not permit attention to all of the points that have been raise, this reply will concentrate on trying to dispell confusion on important matters that might remain.