Estimation of bivariate excess probabilities for elliptical models

Let $(X,Y)$ be a random vector whose conditional excess probability $\theta(x,y):=P(Y\leq y | X>x)$ is of interest. Estimating this kind of probability is a delicate problem as soon as $x$ tends to be large, since the conditioning event becomes an extreme set. Assume that $(X,Y)$ is elliptically distributed, with a rapidly varying radial component. In this paper, three statistical procedures are proposed to estimate $\theta(x,y)$ for fixed $x,y$, with $x$ large. They respectively make use of an approximation result of Abdous et al. (cf. Canad. J. Statist. 33 (2005) 317--334, Theorem 1), a new second order refinement of Abdous et al.'s Theorem 1, and a non-approximating method. The estimation of the conditional quantile function $\theta(x,\cdot)^{\leftarrow}$ for large fixed $x$ is also addressed and these methods are compared via simulations. An illustration in the financial context is also given.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ140 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Probit Models with Binary Endogenous Regressors

Sample selection and endogeneity are frequent causes of biases in non-experimental empirical studies. In binary models a standard solution involves complex multivariate models. A simple approximation has been shown to work well in bivariate models. This paper extends the approximation to a trivariate model. Simulations show that the approximation outperforms full maximum likelihood while a least squares approximation may be severely biased. The methods are used to estimate the influence of trust in the parliament and politicians on voting- propensity. No previous studies have allowed for endogeneity of trust on voting and it is shown to severely affect the results.endogeneity; multivariate probit; approximation; Monte Carlo simulation

Multivariate probit regression using simulated maximum likelihood

We discuss the application of the GHK simulation method to maximum likelihood estimation of the multivariate probit regression model, and describe and illustrate a Stata program mvprobit for this purpose.