A variable selection approach to monotonic regression with Bernstein polynomials

One of the standard problems in statistics consists of determining the relationship between a response variable and a single predictor variable through a regression function. Background scientific knowledge is often available that suggests that the regression function should have a certain shape (e.g. monotonically increasing or concave) but not necessarily a specific parametric form. Bernstein polynomials have been used to impose certain shape restrictions on regression functions. The Bernstein polynomials are known to provide a smooth estimate over equidistant knots. Bernstein polynomials are used in this paper due to their ease of implementation, continuous differentiability, and theoretical properties. In this work, we demonstrate a connection between the monotonic regression problem and the variable selection problem in the linear model. We develop a Bayesian procedure for fitting the monotonic regression model by adapting currently available variable selection procedures. We demonstrate the effectiveness of our method through simulations and the analysis of real data.     

Csiszar’s ϕ-divergences for testing the order in a Markov chain

x ₁, ..., x _n+r can be treated as the sample values of a Markov chain of order r or less (chain in which the dependence extends over r+1 consecutive variables only), and consider the problem of testing the hypothesis H ₀ that a chain of order r− 1 will be sufficient on the basis of the tools given by the Statistical Information Theory: _ϕ-Divergences. More precisely, if p _{a 1 ....., a r}: _{a r +1} denotes the transition probability for a r ^th order Markov chain, the hypothesis to be tested is H ₀:p _{a 1 ....., a r}: _{a r +1} = p _{a 2 ....., a r}: _{a r +1}, a _i ∈{1, ..., s}, i = 1, ..., r + 1 The tests given in this paper, for the first time, will have as a particular case the likelihood ratio test and the test based on the chi-squared statistic. Received: August 3, 1998; revised version: November 25, 1999     

Modeling Endogenous Mobility in Earnings Determination

ABSTRACT

We evaluate the bias from endogenous job mobility in fixed-effects estimates of worker- and firm-specific earnings heterogeneity using longitudinally linked employer–employee data from the LEHD infrastructure file system of the U.S. Census Bureau. First, we propose two new residual diagnostic tests of the assumption that mobility is exogenous to unmodeled determinants of earnings. Both tests reject exogenous mobility. We relax exogenous mobility by modeling the matched data as an evolving bipartite graph using a Bayesian latent-type framework. Our results suggest that allowing endogenous mobility increases the variation in earnings explained by individual heterogeneity and reduces the proportion due to employer and match effects. To assess external validity, we match our estimates of the wage components to out-of-sample estimates of revenue per worker. The mobility-bias-corrected estimates attribute much more of the variation in revenue per worker to variation in match quality and worker quality than the uncorrected estimates. Supplementary materials for this article are available online.     

Estimation of Dynamic Bivariate Mixture Models

This note compares a Bayesian Markov chain Monte Carlo approach implemented by Watanabe with a maximum likelihood ML approach based on an efficient importance sampling procedure to estimate dynamic bivariate mixture models. In these models, stock price volatility and trading volume are jointly directed by the unobservable number of price-relevant information arrivals, which is specified as a serially correlated random variable. It is shown that the efficient importance sampling technique is extremely accurate and that it produces results that differ significantly from those reported by Watanabe.     

Phase-shifting common cycles and common trends

In this paper, we outline a framework for modelling and analysing economic fluctuations and dynamics. It is assumed that there may exist common trends and common cycles in the time series to be analysed It is further generalised that common cycles may have non-coincident, or phase-shifting attributes These attributes are examined via the Markov transition matrix in a VAR system, revealing the way in which the phase-shifting works with the reduced rank Markov transition matrix. The links with the structural common trend model are also presented.     

A Discrete Variable Chain Graph for Applicants for Credit

A bank offering unsecured personal loans may be interested in several related outcome variables, including defaulting on the repayments, early repayment or failing to take up an offered loan. Current predictive models used by banks typically consider such variables individually. However, the fact that they are related to each other, and to many interrelated potential predictor variables, suggests that graphical models may provide an attractive alternative solution. We developed such a model for a data set of 15 variables measured on a set of 14 000 applications for unsecured personal loans. The resulting global model of behaviour enabled us to identify several previously unsuspected relationships of considerable interest to the bank. For example, we discovered important but obscure relationships between taking out insurance, prior delinquency with a credit card and delinquency with the loan.     

Random Bernstein Polynomials

Random Bernstein polynomials which are also probability distribution functions on the closed unit interval are studied. The probability law of a Bernstein polynomial so defined provides a novel prior on the space of distribution functions on [0, 1] which has full support and can easily select absolutely continuous distribution functions with a continuous and smooth derivative. In particular, the Bernstein polynomial which approximates a Dirichlet process is studied. This may be of interest in Bayesian non-parametric inference. In the second part of the paper, we study the posterior from a "Bernstein–Dirichlet" prior and suggest a hybrid Monte Carlo approximation of it. The proposed algorithm has some aspects of novelty since the problem under examination has a "changing dimension" parameter space.     

Some remarks on the duality of continuous time markov chains

A simple proof of Siegmund's result on the relation between dual q-matrices and dual transition functions for Markov chains is given. A byproduct is a uniqueness result for minimal solutions of stochastically monotone q-matrices.     

Inference on Filtered and Smoothed Probabilities in Markov-Switching Autoregressive Models

ABSTRACT

We derive a statistical theory that provides useful asymptotic approximations to the distributions of the single inferences of filtered and smoothed probabilities, derived from time series characterized by Markov-switching dynamics. We show that the uncertainty in these probabilities diminishes when the states are separated, the variance of the shocks is low, and the time series or the regimes are persistent. As empirical illustrations of our approach, we analyze the U.S. GDP growth rates and the U.S. real interest rates. For both models, we illustrate the usefulness of the confidence intervals when identifying the business cycle phases and the interest rate regimes.     

Further results in the optimal policy for imperfect inspection in discrete time

The paper reports investigations into how the optimal inspection interval for a fallible system with imperfect inspection is best determined. This work can be viewed as an extension of Raz and Ladany's 1992 work on imperfect inspection in discrete time. Here the cost functions are rewritten to include a probability conditioning that was omitted. In addition, an upper bound on the search space for the optimal inspection interval is presented. This bound allows one to find the global optimum with very modest computational effort. Insights gained from this bound and from the computational tests are used to present some general properties of the system.