Murray D. Smith

Stochastic Frontier Models with Dependent Error Components

of the stochastic frontier model are assumed to be independent random variables. By employing the copula approach to statistical modelling, the joint behaviour of U and V can be parametrized thereby allowing the data the opportunity to determine the adequacy of the independence assumption. In this context, three examples of the copula approach are given: the first is algebraic (the Logistic-Exponential stochastic frontier model with margins bound by the Farlie--Gumbel--Morgenstern copula), the second uses a cross-section of cost data sampled from the US electrical power industry and the third constructs a model for panel data that is then used to conduct a Monte Carlo exercise in which estimator bias is examined when the dependence structure is incorrectly ignored. Copyright Royal Economic Society 2007

On dependency in double-hurdle models

In microeconometrics, consumption data is typically zero-inflated due to many individuals recording, for one reason or another, no consumption. A mixture model can be appropriate for statistical analysis of such data, with the Dependent Double-Hurdle model (DDH hereafter) one specification that is frequently adopted in econometric practice. Essentially, the DDH model is designed to explain individual demand through a simultaneous two-step process: a market participation decision (first hurdle), and a consumption level decision (second hurdle)—a non-zero correlation/covariance parameter allows for dependency between the hurdles. A significant feature of the majority of empirical DDH studies has been the lack of support for the existence of dependency. This empirical phenomenon is studied from a theoretical perspective using examples based on the bivariate normal, bivariate logistic, and bivariate Poisson distributions. The Fisher Information matrix for the parameters of the model is considered, especially the component corresponding to the dependency parameter. The main finding is that the DDH model contains too little statistical information to support estimation of dependency, even when dependency is truly present. Consequently, the paper calls for the elimination of attempts to estimate dependency using the DDH framework. The advantage of this strategy is that it permits flexible modelling: some possibilities are proposed.

On the expectation of a ratio of quadratic forms in normal variables

Using relatively recent results from multivariate distribution theory, the expectation of a ratio of quadratic forms in normal variables is obtained. Infinite series expressions involving the invariant polynomials of matrix argument are derived. Convergence of the solution depends upon the choice made for two positive, but upper bounded, constants. The same methodology is used to obtain the expectation of multiple ratios of quadratic forms in normal variables.

Infant mortality and child nutrition in Bangladesh

The excess female infant mortality observed in South Asia has typically been attributed to gender discrimination in the intra-household allocation of food and medical care. However, studies on child nutrition find no evidence of gender differences. A natural explanation could be that in environments of high infant mortality of females, the surviving children are healthier, so that child nutrition cannot be studied independently of mortality. In this paper, we use data from the 2004 Bangladesh Demographic Health Survey to investigate if there are any gender differences in survival probabilities and whether this leads to differences in child nutrition. We argue the importance of establishing whether or not there exists a dependence relationship between the two random variables--infant mortality and child nutrition--and in order to detect this we employ a copula approach to model specification. The results suggest, for example, that while male children have a significantly lower likelihood of surviving their first year relative to female children, should they survive they have significantly better height-for-age Z-scores. From a policy perspective, household wealth and public health interventions such as vaccinations are found to be important predictors of better nutritional outcomes.

Symbolic Maximum Likelihood Estimation with Mathematica

Mathematica is a symbolic programming language that empowers the user to undertake complicated algebraic tasks. One such task is the derivation of maximum likelihood estimators, demonstrably an important topic in statistics at both the research and expository level. In this paper, a Mathematica package is provided that contains a function entitled SuperLog. This function utilises pattern-matching code that enhances Mathematicas ability to simplify expressions involving the natural logarithm of a product of algebraic terms. This enhancement to Mathematicas functionality can be of particular benefit for maximum likelihood estimation.

Comparing the exact distribution of the t-statistic to student's distribution when its constituent normal and chi-squared variables are dependent

The ubiquitous t-statistic is well known to follow Students distribution in a variety of settings. Even when a statistical model is such that a derived t-statistic is not distributed exactly according to Students distribution, it is common practice to use critical values of Students distribution as opposed to critical values of the normal distribution which are often appropriate under asymptotic considerations. This paper examines the effect which dependence between the normal and chi-square constituent variables of the t-statistic has in respect to Students distribution. Situations where the aforementioned dependence occurs, often arise in, for example, econometric practice, especially in respect of analysis of t- statistics obtained from simultaneous equation models and dynamic regression models.

mathStatica: Mathematical Statistics with Mathematica

This paper presents mathStatica (2002), a completely general toolset for doing mathematical statistics with Mathematica (Version 4). mathStatica defines statistical operators for taking expectations, finding probabilities, deriving transformations of random variables and so on. Importantly, mathStatica is not tied to a set of pre-specified statistical distributions. Rather, it is designed to derive statistics such as moments, cumulative distribution functions, characteristic functions, and other generating functions for user-defined distributions. mathStatica supports discrete and continuous distributions — univariate and multivariate. Applications to inference include: estimation (moment unbiased, minimum variance unbiased, best unbiased, maximum likelihood: symbolic and numeric), curve-fitting (Pearson and Johnson systems, non-parametric kernels), asymptotics, decision theory, and moment conversion formulae (for conversion between cumulants, raw moments, and central moments: univariate and multivariate). mathStatica accompanies the book: Rose and Smith (2002), Mathematical Statistics with Mathematica (Springer Texts in Statistics).

Invariance Theorems for Fisher Information

In multivariate and multi-parameter contexts, new expressions for Fisher Information are derived using the copula representation of the joint distribution of random variables. Invariance of Fisher Information to margins of the joint distribution is then demonstrated.

Exact densities for variance estimators of the structural disturbances in simultaneous equations models

Abstract The conditional distribution theory which enables the derivation of exact finite sample results for the density of many variance estimators of the structural disturbances in a simultaneous equations model is presented. The theory is then used to derive exact density and moment formulae when the equation contains an arbitrary number of endogenous variables. Bounds upon the existence of estimators moments are also given.

Estimates of the generalized working model

Abstract This paper presents estimates of the one-parameter generalization of Workings (1943) model proposed by Laitinen et al. (1983).