Thomas S. Shively

ESTIMATING IRREGULAR PRICING EFFECTS : A STOCHASTIC SPLINE REGRESSION APPROACH

Markets respond to prices in complex ways. Multiple factors such as price points, odd pricing, and just-noticeable differences often cause steps and spikes in response. The result is market response functions that are frequently nonmonotonic. However, existing regression-based approaches employ functions that are inherently monotonic, which thereby limits representation of important irregularities. In this article, the authors use a stochastic spline regression approach in the framework of a hierarchical Bayes model that permits the estimation of irregular pricing effects and apply the approach to data sets from several product categories. A simulation study indicates that the stochastic spline approach is flexible enough to accommodate irregular response functions. The empirical results show that there are irregularities in own-price response for most of the brands examined and that there are important profit implications of these irregular response functions in pricing decisions. The authors find that t...

Variable Selection and Function Estimation in Additive Nonparametric Regression Using a Data-Based Prior

Abstract A hierarchical Bayesian approach is proposed for variable selection and function estimation in additive nonparametric Gaussian regression models and additive nonparametric binary regression models. The prior for each component function is an integrated Wiener process resulting in a posterior mean estimate that is a cubic smoothing spline. Each of the explanatory variables is allowed to be in or out of the model, and the regression functions are estimated by model averaging. To allow variable selection and model averaging, data-based priors are used for the smoothing parameter and the slope at 0 of each component function. A two-step Markov chain Monte Carlo method is used to efficiently obtain the data-based prior and to carry out variable selection and function estimation. It is shown by simulation that significant improvements in the function estimators can be obtained over an approach that estimates all the unknown functions simultaneously. The methodology is illustrated for a binary regressio...

Point process approach to modeling trends in tropospheric ozone based on exceedances of a high threshold

A major issue with the analysis of data on tropospheric ozone is to establish whether observed trends in the data are real, meaning that they could be attributed to actual changes in the emissions of toxic gases into the atmosphere, or whether they are the result of meteorological changes affecting the conditions under which ozone is generated. One way of investigating this question is to construct a regression model in which the level of ozone is represented as a function of both meteorological variables and time, in order to determine the significance of the time component when the meteorological variables are taken into account. However, the conventional methods of regression analysis do not make any distinction between low and high levels of the series, whereas with ozone it is largely the high levels that are of interest and concern. This paper proposes a method of regression analysis that is based entirely on the exceedances over a high threshold, and applies the method to data from the Houston area.

Fast Evaluation of the Distribution of the Durbin-Watson and other Invariant Test Statistics in Time Series Regression

Abstract A method is given for evaluating p values in O(n) operations for a general class of invariant test statistics that can be expressed as the ratio of quadratic forms in time series regression residuals. The best known of these is the Durbin-Watson statistic, although several others have been discussed in the literature. The method is numerically exact in the sense that the user specifies the error tolerance at the outset. As with existing exact methods, the problem is reexpressed in terms of the distribution function of a single quadratic form in independent normals, which is evaluated by numerically inverting its characteristic function. Existing methods, however, calculate the characteristic function by reducing the matrix defining the quadratic form to either eigenvalue or tridiagonal form, each of which requires O(n 3) operations for sample size n, whereas the new method uses a modification of the Kalman filter to do it in O(n) operations. Moreover, the new method has minimal storage requiremen...

Model selection in spline nonparametric regression

A Bayesian approach is presented for model selection in nonparametric regression with Gaussian errors and in binary nonparametric regression. A smoothness prior is assumed for each component of the model and the posterior probabilities of the candidate models are approximated using the Bayesian information criterion. We study the model selection method by simulation and show that it has excellent frequentist properties and gives improved estimates of the regression surface. All the computations are carried out efficiently using the Gibbs sampler.

An analysis of tests for regression coefficient stability

Abstract An exact small-sample point-optimal test is developed for testing the hypothesis that a regression coefficient is constant against the alternative that it is generated by a first-order autoregressive process. In addition, it is shown how Watson and Engles (1985) asymptotic test can be modified so that its small-sample level can be computed and its small-sample power properties investigated. Simulation results show that the point-optimal test developed in this paperoutperforms the small-sample Watson–Engle test as well as other tests given in the literature. A point-optimal test is also constructed for a stochastic coefficient generated by an ARIMA (1,1,0) process.

Computing p-values for the generalized Durbin-Watson and other invariant test statistics

Abstract Shively, Ansley, and Kohn (1990) give an O( n ) algorithm for computing the p -values of the Durbin-Watson and other invariant test statistics in time series regression. They do so by evaluating the characteristic function of a quadratic form in standard normal random variables and then numerically inverting it. In this paper we obtain a new expression for the characteristic function which simplifies the handling of the independent regressors and so is easier to evaluate. We also obtain general, easily computable bounds on the integration and truncation errors which arise in the numerical inversion of the characteristic function. Empirical results are presented on the speed and accuracy of our algorithm.

An analysis of the trend in ground-level ozone using non-homogeneous poisson processes

Abstract This paper provides a method for measuring the long-term trend in the frequency with which ground-level ozone present in the ambient air exceeds the U.S. Environmental Protection Agencys National Ambient Air Quality Standard (NAAQS) for ozone. A major weakness of previous studies that estimate the long-term trend in the very high values of ozone, and therefore the long-term trend in the probability of satisfying the NAAQS for ozone, is their failure to account for the confounding effects of meterological conditions on ozone levels. Meteorological variables such as temperature, wind speed, and frontal passage play an important role in the formation of ground-level ozone. A non-homogenous Poisson process is used to account for the relationship between very high values of ozone and meteorological conditions. This model provides an estimate of the trend in the ozone values after allowing for the effects of meteorological conditions. Therefore, this model provides a means to measure the effectiveness of pollution control programs after accounting for the effects of changing weather conditions. When our approach is applied to data collected at two sites in Houston, TX, we find evidence of a gradual long-term downward trend in the frequency of high values of ozone. The empirical results indicate how possibly misleading results can be obtained if the analysis does not account for changing weather conditions.

A Bayesian approach to model selection in stochastic coefficient regression models and structural time series models

Abstract A Bayesian model selection procedure is proposed for a stochastic coefficient regression model to determine which coefficients are fixed and which are time-varying. The posterior probabilities are computed by Gaussian quadrature using the Kalman filter. It is shown empirically that the model selection approach works well on both simulated and real data. A similar approach can be used to select a model from a class of state space models. In particular, for a trend plus seasonal structural time series model we show how to determine if the trend and/or seasonal component is deterministic or stochastic.

A Semiparametric Stochastic Spline Model as a Managerial Tool for Potential Insolvency

This study introduces a flexible nonlinear semiparametric spline model, new to solvency studies, as a tool for managerial discretion and regulatory oversight. The model has a linear component and a nonlinear component that uses stochastic splines. The study focuses on the functional relationship between regressors and the probability of financial distress as an object for managerial action. Leverage plots are provided to analyze the potential effect of decisions to modify firm levels of financial variables. If the true relationship between regressors and the response is not linear, then managerial efforts to rectify deteriorating financial conditions can be misinformed by reliance on a linear solvency model. The leverage plots adjust to the firms position within the industry and its specific levels of various financial variables. A five-regressor semiparametric spline model is shown to yield insights into the behavior of the risk of financial distress probabilities that linear parametric models suppress. The model also classifies and validates well in comparison with recent insolvency studies and as well as parametric logit and probit models on the same data.