William E. Wecker

The Signal Extraction Approach to Nonlinear Regression and Spline Smoothing

Abstract This article shows how to fit a smooth curve (polynomial spline) to pairs of data values (yi, xi ). Prior specification of a parametric functional form for the curve is not required. The resulting curve can be used to describe the pattern of the data, and to predict unknown values of y given x. Both point and interval estimates are produced. The method is easy to use, and the computational requirements are modest, even for large sample sizes. Our method is based on maximum likelihood estimation of a signal-in-noise model of the data. We use the Kalman filter to evaluate the likelihood function and achieve significant computational advantages over previous approaches to this problem.

Asymmetric Time Series

Abstract Asymmetric time series respond to innovations with one of two different rules according to whether the innovation is positive or negative. Quoted industrial prices are apparently such a series. It has been observed that when market conditions change, quoted prices are not revised immediately. This delay operates more strongly against reductions in price quotations than against increases. A statistical model for such asymmetric times series is developed and analyzed. An estimation procedure is given as well as a statistical test of the hypothesis of symmetry versus the alternative of asymmetry. Asymmetric time series models are fit to several economic time series.

Predicting a Multitude of Time Series

Abstract Principles for constructing estimators of the Stein type (shrinkage estimators) are discussed in general terms, with emphasis on underlying assumptions. The problem of parameter estimation and prediction for multiple time series is examined with these principles in mind, particularly for the case in which the number of time series is large and the number of observations from each series is small. Our results are applied to the problem of demand estimation in an inventory control setting.

Correcting for Omitted-Variables and Measurement-Error Bias in Regression with an Application to the Effect of Lead on IQ

Abstract Ordinary least squares (OLS) regression estimates are biased, in general, when relevant variables are omitted from the regression equation or when included variables are measured with error. The errors-in-variables bias can be corrected using auxiliary information about unobservable measurement errors. In this article we demonstrate how auxiliary information can also be used to correct for omitted-variables bias. We illustrate our methods with an application to four published studies of the effect on IQ of childhood exposure to lead. Each of the published studies used OLS methods (or equivalent). None of the studies includes a father IQ variable, and none accounts for the biasing effect of measurement error in the right-side variables. For each of the studies we demonstrate that bias-corrected estimates of the effect of lead on IQ are much reduced in size and are not significantly different from 0. Our methods can be used in other applications involving omitted variables or errors of measurement ...

A note on the time series which is the product of two stationary time series

The time series [...,x-1y-1,x0y0,x1y1,...]> which is the product of two stationary time series xt and yt is studied. Such sequences arise in the study of nonlinear time series, censored time series, amplitude modulated time series, time series with random parameters, and time series with missing observations. The mean and autocovariance function of the product sequence are derived.

Bounds on Absorption Probabilities for the m -Dimensional Random Walk

Abstract Simple procedures are given for computing an upper bound on the probability that an m-dimensional random walk has not been absorbed by step n. The increments of the walk are multivariate normal, independent, but not necessarily identical. The upper bound may be computed without knowledge of the means of the increment, the shapes of the nonabsorbing regions, or the starting point of the walk. The absorbing regions may also change with time. Under certain regularity conditions the upper bound is shown to be a geometrically decreasing sequence. An application to the nonstationary stochastic cash balance problem is suggested.