Wayne A. Fuller

Distribution of the Estimators for Autoregressive Time Series with a Unit Root

Abstract Let n observations Y 1, Y 2, ···, Y n be generated by the model Y t = pY t−1 + e t , where Y 0 is a fixed constant and {e t } t-1 n is a sequence of independent normal random variables with mean 0 and variance σ2. Properties of the regression estimator of p are obtained under the assumption that p = ±1. Representations for the limit distributions of the estimator of p and of the regression t test are derived. The estimator of p and the regression t test furnish methods of testing the hypothesis that p = 1.

A Semiparametric Transformation Approach to Estimating Usual Daily Intake Distributions

Abstract The distribution of usual intakes of dietary components is important to individuals formulating food policy and to persons designing nutrition education programs. The usual intake of a dietary component for a person is the long-run average of daily intakes of that component for that person. Because it is impossible to directly observe usual intake for an individual, it is necessary to develop an estimator of the distribution of usual intakes based on a sample of individuals with a small number of daily observations on a subsample of the individuals. Daily intake data for individuals are nonnegative and often very skewed. Also, there is large day-to-day variation relative to the individual-to-individual variation, and the within-individual variance is correlated with the individual means. We suggest a methodology for estimating usual intake distributions that allows for varying degrees of departure from normality and recognizes the measurement error associated with one-day dietary intakes. The est...

Testing for Unit Roots in Seasonal Time Series

Abstract Regression estimators of coefficients in seasonal autoregressive models are described. The percentiles of the distributions for time series that have unit roots at the seasonal lag are computed by Monte Carlo integration for finite samples and by analytic techniques and Monte Carlo integration for the limit case. The tabled distributions may be used to test the hypothesis that a time series has a seasonal unit root.

An Error-Components Model for Prediction of County Crop Areas Using Survey and Satellite Data

Abstract Knowledge of the area under different crops is important to the U.S. Department of Agriculture. Sample surveys have been designed to estimate crop areas for large regions, such as crop-reporting districts, individual states, and the United States as a whole. Predicting crop areas for small areas such as counties has generally not been attempted, due to a lack of available data from farm surveys for these areas. The use of satellite data in association with farm-level survey observations has been the subject of considerable research in recent years. This article considers (a) data for 12 Iowa counties, obtained from the 1978 June Enumerative Survey of the U.S. Department of Agriculture and (b) data obtained from land observatory satellites (LANDSAT) during the 1978 growing season. Emphasis is given to predicting the area under corn and soybeans in these counties. A linear regression model is specified for the relationship between the reported hectares of corn and soybeans within sample segments in...

A Comparison of Unit-Root Test Criteria

During the past fifteen years, the ordinary least squares estimator and the corresponding pivotal statistic have been widely used for testing the unit root hypothesis in autoregressive processes. Recently, several new criteriia, based on the maximum likelihood estimators and weighted symmetric estimators, have been proposed. In this article, we describe several different test criteria. Results from a Monte Carlo study that compares the power of the different criteria indicates that the new tests are more powerful against the stationary alternative. Of the procedures studied, the weighted symmetric estimator and the unconditional maximum likelihood estimator provide the most powerful tests against the stationary alternative. As an illustration, we analyze the quarterly change in busine;ss investories.

Estimation of linear models with crossed-error structure

Abstract Sufficient conditions are presented under which the generalized least-squares estimator, with estimated covariance matrix, is unbiased for the parameters in the crossed-error model and has the same asymptotic distribution as the generalized least-squares estimator. The model permits the presence of independent variables that are constant over cross sections or time periods. The model does not require that the variance components associated with cross sections or time periods be positive.

Survey Design under the Regression Superpopulation Model

Abstract The construction of sample designs and estimators under a linear regression superpopulation model is considered. The anticipated variance, the variance of the predictor computed with respect to the sampling design and the superpopulation model, is used as a criterion for evaluating probability designs and model-unbiased predictors. Regression predictors that are model unbiased and design consistent are constructed.

Fitting Segmented Polynomial Regression Models Whose Join Points Have to Be Estimated

Abstract The study considers the problem of finding the least squares estimates for the unknown parameters of a regression model which consists of grafted polynomial submodels. The abscissae of the join points are a subset of the unknown parameters. Examples are given to illustrate how continuity and differentiability conditions on the model can be used to reparameterize the model so as to allow Modified Gauss-Newton fitting. A slightly generalized version of Hartleys theorem is stated to extend the Modified Gauss-Newton method to this problem.

Properties of Predictors for Autoregressive Time Series

Abstract The prediction of the (n + s)th observation of the pth order autoregressive process is investigated. The mean square of the predictor error through terms of order n —1, conditional on Yn, Y n — 1, …, Y n — p + 1, is obtained for the stationary normal process. The mean squared error expression is similar to the usual regression formula for the variance of the predictor error. The usual regression formula for the estimated variance of a predictor error and its generalization to s-period prediction is shown to provide a consistent estimator of the mean squared error of the least squares predictor for both stationary and non-stationary processes.

Least Squares Estimation When the Covariance Matrix and Parameter Vector are Functionally Related

Abstract Estimation for the linear model y = Xβ + e with unknown diagonal covariance matrix G is considered. The diagonal elements of G are assumed to be known functions of the explanatory variables X and an unknown parameter vector Θ, where Θ is permitted to contain elements of β. A weighted joint least squares estimator is developed that is asymptotically equivalent to the maximum likelihood estimator. Asymptotic properties of the simple least squares estimator and of the weighted joint least squares estimator are obtained. A sampling experiment is used to compare the estimators.