Regression Error Term Research Articles

REGCMPNT- AFortranProgram for Regression Models with ARIMA Component Errors

RegComponent models are time series models with linear regression mean functions and error terms that follow ARIMA (autoregressive-integrated-moving average) component time series models. Bell (2004) discusses these models and gives some underlying theoretical and computational results. The <b>REGCMPNT</b> program is a Fortran program for performing Gaussian maximum likelihood estimation, signal extraction, and forecasting with RegComponent models. In this paper we briefly examine the nature of RegComponent models, provide an overview of the <b>REGCMPNT</b> program, and then use three examples to show some important features of the program and to illustrate its application to various different RegComponent models.

Teaching Causal Inference in Undergraduate Econometrics

This paper argues that the current way in which the undergraduate introductory econometrics course is taught is neither inline with current empirical practice nor very intuitive. It proposes a shift in focus of the course on causal inference using the Roy-Rubin Causal Model (RRCM). A second theme of the paper is the suggestion to use random regressors from the start to improve the ability of students to intuitively relate to the regression model and to enable the teacher to present many regression pitfalls under the umbrella of the non-zero covariance between regressors and the regression error term. Finally, the paper discusses how to make room for the new material and suggests ways of dealing with the added complexity of the approach.

A Bayesian Reliability Approach to Multiple Response Optimization with Seemingly Unrelated Regression Models

This paper presents a Bayesian predictive approach to multiresponse optimization experiments. It generalizes the work of Peterson [33] in two ways that make it more flexible for use in applications. First, a multivariate posterior predictive distribution of seemingly unrelated regression models is used to determine optimum factor levels by assessing the reliability of a desired multivariate response. It is shown that it is possible for optimal mean response surfaces to appear satisfactory yet be associated with unsatisfactory overall process reliabilities. Second, the use of a multivariate normal distribution for the vector of regression error terms is generalized to that of the (heavier tailed) multivariate t-distribution. This provides a Bayesian sensitivity analysis with regard to moderate outliers. The effect of adding design points is also considered through a preposterior analysis. The advantages of this approach are illustrated with two real examples.

Bayesian Analysis of a Doubly Truncated ARMA-GARCH Model

We develop a new Markov Chain Monte Carlo procedure for a time series regression model truncated by upper and lower bounds. The regression error term is assumed to follow an ARMA--GARCH process. We use a convergence diagnostics with a simultaneous test of mean and covariance stationarity and discuss model selection criteria. Using MCMC procedure we test the purchasing power parity theory for the Japanese yen controlled to fluctuate in a narrow band and find that the theory is supported if double truncation is incorporated in estimation.

Associations between stomach cancer incidence and drinking water contamination with atrazine and nitrate in Ontario (Canada) agroecosystems, 1987-1991.

Nitrate and atrazine are two chemicals that are heavily used in certain sectors of agriculture. They are suspected to be associated with the development of certain types of tumours. Existing data were obtained on the incidence of specific types of cancers, contamination of drinking water with atrazine and nitrate, and related agricultural practices for the 40 ecodistricts in the province of Ontario. The data were merged into a georelational database for geographical and statistical analyses. Weighted (by population size) least squares regression analyses were conducted while controlling for confounding socioeconomic and lifestyle factors. Maximum likelihood spatial error models were estimated when least square regression error terms were found to be spatially autocorrelated using the Moran's I statistic. Atrazine contamination levels (range 50-649 ng/l, maximum acceptable concentration [MAC] = 60000 ng/l) were positively associated (P < 0.05) with stomach cancer incidence and negatively associated with colon cancer incidence. Nitrate levels, (range 0-91 mg/l, MAC = 10 mg/l) were negatively associated with stomach cancer incidence. The associations found at the ecodistrict level, both positive and negative, if confirmed by other studies, raise serious questions about maximum allowable limits for atrazine, as well as possibilities of complex trade-offs among disease outcomes, and interactions of biophysical and social mechanisms which might explain them. Although the negative associations appear to have no direct biological explanations, such counter-intuitive outcomes may occur in complex systems where social and biological variables interact.

Spatial Autocorrelation: A Primer

Regression error terms are likely to be spatially autocorrelated in any situation in which “location matters.” While both the precision of the estimates and the reliability of hypothesis testing can be improved by making a correction for spatial autocorrelation, the techniques for making such a correction are not widely understood. The purpose of this paper is to explore some of the issues involved in estimating models with spatially autocorrelated error terms. One of the two most common methods of handling spatial autocorrelation is the weight matrix approach, in which the process generating the errors is modeled. The resulting correlation structure is then derived from the hypothesized process. The second method models the correlation structure itself, rather than the underlying process. The bulk of this paper is concerned with comparing these two methods and their resulting correlation structures. Other issues are discussed at the end of the paper.

Conditional independence in sample selection models

In most sample selection models, the conditional mean function for regression error terms is independent of the regressors conditional on the selection-propensity score. Conditioning on the selection-propensity score therefore controls selection bias, a fact that underlies much of the literature on semi-parametric selection models. This note provides a general formulation of the conditional-independence properties that make it possible to control selection bias using the selection-propensity score.

Testing the distributional assumptions of least squares linear regression

The error terms in least squares linear regression are assumed to be normally distributed with equal variance (homoskedastic), and independent of one another. If any of these distributional assumptions are violated, several of the desirable properties of a least squares fit may not hold. A variety of statistical tests of the assumptions is available. The following are recommended for reasons of ease of use and discriminating power: the K2 test for testing for non-normality, either the Durbin-Watson test or the Q-test for testing for autocorrelation, and either Szroeter's or White's test for testing for heteroskedasticity. The assumptions should be tested in this order; violating one of the assumptions may invalidate the results of subsequent tests. A microcomputer-based software package for least squares linear regression that incorporates the above tests is introduced. Key words: normality, homoskedasticity, independence

Application of a Variance-Stabilizing Transformation Approach to Linear Regression of Calibration Lines

A variance-stabilizing transformation (VST) was applied to the linear regression of calibration standards of different drugs in plasma. This transformation involved the normalization of the dependent variable peak height or peak area ratio (Y), and the independent variable, plasma drug concentration (C). This transformation led to a constant variance in the regression error term across the measured concentration range and allowed the evaluation of the unbiased slope and y intercept with minimum variance. The utility of the VST procedure in comparison with the ordinary least squares (OLS) approach, routinely used in pharmaceutical studies for constructing calibration lines, is described. The principal advantage of the VST approach is allowing a lower minimum level of drug quantification while using a single calibration line over a wide range of drug concentrations. The VST method is especially useful to quantify drug plasma levels in pharmacokinetic evaluation of sustainedrelease dosage forms, where the precise quantification of low levels of drug is critical. The application of the VST method was explored and evaluated in comparison with the OLS method for pharmacokinetic assays of diltiazem, gallopamil, nitroglycerin, and nicotine.

Probability calculation for the Durbin-Watson correlation test

One of the assumptions of least squares regression models is that the regression error terms are independent of each other. This condition is often violated when analyzing data taken from observations over time; an important type of data in environmental studies. The statistic used to estimate whether the lag 1 coefficient of autocorrelation is equal to zero is known as the Durbin-Watson statistic. This article develops a methodology for the implementation of the probability calculation for the Durbin-Watson test for serial autocorrelation in least squares regression models. The algorithms defined by Durbin and Watson in their 1971 paper were examined, essential calculations were defined and a computer implementation using FORTRAN 77 was developed. The computer code was verified via the simulation of a simple known regression with no autocorrelation and confirmation that the distribution of the results is roughly uniform. Timing curve for runs using a DEC VAX-11/750 and a full listing of the subroutine source code are also included.

Determinants of Household Electricity Demand: A Two-Stage Probit Approach

Considerable interest on the part of policy makers in both the public and private sectors has spawned numerous studies of the determinants of residential electricity demand. As a result of increased attention given to the possibility of adjusting household electricity consumption patterns through price manipulation (as in the various peak-load pricing experiments), particular emphasis has been placed on obtaining an accurate estimate of the price elasticity of demand. Unfortunately, OLS applied to the conventional linear (or log-linear) demand specification is biased and inconsistent when the sample includes households that face declining-block rate schedules. The inconsistency results from the fact that, in the presence of declining-block rates, observed marginal price is a stochastic regressor that is negatively correlated with the regression error term. Some researchers have suggested that consistent OLS estimation can be achieved by merely respecifying the demand equation to include particular combinations of marginal price, average price, rate of decline of the price schedule, and real income corrected for intra-marginal price differences [1; 7; 8]. Others, realizing the futility of such an approach, have developed and applied instrumental variable (IV) techniques [4; 6; 11]. These IV methods, however, all involve an artificial linearization of the rate schedule which produces spurious negative correlation between the price variable and the random error term. As a result, such IV methods tend to overcompensate for actual correlation. Consequently, the IV estimated coefficients of the various demand determinants will exhibit biases opposite in sign to those of their OLS counterparts. For example, in the presence of declining-block rates, OLS tends to overestimate the price elasticity of demand (in absolute value), while IV methods produce elasticity estimates that are downward biased. A Two-Stage Probit (TSP) method developed by Terza and Welch [10] is not subject to the bias that plagues IV methods because TSP does not require a linear approximation to the rate schedule. Instead, the discrete and discontinuous nature of household responses to declining-block pricing is captured by incorporating an ordinal probit model that is based on a theory of consumer choice. In the present paper a linear electricity demand model is estimated using the TSP, OLS, and IV methods to demonstrate the effectiveness of the TSP method.

A Simultaneous Equation Demand Model for Block Rates

This paper examines the problem of simultaneous‐equations bias in estimation of the water demand function under an increasing block rate structure. The Hausman specification test is used to detect the presence of simultaneous‐equations bias arising from correlation of the price measures with the regression error term in the results of a previously published study of water demand in Tucson, Arizona. An alternative simultaneous equation model is proposed for estimating the elasticity of demand in the presence of block rate pricing structures and availability of service charges. This model is used to reestimate the price and rate premium elasticities of demand in Tucson, Arizona for both the usual long‐run static model and for a simple short‐run demand model. The results from these simultaneous equation models are consistent with a priori expectations and are unbiased.

METHODOLOGICAL IMPLICATIONS OF NON‐NORMALLY DISTRIBUTED FINANCIAL RATIOS: A COMMENT

This comment argues that proposals to replace financial ratio analysis with regression analysis of the separate ratio components are unnecessary and misguided. Financial ratios are used to predict other variables and not to predict their own components. These ratios also help deal with the size scale problem inherent in regular accounting data, but regression error terms would suffer from the same size effect as the regular data. The actual statistical distribution of financial ratios is an open, important question, but that question in itself does not call for an abandonment of fmancial ratios.

Bayesian Analysis of Regression Error Terms

Abstract Properties of realized regression error terms are studied wherein they are treated as unknown parameters. Posterior distributions for individual realized error terms and for linear and certain quadratic functions of them are derived. Posterior moments of measures of skewness, kurtosis, autocovariance, and autocorrelation, defined in terms of the realized errors, are presented, and selected uses of these results indicated in characterizing properties of realized error terms and in suggesting possible departures from standard assumptions.

Bayesian Analysis of Regression Error Terms

Bayesian Analysis of Regression Error Terms