Showing papers on "Heteroscedasticity published in 1986"

Generalized autoregressive conditional heteroscedasticity

Abstract: A natural generalization of the ARCH (Autoregressive Conditional Heteroskedastic) process introduced in Engle (1982) to allow for past conditional variances in the current conditional variance equation is proposed. Stationarity conditions and autocorrelation structure for this new class of parametric models are derived. Maximum likelihood estimation and testing are also considered. Finally an empirical example relating to the uncertainty of the inflation rate is presented.

Modelling the persistence of conditional variances

Abstract: This paper will discuss the current research in building models of conditional variances using the Autoregressive Conditional Heteroskedastic (ARCH) and Generalized ARCH (GARCH) formulations. The discussion will be motivated by a simple asset pricing theory which is particularly appropriate for examining futures contracts with risk averse agents. A new class of models defined to be integrated in variance is then introduced. This new class of models includes the variance analogue of a unit root in the mean as a special case. The models are argued to be both theoretically important for the asset pricing models and empirically relevant. The conditional density is then generalized from a normal to a Student-t with unknown degrees of freedom. By estimating the degrees of freedom, implications about the conditional kurtosis of these models and time aggregated models can be drawn. A further generalization allows the conditional variance to be a non-linear function of the squared innovations. Throughout empirical...

Jackknife, Bootstrap and Other Resampling Methods in Regression Analysis

Abstract: Motivated by a representation for the least squares estimator, we propose a class of weighted jackknife variance estimators for the least squares estimator by deleting any fixed number of observations at a time. They are unbiased for homoscedastic errors and a special case, the delete-one jackknife, is almost unbiased for heteroscedastic errors. The method is extended to cover nonlinear parameters, regression $M$-estimators, nonlinear regression and generalized linear models. Interval estimators can be constructed from the jackknife histogram. Three bootstrap methods are considered. Two are shown to give biased variance estimators and one does not have the bias-robustness property enjoyed by the weighted delete-one jackknife. A general method for resampling residuals is proposed. It gives variance estimators that are bias-robust. Several bias-reducing estimators are proposed. Some simulation results are reported.

Residual variance and residual pattern in nonlinear regression

Abstract: SUMMARY A nonparametric estimator of residual variance in nonlinear regression is proposed. It is based on local linear fitting. Asymptotically the estimator has a small bias, but a larger variance compared with the parametric estimator in linear regression. Finite sample properties are investigated in a simulation study, including a comparison with other nonparametric estimators. The method is also useful for spotting heteroscedasticity and outliers in the residuals at an early stage of the data analysis. A further application is checking the fit of parametric models. This is illustrated for longitudinal growth data.

Symmetrically Trimmed Least Squares Estimation for Tobit Models

Abstract: This paper proposes alternatives to maximum likelihood estimation of the censored and truncated regression models (known to economists as "Tobit" models). The proposed estimators are based upon symmetric censoring or truncation of the upper tail of the distribution of the dependent variable. Unlike methods based on the assumption of identically distributed Gaussian errors, the estimators are semiparametric, in the sense that they are consistent and asymptotically normal for a wide class of (symmetric) error distributions with heteroskedasticity of unknown form. The paper gives the regularity conditions and proofs of these large sample properties, demonstrates how to construct consistent estimators of the asymptotic covariance matrices, and presents the results of a simulation study for the censored case. Extensions and limitations of the approach are also considered.

An Approach to Statistical Inference in Cross-Sectional Models with Security Abnormal Returns As Dependent Variable

Abstract: Several recent event studies test hypotheses about the association between event-related abnormal security returns and firm characteristics using a three-step procedure.' First, forecast model parameters are estimated, usually parameters of a market model; second, prediction errors or residuals are computed over an event period; third, prediction errors are regressed cross-sectionally on firm characteristics hypothesized to influence the impact of the event on share values. Results of these regressions are used to draw inferences about the relation between abnormal returns and firm characteristics. Researchers who employ three-step procedures acknowledge that crosssectional (OLS) regressions lead to valid inferences if the disturbances are IID (normal) in cross-section; see, for example, Leftwich [1981, p. 23] and Lustgarten [1982, p. 138]. While sufficient assumptions to draw valid inferences from OLS regressions are clear, these assumptions are violated if there is cross-correlation and cross-sectional heteroscedasticity in the firm return processes from which the prediction errors are estimated. Assumptions about the processes generating these prediction

How to Use the Two Sample t-Test

Abstract: This work discusses how two sample t-tests behave when applied to data that may violate the classical statistical assumptions of independence, heteroscedasticity and Gaussianity. The usual two sample t-statistic based on a pooled variance estimate and the Welch-Aspin statistic are treated in detail. Practical “rules-of-thumb” are given along with their applications to various examples so that readers will easily be able to use such tests on their own data sets.

The Linear Regression Model Under Test

Abstract: 1. Introduction.- 2. Technical Preliminaries.- a) The Linear Regression Model.- Notation and assumptions.- Regression residuals.- b) LR-, Wald- and LM-Tests.- Basic principles.- A simple example.- Estimating the information matrix.- c) Miscellaneous Terminology.- 3. Testing Disturbances.- a) Autocorrelation.- The Durbin-Watson test.- The power of the Durbin-Watson test.- The DW inconclusive region.- Relationship to the LM principle.- Dynamic regressions.- Simultaneous equations.- Other alternatives to the DW test.- b) Heteroskedasticity.- Comparing empirical variances.- A general LM test.- Testing for arbitrary heteroskedasticity.- c) Normality.- Comparing moments.- Residuals vs. true disturbances.- 4. Testing Regressors.- a) Structural Change.- The Chow test and related methods.- Testing subsets of the regression coefficients.- Coefficient stability vs. stability of the variance.- Shift point unknown.- The CUSUM and CUSUM of squares tests.- The CUSUM and CUSUM of squares tests with dummy exogenous variables.- Alternatives to the CUSUM and CUSUM of squares tests.- The Fluctuation test.- Local power.- A Monte Carlo comparison of the CUSUM and Fluctuation tests.- Dynamic models.- A modification of the CUSUM test.- A modification of the CUSUM of squares test.- b) Functional Form.- Ramsey's RESET.- The Rainbow test.- Convex or concave alternatives.- Linear vs. log-linear functional form.- Outliers.- c) General Misspecification.- Hausman-type specification tests.- Errors in variables.- Computing the Hausman test.- Exogeneity.- A Hausman test with trending data.- Hausman tests versus classical tests.- The differencing test.- First differences as an IV estimator.- The information matrix test.- 5. Unifying Themes.- a) Variable Transformation.- Linear transformations.- Examples.- Relationship to IV estimation.- b) Variable Addition.- Data transformation tests as RESETs.- Hausman tests as RESETs.- Power comparisons.- Asymptotic independence.- 6. Diagnostic Checking in Practice.- a) Empirical Results.- Demand for money.- Currency substitution.- Bond yield.- Growth of money supply.- The Value of stocks.- Wages.- Unemployment.- Discussion.- b) Issues in Multiple Testing.- Controlling the size.- Ruger's test.- Global vs. multiple significance level.- Holm's procedure.- Robustness.- Further reading.- a) Sample Data.- b) The IAS-SYSTEM.- References.- Author Index.

Improved methods of inference in econometrics

Abstract: Introduction. The Inferential and Decision Framework. The Measure of Performance. Some Alternative Statistical Models, Estimators and Tests. Inequality Estimation and Hypothesis Testing. Inequality Estimation and Hypothesis Testing of the Location Parameter of a Normal Random Variable. General Linear Statistical Model and a Single Linear Inequality Constraint. Inequality Restricted Estimation of Two or More Location Parameters: The Orthonormal Case. An Inequality Restricted Stein Rule Estimator. Hypothesis Testing for Two or More Inequalities: The Orthogonal Case. Inequality Restricted Estimation: General Design and Restriction Matrices. Inequality Hypothesis Testing: General Case. Some Sampling Results for the Stein Family of Estimators. Assessing the Precision of Stein's Estimator. Some Evaluations of the Sampling Performance of the Limited Translation and New Stein Estimators. Estimation Under Non-Normal Errors and Quadratic Loss. Estimation and Hypothesis Testing in the Case of Possible Heteroskedasticity. Subject Index.

ARCH and Bilinear Time Series Models: Comparison and Combination

Abstract: Two extensions to the ARMA model, bilinearity and ARCH errors are compared, and their combination is considered. Starting with the ARMA model, tests for each extension are discussed, along with various least squares and maximum likelihood estimates of the parameters and tests of the estimated models based on these. The effects each may have on the identification, estimation, and testing of the other are given, and it is seen that to distinguish between the two properly, it is necessary to combine them into a bilinear model with ARCH errors. Some consequences of the misspecification caused by considering only the ARMA model are noted, and the methods are applied to two real time series.

Correcting for Heteroscedasticity in Tests for Market Timing Ability

Abstract: Evaluating the performance of portfolio managers has received wide attention in the finance literature. The common practice is to divide performance into (a) market timing ability and (b) security selection ability. The former is the ability of a fund manager to produce a better return distribution by forecasting marketwide movements. The latter is the ability to produce more favorable return distributions based on superior information about individual stocks. Several methods have been proposed in the literature for the evaluation of the selection and the timing abilities of portfolio managers, using only the observed time series of realized returns on the managed portfolios. Among the approaches proposed, the multiple regression approach suggested by Treynor and Mazuy (1966), Henriksson and Merton (1981), and Pfleiderer and Bhattacharya (1983) are particularly attractive since they are simple and easy to apply. While the above multiple regression methods are easy to apply, statistical inference requires care. 1 In this paper we examine the parametric test proposed by Henriksson and Merton for evaluating the market timing ability of portfolio managers. Using simulation techniques we show that correction for heteroscedasticity can significantly affect the conclusions. We find that the heteroscedasticity corrections suggested by Hansen and by White are particularly effective.

Score test for the first-order autoregressive model with heteroscedasticity

Abstract: SUMMARY A score test is proposed for simultaneous testing of independence and homoscedasticity in the first-order autoregressive model with nonconstant variance. The relationships between the score test statistic and the local influence of minor perturbations on a statistical model are examined.

Some new estimation methods for weighted regression where there are possible outliers

Abstract: The problem considered is the robust estimation of the variance parameter in a heteroscedastic linear model. We treat the situation in which the variance is a function of the explanatory variables. To estimate robustly the variance in this case, it is necessary to guard against the influence of outliers in the design as well as outliers in the response. By analogy with the homoscedastic regression case, we propose two estimators that do this. Their performances are evaluated on a number of data sets. We had considerable success with estimators that bound the “self-influence”—that is. the influence an observation has on its own fitted value. We conjecture that in other situations (e.g., homoscedastic regression) bounding the selfinfluence will lead to estimators with good robustness properties.

Bias of S 2 in Linear Regressions with Dependent Errors

Abstract: Bounds are given for the expected value of the estimator of the error variance in linear regressions, when the errors are dependent or heteroscedastic. The bounds are valid irrespective of the covariance structure between the errors. Necessary and sufficient conditions to attain the bounds are supplied.

A bounds test for equality between sets of coefficients in two linear regression models under heteroscedasticity

Abstract: This article proposes a small sample bounds test for equality between sets of coefficients in two linear regressions with unequal disturbance variances. The probability that our test is inconclusive is given under the null hypothesis. It is also shown that our test is more powerful than the Jayatissa test when the regression coefficients differ substantially.

Generalized autoregressive conditional heteroskedasticity

Abstract: The present paper proposes a generalization of the canonical AutoRegressive Conditional Heteroskedasticity (ARCH) model by extending the conditional variance equation toward past conditional variances. The stationarity conditions and autocorrelation structure of the Generalized AutoRegressive Conditional Heteroskedastic (GARCH) model are derived. Using an empirical example of uncertainty of the inflation rate the paper demonstrates that the GARCH model provides a better fit and a more plausible learning mechanism than the ARCH model.

Pooling Under Misspecification: Some Monte Carlo Evidence on the Kmenta and the Error Components Techniques

Abstract: Two different methods for pooling time series of cross section data are used by economists. The first method, described by Kmenta, is based on the idea that pooled time series of cross sections are plagued with both heteroskedasticity and serial correlation.The second method, made popular by Balestra and Nerlove, is based on the error components procedure where the disturbance term is decomposed into a cross-section effect, a time-period effect, and a remainder.Although these two techniques can be easily implemented, they differ in the assumptions imposed on the disturbances and lead to different estimators of the regression coefficients. Not knowing what the true data generating process is, this article compares the performance of these two pooling techniques under two simple setting. The first is when the true disturbances have an error components structure and the second is where they are heteroskedastic and time-wise autocorrelated.First, the strengths and weaknesses of the two techniques are discussed. Next, the loss from applying the wrong estimator is evaluated by means of Monte Carlo experiments. Finally, a Bartletfs test for homoskedasticity and the generalized Durbin-Watson test for serial correlation are recommended for distinguishing between the two error structures underlying the two pooling techniques.

Variable Selection in Heteroscedastic Discriminant Analysis

Abstract: The likelihood ratio test statistic for the identity in means and covariance matrices of k normal populations has a well-known step-down decomposition measuring the contribution of each component of the vector observation. This decomposition in turn gives rise to three components testing the residual homo-scedasticity of each variable, the parallelism of its regression on its predecessors, and the identity of location. A variety of uses of this decomposition in selecting variables is proposed.

Small-Sample Evaluation of Mean-Variance Production Function Estimators

Abstract: Production functions have been shown useful for characterizing input effects on both the mean and variability of yield. Monte carlo experiments are used here to investigate small-sample properties of selected mean-variance production function estimators. Estimation efficiency in the mean is found to improve with iteration on the mean and variance components. Although efficiency in the variance is greatest at the first stage, bias in the variance diminishes through at least the second stage. These effects are influenced by the degree and sign of heteroscedasticity and by sample size.

Some Monte Carlo Studies on the Comparison of Several Means Under Heteroscedasticity and Robustness with Respect to Departure from Normality

Abstract: By using Monte Carlo studies, this paper compares the Welch test, the James test and the Brown-Forsythe test for comparing several means under heteroscedasticity. It appears that all the tests are quite robust with respect to departure from normality. The Brown-Forsythe test is at least as good as the James test and the Welch test; but the differences are so small that the choice is immaterial for practical purposes.

The Effects of Variance Function Estimation on Prediction and Calibration: An Example.

Abstract: : This document considers fitting a straight line to data when the variances are not constant. In most fields, it is fairly common folklore thats how one estimates the variances does not matter too much when estimating the regression function. While this may be true, most problems do not stop with estimating the slope and intercept. Indeed, the ultimate goal of a study may be a prediction or a calibration. It is shown by an example that how one handles the variance function can have large effects. The point is almost trivial, but so often ignored that it is worth documenting. Additionally, this points out that one ought to spend time trying to understand the structure of the variability, a theoretical field that is not particularly well developed. Keywords: weighted least squares; heteroscedasticity.

A robust procedure for comparing several means under heteroscedasticity and nonnormality

Abstract: By applying Tiku's MML robust procedure to Brown and Forsythe's (1974) statistic, this paper derives a robust and more powerful procedure for comparing several means under hetero-scedasticity and nonnormality. Some Monte Carlo studies indicate clearly that among five nonnormal distributions, except for the uniform distribution, the new test is more powerful than the Brown and Forsythe test under nonnormal distributions in all cases investigated and has substantially the same power as the Brown and Forsythe test under normal distribution.

An Approach to Upper Bound Problems for Risks of Generalized Least Squares Estimators

Abstract: First, an approach to an upper bound for the risk matrix of GLSE's is established when the information on the parameter space of the structural parameter in the covariance matrix of the error can be utilized. Second, this result is applied to regression with (i) serial correlation and (ii) heteroscedastic covariance structure. In the heteroscedastic regression, the problem of estimating the common mean of two normal populations is studied in detail.

Two Pitfalls of Using Standard Regression Diagnostics When Both X and Y Have Measurement Error

Abstract: Modern exploratory data analysis produces models that are not based on physical theory but that are consistent with pictures of the data. When both X and Y have error this can be risky, because important features are hidden. Two examples are given that show that systematic model departures and heteroscedasticity may not be detectable with standard regression diagnostics.