Non-spherical Disturbances Research Articles

Bayesian Restricted Stein-rule Least Squares with non-spherical Disturbances

Bayesian restricted stein-rule least squares is a novelty estimator introduced to obtain parameters of the family of restricted least squares with intertwined heteroscedastic and autocorrelated disturbance errors. Errors in regression modelling set to measure the quality of the data and or model of which intertwining of most prominent errors disturbances play a significant role in determining the quality of the aforementioned. Efforts were geared towards the comparison of the performances and relative efficiency of the small sample property of Bayesian and classical family of restricted stein-rule least squares. The sample size was set at 25 to capture the intertwined disturbance errors, and the iteration of the Monte Carlo simulation was set at 10000 for both classical and Bayesian paradigms. and for both Autoregressive of order one (AR(1)) and Moving average of order one (MA(1)) processes respectively were set asymmetrically as -0.8,-0.5,-0.3, 0,0.3,0.5 and 0.8 while as a scale of heteroscedasticity was set at 0-homoscedasticity, -0.3-mild,-0.5-moderate and (0.7,0.9)-severe. The outcome of the study pointed out that Bayesian estimation (both posterior mean and Bayes estimates) for both restricted and restricted stein-rule estimators outperformed classical restricted and restricted stein-rule estimators in both performances and relative efficiency. It is therefore recommended to make use of the Bayesian framework when encountering similar disturbances in a small sample size.

Bayesian Hetero-Elasticnet (A Gibbs Sampler Approach)

Combined heteroscedasticity and multicollinearity as dual non-spherical disturbances were experimented asymptotically. A Gibbs Sampler technique was used to investigate the asymptotic properties of hetero-elasticnet estimator with mean squares error (MSE) and bias as performance metrics. The seed was set to 12345; is set at ; Xs variables were generated as follow: the design matrix was generated from the multivariate normal distribution with mean > 0 and variance . and are truncated with Harvey (1976) heteroscedastic error structure; are collinear covariate with pairwise correlation between 0.6 and 0.9, the sample sizes were 25, 100 and 1000. The number of replications of the experiment was set at 10,000 with burn-in of 1000 which specified the draws that were discarded to remove the effects of the initial values. The thinning was set at 5 to ensure the removal of the effects of autocorrelation in the MCMC simulation. The study found that there is consistency of estimator asymptotically as the sample sizes increases from 25 to 50 so also to 1000, the larger sample size depicted least bias. The estimator exhibited efficiency asymptotically as larger sample sizes depicted least mean squares error. The study therefore recommended Bayesian hetero-elasticnet when data exhibit both heteroscedasticity and multicollinearity.
 Keywords: Elasticnet; Bayesian Inference and Gibbs sampler

Performance of double [formula omitted]-class estimators for coefficients in linear regression models with non-spherical disturbances under asymmetric losses

The risk associated with the estimators of the family of feasible generalized double k-class estimators under the LINEX loss function is derived in a linear regression model. The disturbances are assumed to be non-spherical and their variance–covariance matrix is unknown. A simulation study is conducted to study the risk of some important estimators in the family of feasible generalized double k-class estimators.

Estimation of a subset of regression coefficients of interest in a model with non-spherical disturbances

This paper considers the estimation of a subset of regression coefficients in a linear regression model with non-spherical disturbances, when other regression coefficients are of no interest. A family of estimators is considered and its asymptotic distribution is derived. This proposed family of improved estimators is compared with the usual unrestricted FGLS estimator, and dominance conditions are obtained with respect to risk under quadratic loss as well as the Pitman nearness criterion. The results of a numerical simulation are presented to illustrate the risk performance of various estimators.

Confidence ellipsoids based on a general family of shrinkage estimators for a linear model with non-spherical disturbances

This paper considers a general family of Stein rule estimators for the coefficient vector of a linear regression model with nonspherical disturbances, and derives estimators for the Mean Squared Error (MSE) matrix, and risk under quadratic loss for this family of estimators. The confidence ellipsoids for the coefficient vector based on this family of estimators are proposed, and the performance of the confidence ellipsoids under the criterion of coverage probability and expected volumes is investigated. The results of a numerical simulation are presented to illustrate the theoretical findings, which could be applicable in the area of economic growth modeling.

Forecasting the Maximum Compensation Offer in the Automobile BI Claims Negotiation Process

Most motor bodily injury (BI) claims are settled by negotiation, with fewer than 5% of cases going to court. A well-defined negotiation strategy is thus very useful for insurance companies. In this paper we assume that the monetary compensation awarded in court is the upper amount to be offered by the insurer in the negotiation process. Using a real database, a log-linear model is implemented to estimate the maximal offer. Non-spherical disturbances are detected. Correlation occurs when various claims are settled in the same judicial verdict. Groupwise heteroscedasticity is due to the influence of the forensic valuation on the final compensation amount. An alternative approximation based on generalized inference theory is applied to estimate confidence intervals on variance components, since classical interval estimates may be unreliable for datasets with unbalanced structures.

Weighted average least squares estimation with nonspherical disturbances and an application to the Hong Kong housing market

The recently proposed ‘weighted average least squares’ (WALS) estimator is a Bayesian combination of frequentist estimators. It has been shown that the WALS estimator possesses major advantages over standard Bayesian model averaging (BMA) estimators: the WALS estimator has bounded risk, allows a coherent treatment of ignorance and its computational effort is negligible. However, the sampling properties of the WALS estimator as compared to BMA estimators are heretofore unexamined. The WALS theory is further extended to allow for nonspherical disturbances, and the estimator is illustrated with data from the Hong Kong real estate market. Monte Carlo evidence shows that the WALS estimator performs significantly better than standard BMA and pretest alternatives.

Modeling Spatial Externalities: A Panel Data Approach

In this paper we argue that the Spatial Durbin Model (SDM) is an appropriate framework to empirically quantify different kinds of externalities. Besides, it is attractive from an econometric point of view as it nests several other models frequently employed. Up to now the SDM has been applied in cross-sectional settings only, thereby ignoring individual heterogeneity. This paper extends the SDM to panel data allowing for non-spherical disturbances and proposes an estimator based on ML techniques. Results from a Monte Carlo study reveal that the estimator has satisfactory small sample properties also in cases when neither heteroskedasticity nor serial correlation is present in the data generating process. Moreover, we show that conventional testing procedures may wrongly reject the existence of spatial externalities. In particular, we show that the incidence of a type II error increases as the spatial weight matrix becomes denser.

On the sensitivity of the restricted least squares estimators to covariance misspecification

Summary Traditional econometrics has long stressed the serious consequences of non-spherical disturbances for the estimation and testing procedures under the spherical disturbance setting, that is, the procedures become invalid and can give rise to misleading results. In practice, it is not unusual, however, to find that the parameter estimates do not change much after fitting the more general structure. This suggests that the usual procedures may well be robust to covariance misspecification. Banerjee and Magnus (1999) proposed sensitivity statistics to decide if the Ordinary Least Squares estimators of the coefficients and the disturbance variance are sensitive to deviations from the spherical error assumption. This paper extends their work by investigating the sensitivity of the restricted least squares estimator to covariance misspecification where the restrictions may or may not be correct. Large sample results giving analytical evidence to some of the numerical findings reported in Banerjee and Magnus (1999) are also obtained.

Time Scale Transformations of Discrete Time Processes

This paper investigates the effects of temporal aggregation when the aggregation frequency is variable and possibly stochastic. The results that we report include, as a particular case, the well-known results on fixed-interval aggregation, such as when monthly data are aggregated into quarters. A variable aggregation frequency implies that the aggregated process will exhibit time-varying parameters and non-spherical disturbances, even when these characteristics are absent from the original model. Consequently, we develop methods for specification and estimation of the aggregate models and show with an example how these methods perform in practice.

Risk and Pitman closeness properties of feasible generalized double k-class estimators in linear regression models with non-spherical disturbances under balanced loss function

In this article, a family of feasible generalized double k-class estimator in a linear regression model with non-spherical disturbances is considered. The performance of this estimator is judged with feasible generalized least-squares and feasible generalized Stein-rule estimators under balanced loss function using the criteria of quadratic risk and general Pitman closeness. A Monte-Carlo study investigates the finite sample properties of several estimators arising from the family of feasible double k-class estimators.

Testing poolability in a system of dynamic regressions with nonspherical disturbances

This study analyzes the testing of cross-equation restrictions within a set of regression equations. Through Monte Carlo experiments we examine the actual size of various asymptotic procedures for testing the poolability hypothesis, i.e., equal slope vectors across individual equations. Regression models with both lagged dependent variable regressors and nonspherical disturbances are considered. In these models we find that the performance in finite samples of classical asymptotic test procedures using critical values from either ℱ or χ2 approximations is often rather poor. However, employing the original test statistics with bootstrapped critical values leads to much more accurate inference in finite samples. In an empirical analysis of panel data on GDP growth and unemployment rates in OECD countries it is shown that classical asymptotic tests and bootstrap procedures may lead to conflicting test outcomes.

Double k-Class Estimators in Regression Models with Non-spherical Disturbances

In this paper, we consider a family of feasible generalised double k-class estimators in a linear regression model with non-spherical disturbances. We derive the large sample asymptotic distribution of the proposed family of estimators and compare its performance with the feasible generalized least squares and Stein-rule estimators using the mean squared error matrix and risk under quadratic loss criteria. A Monte-Carlo experiment investigates the finite sample behaviour of the proposed family of estimators.

STEIN-RULE RESTRICTED REGRESSION ESTIMATOR IN A LINEAR REGRESSION MODEL WITH NONSPHERICAL DISTURBANCES

In the present paper, we propose a Stein-rule estimator for the general linear regression model with nonspherical disturbances and a set of linear restrictions binding the regression coefficients. The asymptotic risk properties of the proposed estimator under a quadratic loss structure are derived, and a sufficient condition for the proposed estimator to dominate the feasible generalized restricted least squares estimator in large samples is presented. The small sample behavior of the proposed estimator is studied via a Monte-Carlo experiment.

Exact Results on the Inadmissibility of the Feasible Generalized Least Squares Estimator in Regression Models with Non-Spherical Disturbances

In various guises, feasible generalized least squares (FGLS) estimation has occupied an important place in regression analysis for more than 35 years. Past studies on the characteristics of the FGLS estimators are largely based on large sample evaluations, and the important issue of admissibility remains unexplored in the case of the FGLS estimator. In this paper, an exact sufficient condition for the dominance of a Stein-type shrinkage estimator over the FGLS estimator in finite samples based on squared error loss is given. In deriving the condition, we assume that the model's disturbance covariance matrix is unknown except for a scalar multiple. Further, for models with AR(1) disturbances, it is observed that the dominance condition reduces to one that involves no unknown parameter. In other words, in the case of AR(1) disturbances and where the condition for risk dominance is met, the FGLS estimator is rendered inadmissible under squared error loss.

Operational Variants of the Minimum Mean Squared Error Estimator in Linear Regression Models with Non-Spherical Disturbances

There is a good deal of literature that investigates the properties of various operational variants of Theil's (1971, Principles of Econometrics, Wiley, New York) minimum mean squared error estimator. It is interesting that virtually all of the existing analysis to date is based on the premise that the model's disturbances are i.i.d., an assumption which is not satisfied in many practical situations. In this paper, we consider a model with non-spherical errors and derive the asymptotic distribution, bias and mean squared error of a general class of feasible minimum mean squared error estimators. A Monte-Carlo experiment is conducted to examine the performance of this class of estimators in finite samples.

Stochastic Processes Subject to Time-Scale Transformations: An Application to High-Frequency FX Data

This paper is a general investigation of temporal aggregation in time series analysis. It encompasses traditional research on time aggregation as a particular case and extends the analysis to irregular intervals of aggregation. The Data Generating Process is allowed to evolve at regular, deterministic-irregular or even stochastic intervals of time (operational time). The time scale of this process is then transformed to generate the observational time process. This transformation can be deterministic (such as the familiar aggregation of monthly data into quarters) or more generally, stochastic (such as aggregating stock market quotes by the hour). In general, the observational time model exhibits persistence, time-varying parameters and non-spherical disturbances. Consequently, we review detection, specification, estimation and structural inference in this context, provide new solutions to these issues, and apply our results to high frequency, FX data.

Confidence Sets for the Coefficients Vector of a Linear Regression Model with Nonspherical Disturbances

In this present paper, considering a linear regression model with nonspherical disturbances, improved confidence sets for the regression coefficients vector are developed using the Stein rule estimators. We derive the large-sample approximations for the coverage probabilities and the expected volumes of the confidence sets based on the feasible generalized least-squares estimator and the Stein rule estimator and discuss their ranking.

The frisch-waugh theorem and generalized least squares

We discuss the standard linear regression model with nonspherical disturbances, where some regressors are annihilated by considering only the residuals from an auxiliary regression, and where, analogous to the Frisch-Waugh procedure, the original GLS procedure is applied to the transformed data. We call this procedure pseudo-GLS and give conditions for pseudo-GL5 to be equal to genuine GLS. We also show via examples that these conditions are often violated in empirical applications, and that the Frisch-Waugh Theorem still “works” with nonspherical disturbances if efficient estimation is applied to both the original and the transformed data.

A convenient test of functional form for pooled econometric models

This paper demonstrates a parametric test of specification that can be used in validating econometric models employing pooled time series and cross-section data with fixed effects. This Lagrange multiplier test allows for the simultaneous testing of proper model functional form and the presence of nonspherical disturbances, using a combination of the Box-Cox transformation, the double-length regression of Davidson and MacKinnon, and the Bonferroni inducedt-test. Testing procedures are demonstrated using a model of long distance telephone demand in the United States. The illustrative model used is representative of models filed as direct testimony by telephone companies in administrative law proceedings, which usually require rigorous model validation and defenses of model results in a formal hearing room setting. The tests presented in this paper are useful to a wide variety of researchers who use pooled econometric models with fixed effects in their work.