Weighted Least Squares Estimator

Background

The weighted least squares estimator (WLS) is a refinement of the ordinary least squares estimator (OLS) used primarily in econometrics and statistics to estimate the parameters of a linear model. Unlike OLS, which treats all observations equally regardless of variance, WLS gives more weight to observations with lower variance, thereby improving efficiency and reliability in estimation.

Historical Context

Weighted least squares has its origins in the broader field of least squares estimation introduced by Carl Friedrich Gauss. As econometric analysis expanded in complexity and application in the 20th century, it became clear that addressing heteroskedasticity—where error term variances differ across observations—required modifications to the standard least squares approach, which led to the development of WLS.

Definitions and Concepts

The weighted least squares estimator is a version of the generalized least squares estimator used when the covariance matrix of the random error is known to be diagonal. In other words, while each error term may have a different variance, they are not correlated with each other. The WLS approach minimizes the sum of the squared residuals weighted by the inverse of their variances, assigning more importance to observations with smaller error variances.

Major Analytical Frameworks

Classical Economics

Classical economics relies more on aggregate data less affected by heteroskedasticity. Thus, WLS was not as prominent within classical analysis.

Neoclassical Economics

Used extensively in econometric models within neoclassical economics to ensure that parameter estimates are efficient and unbiased by differing levels of observation reliability.

Keynesian Economics

Keynesian models that employ time series or panel data may use WLS to account for periods of differing economic volatility.

Marxian Economics

Less commonly directly used but mathematically useful in empirical testing of historical economic hypothesis where data variance is nonuniform.

Institutional Economics

Provides valuable adjustments in micro-level institutional data to account for inconsistent data reliability across observations.

Behavioral Economics

Applied where survey data variances differ by demographic factors, leading to more refined modeling of individual or group behavior.

Post-Keynesian Economics

Used to improve the precision of models analyzing periods of differing economic interventions or policies in explaining variable-dependent scenarios.

Austrian Economics

Utilized in empirical testing although less common given the school’s preference towards qualitative assessment.

Development Economics

In valuable in analyzing data from emerging economies where measurement error and data quality can vary substantially across observations.

Monetarism

Employs WLS for analyzing econometric models particularly useful in time series with volatile periods affecting prediction accuracy.

Comparative Analysis

The WLS approach contrasts with OLS in how residual errors are treated. While OLS assumes constant variance (homoskedasticity) across all observations, WLS acknowledges and adjusts for heteroskedasticity, making it preferable for data sets with non-uniform variabilities. WLS brings more efficient and unbiased parameter estimates by incorporating the known variances of each data point whereas OLS could lead to misleading interpretations if these variances are ignored.

Case Studies

Case studies often showcasing applications of WLS involve econometric analyses where heteroskedasticity is present. Examples can include cross-sectional studies in socio-economic surveys, financial econometrics analyzing market volatility impacts, and regional economic growth disparity assessments.

Suggested Books for Further Studies

1. Econometric Analysis by William H. Greene
2. Introduction to Econometrics by James H. Stock and Mark W. Watson
3. Applied Regression Analysis by Norman R. Draper and Harry Smith

Related Terms with Definitions

• Ordinary Least Squares Estimator (OLS): A method for estimating the unknown parameters in a linear regression model, assuming homoskedastic noise.
• Generalized Least Squares Estimator (GLS): A method extending OLS to account for situations where error terms have not only different variances (heteroskedasticity) but also contain some degree of correlation.
• Heteroskedasticity: A condition in regression analysis where the variance of the residuals is not constant across observations.
• Econometrics: The application of statistical and mathematical models to data in order to test theories, develop econometric models, and forecast future trends.

This entry provides a thorough explanation and context for understanding the weighted least squares estimator imperative for applications in econometrics and statistical analysis.