WLS (weighted least squares)

Weighted Least Squares (WLS) is a statistical method used in regression analysis to handle heteroscedasticity, which is a situation where the variance of the error terms is not constant across all levels of the independent variable(s). WLS adjusts the ordinary least squares (OLS) estimation by giving different weights to the data points based on their estimated variances. The purpose of WLS is to provide more accurate and efficient estimates when dealing with heteroscedastic data. Let's explore the concept and application of Weighted Least Squares in more detail.

Heteroscedasticity and Ordinary Least Squares (OLS):

In simple linear regression, the goal is to find the best-fitting line (a straight line) that minimizes the sum of the squared differences between the observed dependent variable (Y) and the predicted values (Ŷ) by the regression model. This method is known as Ordinary Least Squares (OLS).

However, OLS assumes that the variance of the error terms (ε) is constant across all levels of the independent variable(s). In real-world data, it is common to encounter situations where the variance of the errors changes as the values of the independent variable(s) change. This phenomenon is known as heteroscedasticity.

Heteroscedasticity can lead to biased and inefficient estimates of the regression coefficients. When the assumption of constant variance is violated, the OLS estimates may not be the best estimates of the true coefficients.

Weighted Least Squares (WLS) Approach:

To address heteroscedasticity, the WLS approach assigns different weights to each data point in the regression analysis based on their estimated variances. The idea is to give more weight to the observations with smaller variances (i.e., more reliable data) and less weight to observations with larger variances (i.e., less reliable data).

The weights are determined by estimating the variance of the error terms for each observation, typically based on some external information or assumptions about the error variance. For example, the weights may be inversely proportional to the variance, meaning that observations with higher variance receive lower weights, and vice versa.

WLS Estimation Process:

1. Estimate Variances: The first step in WLS is to estimate the variances of the error terms for each observation.
2. Compute Weights: The weights for each data point are obtained by taking the reciprocal of the estimated variances. If the variance is large, the weight will be small, and if the variance is small, the weight will be large.
3. Fit Weighted Regression: The weighted regression model is then fitted using the computed weights. This involves finding the regression coefficients that minimize the weighted sum of squared differences between the observed Y and the predicted Ŷ.
4. Assess Residuals: After fitting the weighted regression, the residuals (the differences between the observed Y and the predicted Ŷ) are examined to check for homoscedasticity (constant variance) and model adequacy.

Applications of Weighted Least Squares:

1. Economics and Finance: WLS is commonly used in economics and finance to handle heteroscedasticity in financial and economic data, where the variance of the errors may change with the level of economic variables.
2. Environmental Studies: Environmental data often exhibit heteroscedasticity due to variations in environmental factors. WLS is used to obtain more reliable regression estimates in these scenarios.
3. Biostatistics: In medical and biological research, WLS is applied when analyzing data with heteroscedasticity, such as in dose-response studies.
4. Social Sciences: WLS is used in social sciences research, such as in educational studies, to deal with heteroscedasticity in survey and behavioral data.

Advantages and Limitations of WLS:

Advantages:

1. WLS provides more efficient and accurate estimates of regression coefficients when dealing with heteroscedastic data.
2. It helps to reduce the impact of influential observations with large variances on the regression estimates.
3. WLS improves the goodness-of-fit of the regression model when heteroscedasticity is present.

Limitations:

1. WLS requires knowledge or estimation of the error variances, which may not always be readily available.
2. The weights are sensitive to the choice of variance estimation method, and incorrect weight assignment can lead to biased estimates.
3. If the variance estimation is inaccurate, WLS may perform worse than OLS, especially with small sample sizes.

In conclusion, Weighted Least Squares (WLS) is a statistical method used to address heteroscedasticity in regression analysis. By assigning different weights to data points based on their estimated variances, WLS provides more accurate and efficient estimates of regression coefficients, particularly in situations where the variance of the errors varies with the level of the independent variable(s). WLS is widely used in various fields to handle heteroscedastic data and improve the robustness of regression models.