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How Good Is Your Model Fit: Weighted Least Squares and Diagnostic Measures

This tutorial delves into evaluating the goodness of fit of a statistical model, specifically focusing on *Weighted Least Squares (WLS)* regression and the various diagnostic measures you can use to assess how well your model represents your data. We'll cover the concepts, underlying principles, code examples (using Python with `statsmodels`), and interpretation of results to provide a comprehensive understanding.

*Why Weighted Least Squares?*

Before we dive into model fit, let's understand why WLS is often preferred over Ordinary Least Squares (OLS) in certain situations.

*Heteroscedasticity:* OLS assumes that the variance of the errors (residuals) is constant across all levels of the predictor variables. This assumption is called homoscedasticity. When this assumption is violated (heteroscedasticity), the standard errors of the OLS estimates are biased, leading to incorrect hypothesis tests and confidence intervals.
*Differing Data Quality:* Sometimes, some data points are considered more reliable or accurate than others. For example, data collected using a more precise instrument, or data points based on larger sample sizes.

WLS addresses these issues by assigning weights to each observation, effectively giving more influence to the observations with lower variance or higher reliability.

*I. Weighted Least Squares Regression: A Quick Overview*

WLS minimizes the weighted sum of squared residuals, where the weights are inversely proportional to the variance of the error for each observation. The objective function to be minimized is:



Where:

`wᵢ` is the weight for the i-th observation.
`yᵢ` is the observed value of the dependent variable for the i-th observation.
`ŷᵢ` is the predicted value of the dependent variable for the i-th observation.

The weights are usually derived from an estimate of the variance of the error term for each observat ...

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