Interpreting a large difference in results after applying weighted least squares

External question — Cross Validated Stack Exchange Author: hanna-oui Original post: https://stats.stackexchange.com/questions/672325 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I am coming here looking for guidance on how to interpret a noticeable difference in results when estimating an OLS versus a weighted OLS model. Let me first provide some context on my problem. I running a panel regression: $\begin{aligned} y_{c,t} = \beta^T x_{c,t} + \epsilon_{c,t} \end{aligned}$ . Here we'll denote $c$ a country and $t$ a year. The panel is imbalanced, although I can balance the sample at the expense of going from ~1.2k total observations to ~550. I am interested in understanding the dynamics of an aggregate phenomenon. Hence, rather than weighting each country uniformly in my sample, I decide to weight each country-year observation by the country's relative importance in the world economy. Of course, the normalization is relative to the countries in my sample, so the weights are derived in-sample and for each year, the weights add up to 1 across all the countries in the sample. As many of you know, this is equivalent to minimizing the $\text{WMSE}$ , that is, $ \begin{aligned} \sum_{c,t} w_{c,t}(y_{c,t} - \beta^T\mathbf{x}_{c,t})^2. \end{aligned} $ This is algebraically equivalent to estimating the following model, $ \begin{aligned} \tilde{y}_{c,t} = \beta^T \tilde{\mathbf{x}}_{c,t} + \tilde{\epsilon}_{c,t}, \end{aligned} $ with $\tilde{y}_{c,t} := \sqrt{w}_{c,t} y_{c,t}$ and so fourth. I should reiterate, we also have $w_{c,t} > 0, \forall c,t$ . Furthermore, there is significant concentration in economic importance, that is, on a given year between 5-10 countries will dominate the weights (accounting for 75%+). I am okay with this, since I am interested in understanding aggregate behavior, so I'd like to capture the behavior of the most important countries. As a sanity check on ensuring my OLS specifications are well-specified, I always like to look at partial residual plots . Reminding that the partial residual plot plots the tuple, ( $\epsilon_{c,t} + \hat{\beta}_j X_j, X_j$ ) where $X_j$ is a predictor and $\hat{\beta}_j$ it's fitted coefficient. Comparing these partial residual plots between the regular and weighted OLS, I notice a striking difference - a linear relationship is much much stronger after weights are applied. I am wondering, is this cause for any concern or suspicion? Of course, the actual regression estimates reflect this trend. The coefficients are large in magnitude and more precisely estimated after weighting. But I'd like to make sure I am interpreting the shift correctly. My interpretation of this change induced by weighting is that the more economically important (higher weighted) observations behave more similarly than the universe of countries do (unweighted OLS), reducing the variation and hence allowing for the linear trend to be more pronounced. Is this a correct interpretation? Thanks for the help! Plots : First, for regular OLS (no weights). These are the partial residuals for a particular covariate whose behavior was noticeably different, US trade: Then, for weighted OLS: