Suppose we have a linear model $Y = X\beta + \varepsilon$, where $Y \in \mathbb{R}^n, \beta \in \mathbb{R}^p, X \in \mathbb{R}^{n \times p},$ and $\varepsilon \sim N_n(0,\sigma^2I_n)$ (multivariate normal). Are the residuals $e$ and fitted values $\hat{Y}$ uncorrelated if only the Gauss-Markov conditions are met?
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To answer this question, consider the properties of the least squares estimators and remember that under the Gauss-Markov conditions, the ordinary least squares (OLS) estimator is the best linear unbiased estimator (BLUE). With this in mind, you can assess whether the assumption of homoscedasticity and no perfect multicollinearity guarantees the uncorrelation between residuals and fitted values or if additional conditions are required.
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