Matrix Expression For The OLS Residual Sum of Squares
The general regression model with n observations and k explanators, the first of which is a constant unit vector whose coefficient is the regression intercept, is
where y is an n × 1 vector of dependent variable observations, each column of the n × k matrix X is a vector of observations on one of the k explanators, is a k × 1 vector of true coefficients, and e is an n× 1 vector of the true underlying errors. The ordinary least squares estimator for is
The residual vector is, so the residual sum of squares is, after simplification,
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