28 Mechanical properties 1-5 in matrix notation

This video dives into the matrix notation of econometrics, transforming familiar mechanical properties into a more powerful framework. Learn how OLS residuals become orthogonal to regressors, fitted values represent projections, and the geometry of OLS is revealed through a few key equations. We'll explore five core properties, building on knowledge from previous videos about matrix notation tricks. The first mechanical property states that OLS residuals are orthogonal to the regressors, expressed as x transpose times the residual equals zero. The second property identifies fitted values as projections of y, written as y hat equals the projection matrix p times y. We then examine that the projection matrix p is idempotent, meaning p squared equals p. The fourth property shows residuals originating from the residual maker matrix m, where e equals m times y, with m being the identity matrix minus p. Finally, we prove that fitted values and residuals are orthogonal, y hat transpose times the residual equals zero. These properties, grounded in linear algebra, illustrate the geometric interpretation of OLS without requiring probabilistic assumptions. Subscribe to @AxiomTutoringCourses for more econometrics insights.