24 The matrix notation and projections revisiting geometry

This video explains the connection between Ordinary Least Squares (OLS) as a projection and its matrix notation. It revisits the concept of a fitted line being the closest possible to the data cloud, with residuals forming a right angle. The explanation then delves into why OLS matrix notation is crucial for understanding OLS as a projection. The video demonstrates how projecting the vector y into the subspace defined by x1 and x2 results in the fitted value y-hat, which is the closest point in that subspace to y. It also highlights that the residual vector is orthogonal to this subspace. The discussion moves to the OLS matrix form, showing the beta-hat solution and the resulting y-hat, which leads to the identification of the projection matrix. This matrix takes any vector, like y, and projects it into the subspace spanned by the columns of x. The video also clarifies the orthogonality condition of the residuals in matrix form, emphasizing that the residual is perpendicular to every column of x and thus the entire subspace. This understanding helps explain why OLS residuals meet the fitted line at a 90-degree angle. Ultimately, the matrix notation reveals the underlying structure of OLS, confirming that fitted values are a projection and that the regression line is the closest point to y in the space spanned by x. Subscribe to @AxiomTutoringCourses for more economics tutorials.