Question: Question 2 Consider the multivariate linear regression model that is described in Turkington (2007, p.71- 72.) y = XB +u The ordinary least squares estimator

Question 2 Consider the multivariate linear regression model that is described in Turkington (2007, p.71- 72.)

y = XB +u The ordinary least squares estimator (OLS) of the vector of coefficients B is B = (X'x) - X'y a) Must the matrix X'X be square? Must X be square? b) The matrix X'X is invertible if X has full column rank (i.e. the columns of X are linearly independent). What is the minimum number of rows (observations) if the model includes k regressors and a constant? c) For a given X, the predicted values of y are: y = X B = X ( X'X ) - * X'y Show that the matrix N = X(X'X)-1X' is symmetric and idempotent. Is / invertible? d) the prediction errors (residuals) are: e =y -y =y- xB=y - x(X'X)-1X'y = y -Ny = (1-N)y = My Show that the matrix M is symmetric and idempotent

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