1. Consider a model of the form Y = X + where X is an n...

   

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1. Consider a model of the form Y = X + where X is an n (p + 1) full rank matrix. Let the vector/ solve the equation, X' (Y - XB) = 0, where 0 is a (p + 1) 1 vector of zeros, and let = X. Show the following equations are true: 1. ' (Y - ) = 0. 2. Y' = '. 3. (Y X)' (Y XB) = Y' Y ' . - 2. (Continued) Let H denote a hat matrix induced by X. Let J denote an n x n matrix of J'H = H'J = J. Show the followings are true: 1. Jis an orthogonal projection matrix. 2. SST = n 's. It can be shown that Y' (IJ)Y and RSS = Y' (I - H)Y where I denotes an n x n identity matrix. 3. SSreg=Y' (H - J)Y. 4. SST = SSreg + RSS. 3. (Continued) Vectors of fitted values and residuals can be rewriiten as =HY and = Y - HY. 1. Show that ' = 0. 2. Claim a sample covariance of the fitted value and the residual is zero. 1. Consider a model of the form Y = X + where X is an n (p + 1) full rank matrix. Let the vector/ solve the equation, X' (Y - XB) = 0, where 0 is a (p + 1) 1 vector of zeros, and let = X. Show the following equations are true: 1. ' (Y - ) = 0. 2. Y' = '. 3. (Y X)' (Y XB) = Y' Y ' . - 2. (Continued) Let H denote a hat matrix induced by X. Let J denote an n x n matrix of J'H = H'J = J. Show the followings are true: 1. Jis an orthogonal projection matrix. 2. SST = n 's. It can be shown that Y' (IJ)Y and RSS = Y' (I - H)Y where I denotes an n x n identity matrix. 3. SSreg=Y' (H - J)Y. 4. SST = SSreg + RSS. 3. (Continued) Vectors of fitted values and residuals can be rewriiten as =HY and = Y - HY. 1. Show that ' = 0. 2. Claim a sample covariance of the fitted value and the residual is zero.