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Unformatted text preview: Recall our recent Reading Assignments… Read and review: (a) the technical appendix in your textbook on Matrix approach to LS regression… Basic Econometrics by D. Gujarati, 2007, 4 th edition. and/or (b) the posted Matrix Algebra review and the Matrix Approach to Regression overview posted on my website. Matrix Approach to Ordinary LeastSquares Regression Tim Fik w/mullet (Spring 1993) OLS Model in Scalar Form Consider the bivariate regression model Y i = β 0 + β 1 X i + ξ i where ξ i ’s are independent normally distributed random error terms with mean 0 and variance σ 2. We may rewrite the individual observations: Y 1 = β 0 + β 1 X 1 + ξ 1 Y 2 = β 0 + β 1 X 2 + ξ 2 : : : : : Y n = β 0 + β 1 X n + ξ n …for i=1,…n sample observations. … and in Matrix Form n 2 1 n 1 2 1 1 1 n 2 1 X X X Y Y Y …in Matrix Form II n 2 1 1 n 2 1 n 2 1 X 1 X 1 X 1 Y Y Y The Design or Regressor Matrix n 2 1 2 n X 1 X 1 X 1 X Vector of Responses (Dependent Variable Vector) n 2 1 1 n Y Y Y Y Vector of Parameters (to be estimated) 1 1 2 Vector of “error terms” (residuals) n 2 1 1 n X Y … which may be generalized for k explanatory variables in the (nxk*) matrix X , and a (k*x1) vector β , where k* = k+1, allowing the estimation of an intercept or constant term. Formally, the model may be expressed as 1 n 1 2 2 n 1 n Y VarianceCovariance Matrix of Y } {Y } Y , {Y } Y , {Y } Y , Y { } {Y } Y , {Y } Y , {Y } Y , {Y } Y { ) Y ( n 2 1 n n 1 n n 1 n 2 2 1 2 n 1 2 1 1 2 2 Main diagonal values are the variances and offdiagonal values are the “covariances”. Covariance Matrix of Y n n 2 n 2 1 n n 2 I Y Y Y Cov {Y} Error Covariance Matrix: Cov( ξ) n n 2 n 2 1 n n 2 I Cov } ξ { Independent errors means that the covariance of any two residuals is zero....
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This note was uploaded on 02/15/2012 for the course GEO 4167 taught by Professor Staff during the Spring '12 term at University of Florida.
 Spring '12
 Staff

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