Week 4&amp;5

# Week 4&amp;5 - Recall our recent Reading Assignments...

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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.

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Matrix Approach to Ordinary Least-Squares Regression Tim Fik w/mullet (Spring 1993)
OLS Model in Scalar Form Consider the bi-variate 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.

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… and in Matrix Form n 2 1 n 1 0 2 1 0 1 1 0 n 2 1 X X X Y Y Y
…in Matrix Form II n 2 1 1 0 n 2 1 n 2 1 X 1 X 1 X 1 Y Y Y

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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

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Vector of Parameters (to be estimated) 1 0 1 2
Vector of “error terms” (residuals) n 2 1 1 n

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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
Variance-Covariance 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 off-diagonal values are the “covariances”.

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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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