ECO375H_Slides_4 - Lecture 4: Multiple Regression Analysis:...

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Lecture 4: Multiple Regression Analysis: Statistical Properties Junichi Suzuki University of Toronto October 1st, 2009
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Announcement I Important change: 2nd problem set is due October 16th. I Will be available by the next lecture I tomorrow. I See comments from Rebecca (available at the Blackboard) I Information Session on Applying to Graduate School I October 15th (Th) 4:10-5:30pm I SS2118
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I Goal: Wrap up Chapter 3 I Response to a question from a student I Proving "Partialling Out Interpretation of MR" I Statistical properties of OLSE I Assumptions I Expected value I Variance I E¢ ciency (Gauss-Markov Theorem) I Consider both I why some good properties hold I what happen if some assumptions do not hold I especially pay attention to omitted variable problem
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What OLS Estimates Imply: Question I Consider the following regression and the corresponding sample regression: ln w = β 0 + β 1 educ + u d ln w = ˆ β 0 + ˆ β 1 educ I β 1 represents a percent increase in her salary when one gets additional one (year) education dw deduc = β 1 w I This implies that a percent increase of additional two-year education is dw deduc = 2 β 1 w I Why not dw deduc = [( 1 + β 1 ) 2 ± 1 ] w
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What OLS Estimates Imply: Answer I Go back to the original equation w = exp ( β 0 + β 1 educ + u ) w 1 w 0 ± 1 = exp ( β 0 + β 1 ( educ + 2 ) + u ) exp ( β 0 + β 1 educ + u ) ± 1 = exp ( 2 β 1 ) ± 1 I When β 1 = 0 . 05 , 8 < : exp ( 2 β 1 ) ± 1 = 10 . 52 % 2 β 1 = 10 . 00 % ( 1 + β 1 ) 2 ± 1 = 10 . 25 % I I The last two are good approximations
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Proving "Partialling Out Interpretation of MR" I Consider a regression of y on x = ( x 1 , x 2 , , x k ) y = β 0 + β 1 x 1 + β 2 x 2 + + β k x k + u I Want to show ˆ β j = n i = 1 ˆ r ij y i n i = 1 ˆ r 2 ij I ˆ r ij is the residual of the regression of x j on other regressors ˆ x j = ˆ δ 1 + l 6 = j ˆ δ l x l ˆ r ij = x ij ± ˆ x ij I Note that FOC implies n i = 1 ˆ r ij ˆ x ij = n i = 1 ˆ r ij ˆ δ 1 + l 6 = j ˆ δ l x l ! = 0
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Proving "Partialling Out Interpretation of MR" Consider the FOC 0 = n i = 1 x ij y i ˆ β 0 ˆ β 1 x i 1 ˆ β k x ik ± = n i = 1 ( ˆ x ij + ˆ r ij ) y i ˆ β 0 ˆ β 1 x i 1 ˆ β k x ik ± = n i = 1 ˆ δ 1 + l 6 = j ˆ δ l x l ! ˆ u ij + n i = 1 ˆ r ij y i ˆ β j n i = 1 ˆ r ij x ij = n i = 1 ˆ r ij y i ˆ β j n i = 1 ˆ r ij ( ˆ x ij + ˆ r ij ) = n i = 1 ˆ r ij y i ˆ β j n i = 1 ˆ r 2 ij
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Statistical Properties of OLSE
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Gauss-Markov Assumptions MLR.1 (Linear in parameters): y = β 0 + β 1 x 1 + β 2 x 2 + + β k x k + u MLR.2 (Random Sampling): ( x i 1 , x i 2 , . . . , x ik , y i ) n i = 1 is a random sample from the population regression MLR.3 (No perfect Collinearity): In the sample , none of the ind var is constant, and no exact linear relationships among them
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This note was uploaded on 10/16/2009 for the course ECON ECO375 taught by Professor Junichisuzuki during the Fall '09 term at University of Toronto- Toronto.

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ECO375H_Slides_4 - Lecture 4: Multiple Regression Analysis:...

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