Econ-103-Lecture-03

# Econ-103-Lecture-03 - Moshe Buchinsky Department of...

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Moshe Buchinsky Economics 103 Department of Economics Introduction to Econometrics UCLA Fall, 2011 Lecture Note 3 The Simple Linear Regression Model II 1 Topics to be Covered 1. Estimating the Variance of the Error Term 2. Estimating Nonlinear Relationships 3. Regressions with Indicator Variables (if time permits) 2 Estimating the Variance of the Error Term The variance of the random error is: Var ( )= 2 = h ( [ ]) 2 i = £ 2 ¤ [ ] | {z } =0 2 = £ 2 ¤ Since the “expectation” is an average value we might consider estimating 2 as the average of the squared errors, namely: b 2 = 1 X =1 2 where the error terms are = 1 2 . Now, the least squares residuals are obtained by replacing the unknown parameters by their least squares estimates: b = b = 1 2 so that b 2 = 1 X =1 b 2 There is a simple modi f cation that produces an unbiased estimator, and that is: b 2 = 1 2 X =1 b 2 (2.19) 1

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so that: £ b 2 ¤ = 2 Whydowed iv ideby 2 ? Because we estimate two parameters 1 and 2 . 2.1 Estimating the Variances and Covariances of the Parameter Estimates To obtain estimates for Var ( 1 ) ,Var ( 2 ) ,Cov ( 1  2 ) , just replace the unknown error variance 2 in Eq. 2.14—2.16 by b 2 to obtain: Var ( 1 )= b 2 P =1 2 P =1 ( ) 2 (2.20) Var ( 2 )= b 2 1 P =1 ( ) 2 (2.21) Cov ( 1  2 )= b 2 P =1 ( ) 2 (2.22) The square roots of the estimated variances are the “standard errors” of 1 and 2 : Se ( 1 )= p Var ( 1 ) (2.23) Se ( 2 )= p Var ( 2 ) (2.24) Table 2.3 Least Squares Residuals b 2 = 1 2 X =1 b 2 = 304 505 2 38 =8013 29 The estimated variances and covariances for a regress ionarearrayedinarectangu lararray , or matrix, with variances on the diagonal and covariances in the “o f -diagonal” positions. Ã Var ( 1 ) Cov ( 1  2 ) Cov ( 1  2 ) Var (
For the food expenditure data the estimated covariance matrix is: Ã 1 884 44 85 9032 85 9032 4 38175 ! The standard errors of 1 and 2 are measures of the sampling variability of the least squares estimates 1 and 2 in repeated samples. The estimators are random variables. As such, they have probability distributions, means, and variances. In particular, if assumption SR6 holds, and the random error terms

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## This note was uploaded on 10/01/2011 for the course ECON 103 taught by Professor Sandrablack during the Fall '07 term at UCLA.

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Econ-103-Lecture-03 - Moshe Buchinsky Department of...

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