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Unformatted text preview: Chapter 2 Properties of the regression coefficients and hypothesis testing Overview Chapter 1 introduced least squares regression analysis, a mathematical technique for fitting a relationship given suitable data on the variables involved. It is a fundamental chapter because much of the rest of the text is devoted to extending the least squares approach to handle more complex models, for example models with multiple explanatory variables, nonlinear models, and models with qualitative explanatory variables. However, the mechanics of fitting regression equations are only part of the story. We are equally concerned with assessing the performance of our regression techniques and with developing an understanding of why they work better in some circumstances than in others. Chapter 2 is the starting point for this objective and is thus equally fundamental. In particular, it shows how the three criteria for assessing the performance of estimators, namely unbiasedness, efficiency, and consistency, are applied in the context of a regression model. Further material Derivation of the expression for the variance of the naïve estimator in Section 2.5 The variance of the naïve estimator in Section 2.5 and Exercise 2.10 is not of any great interest in itself but its derivation provides an example of how one obtains expressions for variances of estimators in general. In Section 2.5 we considered the naïve estimator of the slope coefficient derived by joining the first and last observations in a sample and calculating the slope of that line: 1 1 2 X X Y Y b n n − − = . It was demonstrated that the estimator could be decomposed as 1 1 2 2 X X u u b n n − − + = β and hence that E ( b 2 ) = β 2 . The population variance of a random variable X is defined to be E ([ X – μ X ] 2 ) where μ X = E ( X ). Hence the population variance of b 2 is given by [ ] ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − − + = − = 2 1 1 2 2 1 1 2 2 2 2 2 2 X X u u E X X u u E b E n n n n b β β β σ On the assumption that X is nonstochastic, this can be written October 2007 2 [ ] ( ) 2 1 2 1 2 1 2 u u E X X n n b − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = σ Expanding the quadratic, we have ( ) ( ) ( ) ( ) [ ] 1 2 1 2 2 1 1 2 1 2 2 1 2 2 1 2 1 2 u u E u E u E X X u u u u E X X n n n n n n b − + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = − + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = σ Each value of the disturbance term is drawn randomly from a distribution with mean 0 and population variance , so ( ) ( ) 2 u σ 2 n u E and 2 2 1 u E are both equal to . u u σ n and u 1 are drawn independently from the distribution, so E ( u n u 1 ) = E ( u n ) E ( u 1 ) = 0. Hence ( ) ( ) 2 1 2 2 1 2 2 2 1 2 2 X X X X n u n u b − = − = σ σ σ Define ( n X X A + = 1 2 1 ) , the average of X 1 and X n . and 1 X A A X D n − = − = Then ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ] ( )( )...
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This note was uploaded on 05/26/2010 for the course ECON 301 taught by Professor Öcal during the Spring '10 term at Middle East Technical University.
 Spring '10
 öcal
 Econometrics

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