Lecture 2

Lecture 2 - Economics 513 1 Lecture 2: The Classical Linear...

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Unformatted text preview: Economics 513 1 Lecture 2: The Classical Linear Regression Model Introduction In lecture 1 we • Introduced concept of an economic relation • Noted that empirical economic relations do not resemble textbook relations • Introduced a method to find the best fitting linear relation No reference to mathematical statistics in all of this. Initially econometrics did not use the tools of mathematical statistics. Mathematical statistics develops methods for the analysis of data generated by a random experiment in order to learn about that random experiment. Economics 513 2 Is this relevant in economics? Consider • Wage equation: relation between wage and education, work experience, gender, … • Macro consumption function: relation between (national) consumption and (national) income What is the random experiment? To make progress we start with the assumption that all economic relations are essentially deterministic, if we include all variables ) , , f( 1 W x x y … = Economics 513 3 Hence, if we have data n i x x y iW i i , , 1 , , , , 1 … … = then n i x x y iW i i , , 1 , ) , , f( 1 … … = = Let W x x , , 1 … be the sample averages of the variables and assume that f is sufficiently many times differentiable to have a Taylor series expansion around W x x , , 1 … , i.e. a polynomial approximation " " + − + + − + ′ = ) ( ) ( 1 1 1 W iW W i i x x x x y β β β " " " + − + + − + 2 2 1 1 1 ) ( ) ( W iW i x x x x γ " " + − − + ) )( ( 2 2 1 1 1 x x x x i i δ Economics 513 4 We divide W x x , , 1 … into three groups 1. Variables that do not vary in the sample (take this to be last V W − variables), i.e. for n i , , 1 … = , W W i V V i x x x x = = + + , 1 1 , , , … . Example: gender if we consider a sample of women. 2. Variables in the relation that are omitted or cannot be included because they are unobservable. Let this be the next 1 + − K V variables. 3. Variables included in the relation, i.e. 1 1 , , + K x x … . Economics 513 5 To keep the relation simple we concentrate on the linear part. Hence the observations satisfy i K i K i i x x y ε β β β + + + + = − − 1 , 1 1 1 " with j K j j x ∑ = = − ′ = 1 1 β β β The remainder term contains all the omitted terms + − + + − = ) ( ) ( V iV V K iK K i x x x x β β ε " + − + + − + 2 2 1 1 1 ) ( ) ( V iV V i x x x x γ γ " " + − − + ) )( ( 2 2 1 1 1 x x x x i i δ We call i ε the disturbance (of the exact linear relation). Economics 513 6 Note that 1 , , 1 , ) , , ( x f 1 j − = ∂ ∂ = K j x x W j … … β ∑ − = − = 1 1 1 ) , , f( K j j j W x x x β β … Conclusions: 1. The slope coefficient j β is the partial effect of j x on y ....
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This note was uploaded on 07/22/2008 for the course ECON 513 taught by Professor Rashidian during the Fall '07 term at USC.

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Lecture 2 - Economics 513 1 Lecture 2: The Classical Linear...

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