Lecture%20Notes%202

Lecture%20Notes%202 - 1 1. Simple Regression Analysis Two...

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Unformatted text preview: 1 1. Simple Regression Analysis Two variables: Y and X 1. Y: dependent variable (the regressand) 2. X: independent variable (the regresor) Regression: Regression of Y on X Model: The linear regression model is u X Y + + = 1 , where 1 , are parameters (unknown and to be estimated) and u is the (stochastic) error term/disturbance term. Note that, is the intercept and 1 is the slope of the population regression line. Example: Y X Consumption Income Money demand Interest rate Poverty rate Unemployment rate Data : n observations are available on variable X and Y. Write i i i u X Y + + = 1 , (1) Where i Y : i-th observation of Y and i X : i-th observation of X. Note that, ) ,.... 2 , 1 ( n i u i = are not observable. Based on ( i Y , i X ) (i=1,2,..n) we have to estimate 1 , . Properties of i u : (Assumptions) 1. E( i u ) = 0, E( i u | i X ) = 0 2. Cov( i u , j u ) = 0 for j i 3. Var( i u ) = 2 , constant 4. i u s are normally distriuted, i u | i X , ( ~ N 2 ) Implications: E( i Y | i X ) = i X 1 + and Var( i Y | i X ) = 2 . 2 2. Ordinary Least Squares (OLS) Estimation 1 Idea: Plot ( i Y , i X ), i= 1,2,.n Error : i Y- i Y = i u Principle of OLS: Choose ) , ( 1 b b such that the sum of squares of the errors is minimum., i.e., choose ) , ( 1 b b such that, = = =-- =- = n i n i i i i i n i i X b b Y Y Y u 1 1 2 1 2 1 2 ) ( ) ( is minimum. Denote: S = 2 1 1 ) ( i i n i X b b Y-- = , which is a quadratic function in ) , ( 1 b b ; so, there is unique minimum 2 . We can obtain ) , ( 1 b b from the solution of 1 , = b b o b S and 1 , 1 = b b o b S By differentiating S with respect to ) , ( 1 b b , we have = = =--- = =--- = n i i n i i n i i i X b nb Y X b b Y b S 1 1 1 1 1 ) ( 2 ) 1 )( ( 2 = = = =--- = =--- = n i i n i i n i i i n i i i i X b X b X Y X X b b Y b S 1 2 1 1 1 1 1 1 ) ( 2 ) )( ( 2 Therefore,...
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Lecture%20Notes%202 - 1 1. Simple Regression Analysis Two...

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