Lecture%20Notes%202

# Lecture%20Notes%202 - 1 1 Simple Regression Analysis •...

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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 β β , where 1 , β β are parameters (unknown and to be estimated) and u is the (stochastic) error term/disturbance term. Note that, β is the intercept and 1 β 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 β β , (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 , β β . • 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 σ , constant 4. i u ’s are normally distriuted, i u | i X , ( ~ N 2 σ ) Implications: E( i Y | i X ) = i X 1 β β + and Var( i Y | i X ) = 2 σ . 2 2. Ordinary Least Squares (OLS) Estimation 1 • Idea: Plot ( i Y , i X ), i= 1,2,….n Error : i Y- i Y ˆ = 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 ) ( ) ˆ ( 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 •...

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