week4 - Econ 508 Simple Regression—Inference and...

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Unformatted text preview: Econ 508: Simple Regression—Inference and prediction Juan Fung MSPE February 25, 2011 Juan Fung (MSPE) Econ 508: Simple Regression—Inference and prediction February 25, 2011 1 / 19 From estimation to inference Simple regression model Recall from last time that the objective of an econometric model, Y = α + β X + is to explain the association between two variables, X and Y , by a deterministic component, E [ Y | X ] = α + β X , and a stochastic component, = Y- E [ Y | X ] — which accounts for deviations from mean behavior. Juan Fung (MSPE) Econ 508: Simple Regression—Inference and prediction February 25, 2011 2 / 19 From estimation to inference OLS The method of ordinary least squares (OLS) finds α,β to minimize ESS = X i 2 i = X i ( Y i- ( α + β X i )) 2 . The OLS estimators for the single regression model are ˆ β = S xy S xx , ˆ α = Y- ˆ β X , where S xy = ∑ i ( X i- X )( Y i- Y ) and S xx = ∑ i ( X i- X ) 2 . The next step is to use the OLS estimators to make inferences about α,β . Juan Fung (MSPE) Econ 508: Simple Regression—Inference and prediction February 25, 2011 3 / 19 From estimation to inference Test statistics In order to test hypotheses, or construct confidence intervals, we must characterize the sampling distributions for ˆ α, ˆ β . What distribution do ˆ α, ˆ β follow? Normal. Why? We assume i ∼ N . What are their expected values? They are unbiased estimators : E [ ˆ β ] = β, E [ˆ α ] = α . You also derived their variance: Var ( ˆ β ) = σ 2 S xx Var (ˆ α ) = σ 2 ∑ i X 2 i S xx . Juan Fung (MSPE) Econ 508: Simple Regression—Inference and prediction February 25, 2011 4 / 19 From estimation to inference Then since ˆ β ∼ N ( β,σ 2 / S xx ), Z = ˆ β- β p σ 2 / S xx ∼ N (0 , 1) . What is the problem with using Z as a test statistic? σ 2 unknown!...
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This note was uploaded on 05/23/2011 for the course ECON 508 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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week4 - Econ 508 Simple Regression—Inference and...

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