Lecture 5

# Lecture 5 - Multiple Regression Suppose yi = xi zi wi ui In...

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1 Multiple Regression Suppose y i = α + β x i + θ z i + φ w i + u i In science (experimental control) and economic theory (ceteris paribus) we can hold z, w constant so y i = ( α + θ z * + φ w * ) + β x i + u i which is a simple regression. This simplification is not possible in applied economics Multiple regression divides y i into two components α + β x i + θ z i + φ w i deterministic, E[y i | X,Z,W] u i stochastic disturbance Formulae for the OLS estimators Calculated by minimising SSR = Σ (u i ) 2 = Σ (y i - α - β x i - θ z i - φ w i ) 2 The first-order conditions imply normal equations y * = α e + β e x * + θ e z * + φ e w * (x * = Mean(X), etc) β e V(X) + θ e C(X,Z) + φ e C(X,W) = C(X,Y) β e C(Z,X) + θ e V(Z) + φ e C(Z,W) = C(Z,Y) β e C(W,X) + θ e C(W,Z) + φ e V(W) = C(W,Y) (V(X) = Var(X), C(X,Z) = Cov(X,Z) etc) a set of linear simultaneous equations Properties of OLS estimators in a multiple regression The OLS regression line makes full use of data Σ x i u e i = Σ x i (y i - α e - β e x i - θ e z i - φ e w i ) = 0

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Lecture 5 - Multiple Regression Suppose yi = xi zi wi ui In...

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