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Lecture 5

# Lecture 5 - Introduction to Econometrics Lecture 5 Multiple...

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Introduction to Econometrics 5-1 Technical Notes Introduction to Econometrics Lecture 5: Multiple Regression - Orthogonality and Collinearity Ordinary Least Squares Estimation for Multiple Regression Models Multiple regression analysis is needed in economics partly because data are not the outcome of a controlled experiment, and cannot be selected to satisfy the ceteris paribus condition used for the thought- experiments of theory. In simple regression analysis we assumed that each observation on the dependent variable y i could be written as the sum of explained and unexplained components y i = α + β x i + u i Multiple regression analysis extends the scope of the explained component so that it becomes the sum of a number of distinct effects. So y i = α + β x i + θ z i + φ w i + u i The explained component of y i (its conditional mean) is given by E[y i |X,Z,W] = α + β x i + θ z i + φ w i The OLS estimators are the values of α , β , θ and φ which minimise the residual sum of squares, so we form the minimand Σ (u i ) 2 = Σ (y i - α - β x i - θ z i - φ w i ) 2 and differentiate with respect to each of these parameters 0 = -2 Σ (y i - α - β x i - θ z i - φ w i ) { Σ (u i ) 2 }/ ∂α 0 = -2 Σ x i (y i - α - β x i - θ z i - φ w i ) { Σ (u i ) 2 }/ ∂β 0 = -2 Σ z i (y i - α - β x i - θ z i - φ w i ) { Σ (u i ) 2 }/ ∂θ 0 = -2 Σ w i (y i - α - β x i - θ z i - φ w i ) { Σ (u i ) 2 }/ ∂φ The first equation implies that the regression ‘line’ passes through the point of means, so that mean (Y) = α e + β e mean (X) + θ e mean (Z) + φ e mean (W) mean (U) = 0 where α e is the solution for α , and so on. Subtracting N mean (X){ mean (Y) - α e - β e mean (X) - θ e mean (Z) - φ e mean (W)} which is zero from the second equation, and multiplying by (1/2(N-1)) β e Var(X) + θ e Cov(X,Z) + φ e Cov(X,W) = Cov(X,Y) Similarly the third and fourth equations can be written as β e Cov(Z,X) + θ e Var(Z) + φ e Cov(Z,W) = Cov(Z,Y) β e Cov(W,X) + θ e Cov(W,Z) + φ e Var(W) = Cov(W,Y)

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Introduction to Econometrics 5-2 Technical Notes This is a set of three linear equations in three unknowns: the OLS regression coefficients β e , θ e and φ e are functions of the variances and covariances of the data variables, in exactly the same way as in the case of simple regression. The regression ‘line’ makes full use of the data contained in the explanatory variables, so the sample covariance between the residual and each of the explanatory variables is zero. Thus for the variable X Σ x i (y i - α e - β e x i - θ e z i - φ e w i ) = Σ x i u e i = 0 with similar equations for Z and W. Since the predicted values y p i are just linear combinations of the independent variables, we also have zero covariance between the prediction y p i and the residual u e i . Σ
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Lecture 5 - Introduction to Econometrics Lecture 5 Multiple...

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