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EMET2007 Lecture 7

# Large number of estimated parameters eg k 6 leads to

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Unformatted text preview: ares and interactions leads to a large number of estimated parameters (e.g. k = 6 leads to 27 parameters to be estimated) Lecture 7 (heteroscedasticity) EMET2007/6007 24 th April 2013 19 / 34 Alternative form of the White test b2 = δ0 + δ1 y + δ2 y 2 + ν ε b b save 2 Rb2 ε This regression indirectly tests the dependence of the squared residuals on the explanatory variables, their squares, and interactions, because the predicted value of y and its square implicitly contain all of these terms. H0 : Var (εi jx1 , x2 , becomes H0 LM : , xk ) = Var (εi jxi ) = σ2 δ1 = δ2 = 0 2 = nRb2 s χ2 2 ε Example: Heteroscedasticity in (log) housing price equations 2 Rb2 = 0.0392 LM = 88 (0.0392) t 3.45 p ε Lecture 7 (heteroscedasticity) EMET2007/6007 value = 0.178 24 th April 2013 20 / 34 What can I do about it (I)? Adjust the data to remove it - Weighted least squares (WLS) estimation To use this form of WLS, we …rst assume that the form of the heteroscedasticity is known up to a multiplicative constant Var (εi jx1 , x2 , , xk ) = σ2 h (xi ) , h (xi ) = hi > 0 That is, The functional form of the heteroscedasticity, hi , is known The transformation we apply will result in an error with a constant variance, σ2 Lecture 7 (heteroscedasticity) EMET2007/6007 24 th April 2013 21 / 34 With this function we transform the variables as follows. yi where εi yi p hi yi where εi = β0 + β1 x1,i + β2 x2,i + + βk xk ,i + εi 2 s N 0, σ h (xi ) 1 x1,i x2,i xk εi = β0 p + β1 p + β2 p + + β k p ,i + p hi hi hi hi hi = β0 x0,i + β1 x1,i + β2 x2,i + + βk xk ,i + εi 2 s N 0, σ The Transformed model (Note that this regression model has no intercept) yi = β0 x0,i + β1 x1,i + β2 x2,i + + βk xk ,i + εi has the new regressand yi and the regressors xj ,i Lecture 7 (heteroscedasticity) EMET2007/6007 24 th April 2013 22 / 34 Example: Savings and income savi = β0 + β1 inci + εi Var (εi jinci ) = σ2 inci p p ) hi = inci ) hi = inci Transform (Note that this regression mod...
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