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C7D3 - HETEROSCEDASTICITY WEIGHTED AND LOGARITHMIC...

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1 HETEROSCEDASTICITY: WEIGHTED AND LOGARITHMIC REGRESSIONS u X Y + + = 2 1 β β 2 of variance i i u σ = This sequence presents two methods for dealing with the problem of heteroscedasticity. We will start with the general case, where the variance of the distribution of the disturbance term in observation i is σ i 2 .
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2 HETEROSCEDASTICITY: WEIGHTED AND LOGARITHMIC REGRESSIONS u X Y + + = 2 1 β β i i i i i i i u X Y σ σ β σ β σ + + = 2 1 1 2 of variance i i u σ = If we knew σ i in each observation, we could derive a homoscedastic model by dividing the equation through by it.
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3 HETEROSCEDASTICITY: WEIGHTED AND LOGARITHMIC REGRESSIONS u X Y + + = 2 1 β β i i i i i i i u X Y σ σ β σ β σ + + = 2 1 1 2 of variance i i u σ = 1 of variance 1 of variance 2 2 2 = = = i i i i i i u u σ σ σ σ The population variance of the disturbance term in the revised model is now equal to 1 in all observations, and so the disturbance term is homoscedastic.
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4 HETEROSCEDASTICITY: WEIGHTED AND LOGARITHMIC REGRESSIONS u X Y + + = 2 1 β β i i i i i i i u X Y σ σ β σ β σ + + = 2 1 1 ' ' ' 2 1 u X H Y + + = β β i i i i i i i u u X X H Y Y σ σ σ σ = = = = ' , ' , 1 , ' 2 of variance i i u σ = 1 of variance 1 of variance 2 2 2 = = = i i i i i i u u σ σ σ σ In the revised model, we regress Y ' on X ' and H , as defined. Note that there is no intercept in the revised model. β 1 becomes the slope coefficient of the artificial variable 1/ σ i .
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5 HETEROSCEDASTICITY: WEIGHTED AND LOGARITHMIC REGRESSIONS The revised model is described as a weighted regression model because we are weighting observation i by a factor 1/ σ i . Note that we are automatically giving the highest weights to the most reliable observations (those with the lowest values of σ i ). u X Y + + = 2 1 β β i i i i i i i u X Y σ σ β σ β σ + + = 2 1 1 ' ' ' 2 1 u X H Y + + = β β i i i i i i i u u X X H Y Y σ σ σ σ = = = = ' , ' , 1 , ' 1 of variance 1 of variance 2 2 2 = = = i i i i i i u u σ σ σ σ 2 of variance i i u σ =
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HETEROSCEDASTICITY: WEIGHTED AND LOGARITHMIC REGRESSIONS i i Z λ σ = u X Y + + = 2 1 β β 6 Of course in practice we do not know the value of σ i in each observation. However it may be reasonable to suppose that it is proportional to some measurable variable, Z i . 2 of variance i i u σ =
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HETEROSCEDASTICITY: WEIGHTED AND LOGARITHMIC REGRESSIONS i i i i i i i Z u Z X Z Z Y + + = 2 1 1 β β i i Z λ σ = u X Y + + = 2 1 β β 7 If this is the case, we can make the model homoscedastic by dividing through by Z i . 2 of variance i i u σ =
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8 HETEROSCEDASTICITY: WEIGHTED AND LOGARITHMIC REGRESSIONS The disturbance term in the revised model has constant variance λ 2 . We do not need to know the value of λ 2 . The crucial point is that, by assumption, it is constant. i i i i i i i Z u Z X Z Z Y + + = 2 1 1 β β 2 2 2 2 2 2 / 1 of variance λ λ σ σ σ = = = i i i i i i Z Z u i i Z λ σ = u X Y + + = 2 1 β β ' ' ' 2 1 u X H Y + + = β β i i i i i i i Z u u Z X X Z H Z Y Y = = = = ' , ' , 1 , ' 2 of variance i i u σ =
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9 HETEROSCEDASTICITY: WEIGHTED AND LOGARITHMIC REGRESSIONS We will illustrate this procedure with the UNIDO data on manufacturing output and GDP.
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