CORNELL ECON 3120 - Generalized Least Squares and Feasible Generalized Least Squares (2 pages)

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Econ 3120 1st Edition Lecture 18 Outline of Current Lecture I Heteroskdasticity Current Lecture II Generalized Least Squares and Feasible Generalized Least Squares Generalized Least Squares and Feasible Generalized Least Squares As described above we can deal with heteroskedasticity using OLS robust standard errors but this is not the most efficient way to estimate the s This section outlines how to perform more efficient estimation 4 1 Generalized Least Squares Suppose somewhat unrealistically that we know the form of the heteroskedasticity We will consider heteroskedasticity of the form Var u x 2 h x so that the variance can be expressed as some function of x As an example suppose our model is savei 0 1inci ui 1 where inci is the income of household i in a given year and savei is savings in that year What economic parameter does 1 represent 4 With this model one can imagine that Var ui inci is increasing in income The higher someone s income it makes sense that there is higher variance in the unexplained component of the model In particular suppose Var ui inci 2 inci Armed with this information we can transform the model to one with homoskedastic standard errors Suppose we divide every term in 1 by inci savei inci 0 inci 1inci inci ui inci In this equation the final term which is still an error term has variance Var ui inci inci 1 inci Var ui inci 2 The errors are now homoskedastic Our transformed model satisfies assumptions MLR 1 MLR 4 so that we now have a model whose estimates will be best linear unbiased This procedure is typically called generalized least squares The generic form of this starts with the model y 0 1x1 2x2 kxk u that satisfies MLR 1 MLR 4 and that Var u x 2 h x Then the transformed model y h 0 h 1x1 h 2x2 h kxk h u h will satisfy MLR 1 MLR 5 and the OLS estimates will therefore be best linear unbiased 4 2 Feasible Generalized Least Squares The vast majority of the time we don t know the form of the heteroskedasticity In that case we need to