Generalized Least Squares and Feasible Generalized Least Squares

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Generalized Least Squares and Feasible Generalized Least Squares

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18

Lecture number:
18
Pages:
2
Type:
Lecture Note
School:
Cornell University
Course:
Econ 3120 - Applied Econometrics
Edition:
1
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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 theβ’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) = σ 2 h(x) so that the variance can be expressed as some function of x. As an example, suppose our model is savei = β0 +β1inci +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 β1 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) = σ 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 = β0/ √ inci +β1inci/ √ 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) = σ 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 = β0 +β1x1 +β2x2 +...+βkxk +u that satisfies MLR.1-MLR.4, and that Var(u|x) = σ 2 h(x) Then, the transformed model y/ √ h = β0/ √ h+β1x1/ √ h+β2x2/ √ h+...+βkxk/ √ 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 estimate it. But once we do that, we can do a procedure similar to the above. Suppose we think there is heteroskedasticity of the form Var(u|x) = σ 2 h(x) 5 which we model as Var(u|x) = σ 2 exp(δ0 +δ1x1 +δ2x2 +...+δkxk) (2) This is not the only possible model, we could use other functions of the x’s as well. Exponentiating is convenient because it imposes that the variance will always be positive. Since we need to estimate (2), we write the estimating equation as u 2 = σ 2 exp(δ0 +δ1x1 +δ2x2 +... +δkxk)ν where E(ν) = 1 and E(ν|x) = 0. Taking logs, we get log(u 2 ) = α0 +δ1x1 +δ2x2 +...+δkxk +e Now we’re ready to apply Feasible Generalized Least Squares: 1. Estimate the full model using OLS, collect ˆui 2. Create log(uˆ 2 i ...

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