CORNELL ECON 3120 - Heteroskedasticty (2 pages)

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Lecture number:
Lecture Note
Cornell University
Econ 3120 - Applied Econometrics

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Econ 3120 1st Edition Lecture 19 Outline of Last Lecture I Proxy Variables Outline of Current Lecture II Heteroskedasticity Current Lecture Let s start with our basic multivariate regression model under the standard MLR assumptions MLR 1 The model is given by y 0 1x1 2x2 kxk u MLR 2 Random sampling our sample x1 x2 xk y i 1 n is a random sample following the population model in MLR 1 MLR 3 No perfect collinearity In the sample and in the population 1 Each independent variable has sample variation 6 0 2 None of the independent variables can be constructed as a linear combination of the other independent variables MLR 4 Zero conditional mean E u x1 x2 xk 0 MLR 5 Homoskedasticity The error term in the OLS equation described by MLR 1 has constant variance Var u x1 xk 2 1 We ve seen previously MLR 1 through MLR 4 OLS estimators for will be unbiased In this lecture we re going to focus on violations of MLR 5 Violations of homoskedasticity are called unsurprisingly heteroskedasticity To understand what this means it s easiest to illustrate this graphically using just one x i e the bivariate case Violations of MLR 5 affect variance estimation Again using the bivariate regression y 0 1x u recall that the estimator for 1is given by 1 xi x yi xi x 2 and this can be written as 1 1 xi x ui xi x 2 taking the variance of 1 yields Var 1 xi x 2Var ui xi x 2 2 Now under MLR 5 we can use the fact that Var ui for all i and this collapses to Var 1 2 SSTx But suppose that the variances are heteroskedastic such that Var ui 2 i Now we get Var 1 xi x 2 2 i xi x 2 2 and we cannot simplify any further There are two things to note at this point First the estimators for the variance that we had derived previously are wrong under heteroskedasticity Second OLS is no longer Best Linear Unbiased because we have violated one of the the Gauss Markov assumptions 2 2 Robust Inference Under Heteroskedasticity We ll deal with the issue of efficiency later but there is a pretty straightforward way to deal

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