Econ 3120 1st Edition Lecture 17 Outline of Current Lecture I The Linear Probability Model Current Lecture II Heteroskdasticity 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 with heteroskedasticity using OLS estimation Because OLS estimates are unbiased we can still use the estimates to arrive at a heteroskedasticity robust variance estimator Here s what that looks like in the bivariate case Vard 1 xi x 2u 2 i xi x 2 2 Implementation of this is straightforward using the robust option of the regress command in Stata With multiple regression the robust estimator follows the same reasoning but the formula is a bit more complicated so we won t cover it here Note that the heteroskedacitity robust estimates of variance are valid even in the absence of heteroskedasticity In practice a lot of applied work reports robust standard errors rather than These notes represent a detailed interpretation of the professor s lecture GradeBuddy is best used as a supplement to your own notes not as a substitute regular standard errors just in case At the same time robust estimates are usually larger than standard OLS estimates so using robust estimates does come with a loss of precision 3 Testing for Heteroskedasticity To test for heteroskedasticity we can turn MLR 5 into a null hypothesis to be tested H0 Var u x1 xk 2 The basic idea of implementation is to test whether u 2 i depends on a function of the x s Since the OLS derived u 2 i is an unbiased estimator for 2 i we can use it for testing The following steps can be used to implement the test 1 Run the OLS regression on y 0 1x1 2x2 kxk u to obtain the squared residuals u 2 i 2 Regress the squared residuals on some function of the x s Some important examples include u 2 i 0 1x1 2x2 kxk 3 or u 2 i 0 1y or u 2 i 0 1y 2y 2 There are many other options here 3 Perform the F test that all coefficients of the regression in step 2 are zero This is typically called the Breusch Pagan or White test for heteroskedasticity Implementation is straightforward in Stata by doing each step individually There is also a somewhat tricky postestimation command called estat hettest that you can use but it has so many extra options that it is not that helpful for our purposes