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# CORNELL ECON 3120 - Heteroskdasticity

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Econ 3120 1st Edition Lecture 17Outline of Current Lecture I. The Linear Probability ModelCurrent LectureII. HeteroskdasticityLet’s start with our basic multivariate regression model under the standard MLR assumptions • MLR.1: The model is given by y = β0 +β1x1 +β2x2 +...+βkxk +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) = σ 2 1 We’ve seen previously MLR.1 through MLR.4, OLS estimators for β 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 = β0 + β1x+u, recall that the estimator for ˆβ1is given by ˆβ1 = ∑(xi −x¯)yi ∑(xi −x¯) 2 and this can be written as ˆβ1 = β1 + ∑(xi −x¯)ui ∑(xi −x¯) 2 taking the variance ofˆβ1 yields Var( ˆβ1) = ∑(xi −x¯) 2Var(ui) (∑(xi −x¯) 2) 2 Now, under MLR.5, we can use the fact thatVar(ui) = σ for all i, and this collapses to Var( ˆβ1) = σ 2 SSTx . But suppose that the variances are heteroskedastic, such that Var(ui) = σ 2 i . Now, we get Var( ˆβ1) = ∑(xi −x¯) 2σ 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( ˆβ1) = ∑(xi −x¯) 2uˆ 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) = σ 2 The basic idea of implementation is to test whether ˆu 2 i , depends on a function of the x’s. Since the OLS-derived ˆu 2 i is an unbiased estimator for σ 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 = β0 +β1x1 +β2x2 +...+βkxk +u to obtain the squared residuals ˆu 2 i . 2. Regress the squared residuals on some function of the x’s. Some important examples include: uˆ 2 i = δ0 +δ1x1 +δ2x2 +...+δkxk +ν 3 or uˆ 2 i = δ0 +δ1yˆ+ν or uˆ 2 i = δ0 +δ1yˆ+δ2yˆ 2 +ν 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

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