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• Large sample size in this example justifies use of robust standard errors
• Robust standard errors can be larger or smaller than usual ones so it can make a difference to the inferences we draw VER. 10/23/2012. © P. KOLM 49 Standard Errors through Bootstrapping See the separate handout “Computationally Intensive Econometric Techniques:
The Bootstrap” for discussion of the bootstrap
We repeat part (i) of the previous example, calculating the standard errors using
For the results, see the Matlab code VER. 10/23/2012. © P. KOLM 50 Important Questions Why not always use robust inference?
• Researchers often do in cross-sectional applications when n is large
• In small samples robust standard errors may be far out What about the efficiency of OLS under heteroscedasticity?
• OLS not BLUE in presence of heteroscedasticity so a better estimator is potentially available
• GLS, WLS, or Feasible Least Squares can provide us with a more efficient estimator
How do we know that we are dealing with heteroscedasticity? Can we test for it?
• Often, by visually inspecting the residuals we can often get some intuition whether we are dealing with heteroscedasticity
• Yes, we can formally test for the presence of heteroscedasticity VER. 10/23/2012. © P. KOLM 51 Testing for Heteroscedasticity: The Basic Idea We test the null hypothesis of homoscedasticity, that is
H 0 : Var (u | x 1, x 2,…, x k ) = σ 2 From the definition of variance and since E (u | x ) = 0 , this is equivalent to ( ) () H 0 : E u 2 | x 1, x 2,…, x k = E u 2 = σ 2 VER. 10/23/2012. © P. KOLM 52 Assuming the relationship between u 2 and x j is linear, we estimate the model u 2 = δ0 + δ1x 1 +… + δk x k + v
and test H 0 : δ1 = δ2 = … = δk = 0
H1 :H 0 not true using the F-statistic
Ru 2 / k
2 (1 − Ru 2 ) / (n − k − 1)
ˆ ∼ Fk ,n −k −1 • As usual we reject if the observed value of the test statistic exceeds an appropriate critical value
• Rejection implies heteroscedasticity is present VER. 10/23/2012. © P. KOLM 53 Remarks
• We don’t observe the error u, but the residuals from the OLS regression, u , provide an estimate for it
• We estimate a regression of these residuals squared on a set of “suspect regressors” and an intercept and calculate the R 2 from this regression
(denoted by Ru 2 )
ˆ • The suspect regressors can be all (k), or just some of the regressors in the original model (s < k )
• Statistical packages typically provide tests for heteroscedasticity:
o Breusch-Pagan test for heteroscedasticity (as above)
o White test for heteroscedasticity (more general form of heteroscedasticity, squares and cross products of independent variables) VER. 10/23/2012. © P. KOLM 54 Generalized Least Squares (GLS)
• It is always possible to estimate robust standard errors for OLS
• However, if we know something about the specific form of the heteroscedasticity, we can obtain more efficient estimators than OLS
• Basic idea: Transform the heteroscedastic regression model into one that has homoscedastic errors an...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.
- Fall '14