Clustering Two Way Slides

Clustering Two Way Slides - Robust Inference with Multi-way...

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
Robust Inference with Multi-way Clustering Colin Cameron, Jonah Gelbach, Doug Miller U.C. - Davis, U. Arizona, U.C. - Davis February 2010 Colin Cameron, Jonah Gelbach, Doug Miller Multi-way Clustering February 2010 1 / 44
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
1.1 Introduction Moulton (1986, 1990) and Bertrand, Du&o, and Mullainathan (2004) showed the importance of controlling for clustering. Failure to control underestimates OLS standard errors and overstates t statistics. Initially use one-way random e±ects model. Now use cluster-robust standard errors (White (1984), Arellano (1987), Liang and Zeger (1986), Rogers (1993)). Colin Cameron, Jonah Gelbach, Doug Miller Multi-way Clustering February 2010 2 / 44
Background image of page 2
This paper extends to cluster-robust in two (nonnested) dimensions Example 1: More than one grouped regressor. Example 2: Cross-section unit and time for panel data. Outline of presentation Lengthy discussion of one-way cluster-robust. Present method for two-way cluster-robust. Simulation and application. Conclusion. Colin Cameron, Jonah Gelbach, Doug Miller Multi-way Clustering February 2010 3 / 44
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
1.2 OLS with Cluster Errors Model for G clusters with N g individuals per cluster: y ig = x 0 ig β + u ig , i = 1, . .., N g , g = 1, . .., G , y g = X g β + u g , g = 1, . .., G , y = X β + u , OLS estimator b β = ( G g = 1 N g i = 1 x ig x 0 ig ) & 1 ( G g = 1 N g i = 1 x ig y ig ) = ( G g = 1 X 0 g X g ) & 1 ( G g = 1 X g y g ) = ( X 0 X ) & 1 X 0 y . Colin Cameron, Jonah Gelbach, Doug Miller Multi-way Clustering February 2010 4 / 44
Background image of page 4
OLS with Clustered Errors (continued) As usual b β = β + ( X 0 X ) & 1 X 0 u = β + ( X 0 X ) & 1 ( G g = 1 X g u g ) . Assume independence over g with u g ± [ 0 , Σ g = E [ u g u 0 g ]] . Then b β a ± N [ β , V [ b β ]] with V [ b β ] = ( X 0 X ) & 1 ( G g = 1 X g Σ g X 0 g )( X 0 X ) & 1 . Colin Cameron, Jonah Gelbach, Doug Miller Multi-way Clustering February 2010 5 / 44
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
OLS with Clustered Errors (continued) If ignore clustering the default OLS variance estimate should be in&ated by approximately τ j 1 + ρ x j ρ u ( ¯ N g & 1 ) , where ρ x j is the within cluster correlation of x j ρ u is the within cluster error correlation ¯ N g is the average cluster size. Kloek (1981), Scott and Holt (1982). Moulton (1986, 1990) showed that could be large even if ρ u small. e.g. N G = 81, ρ x = 1 and ρ u = 0.1 then τ j = 9! So should correct for clustering - but Σ g is unknown. Colin Cameron, Jonah Gelbach, Doug Miller Multi-way Clustering February 2010 6 / 44
Background image of page 6
1.3 Random e&ects approach The original way to obtain consistent variance matrix estimate. Assume a random e&ects (RE) model u ig = α g + ε ig α g & iid [ 0, σ 2 α ] ε ig & iid [ 0, σ 2 ε ] . Then Σ g = σ 2 u I N g + σ 2 α e N g e 0 N g and we use b V RE [ b β ] = ( X 0 X ) ± 1 ( G g = 1 X g b Σ g X 0 g )( X 0 X ) ± 1 , where b Σ g = b σ 2 u I N g + b σ 2 α e N g e 0 N g , and b σ 2 ε and b σ 2 α are consistent. Weakness is strong distributional assumptions. Colin Cameron, Jonah Gelbach, Doug Miller Multi-way Clustering February 2010 7 / 44
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
1.4 Cluster-Robust Variance Estimates The current method to obtain variance estimates.
Background image of page 8
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 44

Clustering Two Way Slides - Robust Inference with Multi-way...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online