{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Clustering Two Way Slides

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

Info icon This 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
Image of page 1

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

View Full Document Right 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
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
Image of page 3

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

View Full Document Right 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
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
Image of page 5

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

View Full Document Right 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
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
Image of page 7

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

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

{[ snackBarMessage ]}