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Clustering Two Way Slides

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

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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

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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
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

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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
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

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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
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

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1.4 Cluster-Robust Variance Estimates The current method to obtain variance estimates.
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