Cluster-robust standard errors - Cemfi

Cluster-robust standard errorsClass NotesManuel ArellanoOctober 7, 2011Let θ be a parameter vector identified from the moment conditionEψ (w, θ) =0such that dim (θ) =dim(ψ). Consider a sample W = {w 1 ,...,w N } and a consistent estimator b θ thatsolves the sample moment conditions (up to a small order term).The function ψ (w, θ) may denote orthogonality conditions in a just-identified GMM estimationproblem or the first-order conditions in an m-estimation problem. Alternatively ψ (w, θ) may be regardedas a linear combination of a larger moment vector ϕ (w, θ) in an overidentified GMM estimationproblem, so that ψ (w, θ) =B 0 ϕ (w, θ).Assume that conditions for the following asymptotically linear representation of the scaled estimationerror and asymptotic normality hold:√ ³N bθ − θ´= −D0−1 1 XN√ ψ (w i , θ)+o p (1) → d N ¡ 0,D0 −1 V 0D −10 ¢0Ni=1where D 0 = ∂Eψ (w, θ) /∂c 0 is the Jacobian of the population moments and V 0 is the limiting variance:ÃV 0 = lim Var 1N !X1NX NX√ ψ (w i , θ) = limE £ ψ (w i , θ) ψ (w j , θ) 0¤ .N→∞ N N→∞ Ni=1i=1 j=1To get standard errors of b θ we need estimates of D 0 and V 0 .AnaturalestimateofD 0 is D b =N 1 P N b i=1D iwhere d b ki = 12 Nhψ³w i , b θ + N δ k´− ψ³w i , b θ − N δ k´iis the kth column of D b i , for some small N > 0,and δ k is the kth unit vector. As for V 0 the situation is different under independent or cluster sampling.Independent sampling Denote ψ i = ψ (w i , θ) and ψ b ³i = ψ w i , b ´θ . If the observations areindependent E ¡ ψ i ψ 0 j¢=0for i 6= j so that1V 0 = limN→∞ NNXE ¡ ψ i ψ 0 i¢.i=1If they are also identically distributed V 0 = E ¡ ψ i ψ 0 i¢. Regardless of the latter a consistent estimateof V 0 under independence isbV = 1 NX 0bψ bN iψ i.i=1A consistent estimate of the asymptotic variance of b θ under independent sampling is:Ã !³dVar bθ´= 1 DN b−1 1NX 0bψN iψ bibD −1 . (1)i=11

Cluster sampling We now obtain a consistent estimate of the asymptotic variance of b θ whenthesampleconsistsofH groups (or clusters) of M h observations each (N = M 1 + ... + M H ) such thatobservations are independent across groups but dependent within groups, H →∞and M h is fixedfor all h. For convenience we order observations by groups and use the double-index notation w hm sothat W = {w 11 ,...,w 1M1 | ... | w H1 ,...,w HMH }.Under cluster sampling, letting ψ h = P M hm=1 ψ hm and ψ e h = P M h b m=1 ψ hm we haveÃV 0 = lim Var 1H !X1HX ³ ´√ ψ h = lim E ψN→∞ N N→∞ Nh ψ 0 h ,h=1h=1so that E ¡ ψ i ψ 0 ¡j¢=0only if i and j belong to different clusters, or E ψhm ψ 0 ¢h 0 m =0for h 6= h 0 0 .Thus, a consistent estimate of V 0 iseV = 1 NHXeψ e hψ 0h. (2)h=1The estimated asymptotic variance of b θ allowing unrestricted within-cluster correlation is thereforeÃ!³gVar bθ´= 1 DN b−1 1HX 0eψ eN hψ hbD −1 . (3)h=1Note that (3) is of order 1/H whereas (1) is of order 1/N .Examples A panel data example isψ (w hm , θ) =x hm¡yhm − x 0 hm θ¢where h denotes units and m time periods. If the variables are in deviations from means b θ is thewithin-group estimator. In this case D b hm = −x hm x 0 hm and ψ e h = P M hm=1 x hmbu hm where bu hm arewithin-group residuals. The result is the (large H, fixed M h ) formula for within-group standard errorsthat are robust to heteroskedasticity and serial correlation of arbitrary form in Arellano (1987):ó H!XM −1X hHÃX XM h XM hX H M −1X hgVar bθ´= x hm x 0 hmbu hm bu hs x hm x 0 hs x hm xhm! 0 . (4)h=1 m=1h=1 m=1 s=1h=1 m=1Another example is a τ-quantile regression with moments£ ¡ψ (w hm , θ τ )=x hm 1 yhm ≤ x 0 hm θ ¢ ¤τ − τwhere D b hm = bω hm x hm x 0 hm , bω hm = 1³¯¯¯yhm − x 0 b hmθ¯ ≤ ξ N´/ (2ξ N ), 1 ψh e = P M hm=1 x hmbu hm and bu hm =³1 y hm ≤ x 0 b ´hmθ− τ. Cluster-robust standard errors for QR coefficients are obtained from:ó X HgVar bθ´=M h Xh=1 m=1bω hm x hm x 0 hm! −1 HXM h Xh=1 m=1 s=1ÃXM hX Hbu hm bu hs x hm x 0 hsM h Xh=1 m=11 This choice of D corresponds to selecting an (i, k)-specific scaled N given by ξ N /x ik .bω hm x hm x 0 hm! −1. (5)2

Equicorrelated moments within-cluster The variance formula (3) is valid for arbitrary correlationpatterns among cluster members. Here we explore the consequences for variance estimationof assuming that the dependence between any pair of members of a cluster is the same. Under theerror-component structureE ¡ (ψ hm ψ 0 ¢ Ωη + Ω v if m = shs =Ω η if m 6= s³ ´we have E ψ h ψ 0 h = M h Ω v + Mh 2Ω η. Letting M = 1 P HN h=1 M h , the limiting variance V 0 becomesµV 0 = lim Ω v + M 2Ω η .H→∞ M 1The natural estimator of Ω v is the within-group variance Ω b v . When all clusters are of the same size(M h = M for all h), the natural estimator of Ω η is the between-group variance minus the withinvariance times 1/M : 2bΩ η = 1 HHXh=11M 2 e ψ he ψ0h − 1 M b Ω v .Noting that M 2 /M 1 = M, it turns out that the plug-in estimate of V 0 is identical to e V in (2):bΩ v + M b Ω η = 1HMHXeψ e hψ 0h.h=1When cluster sizes differ, a method-of-moments estimator of Ω η is a weighted between-group varianceminus the within variance times a weighted average of 1/M h :bΩ η = 1 HXµ 1 0w heψH Mh2 hψ eh − 1 Ωv bM hh=1for some weights w h such that P Hh=1 w h = H. 3 Once again, using w h = M 2 h /M 2 as weights, theplug-in estimate of V 0 is identical to e V :bΩ v + M 2M 1b Ωη = 1HM 1H Xh=1eψ he ψ0hThe conclusion is that imposing within-cluster equicorrelation is essentially innocuous for thepurpose of calculating cluster-robust standard errors.ReferencesArellano, M. (1987): “Computing Robust Standard Errors for Within-Group Estimators”, OxfordBulletin of Economics and Statistics, 49, 431-434.Arellano, M. (2003): Panel Data Econometrics, OxfordUniversityPress.2 The within-group sample variance is Ω 1v =H Mh0.H(M 1 −1) h=1 m=1ψ hm − ψ hM hψ hm − ψ hM hAs for the betweengroupvariance note that overall means are equal to zero at θ (Arellano, 2003, section 3.1 on error-component estimation).3 For example, w h =1, w h = M h /M 1 or w h = Mh/M 2 2 .3