7/28/2019 Cluster Robust Se

1/3

Cluster-robust standard errorsClass Notes

Manuel Arellano

October 7, 2011

Let be a parameter vector identified from the moment condition

E (w, ) = 0

such that dim() = dim (). Consider a sample W= {w1,...,wN} and a consistent estimator b thatsolves the sample moment conditions (up to a small order term).

The function (w, ) may denote orthogonality conditions in a just-identified GMM estimation

problem or the first-order conditions in an m-estimation problem. Alternatively (w, ) may be re-

garded as a linear combination of a larger moment vector (w, ) in an overidentified GMM estimation

problem, so that (w, ) = B0 (w, ).

Assume that conditions for the following asymptotically linear representation of the scaled estima-

tion error and asymptotic normality hold:

Nb = D1

0

1N

NXi=1

(wi, ) + op (1)d N

0, D1

0V0D

10

0

where D0 = E (w, ) /c0 is the Jacobian of the population moments and V0 is the limiting variance:

V0 = limN

V ar

1N

NXi=1

(wi, )

!= lim

N

1

N

NXi=1

NXj=1

E (wi, ) (wj, )

0

.

To get standard errors ofb we need estimates ofD0 and V0. A natural estimate ofD0 is bD = 1NPNi=1bDiwhere bdki = 12N h wi,b + Nk wi,b Nki is the kth column of bDi, for some small N > 0,and k is the kth unit vector. As for V0 the situation is different under independent or cluster sampling.

Independent sampling Denote i = (wi, ) and bi = wi,b. If the observations areindependent E

i

0

j

= 0 for i 6= j so that

V0 = limN

1

N

NXi=1

Ei

0

i

.

If they are also identically distributed V0 = Ei0i. Regardless of the latter a consistent estimateof V0 under independence is

bV = 1N

NXi=1

bib0i.A consistent estimate of the asymptotic variance of b under independent sampling is:dV ar b = 1

NbD1 1

N

NXi=1

bib0i! bD1. (1)

1

7/28/2019 Cluster Robust Se

2/3

Cluster sampling We now obtain a consistent estimate of the asymptotic variance ofb whenthe sample consists ofH groups (or clusters) of Mh observations each (N = M1 + ... + MH) such that

observations are independent across groups but dependent within groups, H and Mh is fixedfor all h. For convenience we order observations by groups and use the double-index notation whm so

that W= {w11,...,w1M1 | ... | wH1,...,wHMH}.Under cluster sampling, letting h =

PMhm=1 hm and

eh = PMhm=1bhm we haveV0 = lim

NV ar

1N

HXh=1

h

!= lim

N

1

N

HXh=1

Eh

0

h

,

so that Ei

0

j

= 0 only if i and j belong to different clusters, or E

hm

0

h0m0

= 0 for h 6= h0.

Thus, a consistent estimate of V0 is

eV =

1

N

H

Xh=1 ehe

0

h. (2)

The estimated asymptotic variance ofb allowing unrestricted within-cluster correlation is thereforegV ar b = 1

NbD1 1

N

HXh=1

ehe0h! bD1. (3)

Note that (3) is of order 1/H whereas (1) is of order 1/N.

Examples A panel data example is

(whm, ) = xhm

yhm x0hm

where h denotes units and m time periods. If the variables are in deviations from means b is thewithin-group estimator. In this case bDhm = xhmx0hm and eh = PMhm=1 xhmbuhm where buhm arewithin-group residuals. The result is the (large H, fixed Mh) formula for within-group standard errors

that are robust to heteroskedasticity and serial correlation of arbitrary form in Arellano (1987):

gV ar b = HXh=1

MhXm=1

xhmx0

hm

!1HXh=1

MhXm=1

MhXs=1

buhmbuhsxhmx0hs

HXh=1

MhXm=1

xhmx0

hm

!1. (4)

Another example is a -quantile regression with moments

(whm, ) = xhm 1 yhm x0hm where bDhm = bhmxhmx0hm, bhm = 1yhm x0hmb N / (2N),1 eh = PMhm=1 xhmbuhm and buhm =1

yhm x0hmb . Cluster-robust standard errors for QR coefficients are obtained from:gV ar b = HX

h=1

MhXm=1

bhmxhmx0hm!1

HXh=1

MhXm=1

MhXs=1

buhmbuhsxhmx0hs

HXh=1

MhXm=1

bhmxhmx0hm!1

. (5)

1 This choice of eD corresponds to selecting an (i, k)-specific scaled N given by N/xik.

2

7/28/2019 Cluster Robust Se

3/3

Equicorrelated moments within-cluster The variance formula (3) is valid for arbitrary cor-

relation patterns among cluster members. Here we explore the consequences for variance estimation

of assuming that the dependence between any pair of members of a cluster is the same. Under the

error-component structure

Ehm0hs = ( +v if m = s if m 6= s

we have Eh

0

h

= Mhv + M

2

h. Letting M =1

N

PHh=1 M

h, the limiting variance V0 becomes

V0 = limH

v +

M2

M1

.

The natural estimator ofv is the within-group variance bv. When all clusters are of the same size(Mh = M for all h), the natural estimator of is the between-group variance minus the within

variance times 1/M:2

b = 1H

HXh=1

1

M2ehe0h 1Mbv.

Noting that M2/M1 = M, it turns out that the plug-in estimate of V0 is identical to eV in (2):bv + Mb = 1

HM

HXh=1

ehe0h.When cluster sizes differ, a method-of-moments estimator of is a weighted between-group variance

minus the within variance times a weighted average of 1/Mh:

b = 1H

HXh=1

wh 1M2h

ehe0h 1Mhbvfor some weights wh such that

PHh=1 wh = H.

3 Once again, using wh = M2

h/M2 as weights, the

plug-in estimate of V0 is identical to eV:bv + M2

M1b = 1

HM1

HXh=1

ehe0hThe conclusion is that imposing within-cluster equicorrelation is essentially innocuous for the

purpose of calculating cluster-robust standard errors.

References

Arellano, M. (1987): Computing Robust Standard Errors for Within-Group Estimators, Oxford

Bulletin of Economics and Statistics, 49, 431-434.

Arellano, M. (2003): Panel Data Econometrics, Oxford University Press.

2 The within-group sample variance is ev = 1H(M11)

SH

h=1

SMh

m=1

hm

hMh

hm

hMh

0

. As for the between-

group variance note that overall means are equal to zero ate (Arellano, 2003, section 3.1 on error-component estimation).3 For example, wh = 1, wh = Mh/M1 or wh = M

2

h/M2.

3