Cluster-Robust Inference

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1 Cluster-Robust Inference David Sovich Washington University in St. Louis

2 Modern Empirical Corporate Finance Modern day ECF mainly focuses on obtaining unbiased or consistent point estimates (e.g. indentification!) However, obtaining accurate estimates of standard errrors is also important for correct statistical inference Indeed, the Wald t-statistic used for hypothesis testing: t = ˆβ β 0 se( ˆ β) relies on both the correct ˆβ and se( ˆβ) for correct inference

3 Why Correct se s Matter For example, researchers often focus most of their time on achieving identification and think of se s as an afterthought However, incorrectly estimating the se s can cause researchers to commit Type I (more often) and Type II errors Ex: Inference with White se s in BDM (2004):

4 Cluster-Robust Inference This lecture focuses on obtaining correct inference for linear models where data are grouped into clusters - e.g. panel data The main assumption is that model errors can be correlated within clusters, but must be uncorrelated across clusters As we will see, failure to account for within-cluster correlations can lead to misleadingly small se estimates

5 Roadmap - A Lecture on se s 1. One-Way Cluster-Robust Inference: Theory + Examples 2. Two-Way and Multi-Way Cluster-Robust Inference 3. Practical Issues: Does FE Estimation = No Need to Cluster? 4. Practical Issues: When Should we use Cluster-Robust se s? 5. Practical Issues: The Problem of Few Clusters - An Unresolved Question

6 Note: A Lecture focuses on the CRVE I will focus on the Cluster-Robust Variance Estimator (CRVE) approach for cluster-robust inference The other main approaches are Feasible GLS and Block Bootstrap techniques However, FGLS requires that you specify the correct model for the cluster correlations CRVE requires no such specification (data-driven), but requires the assumption that the number of clusters approaches infinity

7 Main Takeaways 1. Not accounting for cluster-specific correlations (when it exists) most often leads to downward biased se estimates (large t s) 2. The need to control for such correlations is increasing in: a. The size of within-cluster error term correlations b. The size of within-cluster correlations of regressors of interest c. The number of observations within clusters 3. Be careful when clustering along dimensions with a small number of clusters

8 ONE-WAY CLUSTERING

9 A Motivating Example Consider a one-regressor linear model with non-stochastic regressors - e.g. E(x i ) = x i and y i = βx i + u i Assume that E(u i x i ) = 0 (e.g. the model is identified) so is an unbiased estimator of β ˆβ = ( x x ) 1 x y = β + i x i u i i x 2 i

10 A Motivating Example The asymptotic variance of ˆβ is given by ( V β + + ) i x i u i i xi 2 = V( i x i u i ) ( ) i xi 2 2 since x i is assumed to be non-stochastic Then if E[u i u j ] = 0 for all i j, then we recover ) V( ˆβ = i xi 2E[u2 i ( ) ] H i xi 2 2 with the heteroskedasticity robust consistent estimator ˆV = i x 2 i û2 i ( i x 2 i ) 2

11 A Motivating Example Clustering revolves around relaxing the assumption that E[u i u j ] = 0 for all i j In the extreme case of completely general error correlations V ( i x i u i ) = i = i = i ) = V( ˆβ = V( ˆβ cov(x i u i,x j u j ) j (E[x i u i x j u j ] E[x i u i ]E[x j u j ]) j x 2 i E[u 2 i ] + i j i x i x j E[u i u j ] )H + i j i x i x j E[u i u j ] ( i x 2 i ) 2

12 A Motivating Example Clustering imposes the structure that E[u i u j ] is non-zero when i and j share a cluster ) V( ˆβ = i j x i x j E[u i u j ]1 {i,jsamecluster} i xi 2 ) = V( ˆβ + i j i x i x j E[u i u j ]1 {i,jsamecluster} ( ) H i xi 2 2 }{{} ) ) We can see that V( ˆβ > V( ˆβ correlations match regressor correlations H when > 0 - e.g. cluster

13 A Motivating Example Moreover, there is a large loss of efficiency if OLS se s are used. The inefficiency and bias are increasing in 1. The corr. of regressors and errors within a cluster (ex: DD) 2. The more observations in the same cluster Intuition: Corr. 0 implies additional obs. in cluster no longer provide completely independent pieces of new info. Need to adjust Failure to account for within-cluster correlations may cause se estimates to be too small, leading to too many Type I errors

14 Linear Model with One Dimensional Clustering Suppose there are i = 1,...,N observations each of which belong to one of g = 1,...,G clusters (one-dimensional) Example: There are j = 1,..,F firms observed at multiple points in time. The clusters g are the firms, and N = F T The linear model has K stochastic regressors y ig = x igβ + u ig y g = X g β + u g for X g R N g K

15 Linear Model with One Dimensional Clustering Assume β is identified, E[u ig x ig ] = 0, and error independence across clusters E [ ] u ig u jg x ig,x jg = 0 unless g = g Stacking the regression equation over the clusters g yields y 1 y = Xβ + u.. = β + y G X 1.. X G u 1.. u G

16 Linear Model with One Dimensional Clustering The OLS estimator of β can be written as (show on board) ˆβ = ( X X ) 1 X Y = β + ( G ) 1 G X gx g gu g=1 g=1x g Therefore, the conditional variance of ˆβ is ( ) V ˆβ X = ( X X ) 1 ( X V[uu X]X )( X X ) 1

17 Linear Model with One Dimensional Clustering Notice that uu can be written as u 1 uu =.. [ u 1.. u ] G = u G u 1 u 1 u 1 u 2. u 1 u G u G u 1.. u G u G And so the key term V[uu X] = E[uu X] depends on the cluster structure E[u 1 u E[uu 1 ]. E[u 1u G ] E[u 1 u 1 ] 0. 0 X] =... =... E[u G u 1 ]. E[u Gu G ]... X 0.. E[u G u G ]

18 Linear Model with One Dimensional Clustering Multiplying through X = [ X 1.. X G ] yields X V[uu X]X = G g=1 X ge[u g u g X]X g Which yields the following expression for the conditional variance of ˆβ ( ) V ˆβ X = ( X X ) 1 ( ) = V ˆβ X ( G N g N g g=1 i=1 j=1 x ig x jgω ig,jg ) (X X ) 1 N g N g + ( X X ) ( 1 G H g=1 i=1 j i x ig x jgω ig,jg ) (X X ) 1

19 Linear Model with One Dimensional Clustering We see that the comparison of the cluster AVAR and the OLS AVAR centers around the unconditional B : G g=1 E[ X gu g u gx g ] Therefore, OLS se s will be biased downwards when B is larger, or when 1. The regressors within a cluster are more correlated 2. The errors within a cluster are more correlated 3. The term N g is large within the cluster (e.g. bigger sum) 4. The correlation between the regressors and errors is the same sign (usually the case - explain)

20 Asymptotic Distribution and CRVE Estimator Asymptotically, CGM (2011) state that G( ˆβ β) has a limit normal distribution as G, with AVAR ( ) G 1 ( ) 1 lim G G E[X gx G 1 ( 1 g ] lim G G E[X gu g u G 1 gx g ] lim G G E[X gx g ] g=1 Current applied studies use the cluster-robust variance matrix estimator (CRVE) ( G ) ˆV[ ˆβ] = (X X) 1 X gû g û gx g (X X) 1 g=1 }{{} CorrelationsOnlyWithinClusters(g) g=1 g=1 ) 1

21 Be Careful When Using the CRVE Estimator CGM (2013) state CRVE estimator is only an unbiased estimate of the AVAR when the number of clusters G se s using the CRVE estimator are biased downwards in finite samples (e.g. toward zero irregardless of correlation structure) Moreover, Wald t-statistics using CRVE se s have unknown finite sample distributions (even with normal disturbances), although Monte Carlo simulations show that T(G 1) approximates it well for large G

22 Example - Panel Structure in Petersen (2009) Example: See the handout given in class and posted online The example highlights the idea of the Moulton Factor between cluster-robust se s and OLS se s For the j th regressor, the default OLS se estimate based on σ 2 j (X X) 1 should be inflated by τ j 1 + ρ xj ρ ug ( N g 1) Note that cluster se s may be smaller than White se s if the signs of the correlations do not match

23 Intuition about Firm Clustered se s in Petersen (2009) In the example, OLS (i,t) se s think that each additional year of data provides N f iid observations (and vice versa), so that se( ˆβ) = constant N f T as either N f (firms) or T approach infinity 0 This implies that given enough data we will always find a result Firm clustering will help alleviate this problem (N g = T), because each year only provides ( ) 1 + ρxj ρ ug (T 1) 1 N f T iid equivalent observations. In the case ρ xj ρ u = 1, we have that se( ˆβ) is independent of T

24 MULTI-WAY CLUSTERING

25 Linear Model with Two Dimensional Clustering In pratice, there may be more than one dimension in which errors may be correlated within clusters Example: Finance panel datasets have both firm and time components. Cluster Dimension One: Within-firm error correlation (e.g. serially correlated investment opportunities) Cluster Dimension Two: Within-time error correlation (e.g. common shocks across firms)

26 Linear Model with Two Dimensional Clustering Consider a linear model with two dimensions of clustering, G and H y igh = x ighβ + u igh We assume that errors are correlated if observation i and j share either a G or H cluster E [ u igh u jg h x igh,x jg h ] = 0 unless g = g or h = h

27 Linear Model with Two Dimensional Clustering The CRVE estimator is a simple extension of the one-dimeonsional CRVE, with ^B = X ( ûû. S GH) X where S GH is an indicator matrix with (i,j) th element equal to 1 if i and j share a G or H cluster, and. is element-by-element multiplication The estimator^b R K K can be written as ^B = i x i x jû i û j1 {i,j share any cluster} j

28 Relation to Single Cluster Estimators Note that we can re-write the selector matrix S GH as S G + S H S G H where S G H is an indicator matrix with (i,j) th element equal to 1 if i and j share the G and H cluster (decomp. of union) Plugging this into the formula for ^Band plugging it into the CRVE estimator ˆV yields ˆV( ˆβ) = V ˆG ( ˆβ) + V ˆH ( ˆβ) V G H ˆ ( ˆβ)

29 Relation to Single Cluster Estimators Thus, double-clustered CRVE is estimated by running OLS three times and clustering on one dimension (G,H, or G H) each time If G H has one observation (as in finance firm year datasets), ˆ V G H ( ˆβ) reduces to the White heteroskedasticity-robust estimator

30 Non-Positive Definite CRVE Estimates The resulting ˆV estimate with two-way clustering may have negative elements on the diagonal, especially when using FE in the same directions as the clusters This implies that ˆV may not be positive-definite, but the sub-component associated with the regressor of interest may be positive-definite This may lead to a reported error in statistical packages even though inference is valid for the parameters of interest (reghdfe does this)

31 Linear Model with D Dimensional Clustering We can handle arbitrary correlations in up to D dimensions using multi-way clustering Let the D dimensional vector δ i = [δ i1,...,δ id ] assign observations to D dimensional cluster identifiers δ : {1,...,N} D }{{} d=1{1,...g d } observations Let r R D and define the set R {r : r d {0,1},d = 1,...,D, r 0} R D + Finally, let I r (i,j) = 1{r d δ jd = r d δ id d}. Then i and j share at least one cluster I r (i,j) = 1 for some r R

32 Linear Model with D Dimensional Clustering By defining the K K matrix B r by B r = i x i x jû i û ji r (i,j) j We have that B in the CRVE is with multi-way clustering is given by (using the L 1 norm) B = r 1 =k,r R,k=1,...,D ( 1) k+1 B r Which can be re-written in the same exact way as the two-way clustering scheme B = i x i x jû i û j1 {i,j share any cluster} j

33 Linear Model with D Dimensional Clustering We express the multi-way B in such a complicated way to ensure that estimation of the CRVE is easily programmable Example: D = 2 implies that R = {(1,0),(0,1),(1,1)} and thus we have B = ( 1) 1+1 [ B (1,0) + B (0,1) ] + ( 1) 2+1 B (1,1) which is equivalent to clustering one-way twice, and at the intersection once Example: D = 3 implies that R = {(1,0,0),(0,1,0),(0,0,1),(1,1,0),(1,0,1),(0,1,1),(1,1,1)} and thus we have B = ( 1) 1+1 [ B (1,0,0) + B (0,1,0) + B (0,0,1) ] + ( 1) 2+1 B (1,1) + = ( 1) 2+1 [ B (1,1,0) + B (1,0,1) + B (0,1,1) ] + ( 1) 3+1 B (1,1,1) which is just a bunch of one-way clusters and intersection clusters

34 Relevant Number of Clusters For Asymptotics Multi-way clustering relies on asymptotics that are in the number of clusters of the dimensions with the fewest number of clusters Thus, the method is most appropriate when each dimension has many clusters - and small clusters are still an ongoing area of research This concern is especially valid for DD studies in finance panel datasets: we prefer a small T to ensure β is estimated consistently, but small T lowers the number of clusters in the time dimension

35 Practical Issues: FE Estimation and Clustering

36 Does FE Estimation = No Need to Cluster? Including fixed effects generally does not control for all potential within cluster correlations Therefore, one should still use the CRVE estimator in the dimension in which you believe correlations to persist Example: AR(1) random shocks within-firms would imply the need to firm-cluster Example: Heterogeneous responses to time shocks would imply the need to time-cluster

37 Example 1: Difference-in-Differences Consider the augmented placebo policy model from BDM (2004) For a state-time policy variable d s,t, suppose the DGP for y is y i,s,t = αd s,t + x i,s,tβ + δ s + γ t + u i,s,t where the true parameters are α = β = 0 and thus there is no correlation between y and (d,x)

38 Example 1: Difference-in-Differences Suppose also that d s,t can be deconstructed as a within-state AR(1) process and a within-time AR(1) process d s,t = d s s,t + 2d t s,t d s s,t = 0.6d s s,t 1 + vs s,t d t s,t, = 0.6d t s 1,t + vt s,t v s s,t iid N(0,1) v t s,t iid N(0,1) Finally, suppose that y i,s,t is correlated across time as well y is,t = ỹ i,s,t u t s,t where u t s,t is simialr to d t s,t and ỹ i,s,t δ s is weakly correlatd within-state

39 Example 1: Difference-in-Differences Correlation of y within-time + state and α = β = 0 = u i,s,t (errors) are correlated within-time and weakly within-states. Moreover, d s,t is correlated both within-states and within-times Therefore, even in the presence of FE estimation, there still exists residual correlations needed to be controlled for

40 Example 2: A Complex DGP Suppose we have the following model y i,t = x i,tβ + ε i,t ε i,t = θ i f t + η i,t + ν i,t η i,t = δ i + ψγ i,t 1 + ζ i,t θ i = [1,θ 1i ] R l f t = [δ t,f 1t ] R l Thus, ε i,t has residual within-i and within-t correlations after FE estimation ε i,t = δ t + δ }{{} i + θ 1if 1t }{{} + ψγ i,t 1 }{{} +v i,t handledbyfes within t corr within i corr

41 Example 2: A Complex DGP The residual within-i correlations exist because of the auto-regressive γ i,t 1 terms The residual within-t correlations may exist because of the heterogenous responses to common shocks even after the within-transformation E[ ε i,t ε j,t X] = θ 1iE [ ḟ 1t ḟ 1t X ] θ 1j R Therefore, if ẋ i,t is correlated within-i, we need to cluster by i. If it is correlated within-t, we also need to cluster by t.

42 Example 3: Correlation from Hetereogenous Coefficients Consider the following (i,t) panel DGP y i,t = α + x i β + z i,t γ i + v i,t where E[v i,t x i,z i, ] = 0 and the coefficient γ i varies over i By definition, γ i = E[γ i ] + d i γ + d i where d i is zero-mean and γ is the average response. Substitution into the DGP yields y i,t = α + x i β + z i,t γ + (v i,t + z i,t d i )

43 Example 3: Correlation from Hetereogenous Coefficients Applying the standard within-transformation over i yields ẏ i,t = z i,t γ + ( v i,t + ż i,t (γ i γ)) }{{} ε i,t and OLS is consistent if E[γ i ż i,t ] = γ because E[ż i,t γż i,t (γ i γ) ż i,t ] = k (E[γ i ż i,t ] γ) = 0 However, there still exists residual correlation within ε i,t over i, so clustering at the i-level is needed if ż i,t is serially correlated

44 A Practical Note One should always use regression methods that perform full within-transformations in all dimensions in order to compute the correct se s STATA normally performs incorrect small-sample and se corrections when using a LSDV estimation technique A good rule-of-thumb is to always use reghdfe whenever possible

45 When Should We Use Cluster-Robust se s?

46 What Should I Cluster Over? The choice of cluster reflects the restrictions you place on the correlation structure of the error term Theoretically, one should cluster along any dimensions in which residual correlations of the error term and regressors exist However, since the CRVE is an estimate of the true se s, it may not always be best to use the most robust se s possible This stems from the class bias-versus-variance tradeoff in econometrics

47 Bias-versus-Variance Tradeoff Practically speaking, more robust se s (e.g. multi-way and coarser ) have less bias but more variance Lower bias improves the performance of test statistics, but higher variance leads us to find statistical significance when it does not exist For example, if we believe regressors and errors are correlated within a dimension, then we should account for this correlation However, this needs to be balanced against the asymptotics, as CRVE estimates only converge as G becomes large in the smallest cluster dimension (small cluster lead to poor estimate)

48 Highlighting Bias Suppose we should double cluster and instead only cluster by firm This omits ˆV t V HET ˆ from the se computation and an expected bias E [ ] ˆV t V HET ˆ (X X) 1 t i j cov(x i,t ε i,t,x j,t ε j,t )(X X) 1 Thus, the bias is increasing in the correlation between the regressors and the error terms across the time dimension Note also that the reduction in bias is larger when clustering in the smaller dimension - e.g. moving from N to T uncorr. obs instead of N to F

49 Highlighting Variance More robust se s usually have higher variance (e.g. V( se) ˆ ), which may lead to too many Type I errors This is a result of Jensen s Inequality and the rejection criteria of ˆt ˆβ β 0 se( ˆβ) t To see this, note that ˆt is a convex function of se( ˆβ) So if ˆβ and se( ˆβ) are independent (usually are), then E[ˆt] is increasing in V(se( ˆβ)) by Jensen s Inequality

50 Highlighting Variance Therefore, in expectation we too often reject the null hypothesis when the variance of the se s increases Furthermore, the variance of the se s increaes as the number of clusters decreases in a dimension Intuitively, fewer clusters implies less LLN averaging and more estimation error If either T or F is small, then double clustered se s may be widly understated without appropriate corrections

51 A Practical Note There is no solution to the bias-variance tradeoff and no formal test over what to cluster over The consensus in CGM (2013) is to be conservative and avoid bias and use bigger and more aggregate clusters (e.g. industry v. firm) when possible, up to the point where there is a concern about too few clusters CGM (2013) then suggest handling the variability by making small-cluster and critical value corrections I suggest taking the most conservative estimate (based on p values)

52 Practical Issues: The Problem of Few Clusters

53 Too Few Clusters Recall that the CRVE estimator is only consistent and ˆt N(0,1) as the number of clusters G However, for finite G, the distribution of ˆt is unknown and the CRVE estimator becomes higher in variance and bias as G 0 This is the few clusters problem: If G is small, then 1. ˆβ may still be a reasonable estimate of β 2. But the asymptotics for the CRVE have not kicked in yet 3. And we are unsure of the distribution of the calculated ˆt

54 Few Clusters Problem If the asymptotics have yet to kick in, then the CRVE has higher variance and E[ˆt] is higher This leads to over-rejecton of the null hypothesis if we use a standard T(N K) distribution CGM (2013) suggest correcting for this via either a CRVE small-g correction or a Bootstrap procedure

55 CRVE Small-G Corrections CGM (2013) suggest correcting for small G problems by adjusting û and adjusting the critical values The small-g error correction is done by replacing û g with cû, where c = G N 1 and calcuating a new se( ˆβ) G 1 N K We then use T(G 1) critical values to test against the new ˆt estimate - reghdfe does this but xtivreg2 does not Note that with few clusters, the hurdles for rejection are quite high - for G 1 = 4 we have(,, ) = (2.13,2.78,4.60) versus (1.65,1.96,2.33) for standard normal

56 A Practical Note There is no consensus as to what a sufficient number of clusters is - except that more is better In simulations with corrected CRVEs, clusters seems to work well Also, remember that the relevant cluster dimension is always that with the fewest number of clusters (t in finance panels) The best way to correct for this is still unresolved - but I suggest applying the small-g corrections and using the most conservative p-values This is because almost all simulations still over-reject after applying corrections (so want to be conservative as possible)

57 Bonus: Persistent Shocks and Spatial Correlation

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