41903: Group-Based Inference

Transcription

1 41903: Clusterin and Fama-MacBeth Notes 4

2 Dependent Data in Economics Many real-world data sets plausibly exhibit complicated dependence structures Consider areate panel data model: y it = x itβ + α i + δ t + ε it i represents somethin like firm, state, country seems likely that ε it will have a complicated structure e.. consider a productivity shock to IBM today likely related to future and past productivity shocks at IBM plausibly spills over to related firms (e.. Goole) today plausibly spills over to related firms in the future (and related to them in the past) Likely that E[ε it ε jt ] 0 even for i j and s t Failin to account for this dependence will lead to invalid inference for parameters of interest

3 Approaches Group-based inference approaches popular methods to estimatin standard errors robust to non-identically distributed and dependent data. Group-based inference procedures partition data into approximately uncorrelated roups clustered standard errors Lian and Zeer (1986), Arellano (1987) really popularized in panel data application by Bertrand, Duflo, and Mullainathan (2004) Fama-MacBeth (1973) popular in finance but much more enerally applicable Both approaches partition data into = 1,..., G roups I denotes the indices of observations in roup N observations per roup

4 Linear Model Aain Illustrate mechanics of clusterin and Fama-MacBeth in linear model: y i = x i β + ε i ; E[x i ε i ] = 0 E[x i x j ε i ε j ] 0 when i and j both belon to I E[x i x j ε i ε j ] = 0 when i I and j I h with h

5 Clusterin Mechanics Recall β a N(β, V /n) with V = Qn 1 Ω nqn 1 for and Ω n = Var[ 1 n Q n = 1 n n x i ε i ] = 1 n i=1 n i=1 E[x i x i ] 1 n =1 i I n [x i x i ] = Q i=1 G E[x i x j ε i ε j ] = 1 n j I G E[(X E )(E X )] for X the N p matrix formed by stackin observations x i for i I and E the N 1 vector formed by stackin ε i for i I Clustered standard errors based on estimatin Ω n with Ω = 1 n G (X Ê)(Ê X ) = 1 G =1 G =1 =1 N N (X Ê/ N )(Ê X / N ) for Ê the N 1 vector formed by stackin sample residuals ε i for i I

6 Fama-MacBeth (FM) Mechanics FM subset data into the G different roups: X the N p matrix formed by stackin observations x i for i I Y the N 1 vector formed by stackin observations y i for i I β = (X X ) 1 (X Y ) - Estimator of parameter of interest within roup FM Estimator: β = 1 G G β =1 FM Variance estimator: ˆV FM = 1 G 1 G 1 G ( β β)( β β) I.e. the β are just treated as observations. The estimator is their sample mean. The estimated variance is just the estimator of the variance of this sample mean. =1

7 Clustered Covariance Matrix Estimator: Lare G Asymptotics Traditional analysis considers lare G asymptotics (G ) have p Ω Ω n 0: Ê = E X ( β β) (X Ê)(Ê X ) = (X E )(E X ) (X E )(( β β)x X ) (X X ( β β))(e X ) + (X X ( β β))(( β β)x X ) G 1 G n G =1 (X E )(E X p ) Ω n 0 by LLN ( ) 1 vec G G =1 (X E )(( β β)x X ) = 1 G ( ) G =1 X X X E ( β β) = O p(1)o p(1) by LLN and consistency of β Other terms handled similarly Applies more enerally than OLS by replacin x i ε i by s i ( θ) (the element of the score vector for estimatin θ for observation i)

8 Comments on Lare G Clustered s.e. Comments: Consistency of s.e. estimator obtains with essentially no restrictions on within roup covariance structure Essentially arbitrary heteroskedasticity and dependence allowed Ω p.s.d. by construction Leveraes G - need a lare number of roups that can be taken to be approximately independent Appropriate notion of approximately independent does not require there be literally no dependence across roups Hansen (2007) verifies that no further restrictions are required on N when G if roups are literally independent Bester, Conley, and Hansen (2011) shows that N and G required if there is small correlation across roups Similar properties can be established for FM

9 MA(1) Example Suppose you want to estimate [ ] 1 n Ω n = Var η i, where E[η i ] = 0, Var[η i ] = σ 2, Cov(η i, η i 1 ) = θ n i=1 and all other covariances are 0. We have Ω n = σ 2 + 2θ n 1 n

10 MA(1) Example Consider clusterin estimators with G n roups of size N G (inore inteer problems) ˆΩ Gn = 1 n G n N G N G =1 s=( 1)N G +1 t=( 1)N G +1 E[ˆΩ Gn ] = 1 Gn(N n Gσ 2 + 2(N G 1)θ) = σ 2 + 2θ n Gn n Bias = 2θ Gn 1 n no asymptotic bias requires Gn n = 1 N G 0 Let ω st = η sη t E[η sη t]. Var[ˆΩ Gn ] = 1 n 2 Gn =1 NG q=( 1)N G +1 NG r=( 1)N G +1 η sη t Gn NG NG h=1 s=( 1)N G +1 t=( 1)N G +1 E[ωqr ωst] order G nng 2 non-zero terms even if observations were independent Usin this optimistic rate ives Var[ˆΩ Gn ] = C GnN2 G n 2 = C N G n for some C No asymptotic variance requires N G n = 1 G 0 Need G for consistency. With literally no dependence across roups, can et away with no restrictions on roup size. Otherwise, need lare roups.

11 Two-Way Clusterin Cameron, Gelbach, and Miller (2011) propose a multiway clusterin estimator Recall eneral HAC estimator for LS coefficient estimator: ˆΩ = 1 n W (d(i, j))x i x j e i e j i j Cluster estimator special case with W (d(i, j)) = 1((i) = (j)) where (i) returns the roup membership of observation i Consider two roupin variables (e.. state and year). Let (i) denote roup membership in first dimension and h(i) denote roup membership in second dimension. Two-way clusterin is HAC with W (d(i, j)) = 1((i) = (j) or h(i) = h(j)) Can obviously be eneralized to more than two roupin variables Simple computational trick - ˆΩ TW = ˆΩ Cluster, + ˆΩ Cluster,h ˆΩ Cluster, h Good properties will require minimum number of roups alon any dimension to be very lare Not necessarily p.s.d. (may return neative variance estimates) - in practice, common fix is to add a lare enouh p.d. matrix to the estimate to maintain positivity Rules out very sensible models of correlation across observations

12 Usin a Small Number of Lare Groups Takeaways from previous section: Usin many roups keeps variance of standard error estimator small Usin lare roups keeps bias of standard error estimator small These two forces work aainst each other - May lead to poor performance in finite samples Revisit testin problem for scalar parameter of interest ( ˆβ) with estimated standard error ( ˆV ) Assume n( ˆβ β) d A = V Z where Z N(0, 1) Assume ˆV d VW for some random variable W Traditional approach that relies on consistency takes W = 1 t = n( ˆβ β) ˆV d A VW = Z W Z Can use this as lon as distribution of can be obtained W Does not require consistent variance estimation - only a notion of low bias so V cancels We know how to keep bias small - Use a small G (and resultin lare roups)

13 Clustered Standard Errors with Small G Bester, Conley, and Hansen (2011) consider clusterin with eneral dependence usin small G Most useful approach relies on N /N h 1 for all, h Given this, simplify and assume N = N x 1x 1 Q 1 1 N. p. with Q = Q for all x Gx G Q G 1 N B 1. B G x 1ε 1. x Gε G d B 1. B G with Ω N 0, 0 Ω with Ω = Ω for all Ω G For inference about scalar parameter, can allow Ω to differ across roups

14 BCH Results (1) 1. Properties of OLS n( β β) = ( 1 G (x x /N)) 1 ( 1 (x ε / N)) G d ( 1 G Q ) 1 ( 1 B ) G = Q 1 1 G Λ Z where Ω = ΛΛ and Z N(0, I) iid across 2. Properties of ˆΩ: (Let Q = Q and S = ΛZ) ˆΩ d W = 1 G ( Λ Z Z Λ Q Q 1 SZ Λ ) Λ Z S Q 1 Q + Q Q 1 SS Q 1 Q [ ( ) ( )] = 1 G Λ Z Z 1 Z Z G Λ

15 BCH Results (2) 3. V = ( 1 G (x x /N)) 1 1 Ω( G (x x /N)) 1 d Q 1 WQ 1 For simplicity, look at Wald test of H 0 : β = β 0 : ˆF = n( β β) 1 V n( β β) [ [ ( ) ( d ( 1 Z )Λ Q 1 1 Q G G Λ Z Z 1 Z G QQ 1 Λ( 1 Z ) G [ [ ( ) ( = ( 1 Z ) 1 Z Z 1 Z G G G = F Z Z )] Λ ] 1 )]] 1 ( 1 Z ) G The asymptotic distribution is pivotal F Gp G p F p,g p (e.. Rao (2002), Linear Statistical Inference and its Application, Chapter 8b)

16 BCH Results (3) Main BCH Results: Under conditions above and for testin hypotheses of the form Rβ = c d 1. F Gq F G q q,g q where c is a q 1 vector 2. If q = 1, the usual t-statistic, t d G 1 t G G 1 Note that usin Ω = G ˆΩ d in formin the t-statistic ives t t G 1 G 1 - This is the scalin used in Stata When usin clustered standard errors, use critical values from a t or F distribution (rather than N(0,1) or χ 2 ) and rescale - super easy Main drawback: Stron homoeneity conditions employed

17 FM with Small G Ibraimov and Müller (2010) consider FM with a small number of roups Approach only allows testin a scalar hypothesis of the form H 0 : β 0,j = c for some j Extended to two-sample t-tests in Ibraimov and Müller (2016) Let ˆθ be the estimator of the parameter of interest obtained within the th subroup, and let n = G N where N = 1 G N. IM really need one condition: ( N(ˆθ1 θ),..., N(ˆθG θ)) d (b 1,..., b G ) N(0, Σ) where Σ is diaonal Note - no homoeneity imposed

18 IM Results (1) 1. t-statistic: t = θ θ n( θ θ) = VFM nvfm 2. FM Estimator: n( θ θ) = 1 N(ˆθ θ) G d 1 G b

19 IM Results (2) 3. FM Variance Estimator: n ˆV FM = n G = 1 G 1 = 1 G 1 1 G 1 G ( θ θ)( θ θ) =1 ( 2 N( θ θ)) d 1 G 1 Puttin 1., 2., and 3. toether ives ( N(ˆθ θ) 1 G (b b) 2 = sg 2 t d t = 1 G b s G / G 2 N(ˆθh θ)) i.e. the FM t-statistic is asymptotically the same as a t-statistic for testin H 0 : µ = 0 usin the sample mean estimated from independent normals with different variances usin the usual standard error estimator h

20 IM Results (3) How does this help? Variances are not homoeneous, so t does not follow a t-distribution. Simplified statement of theorem from Bakirov and Székely (2005): Let cv G 1 (α) be the usual two-sided critical value from a t G 1 random variable. If α <.083, then for all G 2, sup P( t > cv G 1 (α) H 0 ) = P( t G 1 > cv G 1 (α)) = α σ 2 1,σ2 2,...,σ2 G The t-test for a sample mean assumin common variances remains valid for sizes less than 8% or so reardless of heteroeneity in the variances. The FM-based t-test is asymptotically equivalent to a t-test for a sample mean so remains valid with essentially no homoeneity restrictions.

21 FM-based Permutation Inference Robustness of FM approach relies on test of sinle hypothesis bein willin to live with a conservative test Canay, Romano, and Shaikh (2014) consider a randomization test procedure that addresses both concerns Let ˆθ be the estimator of the parameter(s) of interest obtained within the th subroup. Will use ( N(ˆθ 1 θ),..., N(ˆθ G θ) ) d (b 1,..., b G) N(0, Σ) where Σ is block diaonal - i.e. E[b b h] = 0 for h Note - Good properties of CRS can be obtained under weaker conditions Aain no homoeneity imposed

22 CRS Mechanics Let T (X) be a test-statistic that takes data X and rejects if T (X) is lare (e.. a t-statistic) In our application, we will set X = ( N(ˆθ 1 θ),..., N(ˆθ G θ) ) where θ is the value of the parameter under the null hypothesis M = [ 1, 1] G be the set of sin-chanes for G variables mx be an element from M applied to X Note that as lon as the distribution of X is symmetric, mx has the same distribution as X Implementation: Compute T (mx) for all m M. (If G is lare, this will be infeasible - can instead randomly choose elements from M). Let M R be the set of m used where R denotes the number of elements in this set. Calculate p-value: ˆp = 1 R m M R 1(T (mx) T (X)) Reject hypothesized value at level α if ˆp < α

23 Intuition Basic idea for why this should work: Under H 0, distribution of X = ( N(ˆθ1 θ),..., N(ˆθG θ) ) is symmetric and elements of X are independent Multiplyin the elements X by 1 or 1 does not chane the distribution under H 0 but does chane the values of the test-statistic By lookin at the different values of the test-statistic produced we can then et the distribution of our test-statistic

24 Comments Thouhts on these procedures: All are potentially useful and none uniformly dominates in simulation CRS and IM require formally weaker conditions than clusterin CRS is not conservative under heteroeneity while IM is For small α, CRS will have no power unless a moderate number of roups are available Clusterin uses the full sample in formin its point estimator - FM (e.. IM and CRS) uses the averae of subsample estimators - finite-sample bias will behave as if it is from the smallest subsample size while variance will shrink as if parameter estimated from the full sample Use clusterin when finite-sample bias is a bier concern than heteroeneity Use CRS when a moderate number of clusters is available Use IM when usin very few clusters and heteroeneity is a concern All procedure rely on havin formed lare enouh roups to capture important sources of dependence Be careful of fixed effects that cross roup boundaries - e.. includin four-diit industry classification dummies and roupin by state may induce dependence across states

25 Example: Health Insurance Reform (Aain) Kaestner and Simon (ILLR 2002) Individual level data from CPS March Supplement Data: individuals in 51 states (includin DC) and 10 years Outcome: (lo(wae)) Variables of Interest (Treatments): fullr - full HI reform indicator partialr - partial HI reform indicator hihcost - number of mandates to cover hih cost procedures women - number of mandates for coverin larely female specific procedures other - number of mandates coverin other procedures Controls: ae: dummies for ae cateories (a2 a8) ender: dummy for male education: dummies for hih school, some collee, and collee maritalstatus: dummies for married and divorced race: dummies for African American and other race (white excluded cateory) children: number of children < 6 and number of children between 6 and 18 state effects, year effects, state specific trends

26 HI Reform Reression Estimated model: lo(wae) it = α s + κ t + δ st + treatment stγ + x itβ + ε it where treatment st is a 5 1 vector of the state level treatment variables applyin to individual i at time t. How should we roup?... state year? state? year? two-way state-year? 9 census-reion? 4 census-reion?

27 HI Reform Reression: Results OLS Estimation Results: β State 9 Reions 4 Reions Year State X Year Two-Way Full Reform Partial Reform # Hih Cost # Women # Other cv(5%) Fama-MacBeth Estimation Results: β State s.e. β 9Reions s.e β 4Reions s.e. β Year s.e. Full Reform Partial Reform # Hih Cost # Women # Other cv(5%) CRS lb State ub State lb 9Reions ub 9Reions lb 4Reions ub 4Reions lb Year ub Year Full Reform Partial Reform # Hih Cost # Women # Other