14.385Nonlinear EconometricsLecture 7.Theory: Consistency and Accuracy of Bootstrap.Reference: Horowitz, Bootstrap.1Cite as: Victor Chernozhukov, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare(http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Consistency of BootstrapThe idea is that for large nGn(t, Fˆ) ≈ G (∞t, Fˆ) ≈ G (∞t, F0) ≈ Gn(t, F0).Consistency relies on asymptotics.Def. Gn(t, Fˆ) is consistent if under each F0∈ Fsupt|Gn(t, Fˆ) − G ( ) 0∞t, F0| →p,Example 1. (Inference on Mean)Statistic and parameter of interest:Tn=√n(X¯− θ), θ = θ(F0) = EF0X,Want to know:Gn(t, F0) = PF0(Tn≤ t)Bootstrap DGP and Bootstrap “population” parameter:Fn= empirical df, θ(Fn) = EFnX = X¯We create bootstrap samples {Xi∗} by sampling from theoriginal sample {Xi} randomly with replacement.Bootstrap version of Tn:Tn∗=√n(X¯∗− X¯)Bootstrap gives us:2Cite as: Victor Chernozhukov, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare(http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Gn(t, Fn) = PFn(Tn≤ t).Bootstrap should work as long as the limit distributionof Tnvaries smoothly in F and if the (triangular) CLTholds with the same limit for any sequence in F, and{Fˆ} ∈ F with probability one.One formal theorem is as follows (this follows Horowitz):Theorem (Bickel & Ducharme): Gn(t, Fˆ) is consis-tent if for each F0∈ F: (i) ρ(Fˆ, F0) →p0, (ii) G ( ) is∞t, Fcontinuous in t for each F ∈ F, (iii) for any t and Fnsuch that∈ Fρ(Fn, F0) → 0, we have|Gn(t, Fn) − G (∞t, F0)| → 0,for each t, where ρ is some metric.The result follows from the extended continuous map-ping theorem.Proof: Clearly, ρ(Fˆ, F0) →p0 implies|Gn(t, Fn) − G (∞t, F0)| →p0.Next we apply Polya’s Lemma: Pointwise convergenceof a sequence of monotone functions to a boundedmonotone continuous function implies that the sequenceconverges to this function uniformly. Thus, supt|Gn(t, Fn)(−G∞t, F0)| →p0. ¤Remark: In way, the theorem is nearly at tautology, butit really helps organize thinking.Cite as: Victor Chernozhukov, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare(http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Remark∗: Bickel and Friedman formally verify the con-ditions in Example 1 by checking the conditions of theabove theorem using Kantarovich-Wasserstein-Mallowsmetricρ(P, Q) = infX,Y{EkY − Xk2, Y ∼ P, X ∼ Q}.Theoretical Exercise. Supply the details of the proof. Hint: Ifyou let Xn= Φ−1(FYn(Yn)) =dN(0, 1), where FYnis the distributionfunction of Ynand Φ−1is the quantile function of the standardnormal. Then Yn=√1n(X¯− µ)/σ = FY−(Φ(Xn))n≈ Xn+ o(1),conditional on a set S that contains Xnwith a probability close toone, which holds by the central limit theorem.The following is a nice theorem due to Mammen:Theorem (Mammen): Let {Xi, i ≤ n} be an iid samplefrom population. For a sequence of normalizing con-stants tnand σndefine:¯gn=1(gnPn(Xi), Tn=¯gn−tn),σn¯gn∗=1(gnPn(Xi∗), Tn∗=¯gn∗−¯gn).σnNonparametric bootstrap is consistent if and only if Tnis asymptotically normal.Proof: The sufficiency follows from Bickel and Fried-man. For the necessity see Mammen’s article. ¤Cite as: Victor Chernozhukov, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare(http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Example 2. (Inference in Regression)Option (a) Bootstrap. (Asymptotically pivotal op-tion (b) below is more accurate)Data Xi= (Yi, Wi), where Wiis regressor, and Yidepen-dent variable. Model Yi= Wi0β + ²i.Parameter of interest: θ(F0) = βj.Statistic of interest: Tn=√n(βˆ− βj).Want to know Gn(t, F0) = PF0(Tn≤ t).Fn= empirical df.We create bootstrap samples {Xi∗} by sampling from theoriginal sample {Xi} randomly with replacement.Under bootstrap DGP the “population” parameter isθ(Fn) = βˆj.For each bootstrap sample we compute Tn∗=√n(βˆ∗βˆj)(bootstrap realization of−Tn).Bootstrap gives us Gn(t, Fn) = PFn(Tn≤ t).3Cite as: Victor Chernozhukov, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare(http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Example 2. (Inference in Regression)Option (b) This option bootstraps asymptoticallypivotal statistic.Data Xi= (Yi, Wi), where Wiis regressor, and Yidepen-dent variable. Model Yi= Wi0β + ²i.Parameter of interest: θ(F0) = βj, the j-th componentof β.Statistic of interest:Tn=√n(βˆj− βj)/s.e.(βˆj).Want to know exact law Gn(t, F0) = PF0(Tn≤ t).Fn= empirical df.We create bootstrap samples {Xi∗riginal} by sampling from theo sample {Xi} randomly with replacement.Under bootstrap DGP the “population” parameter isθ(Fn) = βˆj.For each bootstrap sample we computeTn∗=√n(βˆj∗− βˆj)/s.e.∗(βˆj)where s.e.∗(βˆj) denotes the recomputed value of thestandard error using bootstrap samples.Bootstrap gives us Gn(t, Fn) = PFn(Tn≤ t).Option (b) is more accurate than the “natural”option (a).4Cite as: Victor Chernozhukov, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare(http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Accuracy of Bootstrap.This example has an interesting feature:Tn→dN(0, 1),that is Gn(t, F0) ≈ Φ(t) in large samples, and the ex-act law is almost independent of DGP F0. Therefore,Gn(t, Fˆ) ≈ Φ(t) too.Def. A statistic Tnis called asymptotically pivotalrelative to a class of DGPs F, if its limit law does notdepend on DGP F :G (∞t, F ) = G (∞t)for all F ∈ F.Under asymptotic pivotality, the exact law is not verysensitive to the underlying DGP. Replacement of trueDGP F0with Fˆresults in a good approximation of theexact law.“Theorem”: The approximation error of bootstrap ap-plied to asymptotically pivotal statistic is smaller thanthe approximation error of bootstrap applied to an asymp-totically non-pivotal statistic.“Proof”: A simple example is the case of the exactpivotality, where the bootstrap makes no error at all.For mean-like stat this claim as well as regularity condi-tions for it are made formal by appealing to