Introduction Preliminaries The uniform delta-method Applications Uniformity and the delta-method Maximilian Kasy Jos´ e L. Montiel Olea October 27, 2014 Maximilian Kasy Harvard Uniformity 1 of 31
Introduction Preliminaries The uniform delta-method Applications Introduction ◮ Many procedures for estimation and inference: ◮ motivated by asymptotic behavior ◮ for fixed parameter values. ◮ Often, such procedures behave poorly ◮ in finite samples ◮ for some parameter regions. ◮ Such problems can arise, if approximations are not uniformly valid. ◮ Can lead to 1. large mean squared error for estimators, 2. undercoverage for confidence sets, 3. distorted rejection rates for tests, 4. ... Maximilian Kasy Harvard Uniformity 2 of 31
Introduction Preliminaries The uniform delta-method Applications Examples in econometrics 1. Instrumental variables: poor behavior for weak instruments 2. Inference under partial identification: poor behavior near point-identification 3. Estimation after model selection: poor behavior around the critical values for model selection 4. Time series: poor behavior near unit roots Maximilian Kasy Harvard Uniformity 3 of 31
Introduction Preliminaries The uniform delta-method Applications ◮ Unifying theme? ◮ One important tool in asymptotics: Delta-method ◮ Taylor expansions to approximate functions of random variables ◮ Problems ⇔ Large remainder for some parameter values ◮ This paper: ◮ A sufficient and necessary condition ◮ for uniform negligibility ◮ of the remainder. Maximilian Kasy Harvard Uniformity 4 of 31
Introduction Preliminaries The uniform delta-method Applications Roadmap Literature ◮ ◮ Preliminaries: ◮ Definitions ◮ Uniformity and inference ◮ Uniform continuous mapping theorem ◮ Uniform delta method: ◮ Necessary and sufficient condition ◮ Simpler sufficient conditions ◮ Applications: √ t , cos ( t 2 ) ◮ Stylized examples: | t | , 1 / t , ◮ Weak instruments, moment inequalities Maximilian Kasy Harvard Uniformity 5 of 31
Introduction Preliminaries The uniform delta-method Applications Preliminaries Notation: ◮ θ ∈ Θ indexes the distribution of the observed data ◮ µ = µ ( θ ) is some finite dimensional function of θ ◮ asymptotics wrt n ◮ F : cumulative distribution functions ◮ S , T , X , Y and Z : random variables / vectors Maximilian Kasy Harvard Uniformity 6 of 31
Introduction Preliminaries The uniform delta-method Applications Definition (bounded Lipschitz metric) ◮ BL 1 : set of all functions h on R k such that 1. | h ( x ) | ≤ 1 and 2. | h ( x ) − h ( x ′ ) | ≤ � x − x ′ � for all x , x ′ ◮ bounded Lipschitz metric on the set of random variables: � � d θ � E θ [ h ( X 1 )] − E θ [ h ( X 2 )] � . BL ( X 1 , X 2 ) := sup h ∈ BL 1 ◮ van der Vaart and Wellner (1996, section 1.12): convergence in distribution of X n to X ⇔ convergence of d θ BL ( X n , X ) to 0. Maximilian Kasy Harvard Uniformity 7 of 31
Introduction Preliminaries The uniform delta-method Applications Definition (Uniform convergence) 1. X n converges uniformly in distribution to Y n if d θ n BL ( X n , Y n ) → 0 for all sequences { θ n ∈ Θ } . 2. X n converges uniformly in probability to Y n if P θ n ( � X n − Y n � > ε ) → 0 for all ε > 0 and all sequences { θ n ∈ Θ } . Maximilian Kasy Harvard Uniformity 8 of 31
Introduction Preliminaries The uniform delta-method Applications Lemma (Characterization of uniform convergence) 1. X n converges uniformly in distribution to Y n iff d θ BL ( X n , Y n ) → 0 sup θ 2. X n converges uniformly in probability to Y n iff P θ ( � X n − Y n � > ε ) → 0 sup θ for all ε > 0 . Maximilian Kasy Harvard Uniformity 9 of 31
Introduction Preliminaries The uniform delta-method Applications Remarks ◮ Definition of convergence: sequence X n toward another sequence Y n ◮ Special case Y n = X ◮ Uniform convergence in distribution safeguards ◮ for large n ◮ against poor approximation ◮ for some θ . ◮ Next slide: uniform convergence in distribution ⇒ uniform validity of inference procedures Maximilian Kasy Harvard Uniformity 10 of 31
Introduction Preliminaries The uniform delta-method Applications Lemma (Uniform confidence sets) ◮ Suppose Z n = Z n ( µ ) → d Z uniformly, where ◮ Z is continuously distributed and ◮ the distribution of Z does not depend on θ . ◮ Let z be the 1 − α quantile of the distribution of Z. Then C n := { m : Z n ( m ) ≤ z } is such that P θ n ( µ ( θ n ) ∈ C n ) → 1 − α for any sequence θ n . Maximilian Kasy Harvard Uniformity 11 of 31
Introduction Preliminaries The uniform delta-method Applications Theorem (Uniform continuous mapping theorem) Let ψ ( x ) be a Lipschitz-continuous function of x. 1. Suppose X n converges uniformly in distribution to Y n . Then ψ ( X n ) converges uniformly in distribution to ψ ( Y n ) . 2. Suppose X n converges uniformly in probability to Y n . Then ψ ( X n ) converges uniformly in probability to ψ ( Y n ) . Maximilian Kasy Harvard Uniformity 12 of 31
Introduction Preliminaries The uniform delta-method Applications The uniform delta-method Setting ◮ sequence of numbers r n (eg. r n = √ n ) ◮ sequence of random variables T n ◮ such that S n := r n ( T n − µ ) → d S uniformly ◮ all distributions and µ indexed by θ ◮ corresponding sequence X n := r n ( φ ( T n ) − φ ( µ )) ◮ goal: approximate the distribution of X n by the distribution of X := ∂φ ∂ x ( µ ) · S . Maximilian Kasy Harvard Uniformity 13 of 31
Introduction Preliminaries The uniform delta-method Applications ◮ first order Taylor expansion of φ : φ ( t ) = φ ( m )+ ∂φ ∂ m ( m )( t − m )+ o ( t − m ) ◮ normalized remainder � � � � � φ ( t ) − φ ( m ) − ∂φ 1 � � ∆( t , m ) := ∂ m ( m ) · ( t − m ) � . � t − m � ◮ � � ∆( m + ε · s , m ) > ε ′ � p ( ε , ε ′ , m ) := ds . 1 � s �≤ 1 ◮ necessary and sufficient condition for uniform delta-method: bound on p ( ε , ε ′ , m ) ◮ sufficient condition: bound on ∆ Maximilian Kasy Harvard Uniformity 14 of 31
Introduction Preliminaries The uniform delta-method Applications Assumption (Uniform convergence of S n ) Let S n := r n ( T n − µ ) . 1. S n → d S uniformly. 2. S is continuously distributed for all θ . 3. The collection { S ( θ ) } θ ∈ Θ is tight. 4. The density of S satisfies f ≤ f θ ( s ) ∀ s : � s � < s , ∀ θ f θ ( s ) ≤ f ∀ s , ∀ θ Leading example : ◮ S ∼ N ( 0 , Σ( θ )) , with ◮ uniform lower and upper bounds on the eigenvalues of Σ( θ ) . Maximilian Kasy Harvard Uniformity 15 of 31
Introduction Preliminaries The uniform delta-method Applications ◮ Define X n = r n ( φ ( T n ) − φ ( µ )) , T n = µ + 1 � S r n X n = r n ( φ ( � � T n ) − φ ( µ )) X = ∂φ ∂µ ( µ ) · S . ◮ Approximate X n by � X n (uniformly): straightforward under assumption on uniform convergence of S n ◮ Approximate � X n by X (uniformly): requires uniform bound on remainder of Taylor approximation Maximilian Kasy Harvard Uniformity 16 of 31
Introduction Preliminaries The uniform delta-method Applications Theorem (Uniform delta method – part 1) Suppose ◮ assumption on uniform convergence of S n holds, and ◮ φ is continuously differentiable everywhere in µ (Θ) . Then: X n → d � 1. X n uniformly if ∂φ / ∂µ is bounded. X n → p X 2. � uniformly if and only if p ( ε , ε ′ , m ) ≤ δ ( ε , ε ′ ) (1) for all ε , ε ′ , all m ∈ µ (Θ) and some function δ where ε → 0 δ ( ε , ε ′ ) = 0 . lim Maximilian Kasy Harvard Uniformity 17 of 31
Introduction Preliminaries The uniform delta-method Applications Theorem (Uniform delta method – part 2) 3. A sufficient condition for condition (1) : ∆( t , m ) ≤ � δ ( � t − m � ) . (2) for some function � δ where lim ε → 0 � δ ( ε ) = 0 . 4. If ◮ the domain of φ is compact and convex ◮ φ is everywhere continuosly differentiable on its domain then ◮ ∂φ / ∂µ is bounded and ◮ condition (2) holds. Maximilian Kasy Harvard Uniformity 18 of 31
Introduction Preliminaries The uniform delta-method Applications ◮ compact and convex domain of continuously differentiable φ : sufficient for uniformity ◮ too restrictive for most applications ◮ but suggests where problems might occur: 1. neighborhood of boundary points not included in the domain: √ near 0 for | t | , 1 / t , t 2. infinity: cos ( t 2 ) Maximilian Kasy Harvard Uniformity 19 of 31
Introduction Preliminaries The uniform delta-method Applications ◮ One of our assumptions: uniform convergence of S n ◮ Special case: uniform CLT ◮ Follows from CLTs for triangular arrays, eg. Lemma (Uniform central limit theorem) ◮ Let Y i be i.i.d. ◮ with mean µ ( θ ) and variance Σ( θ ) . � � � Y 2 + ε ◮ Assume that E � < M. i Then n 1 ∑ √ S n := ( Y i − µ ( θ )) n i = 1 converges uniformly in distribution to the tight family S ∼ N ( 0 , Σ( θ )) . Maximilian Kasy Harvard Uniformity 20 of 31
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