A more powerful subvector Anderson and Rubin test in linear instrumental variables regression. Patrik Guggenberger Pennsylvania State University

Transcription

1 A more powerful subvector Anderson and Rubin test in linear instrumental variables regression Patrik Guggenberger Pennsylvania State University Joint work with Frank Kleibergen (University of Amsterdam) and Sophocles Mavroeidis (University of Oxford) Indiana University September, 2018

2 Overview Robust inference on a slope coeffi cient(s) in a linear IV regression "Robust" means uniform control of null rejection probability over all "empirically relevant" parameter constellations "Weak instruments" pervasive in applied research (Angrist and Krueger, 1991) adverse effect on estimation and inference (Dufour, 1997; Staiger and Stock 1997)

3 Large literature on "robust inference" for the full parameter vector Here: Consider subvector inference in the linear IV model, allowing for weak instruments First assume homoskedasticity then relax to general Kronecker-Product structure then allow for arbitrary forms of heteroskedasticity Presentation based on two papers; one being "A more powerful subvector Anderson Rubin test in linear instrumental variables regression"

4 Focus on the Anderson and Rubin (AR, 1949) subvector test statistic: "History of critical values": Projection of AR test (Dufour and Taamouti, 2005) Guggenberger, Kleibergen, Mavroeidis, and Chen (2012, GKMC) provide power improvement: Using χ 2 k m W,1 α as critical value, rather than χ2 k,1 α asymptotic size "Worst case" occurs under strong identification still controls HERE: consider a data-dependent critical value that adapts to strength of identification

5 Show: controls finite sample/asymptotic size & has uniformly higher power than method in GKMC One additional main contribution : computational ease Implication: Test in GKMC is "inadmissible"

6 Presentation Introduction: finite sample case a) m W = 1 : motivation, correct size, power analysis (near optimality result) b) m W > 1 : correct size, uniform power improvement over GKMC c) refinement

7 asymptotic case: a) homoskedasticity b) general Kronecker-Product structure c) general case (arbitrary forms of heteroskedasticity)

8 Model and Objective (finite sample case) y = Y β + W γ + ε, Y = ZΠ Y + V Y, W = ZΠ W + V W, y R n, Y R n m Y (end or ex), W R n m W (end), Z R n k (IVs) Reduced form: (y. Y. W ) = Z (Π Y. Π W ) where v y := ε + V Y β + V W γ. ( β γ. I m Y 0. 0 I mw ) + (v y. V Y. V W ), } {{ } V Objective: test H 0 : β = β 0 versus H 1 : β β 0.

9 s.t. size bounded by nominal size & "good" power Parameter space: 1. The reduced form error satisfies: V i i.i.d. N (0, Ω), i = 1,..., n, for some Ω R (m+1) (m+1) s.t. the variance matrix of (Y 0i, V W i ) for Y 0i = y i Y i β 0 = W i γ + ε i, namely Ω (β 0 ) = is known and positive definite. 1 0 β I mw Ω 1 0 β I mw 2. Z R n k fixed, and Z Z > 0 k k matrix.

10 Note: no restrictions on reduced form parameters Π Y and Π W allow for weak IV

11 Several robust tests available for full vector inference H 0 : β = β 0, γ = γ 0 vs H 1 : not H 0 including AR (Anderson and Rubin, 1949), LM, and CLR tests, see Kleibergen (2002), Moreira (2003, 2009). Optimality properties: Andrews, Moreira, and Stock (2006), Andrews, Marmer, and Yu (2018), and Chernozhukov, Hansen, and Jansson (2009)

12 Subvector procedures Projection: "inf" test statistic over parameter not under test, same critical value "computationally hard" and "uninformative" Bonferroni and related techniques: Staiger and Stock (1997), Chaudhuri and Zivot (2011), McCloskey (2012), Zhu (2015), Andrews (2017),Wang and Tchatoka (2018)...; often computationally hard, power ranking with projection unclear Plug-in approach: Kleibergen (2004), Guggenberger and Smith (2005)...Requires strong identification of parameters not under test.

13 GMM models: Andrews, I. and Mikusheva (2016) Models defined by moment inequalities: Gafarov (2016), Kaido, Molinari, and Stoye (2016), Bugni, Canay, and Shi (2017),...

14 The Anderson and Rubin (1949) test AR test stat for full vector hypothesis H 0 : β = β 0, γ = γ 0 vs H 1 : not H 0 AR statistic exploits EZ i ε i = 0 AR test stat: AR n (β 0, γ 0 ) = (y Y β 0 W γ 0 ) P Z (y Y β 0 W γ 0 ) ( 1. β 0. ) ( γ 0 Ω 1. β 0. 0) γ AR stat is distri. as χ 2 k under null hypothesis; critical value χ2 k,1 α

15 Subvector AR statistic for testing H 0 is given by AR n (β 0 ) = min γ R m W where again Y 0 = y Y β 0. (Y 0 W γ) P Z (Y 0 W γ) ( 1. β 0. γ ) Ω ( 1. β 0. γ ), Alternative representation (using κ min (A) = min x, x =1 x Ax): AR n (β 0 ) = ˆκ p, where ˆκ i for i = 1,..., p = 1 + m W be roots of characteristic polynomial in κ κi p Ω (β 0 ) 1/2 ( Y 0. W ) PZ ( Y 0. W ) Ω (β 0 ) 1/2 = 0, ordered non-increasingly

16 When using χ 2 k,1 α correct size; critical values, as for projection, trivially, test has GKMC show that this is also true for χ 2 k m W,1 α critical values

17 Next show: AR statistic is the minimum eigenvalue of a non-central Wishart matrix For par space above, the roots ˆκ i solve where 0 = and M is a k p. ˆκ i I 1+mW Ξ Ξ, i = 1,..., p = 1 + m W, Ξ N (M, I k I p ), Under H 0, the noncentrality matrix becomes M = ( 0 k, Θ W ), where Θ W = ( Z Z ) 1/2 ΠW Σ 1/2 V W V W.ε, Σ VW V W.ε = Σ VW V W Σ εv W σ 1 εε Σ εvw

18 and ( ) σεε Σ εvw Σ εv Σ W VW V W = 1 0 β 0 0 γ I mw Ω 1 0 β 0 0 γ I mw Summarizing, under H 0 the p p matrix Ξ Ξ W ( k, I p, M M ), has non-central Wishart with noncentrality matrix ( ) M 0 0 M = 0 Θ W Θ W and AR n (β 0 ) = κ min (Ξ Ξ)

19 The distribution of the eigenvalues of a noncentral Wishart matrix only depends on the eigenvalues of the noncentrality matrix M M. Hence, distribution of ˆκ i only depends on the eigenvalues of Θ W Θ W, κ i say, i = 1,..., m W and κ = (κ 1,..., κ mw ) When m W = 1, κ = κ 1 = Θ W Θ W is scalar.

20 Figure 1: The cdf of the subset AR statistic with k = 3 instruments, for different values of κ 1 = 5, 10, 15, 100 Theorem: Suppose m W = 1. Then, under the null hypothesis H 0 : β = β 0, the distribution function of the subvector AR statistic, AR n (β 0 ), is monotonically decreasing in the parameter κ 1.

21 New critical value for subvector Anderson and Rubin test: m W = 1 Relevance: If we knew κ 1 we could implement the subvector AR test with a smaller critical value than χ 2 k m W,1 α which is the critical value in the case when κ 1 is "large". Muirhead (1978): Under null, when κ 1 "is large", the larger root κ 1 (which measures strength of identification) is a suffi cient statistic for κ 1 More precisely: the conditional density of AR n (β 0 ) = ˆκ 2 given ˆκ 1 can be approximated by fˆκ2 ˆκ 1 (x) f χ 2 k 1 (x) (ˆκ 1 x) 1/2 g (ˆκ 1 ),

22 where f χ 2 k 1 is the density of a χ 2 k 1 depend on κ 1. and g is a function that does not Analytical formula for g The new critical value for the subvector AR-test at significance level 1 α is given by 1 α quantile of (approximation of AR n given κ 1 ) Denote cv by c 1 α (ˆκ 1, k m W ) Depends only on α, k m W, and ˆκ 1

23 Conditional quantiles can be computed by numerical integration Conditional critical values can be tabulated implementation of new test is trivial and fast They are increasing in ˆκ 1 and converging to quantiles of χ 2 k 1 We find, by simulations over fine grid of values of κ 1, that new test controls size 1(AR n (β 0 ) > c 1 α (ˆκ 1, k m W )) It improves on the GKMC procedure in terms of power

24 Theorem: Suppose m W = 1. The new conditional subvector Anderson Rubin test has correct size under the assumptions above. Proof partly based on simulations; Verified for e.g. α {1%, 5%, 10%} and k m W {1,..., 20}. Summary m W = 1: the cond l test rejects when ˆκ 2 > c 1 α (ˆκ 1, k 1), where (ˆκ 1, ˆκ 2 ) are the eigenvalues of 2 2 matrix Ξ Ξ W ( k, I p, M M ) ; Under the null M M is of rank 1; test has size α

25 4 k = 2 10 k = 5 3 χ 2 k 1, 1 α 2 c 1 α ( ^κ 1,k 1 ) k = k = Critical value function c 1 α ( κ 1, k 1) for α = 0.05.

26 Table of conditional critical values cv=c 1 α (ˆκ 1, k m W ) α = 5%, k m W = 4 ˆκ 1 cv ˆκ 1 cv ˆκ 1 cv ˆκ 1 cv ˆκ 1 cv ˆκ 1 cv Inf 9.5 * For simplicity of implementation we suggest linear interpolation of tabulated cvs; we verify resulting test has correct size

27 0.050 k = 5, m W = 1, α = c GKMC κ 1 Null rejection frequency of subset AR test based on conditional (red) and χ 2 k 1 (blue) critical values, as function of κ 1.

28 Extension to m W > 1 We define a new subvector Anderson Rubin test that rejects when AR n (β 0 ) > c 1 α (κ max ( Ξ Ξ ), k m W ). Note: We condition on the LARGEST eigenvalue of the Wishart matrix. Theorem: The test above has i) correct size and ii) has uniformly larger power than the test in GKMC. Lemma: Under the null H 0 : β = β 0, there exists a random matrix O O(p), such that for Ξ := ΞO R k p, and its upper left submatrix Ξ 11 R k m W +1 2

29 Ξ 11 Ξ 11 is a non-central Wishart 2 2 matrix of order k m W + 1 (cond l on O), whose noncentrality matrix, M 1 M 1 say, is of rank 1; Proof of Theorem: (i) Note that AR n (β 0 ) = κ min ( Ξ Ξ ) = κ min ( Ξ Ξ) κ min ( Ξ 11 Ξ11 ) κmax ( Ξ 11 Ξ11 ) and thus κ max ( Ξ Ξ ) = κ max ( Ξ Ξ ) (1) P (AR n (β 0 ) > c 1 α (κ max ( Ξ Ξ ), k m W )) P (κ min ( Ξ 11 Ξ11 ) > c1 α (κ max ( Ξ 11 Ξ11 ), k mw )) = P (κ 2 ( Ξ 11 Ξ11 ) > c1 α (κ 1 ( Ξ 11 Ξ11 ), k mw )) α,

30 where first inequality follows from (1) and last inequality from correct size for m W = 1 (by conditionning on O) and the lemma Recall summary when m W = 1: new test rejects when ˆκ 2 > c 1 α (ˆκ 1, k 1) where (ˆκ 1, ˆκ 2 ) are the eigenvalues of Ξ Ξ W ( k, I 2, M M ) and M M is of rank 1 under the null (ii) new conditional test is uniformly more powerful than test in GKMC (because c 1 α (, k m W )) is increasing and converging to χ 2 k m W,1 α as argument goes to infinity), i.e. the test in GKMC is inadmissible

31 Power analysis of tests based on (ˆκ 1,..., ˆκ p ) For A = E [ Z (y Y β 0. W ) ] R k p, consider H 0 : ρ (A) m W versus H 1 : ρ (A) = p = m W + 1 H 0 : β = β 0 implies H 0 but the converse is not true: H 0 holds iff [ρ (Π W ) < m W or Π Y (β β 0 ) span(π W )] Under H 0, (ˆκ 1,..., ˆκ p ) are distributed as eigenvalues of Wishart W ( k, I p, M M ) with rank deficient noncentrality matrix - a distribution that appears also under H 0

32 Thus, every test ϕ(ˆκ 1,..., ˆκ p ) [0, 1] that has size α under H 0 must also have size α under H 0 - so cannot have power exceeding size under alternatives H 0 \H 0. In other words, size α tests ϕ(ˆκ 1,..., ˆκ p ) under H 0 can only have nontrivial power under alternatives ρ (A) = p. We use this insight to derive a power envelope for tests of the form ϕ (ˆκ 1,..., ˆκ p ).

33 Power bounds Consider only the case m W = 1. Equivalently, H 0 : κ 2 = 0, κ 1 κ 2 against H 1 : κ 2 > 0, κ 1 κ 2. Obtain point-optimal power bounds using approximately least favorable distribution Λ LF over nuisance parameter κ 1 based on algorithm in Elliott, Müller, and Watson (2015)

34 Power of ϕ c minus power bound 1.0 Power curves when κ 1 = κ power bound ϕ c ϕ G K M C κ 1 κ κ κ 2 Power of conditional subvector AR test ϕ c (ˆκ) = 1 {ˆκ2 >c 1 α (ˆκ 1,k 1)} relative to power bound (left) and power of ϕ c, ϕ GKMC (ˆκ) = 1 {ˆκ 2 >χ 2 k 1,1 α} = 1 {ˆκ2 >c 1 α (,k 1)} and bound at κ 1 = κ 2 (right) for k = 5. Computed using MC replications.

35 Little scope for power improvement over proposed test. scope...: But not zero Refinement: For the case k = 5, m W = 1, and α = 5%, let ϕ adj be the test that uses the critical values in Table above where the smallest 8 critical values are divided by 5

36 Asymptotic case: a) homoskedasticity Define parameter space F under the null hypothesis H 0 : β = β 0. Let U i := (ε i + V W,i γ, V W,i ) and F distribution of (U i, V Y i, Z i ) F is set of all (γ, Π W, Π Y, F ) s.t. γ R m W, Π W R k m W, Π Y R k m Y, E F ( T i 2+δ ) M, for T i {vec(z i U i ), Z i, U i }, E F (Z i (ε i, V W i, V Y i )) = 0, E F (vec(z i U i )(vec(z iu i )) ) = (E F (U i U i ) E F (Z i Z i )), κ min (A) δ for A {E F (Z i Z i ), E F (U i U i )} for some δ > 0, M < Note: no restriction is imposed on the variance matrix of vec(z i V Y i )

37 subvector AR stat equals smallest solution of MZ Y 1+mW ( κi Y n k ) 1/2 (Y P Z Y )( Y M Z Y n k ) 1/2 = 0 where Y := (y Y β 0. W ) R n (1+m W ) Note: Same as in finite sample case with Ω (β 0 ) replaced by Y M Z Y n k critical value is again c 1 α (ˆκ 1, k m W ) the 1 α quantile of (the approximation of) AR n given κ 1

38 Theorem: The new subvector AR test has correct asymptotic size for parameter space F. Again, part of the proof is based on simulations.

39 Asymptotic case: b) general Kronecker Product Structure For U i := (ε i + V W,i γ, V W,i ), p := 1 + m W, and m := m Y + m W let F KP = {(γ, Π W, Π Y, F ) : γ R m W, Π W R k m W, Π Y R k m Y, E F ( T i 2+δ 1) B, for T i {vec(z i U i ), vec(z iz i )}, E F (Z i V i ) = 0k (m+1), E F (vec(z i U i )(vec(z i U i )) ) = G 1 G 2, κ min (A) δ 2 for A {E F ( Zi Z i), G1, G 2 }} for pd G 1 R p p (whose upper left element is normalized to 1) and G 2 R k k and δ 1, δ 2 > 0, B < Covers homoskedasticity, but also cases of (cond) heteroskedasticity

40 Example. Take ( ε i, Ṽ W i ) R p i.i.d. zero mean with pd variance matrix, independent of Z i, and (ε i, V W i ) := f(z i )( ε i, Ṽ W i ) for some scalar valued function f of Z, e.g. f(z i ) = Z i /k 1/2. Then E F (vec(z i U i )(vec(z iu i )) ) = E F ( Ui U i Z iz i ) = E F ( (εi + V W,i γ, V W,i ) (ε i + V W,i γ, V W,i ) Z iz i = E F ( ( ε i + Ṽ W,i γ, Ṽ W,i ) ( ε i + Ṽ W,i γ, Ṽ W,i )) E F ( f(zi ) 2 Z i Z i has KP structure even though E F (U i U i Z i) = f(z i ) 2 E F ( ε i + Ṽ W,i γ, Ṽ W,i ) ( ε i + Ṽ W,i γ, Ṽ W,i ) depends on Z i. ) )

41 Modified AR subvector statistic. Estimate E F (U i U i Z iz i ) by R n := n 1 n i=1 f i f i Rkp kp, where f i := ((M Z (y Y β 0 )) i, (M Z W ) i ) Z i R kp. Let (Ĝ1, Ĝ2) = arg min G 1 G 2 R n F, where the minimum is taken over (G 1, G 2 ) for G 1 R p p, G 2 R k k being pd, symmetric matrices, normalized such that the upper left element of G 1 equals 1. Estimators are unique and given in closed form. The subvector AR statistic, AR KP,n (β 0 ) is defined it as the smallest root ˆκ pn of the roots ˆκ in, i = 1,..., p (ordered nonincreasingly) of the

42 characteristic polynomial ˆκI p n 1Ĝ 1/2 ( Y 0, W ) ZĜ 1 1 ( 2 Z Y 0, W ) Ĝ 1/2 1 = 0. Note: Relative to previous definition, Ĝ 1 replaces Y M Z Y n k and Ĝ2 replaces Z Z n The conditional subvector AR KP test rejects H 0 at nominal size α if AR KP,n (β 0 ) > c 1 α (ˆκ 1n, k m W ), where c 1 α (, ) is defined as above.

43 Theorem: The conditional subvector AR KP test implemented at nominal size α has asymptotic size, i.e. lim sup sup P (β0,γ,π n W,Π Y,F ) (AR AKP,n(β 0 ) > c 1 α (ˆκ 1n, k m W )) (γ,π W,Π Y,F ) F KP equal to α.

44 Asymptotic case: c) General forms of Hetero Perform a Wald type pretest based on Ĝ1 Ĝ2 R n to test the null of Kronecker Product structure If pretest rejects continue with a robust (to hetero and weak IV) subvector procedure, like the AR type tests proposed in Andrews (2017) Otherwise, continue with the test AR KP test Resulting test has correct asymptotic size no matter what the pretest nominal size is

45 Reasons: pretest is consistent against deviations from null for which n 1/2 min G 1 G 2 E F (U i U i Z iz i ) and the AR type tests in Andrews (2017) have correct asymptotic size when n 1/2 min G 1 G 2 E F (U i U i Z iz i ) = O(1) the conditional subvector AR KP test has correct asymptotic size and rejects whenever the AR type test in Andrews (2017) rejects.

46 Asymptotic Size: General theory Distinction between pointwise (asymptotic) null rejection probability and (asymptotic) size Discontinuity in limiting distribution of test statistic Staiger and Stock (1997): simplified version of linear IV model with one IV y 1 = y 2 θ + u, y 2 = Zπ + v Let λ n = (λ 1n, λ 2n, λ 3n ) be sequence of parameters s.t. λ 3n = (F n, π n ) λ 1n = (EZi 2 )1/2 π/σ v and λ 2n = corr(u i, v i )

47 satisfies h n,1 (λ n ) = n 1/2 λ 1n h 1 < and h n,2 (λ n ) = λ 2n h 2. We will denote such a sequence λ n by λ n,h. Work out limiting distribution of 2SLS under λ n,h : σ v σ u ( θ 2SLS θ) = σ v σ u y 2 P Zu y 2 P Zy 2 = (n 1 Z Z) 1/2 n 1/2 Z u/σ u (n 1 Z Z) 1/2 n 1/2 Z y 2 /σ v = d ( zu,h2 z v,h2 (n 1 Z Z) 1/2 n 1/2 Z u/σ u (n 1 Z Z) 1/2 n 1/2 π/σ v + (n 1 Z Z) 1/2 n 1/2 Z v/σ v z u,h2 h 1 + z v,h2, where ) N(0, Σ h2 ) and Σ h2 = ( 1 h2 h 2 1 )

48 Similarly for t test statistic T n (θ 0 ) : T n (θ 0 ) d J h for h = (h 1, h 2 ) under the parameter sequence λ n,h. So, to implement the test, we should take the 1 α-quantile c h (1 α) of J h as the critical value If we implement a test using a Wald statistics with chi-square critical values, the asymptotic size is 1, see Dufour (1997) Problem: we cannot consistently estimate h; we can only estimate consistently λ 1n

49 (h 1, h 2 ) takes on values in H = (R {± }) [ 1, 1] We say the limit distribution of T n (θ 0 ) depends discontinuously on nuisance parameter λ 1 and continuously on λ 2 Continuity: when x x 0 then f(x) f(x 0 ) Here (EZ 2 i )1/2 π/σ v 0, but limit of T n (θ 0 ) does not just depend on 0 Situation arises frequently in applied econometrics and leads to size distortion for various "classical" inference procedures: weak IVs/identification, use of pretests, moment inequalities, (nuisance) parameters on boundary, inference in (V)ARs with unit root(s)

50 General Theory: Asymptotic Size of Tests {ϕ n : n 1} sequence of tests for null hypothesis H 0 λ indexes the true null distribution of the observations Parameter space for λ is some space Λ RP n (λ) denotes rejection probability of ϕ n under λ The asymptotic size of ϕ n for the parameter space Λ is defined as: AsySz = lim sup sup RP n (λ) n λ Λ

51 Formula for Calculation of AsySz Recall relevance of limits of h n,1 (λ n ) = n 1/2 λ 1n = n 1/2 (EZ 2 i )1/2 π/σ v and h n,2 (λ n ) = λ 2n = corr(u i, v i ) for limit distributions of test statistics in weak IV example Generalizing, let {h n (λ) = (h n,1 (λ),..., h n,j (λ)) R J : n 1} be a sequence of functions on Λ, where h n,j (λ) R j = 1,..., J. For any subsequence {p n } of {n} and h (R {± }) J denote a sequence {λ pn Λ : n 1} such that h pn (λ pn ) h by Define λ pn,h H = {h (R {± }) J : there is subsequence {p n } and sequence λ pn,h}.

52 Theorem, Andrews, Cheng, and Guggenberger (2011) Assume that under any sequence λ pn,h for some RP (h) [0, 1]. Then: RP pn (λ pn,h) RP (h) AsySz = sup h H RP (h). Proof. i) Let h H. To show AsySz RP (h). By definition of H, there is λ pn,h. Then AsySz = lim sup n sup λ Λ RP n (λ) lim sup n RP p n (λ pn,h) = RP (h)

53 Proof. (continued) ii) To show AsySz sup h H RP (h). Let {λ n Λ : n 1} be a sequence such that lim sup n RP n(λ n ) = AsySz. Let {p n : n 1} be a subsequence of {n} such that lim n RP pn (λ pn ) exists and equals AsySz and h pn (λ pn ) h. Therefore this sequence is of type λ pn,h, and thus, by assumption, RP pn (λ pn ) RP (h). Because also RP pn (λ pn ) AsySz, it follows that AsySz = RP (h).

54 Specification of λ for subvector Anderson and Rubin test Given F let W F := (E F Z i Z i )1/2 and U F := Ω(β 0 ) 1/2. Consider a singular value decomposition of C F Λ F B F W F (Π W γ, Π W )U F i.e. B F denote a p p orthogonal matrix of eigenvectors of U F (Π W γ, Π W ) W F W F (Π W γ, Π W )U F

55 and C F denote a k k orthogonal matrix of eigenvectors of W F (Π W γ, Π W )U F U F (Π W γ, Π W ) W F Λ F denotes a k p diagonal matrix with singular values (τ 1F,..., τ pf ) on diagonal, ordered nonincreasingly Note τ pf = 0

56 Define the elements of λ F to be λ 1,F : = (τ 1F,..., τ pf ) R p, λ 2,F : = B F R p p, λ 3,F : = C F R k k, λ 4,F : = W F R k k, λ 5,F : = U F R p p, λ 6,F : = F, λ F : = (λ 1,F,..., λ 9,F ). A sequence λ n,h denotes a sequence λ Fn such that (n 1/2 λ 1,Fn,..., λ 5,Fn ) h = (h 1,..., h 5 ) Let q = q h {0,..., p 1} be such that h 1,j = for 1 j q h and h 1,j < for q h + 1 j p 1

57 Roughly speaking, need to compute asy null rej probs under seq s with (i) strong ident n,(ii) semi-strong ident n, (iii) std weak ident n (all parameters weakly ident d) & (iv) nonstd weak ident n strong identification: lim n τ mw,f n > 0 semi-strong ident n: lim n τ mw,f n = 0 & lim n n 1/2 τ mw,f n = weak ident n: lim n n 1/2 τ mw,f n < standard (of all parameters): lim n n 1/2 τ 1,Fn < as in Staiger & Stock (1997) nonstandard: lim n n 1/2 τ mw,f n < & lim n n 1/2 τ 1,Fn = includes some weakly/some strongly ident d parameters, as in Stock & Wright (2000); also includes joint weak ident n

58 Andrews and Guggenberger (2014): Limit distribution of eigenvalues of quadratic forms Consider a singular value decomposition C F Λ F B F of W F D F U F Define λ F, h, λ n,h... as above Let κ jn j = 1,..., p denote jth eigenval of nû n D nŵ nŵn D nûn,

59 where under λ n,h n 1/2 ( D n D Fn ) d D h R k p, Ŵ n W Fn p 0 k k, Û n U Fn p 0 p p, W Fn h 4, U Fn h 5 with h 4, h 5 nonsingular Theorem (AG, 2014): under {λ n,h : n 1}, (a) κ jn p for all j q (b) vector of smallest p q eigenvals of nû n D nŵ nŵn D nûn, i.e., ( κ (q+1)n,..., κ pn ), converges in dist n to p q vector of eigenvals of random matrix M(h, D h ) R (p q) (p q)

60 complicated proof; eigenvalues can diverge at any rate or converge to any number can become close to each other or close to 0 as n

61 We apply this result with W F = (E F Z i Z i )1/2, Ŵn = (n 1 Z i Z i )1/2, U F = Ω(β 0 ) 1/2, Ûn = D F Y M Z Y n k 1/2 = (Π W γ, Π W ), D n = (Z Z) 1 Z Y to obtain the joint limiting distribution of all eigenvalues,

62 Joint asymptotic dist n of eigenvalues Recall: test statistic and critical value are functions of p = 1 + m W roots of MZ Y 1+mW ( κi Y n k ) 1/2 (Y P Z Y )( Y M Z Y n k ) 1/2 = 0 To obtain joint limiting distribution of eigenvalues, we use general result in Andrews and Guggenberger (2014) about joint limiting distribution of eigenvalues of quadratic forms Results: the joint limit depends only on localization parameters h 1,1,..., h 1,mW

63 asymptotic cases replicate finite sample, normal, fixed IV, known variance matrix setup together with above proposition, correct asymptotic size then follows from correct finite sample size