Online Appendix. j=1. φ T (ω j ) vec (EI T (ω j ) f θ0 (ω j )). vec (EI T (ω) f θ0 (ω)) = O T β+1/2) = o(1), M 1. M T (s) exp ( isω)

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1 Online Appendix Proof of Lemma A.. he proof uses similar arguments as in Dunsmuir 979), but allowing for weak identification and selecting a subset of frequencies using W ω). It consists of two steps. Step proves asymptotic normality and Step 2 verifies that the limiting covariance matrix is an identity matrix. Step. First consider ξ. Rewrite it as ξ = + φ ω j ) vec I ω j ) EI ω j )) j= φ ω j ) vec EI ω j ) f θ0 ω j )). j= B.) B.2) he term B.2) is asymptotically negligible. Specifically, EI ω) can be expressed as EI ω) = s= + s / ) Γs) exp isω) with Γs) = /)) f θ 0 ω) exp isω) dω. Using the property of the Cesaro sum and that f θ0 ω) belongs to the Lipschitz class of degree β with respect to ω, we have Hannan, 970, p. 53) sup ω [,π] vec EI ω) f θ0 ω)) = O β ). he term B.2) is therefore bounded by 2 /2 sup φ ω) ω [,π] sup ω [,π] vec EI ω) f θ0 ω)) = O β+/2) = o), where the first equality is because φ ω) is finite by Assumption W and the last follows because β > /2. hus, to derive the limiting distribution of ξ, it suffices to consider B.) only. Let φ M ω) denote the M )-th order Cesaro sum of the Fourier series for φ ω): φ M ω) = M s= M+ with η s) = /) φ ω) exp isω) dω. hen, B.) = + s ) η M s) exp isω) φ M ω j ) vec I ω j ) EI ω j )) j= φ ω j ) φ M ω j )) vec I ω j ) EI ω j )). j= he second term will be asymptotically negligible if, because of conjugacy, B.3) j= φ ω j ) φ M ω j )) vec I ω j ) EI ω j )) = o p ). B.4) Establishing this result faces some difficulty because φ ω) has a finite number of discontinuities within [0, π] due to the presence of W ω), implying φ ω j ) φ M ω j ) does not converge uniformly B-

2 to zero over [0, π] the Gibbs phenomenon). However, results in Hannan 970, p ) imply that φ M ω) converges uniformly to φ ω) over all closed intervals excluding the jumps. At the jumps, the approximation errors remain bounded. Assume the jumps occur at ω k k =,..., K). hen, for any ε > 0 there exist finite constants M > 0 and C > 0 independent of, such that C if ω I K k= φ M ω) φ ω) [ ωk ε, ω k + ε] ε if ω [0, π] but ω / I. Apply the above partition, B.4) can be decomposed into + For the first term: V ar) C2 j= + C2 C2 j= j= ω j I ) φ ω j ) φ M ω j )) vec I ω j ) EI ω j )) ) ω j / I ) φ ω j ) φ M ω j )) vec I ω j ) EI ω j )) 2). ω j I ) V ar {vec I ω j ) EI ω j ))} j= h=,h j j= ω j I ) E {vec I ω j ) EI ω j )) vec I ω h ) EI ω h )) } ω j I ) D + C2 2 j= h= ω j I ) D, where D is some finite constant and the second inequality follows from heorem.7. in Brockwell and Davis 99), i.e., for any ω j and ω h in [0, π], E {vec I ω j ) EI ω j )) vec I ω h ) EI ω h )) O) if h = j, } = O ) otherwise. Because the length of I can be made arbitrarily small by choosing a small ε and a large M, we have V ar) = o). Similar arguments can be applied to 2): V ar2) ε 2 j= ω j / I ) D + ε 2 2 j= h= ω j / I ) D 2Dε 2, which can again be made small by choosing a small ε and a large M. hus, V ar 2) = o). Combining the above results, we have proved B.4). B-2

3 It remains to analyze the first term in B.3). Apply the definition of φ M ω j ): = 4π + 4π φ M ω j ) vec I ω j ) EI ω j )) j= M s= M M s= M s ) { )} η M s) vec ˆΓs) EˆΓs) 3) s ) { )} η M s) vec ˆΓs ) EˆΓs ), 4) where the last equality uses s= exp isω j) = 0 unless s = k k = 0, ±,...) and ˆΓs) s t= = Y t+s μθ 0 )) Y t μθ 0 )) if 0 s. ˆΓ s) if + s 0. erm 4) converges in probability to zero. his is because M is finite, η s) is uniformly bounded and vecˆγs ) EˆΓs )) p 0 for each s < M by the definition of ˆΓs ) note that the summation in the definition of ˆΓs ) involves at most M terms). In 3), vecˆγs) EˆΓs)) satisfies a central limit theorem for each s M, see Hannan 976). hus, 3) converges to a vector of normal random variables because M is finite. herefore, ξ has a multivariate normal limiting distribution. For ξ 2, ψ is finite because of Assumption W. Its asymptotic normality then follows from the central limit theorem. Step 2. For ξ it suffices to examine the covariance matrix of 3). Apply the definition of η s) and use the relationship between the vec and the trace operator. as ξ M l) = 8π 2 where M s= M Its l-th element can be written s ) { W ω)tr Bl, ω) } ˆΓs) EˆΓs)) exp isω) dω, M Bl, ω) = f θ 0 ω) q k= ) f θ0 ω) [Q c θ 0 )Λ c θ 0 ) /2] f θ θ k kl 0 ω) B.5) with [.] kl denoting the k, l)-th element of the matrix inside the bracket. Because Bl, ω) is an n Y -by-n Y matrix, B.5) can be further rewritten as ξ M l) = M 8π 2 s ) n Y W ω) B M ba l, ω) ˆΓab s) EˆΓ ab s)) exp isω) dω, s= M a,b= B-3

4 where B ba l, ω) is the b,a)-th element of the corresponding matrix. herefore, = Covξ M l), ξ M k)) n Y M s ) h ) M M 64π 4 W r)b ba l, r)w λ)b dc k, λ) a,b,c,d= s,h= M+ { ) E ˆΓab s) EˆΓ ab s) ˆΓcd h) EˆΓ cd h))} exp isr) dr exp ihλ) dλ B.6) he only random elements in B.6) are the sample autocovariances, which satisfy see eq. 3) in p.397 in Hannan, 976) { E + ) ˆΓab s) EˆΓ ab s) ˆΓcd h) EˆΓ cd h))} f ac ω)f bd ω) exp ih s)ω) dω f ad ω)f bc ω) exp is + h)ω) dω, where f ac ω) stands for the a,c)-th element of f θ0 ω). Applying 5) to B.6) leads to π q { π M f ac ω)f bd ω) W r)b ba l, r) s ) } ) exp isr ω)) dr 8π a,b,c,d= M s= M+ { π M W λ)b dc k, λ) h ) } ) exp ihω λ)) dλ dω. M h= M+ he two terms inside the two curly brackets are Fejér s kernels. herefore, { π M W r)b ba l, r) s ) } exp isr ω)) dr W ω)b ba l, ω), M { W λ)b dc k, λ) s= M+ M h= M+ h ) exp ihω λ)) M } 5) 6) dλ W ω)b dc k, ω) uniformly over all closed intervals excluding the jumps. At the jumps, the approximation error is finite, therefore it does not interfere with the limiting results. he effect of 6) can be analyzed similarly. Combining the two results, we have 8π = 4π = 2 Covξ M l)ξ M k)) n Y a,b,c,d= W ω) { f ac ω)f bd ω)b ba l, ω)b dc k, ω) + f ad ω)f bc ω)b ba l, ω)b dc k, ω) } dω W ω)tr {f θ0 ω)bk, ω)f θ0 ω)bl, ω)} dω W ω j ) vec Bl, ω j )) f θ0 ω j ) f θ0 ω j ) ) vec Bk, ω j )) + o), j= B-4

5 where the first equality uses B dc k, ω) = B cd k, ω), f bd ω) = f db ω), f bc ω) = f cb ω), B dc k, ω) = B dc k, ω) and the last equality follows because the summand belongs to Lipβ) with β > /2. In matrix notation, the above result can be stated as V ar ξ ) = Λ c θ 0 ) /2 Q c θ 0 ) ) vec fθ0 ω j ) W ω j ) 2 θ f θ 0 ω j ) f θ 0 ω j )) vec f θ 0 ω j ) θ Qc θ 0 )Λ c θ 0 ) /2 + o). j=0 Now consider ξ 2. It is asymptotically independent of ξ, satisfying V ar ξ 2 ) Λc θ 0 ) /2 Q c θ 0 ) {W 0) μθ 0) θ f θ 0 0) μθ 0) θ } Q c θ 0 )Λ c θ 0 ) /2. herefore, V ar ξ + ξ 2 ) = Λ c θ 0) /2 Q c θ 0) M θ 0 ) Q c θ 0)Λ c θ 0) /2 + o) I q +q 2, where the last equality uses the definition of Q c θ 0) and Λ c θ 0). Additional Proof. his proof shows that the confidence band covers the impulse response function with probability at least α) asymptotically. Let C θ α) denote the α) confidence set for θ obtained by inverting S θ) and C IR the confidence band for the impulse response function obtained from Steps to 3. By construction, if θ 0 C θ α), then IR θ 0 ) C IR. hus, if IR θ 0 ) / C IR, then θ 0 / C θ α). Equivalently, Pr IR θ 0 ) / C IR ) Pr θ 0 / C θ α)). As, Pr θ 0 / C θ α)) α. herefore, lim Pr IR θ 0 ) / C IR ) α. Eigenvalue conditions corresponding to other characterizations of weak identification We illustrate that the characterizing conditions for weak identification used in the IV and GMM literature can be stated using the curvatures of the criterion functions used for inference as in Assumption W. Linear IV Staiger and Stock, 997): Consider the model y = Y β + u, Y = ZΠ + v, where y and Y are vectors, Z is a K matrix of instruments and u and v are vectors of disturbances with Euu = σ 2 ui. he objective function is Q β) = y Y β) P Z y Y β). Its first order derivative, normalized by /2, equals D β 0 ) = 2 /2 u ZΠ 2 /2 u Z ) Z Z ) Z v ). If β 0 is strongly identified, i.e., Π is nonzero and independent of, then the first term in D β 0 ) is of exact order O p ) and the second is O p /2 ). herefore, lim E D β 0 ) D β 0 ) ) = lim 4 E Π Z ZΠ ) σ 2 u, B-5

6 consistent with the order of Λ θ 0 ) in Assumption W. If β 0 is weakly identified, i.e., Π = /2 C, then D β 0 ) is of exact order O p /2 ). herefore the eigenvalue of ED β 0 ) D β 0 ) ) is of order O ), consistent with the order of Λ 2 θ 0 ) in Assumption W. Weak identification in a CU-GMM setting Kleibergen, 2005): Consider inference based on the moment restriction Eφ t θ 0 ) = 0 with θ 0 R m. Without loss of generality, assume φ t θ 0 ) is serially uncorrelated. Let f θ) = /2 t= φ t θ). hen the CU-GMM criterion function is given by Q θ) = f θ) ˆVff θ) f θ), where ˆV ff θ) p V ff θ)=lim Varf θ)). Define f θ 0 ) θ = q θ 0 ) = q, θ 0 ),..., q m, θ 0 )). Kleibergen 2005) characterized the strength of identification using the order of Eq θ 0 ). Under strong identification, /2 Eq θ 0 ) has a fixed full rank value while under weak identification /2 Eq θ 0 ) = /2 C. We have ) D θ 0 ) = 2 /2 f θ 0 ) ˆVff θ 0 ) ˆR θ 0 ) Eq θ 0 ) a) +2 /2 f θ 0 ) ˆVff θ 0 ) Eq θ 0 ), b) where the j-th column of ˆR θ 0 ) equals q j, θ 0 ) ˆV θf,j θ 0 ) ˆV ff θ 0 ) f θ 0 ), i.e., the residual from projecting q j, θ 0 ) onto f θ 0 ); ˆVθf,j θ 0 ) is the sample covariance between f θ 0 ) and q j, θ 0 ). hus E D θ 0 ) D θ 0 ) ) = Ea a) + Eb b) + Ea b) + Eb a). he first term Ea a) is of order O ) irrespective of the strength of identification. he second term Eb b) is of exact order O) under strong and O ) under weak identification, respectively. he order of Eb b) + Ea b) is always lower than that of Eb b). herefore, the eigenvalues of ED θ 0 ) D θ 0 ) ) are O) under strong identification and O ) under weak identification, consistent with Assumption W in the paper. Weak identification under a GMM setting Stock and Wright, 2000): Consider the same setup as in the CU-GMM case, but with inference based on the following GMM criterion function Q θ) = f θ) W f θ), where W is some consistent estimate of the optimal weighting matrix that is, without loss of generality, assumed to be non-random. hen, D θ 0 ) = 2 /2 f θ 0 ) W q θ 0 ) ˆR ) θ 0 ) c) ) +2 /2 f θ 0 ) W ˆR θ 0 ) Eq θ 0 ) d) +2 /2 f θ 0 ) W Eq θ 0 ). e) Simple algebra shows that the leading term in ED θ 0 ) D θ 0 ) ) is Ec c) + Ec e) + Ee c) + Ed d) + Ee e). B-6

7 he first four terms are always of order O ) irrespective of the strength of identification. he last term converges to a positive definite matrix under strong identification. eigenvalues of ED θ 0 ) D θ 0 ) ) are of order O) under strong identification. herefore, all the Under weak identification, Stock and Wright 2000) assumed that θ admits a partition: θ = α, β ), such that α is weakly identified while β is strongly identified. Specifically, let m α, β) = Ef α, β) and write m α, β) = m α 0, β 0 ) + /2 m α, β) + m 2 β) with m α, β) = /2 m α, β) m α 0, β)) and m 2 β) = m α 0, β) m α 0, β 0 ). Stock and Wright 2000) assumed c.f. Assumption C in their paper) Let C =Diag /2 I dimα), I dimβ) ), then m α, β) m α, β) and m 2 β) m 2 β). C Ee e)c = 4 C Eq θ 0 ) W E f θ 0 ) f θ 0 ) ) W Eq θ 0 ) C = 4 C Eq θ 0 ) W V ff θ 0 ) W ) Eq θ 0 ) C [ ] [ 4 V ff θ 0) m α 0,β 0 ) α m 2 β 0 ) β ] m α 0,β 0 ) m 2 β 0 ). α β he limit is a positive definite matrix. herefore, in large samples, Ee e) has dimβ) eigenvalues that are O) and dimα) eigenvalues of order O ). So is ED θ 0 ) D θ 0 ) ). Weak identification in a two-equation model he model consists of two equations: r t = γy t + βπ t + u t, π t = ρπ t + v t, with var u t ) = σ 2 u, var v t ) = σ 2 v, covu t, v t ) = σ uv and Eu t u s = Ev t v s = Eu t v s = 0 for all t s. he first equation is a monetary policy rule aylor, 993) with y t and π t being deviations of GDP and inflation from their targets and the second equation describes the inflation dynamics. he parameter of interest is β. o simplify the derivation, we assume ρ, γ and σ 2 v, are known. he unknown parameter vector is therefore θ = β, σ 2 u, σ uv ). Rewrite the model as r t = βπ t + u t, B.7) π t = ρπ t + v t with r t = r t γy t. It can then be viewed as a dynamic version of the limited information simultaneous equation model, in which π t is the endogenous explanatory and π t is the instrument. he parameter β is weakly identified if ρ is small. Intuitively, because there is little persistence in π t, it B-7

8 is difficult to differentiate between systematic policy responses βπ t ) and random disturbances u t ). Geometrically, it is possible to move θ along a certain direction such that the likelihood surface changes little. In the extreme case with ρ = 0, β becomes unidentified. hen, there exists a path along which the likelihood is completely flat It turns out changing θ in the direction given by, 2σ uv, σ 2 v) yields such a non-identification curve). We let ρ = /2 c with c > 0; other parameter values are independent of. Let W ω) = for all ω [, π]. Lemma B. Let θ 0 denote the true value of θ = β, σ 2 u, σ uv ), then M θ 0 ) satisfies:. It has two positive eigenvalues λ and λ 2 satisfying λ and λ he smallest eigenvalue λ 3 satisfies λ 3 6π 2 σ 4 vc 2 + σ 4 v + 4σ 2 uv) σ 2 vσ 2 u σ 2 uv). 3. he elements of vec f θ0 ω) θ Q θ 0 )Λ θ 0 ) /2 are bounded and Lipschitz continuous in ω. Note that Lemma B.. corresponds to Assumption Wi), while B..2 corresponds to Wii). B..3 is a stronger result than Wiv). References Brockwell, P., and R. Davis 99): ime Series: heory and Methods. New York: Springer-Verlag, 2nd Edition. Dunsmuir, W. 979): A Central Limit heorem for Parameter Estimation in Stationary Vector ime Series and its Application to Models for a Signal Observed with Noise, he Annals of Statistics, 7, Hannan, E. J. 970): Multiple ime Series. New York: John Wiley. 976): he Asymptotic Distribution of Serial Covariances, Annals of Statistics, 4, Kleibergen, F. 2005): esting Parameters in GMM without Assuming that hey are Identified, Econometrica, 73, Staiger, D., and J.H. Stock 997): Instrumental Variables Regression with Weak Instruments, Econometrica, 65, Stock, J.H., and J.H. Wright 2000): GMM with Weak Identification, Econometrica, 68, B-8