Department of Economics, Boston University. Department of Economics, Boston University

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1 Supplementary Material Supplement to Identification and frequency domain quasi-maximum likelihood estimation of linearized dynamic stochastic general equilibrium models : Appendix (Quantitative Economics, Vol. 3, No. 1, March 212, Zhongjun Qu Department of Economics, Boston University Denis kachenko Department of Economics, Boston University he spectral density matrix f θ (ω is a Hermitian matrix satisfying f θ (ω = f θ (ω. It is in general not symmetric. he following correspondence is useful for understanding and proving the identification results: [ ] f θ (ω f θ (ω R with f θ (ω R Re(fθ (ω Im(f θ (ω = (A.1 Im(f θ (ω Re(f θ (ω where Re( and Im( denote the real and the imaginary parts of a complex matrix, that is, if C = A + Bi, thenre(c = A and Im(C = B. Becausef θ (ω is Hermitian, f θ (ω R is real and symmetric (see Lemma 3.7.1(v in Brillinger (21. o simplify notation, let R(ω; θ = vec(f θ (ω R he following lemma is crucial for proving the subsequent results. Lemma A.1. We have the identity ( vec(fθ (ω ( vec(fθ (ω = 1 ( R(ω; θ 2 ( R(ω; θ (A.2 Proof. he(j kth element of the term on the left hand side is equal to ( vec(fθ (ω ( vec(fθ (ω θ j { fθ (ω = tr θ j f θ (ω θ k Zhongjun Qu: qu@bu.edu Denis kachenko: tkatched@bu.edu θ k } { ( fθ (ω = tr Re θ j } f θ (ω θ k Copyright 212 Zhongjun Qu and Denis kachenko. Licensed under the Creative Commons Attribution- NonCommercial License 3.. Available at DOI: /QE126

2 2 Qu and kachenko Supplementary Material = 1 {( R } 2 tr fθ (ω f θ (ω = 1 { θ j θ k 2 tr (fθ (ω R (f θ (ω R } θ j θ k = 1 ( vec(fθ (ω R { vec(fθ (ω R } 2 θ j θ k where the first equality is because of the identity vec(a vec(b = tr (AB for generic matrices A and B, the second is because f θ (ω is Hermitian, thus this term is real valued, the third equalityis because of the definition (A.1, the fourth is because, for generic complex matrices, if Z = XY, thenz R = X R Y R (see Lemma 3.7.1(ii in Brillinger (21, and the fifth is because f θ (ω R is real and symmetric. he last term in the display is simply the (j kth element of the right hand side term in (A.2. his completes the proof. Proof of heorem 1. Lemma A.1 implies that G(θ defined by (9 is real, symmetric, positive semidefinite, and equal to 1 2 ( ( R(ω; θ R(ω; θ dω his allows us to adopt the arguments in heorem 1 in Rothenberg (1971 to prove the result. Suppose is not locally identified. hen there exists an infinite sequence of vectors {θ k } k=1 approaching such that, for each k, R(ω; = R(ω; θ k for all ω [ π] For an arbitrary ω [ π] by the mean value theorem and the differentiability of f θ (ω in θ = R j (ω; θ k R j (ω; = R j(ω; θ(j ω (θ k where the subscript j denotes the jth element of the vector and θ(j ω lies between θ k and and in general depends on both ω and j.let hen d k = θ k θ k R j (ω; θ(j ω d k = for every k he sequence {d k } is an infinite sequence on the unit sphere and therefore there exists a limit point d (note that d does not depend on j or ω. As θ k, d k approaches d and we have R j (ω; θ(j ω lim k d k = R j(ω; d =

3 Supplementary Material QML estimation of linearized DSGE models 3 where the convergence result holds because f θ (ω is continuously differentiable in θ (Assumption 3. Because this holds for an arbitrary j, it holds for the full vector R(ω; herefore, which implies R(ω; d = d ( R(ω; θ ( R(ω; θ d = Because the above result holds for an arbitrary ω [ π] it also holds when integrating over [ π] hus { ( ( } R(ω; d θ R(ω; θ dω d = Applying Lemma A.1,becaused, G( is singular. o show the converse, suppose that G(θ has constant rank ρ<qin a neighborhood of denoted by δ(. hen consider the characteristic vector c(θ associated with one of the zero roots of G(θ.Because ( ( R(ω; θ R(ω; θ dω c(θ = we have ( R(ω; θ ( R(ω; θ c(θ c(θ dω = Since the integrand is continuous in ω and always nonnegative, we must have ( ( R(ω; θ R(ω; θ c(θ c(θ = for all ω [ π] and all θ δ(. Furthermore, this implies R(ω; θ c(θ = (A.3 for all ω [ π] and all θ δ(.becauseg(θ is continuous and has constant rank in δ(, the vector c(θ is continuous in δ(. Consider the curve χ defined by the function θ(v which solves, for v v, the differential equation θ(v = c(θ v θ( = hen R(ω; θ(v v = R(ω; θ(v θ(v R(ω; θ(v θ(v = v θ(v c(θ =

4 4 Qu and kachenko Supplementary Material for all ω [ π] and v v, where the last equality uses (A.3. hus, R(ω; θ is constant on the curve χ. his implies that f θ (ω is constant on the same curve and that is unidentifiable. his completes the proof. Proof of Corollary 1. he statement in the subsequent proof applies to all ω [ π]. Using the same argument as in the proof of Lemma A.1, I( can be rewritten as I( = 1 2π ( R(ω; θ ([ fθ (ωr] [ fθ (ωr] R(ω; dω (A.4 Because spectral density matrices are Hermitian and positive semidefinite, f θ (ω R is real, symmetric, and positive semidefinite (cf. Lemma 3.7.1(vii in Brillinger (21. Furthermore, because here f θ (ω has full rank, f θ (ω R is in fact positive definite. hus, ([f θ (ω R ] [f θ (ω R ] is positive definite (cf. heorem 1 in Magnus and Neudecker (1999,p.28. We now prove G( and I( have the same null space. Since they are both q q matrices, the result then implies they have the same rank. First, suppose c R q and I( c =.henc I( c = or, explicitly, ( R(ω; θ ([ c fθ (ωr] [ fθ (ωr] ( R(ω; c dω = Because the integrand is continuous in ω and always nonnegative, we must have ( R(ω; θ ([ c fθ (ωr] [ fθ (ωr] ( R(ω; c = Because ([f θ (ω R ] [f θ (ω R ] is positive definite, this implies R(ω; c = herefore, ( ( R(ω; θ R(ω; θ c = and, consequently, G( c =. Next suppose c R q and G( c =. Applying the same argumentthatleadsto(a.3, we have ( R(ω; θ c = hen, trivially, ( R(ω; θ ([ fθ (ωr] [ fθ (ωr] ( R(ω; c = Upon integration, we have I( c =.

5 Supplementary Material QML estimation of linearized DSGE models 5 Proof of heorem 2. Using Lemma A.1 again, Ḡ( θ can be equivalently represented as Ḡ( θ = 1 ( ( ( R(ω; θ R(ω; θ μ( θ μ( θ 2 dω + with both terms on the right hand side being real, symmetric, and positive semidefinite. Let [ ] R(ω; θ R(ω; θ = 1 π μ( θ hen Ḡ( θ = 1 2 ( ( R(ω; θ R(ω; θ dω Using this representation, the proof proceeds in the same way as in heorem 1, with θ replaced by θ and R(ω; θ replaced by R(ω; θ. he detail is omitted. Proof of Corollary 3. We only prove the first result, as the second can be proven analogously using the formulation in the proof of heorem 2. Suppose the subvector θ s is not locally identified. Write θ = (θs θ r.hereexistsan infinite sequence of vectors {θ k } k=1 approaching such that R(ω; = R(ω; θ k for all ω [ π] and each k By the definition of partial identification, {θk s } can be chosen such that θs k θs / θ k >ε,withε being some arbitrarily small positive number. he values of θk r can either change or stay fixed in this sequence; no restriction is imposed on them besides those in the preceding display. As in the proof of heorem 1, in the limit, we have R(ω; d = with d s (where d s comprises the elements in d that correspond to θ s herefore, on one hand, G( d = ; on the other hand, because d s and by definition θ s / θ =[I dim(θ s dim(θ r ],wehave which implies θ s d = ds G a ( d hus, we have identified a vector that falls into the orthogonal column space of G( but not of G a ( Because the former orthogonal space always includes the latter as a subspace, this result implies G a ( has a higher column rank than G(.

6 6 Qu and kachenko Supplementary Material o show the converse, suppose that G(θ and G a (θ have constant ranks in a neighborhood of denoted by δ(. Because the rank of G(θ is lower than that of G a (θ there exists a vector c(θ such that G(θc(θ = but G a (θc(θ which implies for all ω [ π] and all θ δ( (cf. arguments leading to (A.3, but R(ω; θ c(θ = [ R(ω; θ/ θ θ s / ] [ ] c(θ = c s (θ where c s (θ denotes the elements in c(θ that correspond to θ s.becauseg(θ is continuous and has constant rank in δ(, the vector c(θ is continuous in δ(.asin heorem 1,consider the curve χ defined by the function θ(v which solves, for v v, the differential equation θ(v v = c(θ θ( = On one hand, because c s (θ and c s (θ is continuous in θ points on this curve correspond to different θ s On the other hand, R(ω; θ(v v = R(ω; θ(v θ(v R(ω; θ(v θ(v = v θ(v c(θ = for all ω [ π] and v v implying f θ (ω is constant on the same curve. herefore, θ s is not locally identifiable. Proof of Corollary 5. he proof is essentially the same as in Rothenberg (1971, heorem 2 and is included for the sake of completeness. Suppose Ψ(θ has rank s for all θ in a neighborhood of. hen, by the implicit function theorem, there exists a partition of θ into θ 1 R s and θ 2 R q s such that θ 1 = q(θ 2 for all solutions of ψ(θ = in a neighborhood of with θ 2 being an interior point of that neighborhood. Consequently, the spectral density can be rewritten as f θ (ω = f q(θ 2 θ 2(ω which involves only q s parameters. Let Q(θ 2 = q(θ2 θ 2 and G(θ = [ Q(θ 2 I ] G(θ [ Q(θ 2 hen, by heorem 1, is identified if and only if G( has full rank. I ]

7 Supplementary Material QML estimation of linearized DSGE models 7 Suppose there exists a vector d R q s such that G( d = (A.5 hen the structure of G(θ (cf. Lemma A.1 implies that (A.5 holds if and only if [ Q(θ 2 ] G( d = I Let [ Q(θ 2 ] c = d I hen we have (a c if and only d and (b [ ] G(θ c = Ψ( if and only if (A.5 holds, where Ψ( c = always holds because satisfies the constraint ψ(θ = hus, the preceding matrix has full rank if and only if is identified under the constraints. his completes the proof. Proof of Corollary 6. Without loss of generality, assume n Y = 1. Otherwise,the proof can be carried out by analyzing R(ω; θ. he map θ f θ is infinite dimensional. he proof therefore involves two steps. he first is to reduce it to a finite dimensional problem. he second is to apply a constant rank theorem (a generalization of the implicit function theorem. Consider a positive integer N and a partition of the interval [ π] by ω j = (2πj/ 2 N π with j = 1 2 N hen the map θ (f θ (ω f θ (ω 2 N (A.6 is finite dimensional. o simplify notation, let f θ N = (f θ (ω f θ (ω 2 N.Conventionally, the rank of the above map is defined as the rank of the Jacobian matrix f θ N /, which is of dimension (2 N + 1 q with rank no greater than q 1 at,becauseifthe rank equals q then becomes locally identified, contradicting the assumption in the corollary. Note that, for a given N, its rank can be strictly less than q 1. We now show that there exists a finite N such that f θ N / has rank q 1 at.suppose such an N does not exist. hen the rank of f θ N / is at most q 2 for arbitrarily large N. his implies that the rank of G N ( = 2π 2 N N ( fθ (ω j ( fθ (ω j is at most q 2 for arbitrarily large N because vectors orthogonal to f θ N / are also orthogonal to G N (θ by construction. Let λ N j (j = 1 q be the eigenvalues of G N ( sorted in an increasing order. hen, for any finite N, λ N 1 = λ N 2 =

8 8 Qu and kachenko Supplementary Material On the other hand, because G N ( G( so do its eigenvalues. hus, for any ε> there exists a finite N such that λ 2 λ N 2 <ε,whereλ 2 is the second smallest eigenvalue of G(. Choosing ε = λ 2 /2 leads to λ N 2 >λ 2 /2 Since rank(g( = q 1 by assumption, λ 2 is strictly positive. hus, we reach a contradiction. Because the convergence of G N (θ G(θ is uniform in an open neighborhood of say δ(, the above analysis also implies there exists an N such that f θ N / has constant rank q 1 in that neighborhood. Use such an N and consider again the map θ f θ N which is finite dimensional, is continuously differentiable, and has constant rank q 1 in δ(. Define the level set { } θ δ(θ : f θ N = f θ N hen the rank theorem (Krantz and Parks (22, heorem and the discussion on p. 56 implies that the level set constitutes a smooth, parameterized one dimensional manifold. hus, there exists a unique level curve passing through satisfying f θ N = f θ N herefore, we have established the result for a particular finite N. Further increasing N leads to finer partitions of [ π]. his cannot decrease the rank of the map (A.6 and therefore cannot increase the number of level curves passing through.hus,in the limit, there is at most one level curve passing through. he existence of such a curve for the infinite dimensional case has already been shown in the main text, given by (1. his completes the proof. Proof of Lemma 1. Applying Lemma A.3.3 (1 in Hosoya and aniguchi (1982, for a given θ Θ, wehave plim tr{w(ω j fθ (ω ji (ω j }= 1 2π o prove stochastic equicontinuity, consider for any θ 1 θ 2 Θ, tr { W(ω j ( fθ 1 (ω j fθ 2 (ω j I (ω j } Apply a first order aylor expansion tr { W(ωfθ (ωf (ω } dw tr { W(ω j ( fθ 1 (ω j fθ 2 (ω j I (ω j } = 1 tr{w(ω j f θ (ω ji (ω j } (θ 1 θ 2 (A.7

9 Supplementary Material QML estimation of linearized DSGE models 9 = 1 W(ω j vec(i (ω j {f θ vec(f θ (ω j (θ 1 θ 2 (ω j f θ (ω j} where θ lies between θ 1 and θ 2.henormof(A.7 isboundedby he quantity 1 vec(i (ω j (f θ (ω j f θ {f θ (ω j f θ (ω j vec(f θ (ω j (ω j} vec(f θ (ω j θ 1 θ 2 is uniformly bounded by Assumption 5(ii. he term vec(i (ω j only depends on and is O p (1 because the diagonal elements of I (ω j are positive and satisfy a law of large numbers (Hosoya and aniguchi (1982, Lemma A.3.3 (1, and the norm of the off-diagonal elements can be bounded by the diagonal elements using the Cauchy Schwarz inequality. herefore, the term (A.7 can be made uniformly small by choosing a small θ 1 θ 2. Meanwhile, W(ω j log det f θ (ω j 1 W(ωlog det f θ (ω dw 2π uniformly in θ Θ. hus, the first result holds. For the second result, we first show that maximizes L (θ. Apply the same argument as in Hosoya and aniguchi (1982, p For every ω [ π], W(ω [ log det f θ (ω + tr{fθ (ωf (ω} ] = W(ωlog det f θ (ω + W(ω [ tr{fθ (ωf (ω} log det{fθ (ωf (ω} ] [ ny ] = W(ωlog det f θ (ω + W(ω λ j (ω log λ j (ω 1 + W(ωn Y where λ j (ω is the jth eigenvalue of fθ (ωf (ω Because λ j (ω log λ j (ω 1 and the equality holds if and only if λ j (ω = 1, j = 1 n Y, this implies L (θ 1 W(ω ( log det f θ (ω + n Y dω 2π which holds with equality if and only if λ j (ω = 1 for all ω [ π] (j = 1 n Y. However, λ j (ω = 1 (j = 1 n Y impliesf θ (ω = f θ (ω because the latter are positive definite Hermitian matrices. Hence, is a global maximizer.

10 1 Qu and kachenko Supplementary Material he above result implies that any other parameter vector, say θ 1, is a maximizer if and only if f θ1 (ω = f θ (ω for all ω [ π]. Now suppose the parameters are locally identified. hen there are no parameter values close to satisfying this equality. hus, is the locally unique maximizer. o see the converse, suppose is the locally unique maximizer. hen there cannot be any parameter close to satisfying f θ (ω = f θ (ω for all ω hus, by definition, we have local identification. he argument to establish the result for the global identification proceeds in the same way. he third result follows directly from the uniform weak law of large numbers. Proof of heorem 3. We only prove the second result, which includes the first as a special case. he first order condition (FOC gives 2π /2 W(ω j vec(f θ (ω j + 2 /2 μ( θ f ((Y t μ( θ = θ { f θ (ω j f θ (ω j } vec ( I (ω j f θ (ω j Note that the first summation starts at j = and I ( = (. he above FOC implies I θ 2π /2 W(ω j vec(f (ω j vec ( I (ω j f θ (ω j ( f (ω j f (ω j + 2 /2 μ( fθ ((Y t μ( θ = o p (1 which holds because θ p, f θ (ω j and μ( are continuously differentiable, and fθ (ω j have bounded eigenvalues. Apply a first order aylor expansion around.hen the left hand side of the preceding display is equal to 2π /2 W(ω j vec(f (ω j ( f (ω j f (ω j vec ( I (ω j f θ (ω j (I + 2 /2 μ( fθ ((Y t μ( 2π W(ω j vec(f (ω j ( f θ (ω j fθ (ω j vec(f (ω j 1/2 ( θ (III (II (A.8

11 Supplementary Material QML estimation of linearized DSGE models 11 2 μ( fθ ( μ( 1/2 ( θ (IV + o p (1 First consider term (III. he quantity in front of 1/2 ( θ converges to W(ω vec(f (ω ( f θ (ω fθ (ω vec(f (ω dω whose (h kth element is given by fθ tr {W(ωf θ (ω (ω f θ (ω f } (ω dω h k herefore, the above expansion implies (cf. heorem 3 for the definition of M 1/2 ( θ = M (I + M (II + o p (1 erm (I satisfies a central limit theorem (CL, whose covariance matrix has the (h kth element given by (see heorem 3.1 and Proposition 3.1 in Hosoya and aniguchi (1982; in particular, their formula for U jl f 4π W(ωtr {f θ (ω + n ɛ a b c d=1 (ω h f θ (ω f (ω k [ 1 π κ abcd W(ωH f (ω 2π [ 1 π W(ωH f (ω 2π (ω k (ω h ] H(ωdω cd } dω erm (II also satisfies a CL, with covariance matrix given by ] H(ωdω ab 8π μ( fθ ( μ( o complete the proof, we only need to verify the covariance matrix between (I and (II. Let A = Cov((I (II and consider its (h kth element { ( fθ A hk = 4π Cov tr W(ω j (ω j ( I (ω j f θ (ω h ( } 1 μ( fθ ( (Y t μ( k

12 12 Qu and kachenko Supplementary Material Define φ h (ω j = f (ω j h and ψ k ( = μ( k f ( hen { ( A hk = 4π Cov tr W(ω j φ h (ω j ( I (ω j f θ (ω ( } 1 ψ k ( (Y t μ( { = 4π Cov W(ω j n Y a b=1 1 n Y } ψ k c ( (Y tc μ c ( = 4π c=1 n Y a b c=1 φ h ab (ω j ( I ba (ω j f θ ba(ω { Cov W(ω j φ h ab (ω j ( I ba (ω j f θ ba(ω 1 } ψ k c ( (Y tc μ c ( where I ba (ω j is the (b ath element of I (ω j and other quantities are defined analogously. Consider the two terms inside the curly brackets separately. Applying the same argument as in heorem in Brockwell and Davis (1991, we have W(ω j φ h a b (ω j ( I ba (ω j f θ ba(ω = 1 n ɛ f g=1 H ga (ω j + o p (1 W(ω j φ h ab (ω jh bf (ω j (I ɛ fg (ω j EI ɛ fg (ω j where and Ifg ɛ (ω j denote the (f gth element of the periodogram of ɛ t. Applying heorem in Brockwell and Davis (1991, we have 1 ψ k c ( (Y tc μ c ( = 1 n ɛ ψ k c (H cl(ɛ tl + o p (1 l=1

13 Supplementary Material QML estimation of linearized DSGE models 13 where H( = h j ( (cf. (3. herefore, their covariance is equal to 1 n ɛ f g l=1 = 1 W(ω j φ h ab (ω jh bf (ω j H ga (ω jψ k c (H cl( E { (I ɛ fg (ω j EI ɛ fg (ω jɛ tl } + op (1 = 1 2π n ɛ f g l=1 n ɛ f g l=1 Some algebra shows that A hk = 2 W(ω j φ h ab (ω jh bf (ω j H ga (ω jψ k c (H cl(ξ fgl + o p (1 { } W(ωH (ω ga φ h ab (ωh bf (ω j dω ξ fgl {ψ k c (H cl(}+o p (1 n ɛ f g l=1 [ W (ωh(ω ] (ω H(ωdω h gf f [ μ( ] ξ gf l fθ (H( k l Proof of Corollary 7. We prove the second result. Because the argument is very similar to heorem 3 and aniguchi (1979, heorem 2, we only provide an outline. he estimate θ solves L ( θ = (A.9 and the pseudo-true value θ m satisfies L m ( θ m = (A.1 Consider a aylor expansion of (A.9 around θ m, L ( θ m + 2 L ( θ θ ( θ θ m = where θ lies between θ and θ m. Rearrange terms and apply (A.1: Furthermore, 1/2 ( θ θ m = [ 2π 2 L ( θ θ ] ( 2π /2 L ( θ m 2π 1/2 L m ( θ m 2π 2 L ( θ θ [ 2 W(ω θ log det( f θ m (ω + 2 θ tr{ f θ m (ωf (ω }]

14 14 Qu and kachenko Supplementary Material + 2 μ( θ m f θ m ( μ( θ m because θ p θ m and because of the continuity of the integrand. Also, 2π /2 L ( θ m = 2π /2 2π 1/2 L m ( θ m W(ω j tr{ f θ m (ω j (I (ω j f (ω } + 2 /2 μ( θ m f θ m ((Y t μ + o p (1 = (M1 + (M2 + o p (1 erms (M1and(M2 satisfyacentrallimittheoremandcanbeanalyzedinthesameway as terms (I and (II in (A.8. he limiting covariance matrix can be verified accordingly. he detail is omitted. Proof of heorem 4. It suffices to verify that Assumptions 1 4 in Chernozhukov and Hong (23 hold under our set of conditions. Relabel these assumptions as CH1 CH4. CH1 and CH2 are trivial. CH3 is implied by Lemma 1(i, (ii, and (iv. o verify CH4, applying a second order aylor expansion of L (θ around (cf. CH4(i: with L (θ L ( = (θ L ( θ { R (θ = (θ 2 } L ( θ θ 2 L ( θ (θ (θ 2 L ( θ (θ + R (θ where θ lies between θ and.now /2 L ( θ d N( V herefore, CH4(ii is satisfied (V corresponds to n in CH4. For CH4(iii, note that V is nonrandom and positive definite, and that 2 L ( θ ( = vec(fθ (ω j { W(ω j f (ω j fθ (ω j } ( vec(fθ (ω j (A.11

15 Supplementary Material QML estimation of linearized DSGE models 15 = 1 ( vec(fθ (ω { W(ω j f 2π (ω fθ (ω } ( vec(fθ (ω dω + o(1 where the leading term on the right hand side is nonrandom and positive definite because fθ (ω, and ( vec(fθ (ω ( vec(fθ (ω W(ω j dω are positive definite by Assumption 5 and local identification. It is O(1 because the integrand is bounded; see Assumption 5. herefore, CH4(iii is satisfied. CH4(iv.a holds because R (θ 1/2 (θ 2 2 L ( θ θ 2 L ( θ where the second term can be made arbitrarily small by choosing θ small because of (A.11 and the boundedness and continuity of vec(f θ (ω/ and fθ (ω in θ (Assumptions 3 and 5(ii. CH4(iv.b holds because of the preceding argument and the fact that 1/2 (θ 2 = O(1. he proof for θ involves the same argument and is therefore omitted. References Brillinger, D. R. (21, ime Series: Data Analysis and heory. SIAM, Philadelphia. [1, 2, 4] Brockwell, P. and R. Davis (1991, ime Series: heory and Methods, second edition. Springer-Verlag, New York. [12] Krantz, S. G. and H. R. Parks (22, he Implicit Function heorem: History, heory, and Applications. Birkhäuser, Boston. [8] Magnus, J. R. and H. Neudecker (1999, Matrix Differential Calculus With Applications in Statistics and Econometrics, second edition. Wiley, Chichester. [4] aniguchi, M. (1979, On estimation of parameters of Gaussian stationary processes. Journal of Applied Probability, 16, [13]