Should We Go One Step Further? An Accurate Comparison of One-Step and Two-Step Procedures in a Generalized Method of Moments Framework

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1 Sould We Go One Step Furter? An Accurate Comparison of One-Step and wo-step Procedures in a Generalized Metod of Moments Framework Jungbin Hwang and Yixiao Sun Department of Economics, University of California, San Diego Preliminary. Please do not circulate. Abstract According to te conventional asymptotic teory, te two-step Generalized Metod of Moments (GMM) estimator and test perform as least as well as te one-step estimator and test in large samples. e conventional asymptotics teory, as elegant and convenient as it is, completely ignores te estimation uncertainty in te weigting matrix, and as a result it may not re ect nite sample situations well. In tis paper, we employ te xed-smooting asymptotic teory tat accounts for te estimation uncertainty, and compare te performance of te one-step and two-step procedures in tis more accurate asymptotic framework. We sow te two-step procedure outperforms te one-step procedure only wen te bene t of using te optimal weigting matrix outweigs te cost of estimating it. is qualitative message applies to bot te asymptotic variance comparison and power comparison of te associated tests. A Monte Carlo study lends support to our asymptotic results. JEL Classi cation: C2, C32 Keywords: Asymptotic E ciency, Fixed-smooting Asymptotics, Heteroskedasticity and Autocorrelation Robust, Increasing-smooting Asymptotics, Asymptotic Mixed Normality, Nonstandard Asymptotics, wo-step GMM Estimation Introduction E ciency is one of te most important problems in statistics and econometrics. In te widelyused GMM framework, it is standard practice to employ a two-step procedure to improve te e ciency of te GMM estimator and te power of te associated tests. e two-step procedure requires te estimation of a weigting matrix. According to te Hansen (982), te optimal weigting matrix is te asymptotic variance of te (scaled) sample moment conditions. For time series data, wic is our focus ere, te optimal weigting matrix is usually referred to as te j6wang@ucsd.edu, yisun@ucsd.edu. Correspondence to: Department of Economics, University of California, San Diego, 95 Gilman Drive, La Jolla, CA

2 long run variance (LRV) of te moment conditions. o be completely general, we often estimate te LRV using te nonparametric kernel or series metod. Under te conventional asymptotics, bot te one-step and two-step GMM estimators are asymptotically normal. In general, te two-step GMM estimator as a smaller asymptotic variance. Statistical tests based on te two-step estimator are also asymptotically more powerful tan tose based on te one-step estimator. A driving force beind tese results is tat te two-step estimator and te associated tests ave te same asymptotic properties as te corresponding ones wen te optimal weigting matrix is known. However, given tat te optimal weigting matrix is estimated nonparametrically in te time series setting, tere is large estimation uncertainty. A good approximation to te distributions of te two-step estimator and te associated tests sould re ect tis relatively ig estimation uncertainty. One of te goals of tis paper is to compare te asymptotic properties of te one-step and twostep procedures wen te estimation uncertainty in te weiging matrix is accounted for. ere are two ways to capture te estimation uncertainty. One is to use te ig order conventional asymptotic teory under wic te amount of nonparametric smooting in te LRV estimator increases wit te sample size but at a slower rate. Wile te estimation uncertainty vanises in te rst order asymptotics, we expect it to remain in ig order asymptotics. e second way is to use an alternative asymptotic approximation tat can capture te estimation uncertainty even wit just a rst-order asymptotics. o tis end, we consider a limiting tougt experiment in wic te amount of nonparametric smooting is eld xed as te sample size increases. is leads to te so-called xed-smooting asymptotics in te recent literature. In tis paper, we employ te xed-smooting asymptotics to compare te one-step and twostep procedures. For te one-step procedure, te LRV estimator is used in computing te standard errors, leading to te popular eteroskedasticity and autocorrelation robust (HAR) standard errors. See, for example, Newey and West (987) and Andrews (99). For te two-step procedure, te LRV estimator not only appears in te standard error estimation but also plays te role of te optimal weigting matrix in te second-step GMM criterion function. Under te xed-smooting asymptotics, te weigting matrix converges to a random matrix. As a result, te second-step GMM estimator is not asymptotically normal but rater asymptotically mixed normal. e asymptotic mixed normality re ects te estimation uncertainty of te GMM weigting matrix and is expected to be closer to te nite sample distribution of te second-step GMM estimator. In a recent paper, Sun (24b) sows tat bot te one-step and two-step test statistics are asymptotically pivotal under tis new asymptotic teory. So a nuisance-parameter-free comparison of te one-step and two-step tests is possible. Comparing te one-step and two-step procedures under te new asymptotics is fundamentally di erent from tat under te conventional asymptotics. Under te new asymptotics, te twostep procedure outperforms te one-step procedure only wen te bene t of using te optimal weigting matrix outweigs te cost of estimating it. is qualitative message applies to bot te asymptotic variance comparison and te power comparison of te associated tests. is is in sarp contrast wit te conventional asymptotics were te cost of estimating te optimal weigting matrix is completely ignored. Since te new asymptotic approximation is more accurate tan te conventional asymptotic approximation, comparing te two procedures under tis new asymptotics will give an onest assessment of teir relative merits. is is con rmed by a Monte In tis paper, te one-step estimator refers to te rst-step estimator in te typical two-step GMM framework. It does not refer to te continuous updating GMM estimator tat involves only one step. We use te terms one-step and rst-step intercangingly. 2

3 Carlo study. ere is a large and growing literature on te xed-smooting asymptotics. For kernel LRV variance estimators, te xed-smooting asymptotics is te so-called te xed-b asymptotics rst studied by Kiefer, Vogelsang and Bunzel (22) and Kiefer and Vogelsang (22a, 22b, 25) in te econometrics literature. For oter studies, see, for example, Jansson (24), Sun, Pillips and Jin (28), Sun and Pillips (29), Gonçlaves and Vogelsang (2), and Zang and Sao (23) in te time series setting; Bester, Conley, Hansen and Vogelsang (24) and Sun and Kim (24) in te spatial setting; and Gonçalves (2), Kim and Sun (23), and Vogelsang (22) in te panel data setting. For OS LRV variance estimators, te xed-smooting asymptotics is te socalled xed-k asymptotics. For its teoretical development and related simulation evidence, see, for example, Pillips (25), Müller (27), and Sun (2, 23). e approximation approaces in some oter papers can also be regarded as special cases of te xed-smooting asymptotics. is includes, among oters, Ibragimov and Müller (2), Sao (2) and Bester, Conley, and Hansen (2). e xed-smooting asymptotics can be regarded as a convenient device to obtain some ig order terms under te conventional increasing-smooting asymptotics. e rest of te paper is organized as follows. e next section presents a simple overidenti ed GMM framework. Section 3 compares te two procedures from te perspective of point estimation. Section 4 compares tem from te testing perspective. Section 5 extends te ideas to a general GMM framework. Section 6 reports simulation evidence on te nite sample performances of te two procedures. Proofs of te main teorems are provided in te Appendix. A word on notation: for a symmetric matrix A; A =2 (or A =2 ) is a matrix square root of A suc tat A =2 A =2 = A: Note tat A =2 does not ave to be symmetric. We will specify A =2 explicitly wen it is not symmetric. If not speci ed, A =2 is a symmetric matrix square root of A based on its eigendecomposition. By de nition, A =2 = A =2 and so A = (A =2 ) (A =2 ): For matrices A and B; we use A B to signify tat A B is positive (semi)de nite. We use and O intercangeably to denote a matrix of zeros wose dimension may be di erent at di erent occurrences. For two random variables X and Y; we use X? Y to indicate tat X and Y are independent. 2 A Simple Overidenti ed GMM Framework o illustrate te basic ideas of tis paper, we consider a simple overidenti ed time series model of te form: y t = + u t ; y t 2 R d ; y 2t = u 2t ; y 2t 2 R q () for t = ; :::; were 2 R d is te parameter of interest and te vector process u t := (u t ; u 2t ) is stationary wit mean zero. We allow u t to ave autocorrelation of unknown forms so tat te long run variance of u t : X = lrvar(u t ) = Eu t u t j j= 3

4 takes a general form. However, for simplicity, we assume tat var(u t ) = 2 I d for te moment 2. Our model is just a location model. We initially consider a general GMM framework but later nd out tat our points can be made more clearly in te simple location model. In fact, te simple location model may be regarded as a limiting experiment in a general GMM framework. Embedding te location model in a GMM framework, te moment conditions are E(y t ) q = were y t = (y t ; y 2t ). Let g () = p P t= y t p P t= y 2t! : en a GMM estimator of can be de ned as ^GMM = arg min g () W g () for some positive de nite weigting matrix W : Writing W W W = 2 W 2 W 22 were W is a d d matrix and W 22 is a q q matrix, ten it is easy to sow tat ; ^GMM = X t= (y t W y 2t ) for W = W 2 W 22 : ere are at least two di erent coices of W. First, we can take W to be te identity matrix W = I m for m = d + q: In tis case, W = and te GMM estimator ^ is simply ^ = X y t : t= Second, we can take W to be te optimal weigting matrix W =. Wit tis coice, we obtain te GMM estimator: ~ 2 = X (y t y 2t ) ; t= were = 2 22 is te long run regression coe cient matrix. Wile ^ completely ignores te information in fy 2t g ; ~ 2 takes advantage of tis information. 2 If for any 2 > ; we can let V V 2 var (u t) = V := 6= 2 I V 2 V d 22 V =2 = (V 2) =2 V 2 (V 22) =2 (V 22) =2 en V =2 (y t; y 2t) can be written as a location model wose error variance is te identity matrix I d : e estimation uncertainty in estimating V will not a ect our asymptotic results.! : 4

5 were Under some moment and mixing conditions, we ave p ^ d =) N(; ) and p 2 = : ~2 d =) N (; 2 ) ; So Asym Var( ~ 2 ) < Asym Var(^ ) unless 2 =. is is a well known result in te literature. Since we do not know in practice, ~ 2 is infeasible. However, given te feasible estimator ^ ; we can estimate and construct a feasible version of ~ 2 : e common two-step estimation strategy is as follows. i) Estimate te long run covariance matrix by ^ := ^ (^u) = X X Q ( s ; t ) s= t= ^u t! X ^u = ^u s! X ^u = were ^u t = (y t ^ ; y 2t ). ii) Obtain te feasible two-step estimator ^ 2 = P t= y t ^y2t were ^ = ^ 2 ^ 22. In te above de nition of ^, Q (r; s) is a symmetric weigting function tat depends on te smooting parameter : For conventional kernel estimators, Q (r; s) = k ((r s) =b) and we take = =b: For te ortonormal series (OS) estimators, Q (r; s) = K P K j= j (r) j (s) and we take = K; were j (r) are ortonormal basis functions on L 2 [; ] satisfying R j (r) dr = : We parametrize in suc a way so tat indicates te level or amount of smooting for bot types of LRV estimators. Note tat we use te demeaned process f^u t P = ^u g in constructing ^ (^u) : For te location model, ^ (^u) is numerically identical to ^ (u) were te unknown error process fu t g is used. e moment estimation uncertainty is re ected in te demeaning operation. Had we known te true value of and ence te true moment process fu t g ; we would not need to demean fu t g. Wile ~ 2 is asymptotically more e cient tan ^ ; is ^ 2 necessarily more e cient tan ^ and in wat sense? Is te Wald test based on ^ 2 necessary more powerful tan tat based on ^? One of te objectives of tis paper is to address tese questions. 3 A ale of wo Asymptotics: Point Estimation We rst consider te conventional asymptotics were! as! but at a slower rate, i.e., =! : e asymptotic direction is represented by te curved arrow in Figure. Sun (24a, 24b) calls tis type of asymptotics te Increasing-smooting Asymptotics, as increases wit te sample size. Under tis type of asymptotics and some regularity conditions, we ave ^! p : It can ten be sown tat ^ 2 is asymptotically equivalent to ~ 2, i.e., p ( ~ 2 ^2 ) = o p (). As a direct consequence, we ave p d ^ =) N(; ); p d ^2 =) N ; : So ^ 2 is still asymptotically more e cient tan ^ : 5

6 Figure : Conventional Asymptotics (increasing smooting) vs New Asymptotics ( xed smooting) e conventional asymptotics, as elegant and convenient as it is, does not re ect te nite sample situations well. Under tis type of asymptotics, we essentially approximate te distribution of ^ by te degenerate distribution concentrating on. at is, we completely ignore te estimation uncertainty in ^: e degenerate approximation is too optimistic, as ^ is a nonparametric estimator, wic by de nition can ave ig variation in nite samples. o obtain a more accurate distributional approximation of p (^ 2 ); we could develop a ig order increasing-smooting asymptotics tat re ects te estimation uncertainty in ^. is is possible but requires strong assumptions tat cannot be easily veri ed. In addition, it is also tecnically callenging and tedious to rigorously justify te ig order asymptotic teory. Instead of ig order asymptotic teory under te conventional asymptotics, we adopt te type of asymptotics tat olds xed as! : is asymptotic beavior of and is illustrated by te arrow pointing to te rigt in Figure. Given tat is xed, we follow Sun (24a, 24b) and call tis type of asymptotics te Fixed-smooting Asymptotics. is type of asymptotics takes te sampling variability of ^ into consideration. Sun (23, 24a) as sown tat te xed-smooting asymptotic distribution is ig order correct under te conventional increasing-smooting asymptotics. So te xed-smooting asymptotics can be regarded as a convenient device to obtain some ig order terms under te conventional increasing-smooting asymptotics. o establis te xed-smooting asymptotics, we maintain Assumption on te kernel function and basis functions. 6

7 Assumption (i) For kernel HAR variance estimators, te kernel function k () satis es te following condition: for any b 2 (; ] and, k b (x) and k (x) are symmetric, continuous, piecewise monotonic, and piecewise continuously di erentiable on [ ; ]. (ii) For te OS HAR variance estimator, te basis functions j () are piecewise monotonic, continuously di erentiable and ortonormal in L 2 [; ] and R j (x) dx = : Assumption on te kernel function is very mild. It includes many commonly used kernel functions suc as te Bartlett kernel, Parzen kernel, and Quadratic Spectral (QS) kernel. De ne Q (r; s) = Q (r; s) Z Q (; s)d wic is a centered version of Q (r; s); and Z Q (r; )d + Z Z Q ( ; 2 )d d 2 ~ = X X Q ( s ; t )^u t^u s: s= t= Assumption ensures tat ~ and ^ are asymptotically equivalent. Furtermore, under tis assumption, Sun (24a) sows tat, for bot kernel LRV and OS LRV estimation, te centered weigting function Q (r; s) satis es : Q (r; s) = X j j (r) j (s) j= were f j (r)g is a sequence of continuously di erentiable functions satisfying R j(r)dr = and te series on te rigt and side converges to Q (r; s) absolutely and uniformly over (r; s) 2 [; ] [; ]. e representation can be regarded as a spectral decomposition of te compact Fredolm operator wit kernel Q (r; s) : See Sun (24a) for more discussion. Now, letting () := and using te basis functions f j ()g j= in te series representation of te weigting function, we make te following assumptions. Assumption 2 e vector process fu t g t= satis es: a) =2 P t= j(t= )u t converges weakly to a continuous distribution, jointly over j = ; ; :::; J for every xed J; b) For every xed J and x 2 R m,! X P p j ( t )u t x for j = ; ; ::; J = t=! X P =2 p j ( t )e t x for j = ; ; :::; J + o() as! t= were! =2 = =2 2 2 =2 22 =2 > 22 is a matrix square root of = P j= Eu tu t j and e t s iid N(; I m ) 7

8 Assumption 3 P j= k Eu tu t j k< : Proposition Let Assumptions 3 old. As! for a xed, we ave: (a) ^ =) d were = =2 ~ =2 := ; ;2 ;2 ;22 Z Z ~; ~ = Q (r; s)db ~ m(r)db m (s) ;2 := ~ ;2 ;22 ~ and B m () is a standard Brownian motion of dimension m = d + q; (b) p d ^2 =) I d ; =2 B m () were = (; d; q) := ;2 ;22 is independent of B m () : Conditional on, te asymptotic distribution of p ^2 is a normal distribution wit variance V 2 = I d ; 2 Id 2 22 = : at is, p (^ 2 ) is asymptotically mixed-normal rater tan normal. Since V = = almost surely, te feasible estimator ^ 2 as a large variation tan te infeasible estimator ~ 2 : is is consistent wit our intuition. Under te xed-smooting asymptotics, we still ave p d (^ ) =) N(; ) as ^ pdoes not depend on te smooting parameter : So te asymptotic (conditional) variances of (^ ) and p (^ 2 ) are bot larger tan in terms of matrix de niteness. However, it is not straigtforward to directly compare tem. o evaluate te relative magnitude of te two (conditional) asymptotic variances, we de ne ~ := ~ (; d; q) := ~ ;2 ~ ;22 ; (2) wic does not depend on any nuisance parameter but depends on ; d; q: For notational economy, we sometimes suppress tis dependence. Direct calculations sow tat Using tis, we ave, for any conforming vector a : = =2 2 ~ = : (3) Ea (V 2 ) a = Ea 22 a a a = a =2 2 E~ ~ = a =2 2 =2 2 a a a E ~ ~ = ( =2 2 ) i ( =2 2 ) a: (4) e following lemma gives a caracterization of E ~ (; d; q) ~ (; d; q) : 8

9 Lemma 2 For any d ; we ave E ~ (; d; q) ~ (; d; q) = Ejj ~ (; ; q) jj 2 I d : Let On te basis of te above lemma, we de ne g(; q) := Ejj~ (; ; q)jj 2 + Ejj ~ 2 (; ): (; ; q)jj2 = =2 2 ( =2 22 ) 2 R dq ; wic is te long run correlation matrix between u t and u 2t. Using (4) and rewriting it in terms of ; we obtain te following proposition. Proposition 3 Let Assumptions 3 old. If g(; q)i d is positive (negative) semide nite, ten ^ 2 as a larger (smaller) asymptotic variance tan ^ under te xed-smooting asymptotics. In te proof of te proposition, we sow tat te coices of te matrix square roots =2 and =2 22 are innocuous. o use te proposition, we need only to ceck weter te condition is satis ed for a given coice of te matrix square roots. If = ; ten g(; q)i d olds trivially. In tis case, te asymptotic variance of ^ 2 is larger tan te asymptotic variance of ^ : Intuitively, wen te long run correlation is zero, tere is no information tat can be explored to improve e ciency. If we insist on using te long run correlation matrix in attempt to improve te e ciency, we may end up wit a less e cient estimator, due to te noise in estimating te zero long run correlation matrix. On te oter and, if = I d after some possible rotation, wic olds wen te long run variation of u t is almost perfectly predicted by u 2t ; ten g(; q)i d <. In tis case, it is wortwile estimating te long run variance and using it to improve te e ciency ^ 2 : ere are many scenarios in between were te matrix g(; q)i d is inde nite. In tis case we may still be able to access te de niteness of te matrix along some directions; see eorem. e case wit d = is simplest. In tis case, if > g(; q); ten ^ 2 is asymptotically more e cient tan ^ : Oterwise, it is not asymptotically less e cient. In te case of kernel LRV estimation, it is ard to obtain an analytical expression for Ejj ~ (; ; q)jj 2 and ence g(; q), altoug we can always simulate it numerically. e tresold g(; q) depends on te smooting parameter = =b and te degree of overidenti cation q: ables 3 report te simulated values of g(; q) for b = : : : : :2 and q = 5. It is clear tat g(; q) increases wit q and decreases wit te smooting parameter = =b. Wen te OS LRV estimation is used, we do not need to simulate g(; q) as we can obtain a closed form expression for it. Using tis expression, we can prove te following corollary. Corollary 4 Let Assumptions 3 old. Consider te case of OS LRV estimation. If q K I d is positive (negative) semide nite, ten ^ 2 as a larger (smaller) asymptotic variance tan ^ under te xed-smooting asymptotics. A necessary and su cient condition for q=(k )I d to be positive semide nite is tat te largest eigenvalue of is smaller tan q=(k ): e eigenvalues of are te squared long run correlation coe cients between c u t and c 2 u 2t for some c and c 2 ; i.e., te square long run canonical correlation coe cients between u t and u 2t : So if te largest squared long run 9

10 canonical correlation is smaller tan q=(k ); ten ^ 2 as a larger asymptotic variance tan ^ : Similarly, if te smallest squared long run canonical correlation is greater tan q=(k ); ten ^ 2 as a smaller asymptotic variance tan ^ : Since ^ 2 is not asymptotically normal, asymptotic variance comparison does not paint te wole picture. o compare te asymptotic distributions of ^ and ^ 2, we consider te case tat d = q = and K = 4 as an example: Figure 2 represents te sapes of probability density functions in te case of OS LRV estimation wen ( ; 2 2 ; 22) = (; :; ). In tis case, = :9. e rst grap sows p (^ ) a N(; ) (red solid line) and p (^ 2 ) a N(; :9) (blue dotted line) under te conventional asymptotics. e conventional limiting distributions for p (^ ) and p (^ 2 ) are bot normal but te latter as a smaller variance, so te asymptotic e ciency of ^ 2 is always guaranteed. However, tis is not true in te second grap of Figure 2, wic represents te limiting distributions under te xed-smooting asymptotics. Here, as before te red solid line represents N(; ) but te blue dotted line represents te mixed normal distribution MN[; :9( + ~ 2 )] wit ~ = P K i= i 2i = P K i= 2 2i s MN(; = 2 K ): is can be obtained by using (4) wit a = : More speci cally, V 2 = =2 2 ~ ~ ( =2 2 ) = :9 ~ 2 + :9 = :9( + ~ 2 ): (5) Comparing tese two di erent families of distributions, we nd tat te asymptotic distribution of ^ 2 as fatter tail tan tat of ^ : e asymptotic variance of ^ 2 is :9( + =2) = :35 wic is larger tan te asymptotic variance of ^. More generally, wen d = q = wit ( ; 2 2 ; 22) = (; 2 ; ); we can use (4) to obtain te asymptotic variance of ^ 2 as follows: + 2 E ~ 2 = 2 + K = 2 + E ~ 2 : = 2 K K 2 So, for any coice of K, te asymptotic variance of ^ 2 decreases as 2 increases. Under te quadratic loss function, te asymptotic relative e ciency of ^ 2 relative to ^ is tus equal to 2 (K ) = (K 2) : e second factor (K ) = (K 2) > re ects te cost of aving to estimate : Wen 2 = =(K ); ^ and ^ 2 ave te same asymptotic variance. For a given coice of K, if 2 =(K ), we prefer to use ^. is is not captured by te conventional asymptotics, wic always cooses ^ 2 over ^. 4 A ale of wo Asymptotics: Hypotesis esting We are interested in testing te null ypotesis H : R = r against te local alternative H : R = r + = p for some p d matrix R and p vectors r and : Nonlinear restrictions can be converted into linear ones using te Delta metod. We construct te following two Wald statistics: W := (R^ r) R^ R (R^ r) W 2 := (R^ 2 r) R^ 2 R i (R^2 r)

11 Conventional Asymptoptic Distributions One step estimator: N(,) wo step estimator: N(,.9) Fixed smootign Asymptotic Distributions One step estimator: N(,) wo step estimator: MN(,.9(+ 2 )) Figure 2: Limiting distributions of ^ and ^ 2 based on te OS LRV estimator wit K = 4: were ^ 2 = ^ ^2 ^ 22 2 : Wen p = and te alternative is one sided, we can construct te following two t statistics: : = p R^ r pr^ R (6) p R^ 2 r 2 : = pr^ 2 R : (7) No matter weter te test is based on ^ or ^ 2 ; we ave to employ te long run covariance estimator ^. De ne te p p matrices and 2 according to = R R and 2 2 = R 2 R : In oter words, and 2 are matrix square roots of R R and R 2 R respectively.

12 Under te conventional increasing-smooting asymptotics, it is straigtforward to sow tat under H : R = r + = p : W d =) 2 p( Wen = ; we obtain te null distributions: 2 ), W 2 d =) 2 p( 2 d =) N( ; ), 2 d =) N( 2 ; ): W ; W 2 d =) 2 d p and ; 2 =) N(; ): So under te conventional increasing-smooting asymptotics, te null limiting distributions of W and W 2 are identical. Since 2 2 2, under te conventional asymptotics, te local asymptotic power function of te test based on W 2 is iger tan tat based on W : e key driving force beind te conventional asymptotics is tat we approximate te distribution of ^ by te degenerate distribution concentrating on : e degenerate approximation does not re ect te nite sample distribution well. As in te previous section, we employ te xed-smooting asymptotics to derive more accurate distributional approximations. Let and C pp = C qq = Z Z Z Z Q (r; s)db p(r)db p (s) ; C pq = Q (r; s)db q(r)db q (s) ; C qp = C pq Z Z D pp = C pp C pq C qq C pq 2 ), Q (r; s)db p(r)db q (s) were B p () 2 R p and B q () 2 R q are independent standard Brownian motion processes. Proposition 5 Let Assumptions 3 old. As! for a xed, we ave, under H : R = r + = p : d (a) W =) W ( 2 ) were (b) W 2 (c) (d) 2 d =) W 2 ( W (kk 2 ) = [B p () + ] C pp [B p () + ] for 2 R p : (8) 2 2 ) were W 2 (kk 2 ) = B p () C pq Cqq B q () + D pp d =) d =) 2 2 Bp () C pq C qq B q () + : (9) := Bp ()+ p = Cpp for p = : := Bp () C pq Cqq B q () + 2 p = Dpp for p = : In Proposition 5, we use te notation W (kk 2 ), wic implies tat te rigt and side of (8) depends on only troug kk 2 : is is true, because for any ortogonal matrix H : [B p () + ] C pp [B p () + ] = [HB p () + H] HC pp H [HB p () + H] d = [B p () + H] C pp [B p () + H] : 2

13 If we coose H = (= kk ; ~ H) for some ~ H suc tat H is ortogonal, ten [B p () + ] C pp [B p () + ] d = [B p () + kk e p ] C pp [B p () + kk e p ] : So te distribution of [B p () + ] C pp [B p () + ] depends on only troug kk : Similarly, te distribution of te rigt and side of (9) depends only on kk 2 : Wen = ; we obtain te limiting distributions of W ; W 2 ; and 2 under te null ypotesis: W W 2 2 d =) W := W () = B p () C pp B p () ; d =) W 2 := W 2 () = B p () C pq C d =) := () = B p ()= p C pp ; qq B q () D pp d =) 2 := 2 () = B p () C pq C qq B q () = p D pp : Bp () C pq C qq B q () ; ese distributions are di erent from tose under te conventional asymptotics. For W and ; te di erence lies in te random scaling factor C pp or p C pp : e random scaling factor captures te estimation uncertainty of te LRV estimator. For W 2 and 2 ; tere is an additional di erence embodied by te random location sift C pq Cqq B q () wit a consequent cange in te random scaling factor. e proposition below provides some caracterization of te two limiting distributions W and W 2 : Proposition 6 For any x > ; te following old: (a) W 2 () rst-order stocastically dominates (FSD) W () in tat P [W 2 () x] > P [W () x] : i i (b) P W (kk 2 ) x strictly increases wit kk 2 and lim kk! P W (kk 2 ) x = : i i (c) P W 2 (kk 2 ) x strictly increases wit kk 2 and lim kk! P W 2 (kk 2 ) x = : Proposition 6(a) is intuitive. W 2 FSD W because W 2 FSD B p () Dpp B p (), wic in turn FSD B p () Cpp B p () wic is just W. According to a property of te rst-order stocastic dominance, we ave W 2 d = W + W e for some W e > : Intuitively, W 2 sifts some of te probability mass of W to te rigt. A direct implication is tat te asymptotic critical values for W 2 are larger tan te corresponding ones for W : e di erence in critical values as implications on te power properties of te two tests. For x > ; we ave P ( > x) = 2 P W x 2 and P ( 2 > x) = 2 P W 2 x 2. It ten follows from Proposition 6(a) tat P ( 2 > x) P ( > x) for x > : So for a onesided test wit te alternative H : R > r; critical values from 2 are larger tan tose from 3

14 : Similarly, we ave P ( 2 < x) P ( < x) for x < : is implies tat for a one-sided test wit te alternative H : R < r; critical values from 2 are smaller tan tose from : Let W and W 2 be te ( ) quantile from te distributions W and W 2 ; respectively. e local asymptotic power functions of te two tests are 2 := 2 ; K; p; q; = P W ( i 2 ) > W ; 2 2 := 2 2 ; K; p; q; = P W 2 ( i 2 ) > W 2 : 2 Wile ; we also ave W 2 > W : e e ects of te critical values and te noncentrality parameter move in opposite directions. It is not straigtforward to compare te two power functions. However, Proposition 6 suggests tat if te di erence in te noncentrality parameters is large enoug to o set te increase in critical values, ten te two-step test based on W 2 will be more powerful. o evaluate 2 2 ; we de ne 2 2 R = R R =2 (R2 ) =2 22 ; () wic is te long run correlation matrix R between Ru t and u 2t : In terms of R 2 R pq we ave = R R R R R R = I p R 222 2R i n = Ip R R = ( Ip R =2) R R R Ip R =2 R > : I p o So te di erence in te noncentrality parameters depends on te matrix R R : Let R = P min(p;q) i= i a i b i be te singular value decomposition of R were fa i g and fb i g are ortonormal vectors in R p and R q respectively. Sorted in te descending order, 2 i are te (squared) long run canonical correlation coe cients between Ru t and u 2t : en = min(p;q) X i= 2 i 2 i a i 2 : Consider a special case tat 2 := max p i= 2 i approaces. If a 6= ; ten jj2 jj 2 2 and ence jj2 jj 2 approaces as 2 approaces from below. is case appens wen te second block of moment conditions as very ig long run prediction power for te rst block. In tis case, we expect te W 2 test to be more powerful, as lim j j! 2 ( 2 2 ) = : Consider anoter special case tat max i 2 i = ; i.e., R is a matrix of zeros. In tis case, te second block of moment conditions contains no additional information, and we ave 2 2 = 2 : In tis case, we expect te W 2 test to be less powerful. It follows from Proposition 6(b)(c) tat for any, tere exists a unique () := (; ; p; q; ) suc tat () = 2 ( ()) : 4

15 en 2 ( 2 2 ) < ( 2 ) if and only if 2 2 ( 2 ) 2 = ( 2 ) Ip R =2 R R R 2 f 2 < ( 2 ) 2 : But 2 i Ip I p R =2 R ; were (; ; p; q; ) f () := f(; ; p; q; ) = : (; ; p; q; ) So te power comparison depends on weter f( 2 )I p R R is positive semide nite. Proposition 7 Let Assumptions 3 old and de ne := 2 = (R R ). If f(; ; p; q; )I p R R is positive (negative) semide nite, ten te two-step test based on W 2 as a lower (iger) local asymptotic power tan te one-step test based on W under te xedsmooting asymptotics. Note tat R R = R R =2 R R R R =2 : o pin down R R ; we need to coose a matrix square root of R R : Following te arguments in te proof of Proposition 3, it is easy to see tat we can use any matrix square root. In particular, we can use te principal matrix square root. Wen p = ; wic is of ultimate importance in empirical studies, R R is equal to te sum of te squared long run canonical correlation coe cients. In tis case, f(; ; p; q; ) is te tresold value of R R for assessing te relative e ciency of te two tests. More speci cally, wen R R > f(; ; p; q; ), te two-step W 2 test is more powerful tan te one-step W test. Oterwise, te two-step W 2 test is less powerful. Proposition 7 is in parallel wit Proposition 3. e qualitative messages of tese two propositions are te same wen te long run correlation is ig enoug, we sould estimate and exploit it to reduce te variation of our point estimator and improve te power of te associated tests. However, te tresolds are di erent quantitatively. e two propositions fully caracterize te tresold for eac criterion under consideration. Proposition 8 Consider te case of OS LRV estimation. For any 2 R +, we ave () > 2 () and ence (; ; p; q; ) > and f(; ; p; q; ) > : Proposition 8 is intuitive. Wen tere is no long run correlation between Ru t and u 2t ; we ave 2 2 = 2 : In tis case, te two-step W 2 test is necessarily less powerful. e proof uses te teory of uniformly most powerful invariant tests and te teory of complete and su cient statistics. It is an open question weter te same strategy can be adopted to prove Proposition 8 in te case of kernel LRV estimation. Our extensive numerical work supports tat (; ; p; q; ) > and f(; ; p; q; ) > continue to old in te kernel case. It is not easy to give an analytical expression for f(; ; p; q; ) but we can compute it numerically witout any di culty. In ables 4 and 5, we consider te case of OS LRV estimation and compute te values of f(; K; p; q; ) for = ; K = 8; ; 2; 4, p = 4 and q = 3: Similar to te asymptotic variance comparison, we nd tat tese tresold values increase as te degree of over-identi cation increases and decrease as te smooting parameter K increases. Note tat te continuity of two power functions (; K; p; q; ) and 2 (; K; p; q; ) guarantees 5

16 tat for 2 [; ] te two-step W 2 test as a lower local asymptotic power tan te one-step W test if te largest eigenvalue of R R is smaller tan min 2[;] f(; K; p; q; ). on te oter and, for 2 [; ] te two-step W 2 test as a iger local asymptotic power if te smallest eigenvalue of R R is greater tan max f(; K; p; q; ). For te case of kernel LRV estimator, results not reported ere sow tat f(; ; p; q; ) increases wit q and decreases wit. is is entirely analogous to te case of OS LRV estimation. 5 General Overidenti ed GMM Framework In tis section, we consider te general GMM framework. e parameter of interest is a d vector 2 R d. Let v t 2 R dv denote te vector of observations at time t. We assume tat is te true value, an interior point of te parameter space. e moment conditions E f(v t ; ) = ; t = ; 2; :::; : old if and only if = were f (v t ; ) is an m vector of continuously di erentiable functions. e process f (v t ; ) may exibit autocorrelation of unknown forms. We assume tat m d and tat te rank of E[@ f (v t ; ) =@ ] is equal to d: at is, we consider a model tat is possibly over-identi ed wit te degree of over-identi cation q = m d: 5. One-Step and wo-step Estimation and Inference De ne te m m contemporaneous covariance matrix and te LRV matrix as: = E f(v t ; ) f(v t ; ) and = X j= j were j = E f(v t ; ) f(v t j ; ) : Let g t () = p tx f(v j ; ): j= Given a simple positive-de nite weigting matrix W tat does not depend on any unknown parameter, we can obtain an initial GMM estimator of as ^ = arg min g () W g (): For example, we may set W equal to I m. In te case of IV regression, we may set W equal to Z Z = were Z is te matrix of te instruments. Using or as te weigting matrix, we obtain te following two (infeasible) GMM estimators: ~ : = arg min () g (); 2 () ~ 2 : = arg min () g (): 2 (2) For te estimator ~ ; we use te contemporaneous covariance matrix as te weigting matrix and ignore all te serial dependency in te moment vector process ff(v t ; )g t=. In contrast to tis procedure, te second estimator ~ 2 accounts for te long run dependency. e feasible 6

17 versions of tese two estimators ^ and ^ 2 can be naturally de ned by replacing and wit teir estimates est (^ ) and est (^ ) were est () : = est () : = X f(v t ; ) f(v t ; ) ; (3) t= X s= t= X Q ( s ; t ) f(v t ; ) f(v s ; ) : (4) o test te null ypotesis H : R = r against H : R = r + = p ; we construct two di erent Wald statistics as follows: W : = (R^ r) n R ^V R o (R^ r); (5) W 2 : = (R^ 2 r) n R ^V 2 R o (R^2 r); were ^V = ^V 2 = G est (^ ) G i G est (^ ) est (^ ) est (^ ) G i G est (^ ) G i (6) G2 est (^ 2 ) G i 2 and G = X f(v t ; ; G2 = =^ X f(v t ; : =^2 ese are te standard Wald test statistics in te GMM framework. o compare te two estimators ^ and ^ 2 and associated tests, we maintain te standard assumptions below. Assumption 4 As! ; ^ = +o p () ; ^ = +o p () ; ^ 2 = +o p () for an interior point 2 : Assumption 5 De ne G t () = = tx f(v t ; for t and G () = : For any = + o p (); te following olds: (a) plim! G[r ] ( ) = r G uniformly in r were G = G( ) and G() = E@ f(v t ; )=@ ; (b) est ( ) p! > : Wit tese assumptions and some mild conditions, te standard GMM teory gives us p (^ ) = p X t= G G i G f(vt ; ) + o p (): Under te xed-smooting asymptotics, Sun (24b) establises te representation: p (^2 ) = p X t= G G i G f(v t ; ) + o p (); 7

18 were is de ned in te similar way as in Proposition : = =2 ~ =2 : Due to te complicated structure of two transformed moment vector processes, it is not straigtforward to compare te asymptotic distributions of ^ and ^ 2 as in Sections 3 and 4. o confront tis callenge, we let G = U V mm md dd be a singular value decomposition (SVD) of G, were = A dd ; O dq ; A is a d d diagonal matrix and O is a matrix of zeros. Also, we de ne f (v t ; ) = (f (v t ; ); f 2 (v t ; )) := U f(vt ; ) 2 R m ; were f (v t; ) 2 R d and f2 (v t; ) 2 R q are te rotated moment conditions. e variance and long run variance matrices of ff (v t ; )g are := U U = and := U U. o convert te variance matrix into an identity matrix, we de ne te normalized moment conditions below: were f(v t ; ) = f (v t ; ) ; f 2 (v t ; ) := ( =2 ) f (v t ; ) =2 = ( 2 )=2 2 ( 22 ) =2 ( 22 )=2! : (7) More speci cally, f (v t ; ) : = ( 2) =2 f (v t ; ) i 2 ( 22) f2 (v t ; ) 2 R d ; f 2 (v t ; ) : = ( 22) =2 f 2 (v t ; ) 2 R q : en te contemporaneous variance of te time series ff(v t ; )g is I m and te long run variance is := ( =2 ) ( =2 ) : Lemma 9 Let Assumptions 5 old by replacing u t wit f(v t ; ) in Assumptions 2 and 3. en as! for a xed ; ( 2) =2 AV p (^ ) = p ( 2) =2 AV p (^ 2 ) = p X t= X t= f (v t ; ) + o p () were := ;2 ;22 is te same as in Proposition. d =) N(; ) (8) [f (v t ; ) f 2 (v t ; )] + o p () (9) d =) MN ;

19 e asymptotic normality and mixed normality in te lemma are not new. e limiting distribution of p (^ ) is available from te standard GMM teory wile tat of p (^ 2 ) as been recently establised in Sun (24b). It is te representation tat is new and insigtful. e representation casts te two estimators in te same form and enables us to directly compare teir asymptotic properties. It follows from te proof of te lemma tat ( 2) =2 AV p ( ~ 2 ) = p X t= [f (v t ; ) f 2 (v t ; )] + o p () were = 2 22 as de ned before. So te di erence between te feasible and infeasible twostep GMM estimators lies in te uncertainty in estimating : Wile te true value of appears in te asymptotic distribution of te infeasible estimator ~ 2 ; te xed-smooting limit of te implied estimator ^ := ^ 2 ^ 22 appears in tat of te feasible estimator ^ 2 : It is important to point out tat te estimation uncertainty in te wole weigting matrix est matters only troug tat in ^: If we let (u t ; u 2t ) = (f (v t ; ); f 2 (v t ; )); ten te rigt and sides of (8) and (9) are exactly te same as wat we would obtain in te location model. e location model, as simple as it is, as implications for general settings from an asymptotic point of view. More speci cally, de ne y t = ( 2) =2 AV + u t ; y 2t = u 2t ; were u t = f (v t ; ) and u 2t = f 2 (v t ; ). e estimation and inference problems in te GMM setting are asymptotically equivalent to tose in te above simple location model wit fy t ; y 2t g as te observations. o present our next teorem, we transform R into ~ R using wic as te same dimension as R: We let Z Z Z ~ (; p; q) = Q (r; s)db p(r)db q (s) wic is compatible wit te de nition in (2). We de ne = =2 2 = R dq and R = ~R = RV A ( 2) =2 ; (2) Z Q (r; s)db q(r)db q (s) ; ~R ~ R =2 ~R2 = R pq : Wile is te long run correlation matrix between f (v t ; ) and f 2 (v t ; ), R is te long run correlation matrix between Rf ~ (v t ; ) and f 2 (v t ; ). Finally, we rede ne te noncentrality parameter : = R( G G) ( G G)( G G) R i ; (2) wic is te noncentrality parameter based on te rst-step test. For te location model considered before, te above de nition of R is identical to tat in (). In tat case, we ave G = (I d ; O dq ) and so U = I m, A = I d and V = I d : Given te assumption tat = = I m ; wic implies tat 2 = I d; we ave ~ R = R: In addition, te noncentrality parameter reduces to = (R R ) as de ned in Proposition 7. 9

20 eorem Let te assumptions in Lemma 9 old. (a) If g(; q) I d is positive (negative) semide nite, ten ^ 2 as a larger (smaller) asymptotic variance tan ^ as! for a xed : (b) If g(; q) I p R R is positive (negative) semide nite, ten R^ 2 as a larger (smaller) asymptotic variance tan R^ as! for a xed : (c) If f (; ; p; q; ) I p R R is positive (negative) semide nite, ten te test based on W 2 is asymptotically less (more) powerful tan tat based on W ; as! for a xed : eorem (a) is a special case of eorem (b) wit R = I d. We single out tis case in order to compare it wit Proposition 3. It is reassuring to see tat te results are te same. e only di erence is tat in te general GMM case we need to rotate and standardize te original moment conditions before computing te long run correlation matrix. Part (b) can also be applied to a general location model wit a nonscalar error variance, in wic case ~ R = R ( 2 )=2. eorem (c) reduces to Proposition 7 in te case of te location model. 5.2 wo-step GMM Estimation and Inference wit a Working Weigting Matrix In te previous subsection, we employ two speci c weigting matrices te variance and long run variance estimators. In tis subsection, we consider a general weigting matrix W (^ ); wic may depend on te initial estimator ^ and te sample size ; leading to yet anoter GMM estimator: i ^a = arg min g () W (^ ) g () 2 were te subscript a signi es anoter or alternative. An example of W (^ ) is te implied LRV matrix wen we employ a simple approximating parametric model to capture te dynamics in te moment process. We could also use te general LRV estimator but we coose a large so tat te variation in W (^ ) is small. In te kernel LRV estimation, tis amounts to including only autocovariances of low orders in constructing W (^ ): We assume tat W (^ )! p W, a positive de nite nonrandom matrix under te xed-smooting asymptotics. W may not be equal to te variance or long run variance of te moment process. We call W (^ ) a working weigting matrix. is is in te same spirit of using a working correlation matrix rater tan a true correlation matrix in te generalized estimating equations (GEE) setting. See, for example, Liang and Zeger (986). In parallel to (5), we construct te test statistic W a := (R^ a r) n R ^V a R o (R^a r); were, for G a = P f(v t ; )=@ ; =^a ^V a is de ned according to ^V a = G a W (^ a ) G a i G a W (^ a ) est ^a W (^ a ) G a i G a W (^ a ) G a i ; wic is a standard variance estimator for ^ a : De ne W = U W U and W = =2 W ( =2 ) W W = 2 W 2 W 22 2

21 and a = W 2 W 22 : Using te same argument for proving Lemma 9, we can sow tat ( 2) =2 A V p (^ a ) = p X t= [f (v t ; ) a f 2 (v t ; )] + o p (): (22) e above representation is te same as tat in (9) except tat is now replaced by a : De ne ~ a according to a = =2~ 2 a = : at is, ~ a = =2 2 ( a ) =2 22 = =2 2 a =2 22 I d =2 : (23) e relationsip between a and ~ a is entirely analogous to tat between and ~ ; see (3). In fact, if we let ~W = =2 W =2 ~W W2 ~ = ~W 2 W22 ~ ; ten ~ a = W ~ 2 W ~ 22 ; wic is te long run regression matrix implied by te normalized weigting matrix W ~ : Let V a be te long run variance of R ~ [f (v t ; ) a f 2 (v t ; )] and a;r = Va =2 ~R (2 a 22 )i =2 22 ; be te long run correlation matrix between R ~ [f (v t ; ) R = I d ; a;r reduces to a f 2 (v t ; )] and f 2 (v t ; ) : Wen Let a = + a 22 a a 2 2 =2 a (2 a 22 ) =2 22 : ~R a = RV A ( 2) =2 =2 2 = R ~ =2 2 and D ~ =2 R = ~Ra R ~ ~Ra a = ~R2 R ~ =2 ~R =2 2 : ~D R as te same dimension as ~ R a and R: By construction, ~ DR ~ D R = I p, tat is, eac row of ~ D R is a unit vector in R d and te rows of ~ D R are ortogonal to eac oter. e matrix ~ D R signi es te ortogonal directions embodied in ~ R a : Wit tese additional notations, we are ready to state te teorem below. eorem Let te assumptions in Lemma 9 old. Assume furter tat W (^ )! p positive de nite nonrandom matrix. (a) If E ~ (; d; q) ~ (; d; q) a ~ ~ a is positive (negative) semide nite, ten ^ 2 as a larger (smaller) asymptotic variance tan ^ a ; as! for a xed : (b) If E ~ (; p; q) ~ (; p; q) ~ DR ~ a ~ a ~ D R is positive (negative) semide nite, ten R^ 2 as a larger (smaller) asymptotic variance tan R^ a ; as! for a xed : (c) If g(; d) I d a a is positive (negative) semide nite, ten ^ 2 as a larger (smaller) asymptotic variance tan ^ a ; as! for a xed : (d) If g(; d)i p a;r a;r is positive (negative) semide nite, ten R^ 2 as a larger (smaller) asymptotic variance tan R^ a ; as! for a xed : (e) Let a = jjva =2 jj 2 : If f ( a ; ; p; q; ) I p a;r a;r ten te test based on W 2! for a xed : W, a is positive (negative) semide nite, is asymptotically less (more) powerful tan tat based on W a ; as 2

22 eorem (a) is intuitive. Wile E ~ (; d; q) ~ (; d; q) can be regarded as te variance in ation arising from estimating te long run regression matrix, ~ ~ a a can be regarded as te bias e ect from not using a true long run regression matrix. Wen te variance in ation e ect dominates te bias e ect, te two-step estimator ^ 2 will ave a larger asymptotic variance tan ^a : In te special case wen W = ; we obtain te infeasible two-step estimator ~ 2 and ~ a = : In tis case, E ~ (; d; q) ~ (; d; q) ~ ~ a a olds trivially and te feasible two-step estimator ^2 as iger variation tan its infeasible version ~ 2 : In anoter special case wen W = ; we obtain te rst step estimator ^ in wic case a = and by (23), ~ a = (I d ) =2 : Hence ~ a ~ a = I d =2 I d =2 : e condition in eorem (a) reduces to weter g(; q) I q not. eorem (a) tus reduces to eorem (a). is positive semide nite or eorem (b) is a rotated version of eorem (a). Given tat ~ D R ~ (; d; q) d = ~ (; p; q); te condition in eorem (a) implies tat in eorem (b). So part (b) of te teorem is stronger tan part (a) wen R is not a square matrix. Wen W = ; we ave a = and so ~D R ~ a ~ a ~ D R = = = ~R2 R ~ =2 i i ~R =2 ~a ~ 2 ~R =2 ~R2 ~ =2 a 2 R ~R2 R ~ =2 ~R (a ) 22 f( a )g R ~ i R2 ~ R ~ =2 ~R2 ~ R =2 ~R ~ R i ~ R2 ~ R =2 = I p R R =2 R R Ip R =2 R : As a result, te condition E ~ (; p; q) ~ (; p; q) D ~ R a ~ ~ ~ a DR in eorem (b) is equivalent to E ~ (; p; q) ~ (; p; q) I p R =2 R R R Ip R =2 R ; wic can be reduced to te same condition in eorem (b). eorem (c)-(e) is entirely analogous to eorem (a)-(c). e only di erence is tat te second block of moment conditions is removed from te rst block using te implied matrix coe cient a before computing te long run correlation coe cient. o understand (e), we can see tat te e ective moment conditions beind R^ a are: Ef a (v t ; ) = for f a (v t ; ) = RV A ( 2) =2 [f (v t ; ) a f 2 (v t ; )] : R^ a uses te information in Ef 2 (v t ; ) = to some extent, but it ignores te residual information tat is still potentially available from Ef 2 (v t ; ) = : In contrast, R^ 2 attempts to explore te residual information. If tere is no long run correlation between f a (v t ; ) and f 2 (v t ; ) ; i.e., ar = ; ten all te information in Ef 2 (v t ; ) = as been fully captured by te e ective moment conditions underlying R^ a : As a result, te test based on R^ a necessarily outperforms tat based on R^ 2 : If te long run correlation ar is large enoug in te sense tat is given in (e), te test based on R^ 2 could be more powerful tan tat based on R^ a in large samples. 22

23 5.3 Practical Recommendation eorems and give teoretical comparisons of various procedures. But tere are some unknown quantities in te two eorem. In practice, given te set of moment conditions Ef(v t ; ) = and te data fv t g ; suppose tat we want to test H : R = r against R 6= r for some R 2 R pd We can follow te steps below to evaluate te relative merit of one-step and two-step testing procedures.. Compute te initial estimator ^ P = arg min t= f(v t ; ) 2 : 2. One te basis of ^ ; use a data-driven metod to select te smooting parameter: Denote te data-driven value by ^: 3. Compute est (^ ) and est (^ ) using te formulae in (4) and smooting parameter f(v t;) 4. Compute G (^ ) = P t= =^ and its singular value decomposition were ^ = ( ^A dd ; O dq ) and ^A dd is diagonal. ^U ^ ^V 5. Estimate te variance and te long run variance of te rotated moment processes by ^ := ^U est (^ ) ^U and ^ := ^U est (^ ) ^U. 6. Compute te normalized LRV estimator: ^ ^ = (^ =2 ) ^ (^ =2 ) := ^ 2 ^2 ^22 were ^ =2 = ^ 2 =2 ^ 2 ^ 22 =2 ^ 22 =2 C A : (24) 7. Let ~ R est = R ^V ^A (^ 2 )=2 and compute ^ R^ R = ( ~ R est ^ ~ R est ) =2i ~ Rest ^2 ^ 22 ^ 2 ~ R est i ( ~ R est ^ ~ R est ) =2i and its largest eigenvalue v max ^ R^ R and smallest eigenvalue vmin ^ R^ R : 8. Coose te value of o suc tat P 2 p () > p = 75%. We may also coose a value of to re ect scienti c interest or economic signi cance. 9. (a) If v min ^ R^ R > f o ; ^; p; q;, ten we use te second-step testing procedure based on W 2 : (b) If v max ^ R^ R < f o ; ^; p; q; ; ten we use te rst-step testing procedure based on W : (c) If eiter te condition in (a) or in (b) is satis ed, ten we use te rst-step testing procedure based on W a : 23

24 6 Simulation Evidence is section compares te nite sample performances of one-step and two-step estimators and tests using te xed-smooting approximation. 6. Point Estimation We consider te location model given in () wit te true parameter value = (; :::; ) 2 R d but we allow for a nonscalar error variance. e error fu t g follows a VAR() process: u i t = u i t u i 2t = u i 2t u j 2t j= + p q qx + e i 2t for i = ; :::; q + ei t for i = ; :::; d (25) were e i t iid N(; ) across i and t, ei 2t iid N(; ) across i and t; and fe t ; t = ; 2; :::; g are independent of fe 2t ; t = ; 2; :::; g : Let u t := (u t ; u 2t ) 2 R m, ten u t = u t + e t were mm = I d p q J d;q I q! ; e t = et e 2t iid N (; I m ) : Direct calculations give us te expressions for te long run and contemporaneous variances of fu t g as = = X Eu t u t j = (I m ) (I m ) I ( ) 2 d + 2 ( ) 3 p q J q;d I ( ) 2 q J ( ) 4 d ( ) 3 p q J d;q A and = var(u t ) = I 2 d + 2 (+ 2 ) J ( 2 ) 3 d pq J ( 2 ) 2 dq p q ( 2 ) 2 J qd 2 I q C A ; were J d;q is te d q matrix of ones. In our simulation, we x te value of at :75 i so tat eac time series is reasonably persistent. Let u t = ( 2 ) =2 u t 2 ( 22 ) u 2t and u 2t = ( 22 ) =2 u 2t and be te long run correlation matrix of u t = (u t ; u 2t ) : Wit some algebraic manipulations, we ave = d + ( 2 ) 2 J d : 2 So te maximum eigenvalue of is given by v max ( ) = + ( 2 ) 2 =(d 2 ), wic is also te only nonzero eigenvalue. Obviously, v max ( ) is increasing in 2 for any given and d: Given and d, we coose te value of to get a di erent value of v max ( ): We consider v max ( ) = ; :9; :8; :::; :9; :99 wic are obtained by setting 2 = v max ( )( 2 ) 2 = (d( v max ( )) : 24