Bootstrap Inference for Impulse Response Functions in Factor-Augmented Vector Autoregressions

Transcription

1 Bootstra Inference for Imulse Resonse Functions in Factor-Augmented Vector Autoregressions Yohei Yamamoto y University of Alberta, School of Business February 2010 Abstract his aer investigates standard residual-based bootstra rocedures to obtain con- dence intervals for the imulse resonse functions (IRFs) in factor-augmented vector autoregressions (FAVARs) for which the latent factors are estimated by a two-ste rincial comonents method. he main contribution is that we comare two methods : A. bootstra with factor estimation and B. bootstra without factor estimation. A set of Monte Carlo simulations indicate that the latter has a defect in its coverage roerties while the former yields good results in general. he results suggest that the uncertainty associated with latent factor estimation is not always negligible in ractice. he imortance of identi cation of imulse resonse functions is revisited in order to achieve a right rocedure. A roof of the asymtotic validity of the suggested rocedure A. is rovided. JEL Classi cation Number: C14, C22 Keywords: factor models, identi cation, rincial comonents, coverage ratio I am grateful to Pierre Perron, Zhongjun Qu, Ivan Fernandez-Val, and Simon Gilchrist for helful suggestions. I also thank Adam McClosky and Vladimir Yankov for discussions in detail. All remaining errors are solely my resonsibility. y University of Alberta School of Business, 2-37 Business Building, University of Alberta, Edmonton, AB, Canada 6G 2R6 (yohei.yamamoto@ualberta.ca).

2 1 Introduction Fast-growing attention has recently been aid to factor analysis in the elds of macroeconomics and nance. Factor models essentially cature comovements of large data sets with resect to a handful of common latent factors and these models have become increasingly useful due to the recent usurge of comutation facility and data availability, often referred to as the "data-rich environment" (Bernanke and Boivin, 2003). One of the breakthroughs is the factor-augmented vector autoregression (FAVAR), which uses the vector autoregression (VAR) framework to analyse the time series characteristics of the common factors reresenting such a large number of data series as cannot directly be accommodated by VARs. We have lately seen several theoretical and emirical works directing this research in a romising direction. Stock and Watson (2005) rovide a comrehensive summary of ongoing FAVAR modeling and estimation. In olicy analysis, Bernanke and Boivin (2003), Bernanke, Boivin and Eliasz (2005) and Giannone, Reichlin and Sala (2004) aly the technique to US macroeconomic data series and nd the device is useful to monetary olicy analysis. For nance and asset ricing alications, Ang and Piazzesi (2003) incororate a multifactor a ne term structure model, Ludvigson and Ng (2008) investigate the e ects of macro factors on bond market remia uctuations and Gilchrist, Yankov and Zakraj¼sek (2008) analyze the imact of credit sreads on economic activity. his aer exlores an inference scheme for FAVARs when the latent factors are estimated by the static rincial comonents (PC) method, although one of the most controversial issues in the factor analysis literature has been how to extract the latent factors from the underlying data. Among many cometing methods of factor extraction 1, static PC estimation is one of the simlest algorithms to imlement and a great deal of rogress regarding estimation under large N (the number of equations in the model) and (the number of observations er equation) asymtotics has been made recently. For instance, Stock and Watson (1998, 2002) rovide consistency results for estimated factors and loadings, u to some normalization, Bai (2003) develos asymtotic normal inferential theory for estimated factors and loadings and Bai and Ng (2006) extend these results to dynamic factor models, including FAVAR seci cations. 1 In contrast to the static PC method, Foni, Hallin, Lii and Reichlin (2000, 2005) suggest estimating dynamic factors using a dynamic PC technique. Although the static PC method can deal with dynamic factors by stacking them in one vectors, the relative merits of the two methods remains an oen question. Several maximum likelihood-tye estimation rocedures using state sace models have also been roosed. See Kaetanios and Marcellino (2003) for examle. 1

3 Based on these develoments, the goal of this aer is to investigate widely-used standard residual-based bootstra methods for imulse resonse functions (IRFs). Not to mention, bootstra inference is often believed to generate more reliable con dence intervals for IRFs than normal asymtotic aroximations in the context of VAR analysis (see Runkle, 1987 and Kilian, 1998, 1999, for examle). For factor-augmented versions, the bootstraing framework rovides an additional bene t because an existing Gaussian inference under large N and does not account for factor estimation uncertainty and treats factor estimates as factual. As a result, these inference techniques may lead to overly otimistic con dence intervals. In rincile, true latent factors cannot be extracted when the data set is naturally nite number, so the risk of overlooking this uncertainty, given relevant samle sizes, should not be ignored. In this sense, the advantages of bootstra inference relative to the Gaussian inference are intensi ed in the factor-augmented setting. his aer aims to comensate for some of the ractical shortcomings of existing inferential theory by roosing a relevant bootstra rocedure. On the other hand, the rocedure this aer rooses involves factor estimation in each bootstra relication. his enables us to cature the disersion of the estimates for which the uncertainty associated with factor estimation is fully accounted. o this end, we rst review the models and identi cation roblems for structural FAVAR models. here are two tyes of identi cation issues: 1) regarding to the individual arameters instead of a sace sanned by factors and 2) regarding to the structural relationshis among variables. Several examles of identi cation schemes including what were introduced by Stock and Watson (2005) deal with the second roblem, however, the rst roblem has not been throughly discussed as far as the author knows. More imortantly, we secify the two bootstra methods: A. bootstra with factor estimation and B. bootstra without factor estimation and nite samle roerties of both methods are investigated via Monte Carlo simulations with arti cial and US macroeconomic data. he distinction of this two methods is not often mentioned in literature artly because estimated factors can be asymtotically treated as true factors as shown in Bai (2003) and Bai and Ng (2006) under some conditions. However, our results show that the latter rocedure will have a defect in its coverage roerties and it suggests that neglecting uncertainty associated with factor estimation is can lead overly otimistic inference results. his seemingly indeendent two roblems have a common thread to which a caution has to be rovided. When we conduct bootstraing with factor estimation, the individual factor and arameter estimates must face accidental rotation since the factor models can 2

4 identify only a sace sanned by factors, not individual factors and arameters. Hence if the IRF is not well identi ed in this sense, then estimates of the IRF in each relication of the bootstra rocedure is di erently rotated and this results in senseless con dence intervals. With these caveats in mind, this aer con rms a relevant inference rocedure for identi ed IRFs in terms of coverage roerties. he rest of the aer is structured as follows. Section 2 introduces the models and IRFs and discuss issues of identi cation. Section 3 describes the estimation method and examles of identi cation schemes in ractice introduced by Stock and Watson (2005). In section 4, we roose two bootstra methods while the asymtotic validity of the rocedure is shown in section 5. Section 6 assesses the rocedures nite samle roerties via simulations using arti cial data as well as calibrated models of US macroeconomic data. Section 8 is comosed of concluding remarks and the aendices include some technical derivations and details on a bootstra bias-correction rocedure. 2 he model and identi cation 2.1 Reduced-form models Consider a data generating rocess: X t = F t + u t ; t = 1; :::; (1) where X t is an N 1 vector of observations and N is the (tyically large) number of equations. We assume that X t is driven by much lower dimensional unobservable factors F t (r 1; N r) with time-invariant unobservable factor loadings (N r). u t is an N 1 idiosyncratic shock. We call (1) the observation equation. More generally, autocorrelations of the observables can be accounted by adding lagged values of X t in the right hand side of (1). However, estimates of the arameters attached to the lagged X t are well-behaved 2 and adding this comonent does not change the main result of this aer. Hence we kee this simle model. In addition, the factors F t form a vector autoregression with coe cient arameters (L) of order and an error term e t (r 1) so that F t = (L)F t 1 + e t : (2) 2 hat is, the arameters attached to the lagged X t are identi ed and their consistency and asymtotic normality hold. See Bai and Ng (2006). 3

5 Equation (2) is called the VAR equation. Variables written without their associated "t" subscrit are meant to denote the entire matrix of observations, for examle, X = [X 1 ::X ] 0 is a N matrix and F is a r matrix. hroughout the aer, we also assume that the number of factors r and the VAR lag order are known in order to construct valid asymtotic coverage robabilities and simlify the results. the consistent estimation of the number of factors. 3 here are several methods available for Inference accounting for uncertainty associated with has also been exlored in, for examle, Kilian (2001), but this issue is beyond the scoe of this aer and will not be discussed. 2.2 Structural models Structural VARs are much more oularly used to consider the contemoraneous relationshis between variables of interest in macroeconomic alications. FAVARs are also able to identify most of these relationshis given a articular interretation of the models (see Bernanke, Boivin and Eliasz, 2005 and Kaetanios and Marcellino, 2006 for examles). Stock and Watson (2005) give a comrehensive modeling strategy, hence we take their lead. Using a r r invertible matrix P; let the structural factor model be de ned as X t = F t + u t ; (3) F t = (L)F t 1 + t ; (4) where = P, Ft = P 1 F t, (L) = P 1 (L)P and t is a "structural innovation" for which E( t 0 t) = I is assumed. As we will see later in detail, estimation for the structural model is conducted using the reduced-form reresentation obtained by multilying (4) by the matrix P from left so that we get (2) where e t = P t as is common in standard structural VARs. 2.3 Imulse resonse functions Next we consider the IRF of variable i to the VAR innovations in both reduced-form and structural models. he model comosed of (1) and (2) can be rewritten in vector moving average form so that X i;t = i (L)e t + u t ; 3 Recent examles include Bai and Ng (2002) for static factors and Onatski (2007), Hallin and Li¼ska (2007) and Bai and Ng (2007) for dynamic factors. 4

6 where i is the ith row of and (L) P 1 i=0 hl h is de ned as the moving average olynomial associated with F t such that F t = (L)e t and (L) = [ I (L)] 1. hen First, let the reduced-form IRF of observable i at time horizon h (h = 0; 1; 2; :::) be i;h. t = i h: Second, the structural IRFs ' i;h will be similarly de ned based on the model comosed of (3) and (4). It can be straightforwardly shown that ' t = i h: where (L) P 1 i=0 h L h = [I (L)] = P 1 (L)P: 1 ; 2.4 Princiles of identi cation It is widely known that in factor models, the factors and loadings are identi ed only u to scale and ermutation. In terms of FAVAR analysis, we here discuss rinciles of how to identify the factors and IRFs of the models resented in the revious subsections. Consider a simle two factor model: X t = 1 f 1;t + 2 f 2;t + u t ; f 1;t = 1 f 1;t 1 + e 1;t ; f 2;t = 2 f 2;t 1 + e 2;t ; where only X t (N 1) are observable, factors f i;t and autoregressive arameters i are scalars, loadings i are N 1 vectors, u t is an N 1 error vector and e i;t are scalars (i = 1; 2). his simle model rovides a good illustration of the identi cation roblem shared by models described by (1) and (2). here are two ingredients for this model to be identi ed. he rst relates to an indeterminancy roblem between the factors and loadings. More seci cally, for any non-zero constant a, the counterfactual set (af 1;t ; 1 a 1 ; a 1 a 1 ; ae 1;t ) yields the same data generating rocess as the true set (f 1;t ; 1 ; 1 ; e 1;t ). In this sense, a "scaling" for the latent comonents is necessary to achieve identi cation. We address this roblem 5

7 by restricting [e 1;t ; e 2;t ] to be serially uncorrelated and uncorrelated with each other and have unit variances. he second ingredient relates to the fact that the model involves two (ossibly more) latent rocesses seci ed symmetrically. herefore one must "label" which latent factor is the rst or second to determine which loading, 1 or 2, to attach to which factor. o address this roblem, we imose exclusion restrictions, tyically taking the form of a recursive structure between f 1;t and f 2;t in this setting. For examle, some variables in X t are a ected by the rst factor and not the second. Once these two roblems have been addressed, the arameters, factors and IRFs can be identi ed. For the reduced-form model (1) and (2), the indeterminancy roblem arises since, for any invertible r r matrix a set (HF t ; H 1 ; HH 1 ; He t ) yields exactly the same observations X t, hence one needs to set H a riori to some value. However, in structural models, the orthogonality condition of the errors E( t 0 t) = I restricts them to be freely scaled so that H is uniquely determined as the identity matrix. We also assume the existence of some recursive structure, which corresonds to labeling, to identify the structural relationshis. 3 Estimation Just like standard VARs, structural FAVAR models are estimated based on reduced-form models. Once the arameters and residuals of the reduced-form model are obtained, the matrix P can be identi ed under the assumtions of orthogonal structural innovations and recursive structures on the variables. his section revisits how the arameters and matrix P are estimated in ractice. 3.1 Estimation of reduced-form models he reduced-form model is estimated by the following two-ste PC rocedure. In the rst ste, extract the factors using the PC method. his is imlemented by nding the solutions ( ^F ; ^) P = arg min N P ;F i=1 t=1 [X i;t i F t ] 0 [X i;t i F t ]: (5) Once again, this roblem is not uniquely solvable since with any r r invertible matrix H, H 1 and HF are also solutions of (5). However, with the normalization F 0 F= = I, the rincial comonents method yields that the estimates of F t which are the eigenvectors of X 0 X=(N ) corresonding to the r largest eigenvalues (multilied by ). hen ^ and ^u t are obtained by regressing X t on ^F t. In the second ste, the VAR equation for ^F t is estimated using standard least squares rocedures. his yields the estimates ^ and ^e t. 6

8 Although the normalization F 0 F= = I xes the scale of the latent factors, this restriction should not be satis ed. Also the labelling is not seci ed so that the factors are identi ed only u to ermutation. his results in the fact that estimating the factors F as the rincial comonents only allows one to estimate the sace sanned by latent factors. he articular H after normalizing F 0 F= = I is H N = V 1 N ( ^F 0 F= )( 0 =N) where V N is a diagonal matrix with the diagonal elements being the r largest eigenvalues of X 0 X=(N ) in descending order 4. Hence the following facts directly obtained from Bai and Ng (2006) are worth revisiting. Lemma 1 Using two-ste PC estimation method with the normalization F 0 F= = I, the arameters, factors and IRFs of reduced-form models (1) and (2) yield estimates such that ^ i i H 1 N = o (1); ^ H N H 1 N = o (1); ^F t H N F t = o (1); and ^i;h i;hh 1 = o (1); 8i; t and uniformly in h = 0; 1; 2; ::: where H N = V 1 N ( ^F 0 F= )( 0 =N). N It is noted that, although we know the form of H N uon the normalization F 0 F= = I, the actual value deends on the realized unobservable rocesses. his results in a situation where the researcher does not exactly know the true counterarts of their estimates 5. A di erent roblem with the same root cause is sometimes called "re ection" or "axis reversal" for which researchers cannot control the signs of the estimated rincial comonents. Under common but more strict assumtions ( 0 =N = I and a diagonal F ), Stock and Watson (2002) show that the limit of H N becomes a diagonal matrix with its diagonal comonents +1 or 1. One way or another, the fact that H N deends on articular realizations of the data creates a roblem for bootstraing when PC estimation of latent factors is conducted in each relication. his is because the entries of H N are randomly varied in each relication, creating additional variability in arameter estimates and in IRF estimates. In this sense, the imortance of identi cation of individual arameters cannot be overemhasized. 4 See Bai (2003). 5 A classic work of Cattell (1978) called it "accidental" rotation. 7

9 Remark 1 In a secial case of orthogonal factor analyses, the restriction F 0 F= = I is imosed in the DGP so that the normalization F 0 F= = I is consistent with the true DGPs. However, in VAR analyses, assuming orthogonal factors should be too restrictive to accommodate general time series attern of shocks and the common factors are to be allowed to follow any autoregressive rocesses of researchers interest. Hence we do not assume F 0 F= = I in the DGP and it is the case in most recent literature including Bai and Ng (2002) among others (see Assumtion 1). 3.2 Estimation of identi ed structural models he task here is to identify the arameters P in addition to reduced-form arameters. Before we discuss the details of the identi cation schemes, the following reasoning is articularly imortant to note. he two-ste rocedure yields factor and arameter estimates which have the roerties given in Lemma 1 and residuals have the roerty that ^e t H N e t ( means that the normed di erence between them is o (1) here and hereafter). his fact indicates that secifying an estimate ^P so that ^P H N P is required to estimate the factors and arameters in the structural models. We thus obtain the following lemma. Lemma 2 Let P be de ned in (3) and (4) and H N is an r r matrix such that ^P HN P = o (1); as de ned in section 3-1. Suose ^P holds when N;! 1. hen the arameters and factors in the structural model are consistently estimated. Moreover, the structural IRF is consistently estimated. hat is, ^ i i = o (1); ^ = o (1); ^F t Ft = o (1); and ^' i;h ' i;h = o (1); 8 i; t and uniformly in h = 0; 1; 2; ::: where ^ i = ^ i ^P, ^ = ^P 1 ^ ^P, ^F t ^' i;h = ^ ^P i;h. = ^P 1 ^Ft and 8

10 Note that as long as H N is obtained, the arameter and IRF estimates do not have rotation terms. In ractice, the matrix P together with H N can be estimated in several ways. Now we review two identi cation schemes for H N P in factor models. hese are considered by Stock and Watson (2005). Examle 1: Cholesky factorization tye identi cation First, consider the case where it is known a riori that some observations are not a ected by certain factors at the same time and that the uer r (number of factors) rows of ' 0 form a lower triangular matrix such that 2 3 x 0 0 x x 0 ' 0 = = P = x x x : x x x he assumtion is justi ed, for examle, when the rst factor is a main source of stock rice uctuations but the factor does not contemoraneously a ect non- nancial variables, such as consumtion or roduction. Given this restriction, estimation is relatively easily imlemented. he reduced-form VAR residuals are the estimates of H N e t H N P t so that ^ 1:r^e t simly becomes consistent estimates of 1:r P t, where 1:r denotes the uer r rows of. With the assumtion E( t 0 t) = I in mind, the Cholesky factorization of ^ 1:r [( P ^e t^e 0 t)= ] ^ 0 1:r gives a consistent estimate of 1:r P because where ^ = = = P t=1 ^e t^e 0 t = and ^P = ^ 1 1:r[Chol(^ 1:r ^^0 1:r )]; ^ 1:r[Chol(^ 1 1:r ^^0 1:r )] H N 1:r[Chol( 1 1:r H 1 N H N P P 0 HN 0 H 10 ^ 1:r[Chol(^ 1 1:r ^^0 1:r )] ^ 1 1:r[Chol(^ 1:r ^^0 1:r )] H N 1 1:r[Chol( 1:r P {z } H N P = o (1): triangular Hence, this will yield an estimate ^P which has a roerty P 0 0 1:r)] ; ^P N 0 1:r)] ; H N P = o (1). 9

11 Remark 2 If a recursive structure among the factors is assumed so that the rst factor is not contemoraneously a ected by the second factor and so on, then P is assumed to be lower triangular and ^P = Chol(^) Chol(H N P P 0 HN 0 ). However, it is not always ossible to recover ^P in this way as an estimate of H N P since it is not tyically a triangular matrix even though P is known to be triangular. Examle 2: Block triangular tye identi cation Consider now an r factor model and assume that the observations X t are divided into r grous [X 1;t ; : : : ; X r;t ]; where X i;t is contemoraneously a ected only by F j;t (j i). In a two-factor model, the observation equation takes the form X 1;t 5 = F 1;t u 1;t 5 ; X 2;t F2;t u 2;t = F 1;t ;t5 + 4 u 1;t 5 : 22 F2;t ;t u 2;t his assumtion is relevant to many economic situations for which the observations can be classi ed into two (or ossibly more) grous, for examle, nancial variables and non- nancial variables. In these cases, the following rocedure can be used to estimate H N P. First generate residuals v 1;t and v 2;t by regressing [X 1;t ; X 2;t ] on the rst lag of the estimated factors [ ^F 1;t 1 ; ^F 2;t 1 ] yields the following relationshis: v 1;t = 11 1;t + u 1;t ; (6) v 2;t = 21 1;t ;t + u 2;t : (7) where the structural innovations are ;t5 = 4 B 1 5 e t ; with 2;t B B 1 5 B 2 1 = P; hen (6) can be rewritten as v 1;t = 11B 1 e t + u 1;t : Since the estimates ^v 1;t v 1;t and ^e t H e t are attainable via reduced-form estimation, the reduced rank regression of ^v 1;t on ^e t will yield estimates of the structural innovations such that d B 1 e t 1;t ( 1;t is of unit rank by de nition) and ^ 11 ^B1 is consistent for 11 B 1. Once ^ 1;t is obtained, ^ 2;t can also be subsequently estimated using (7). Finally, regressing ^e t on ^ t yields an estimate ^P H N P since ^e t H N e t and ^ t t resectively. 10

12 4 Bootstra inferences his section introduces two residual-based bootstra rocedures to construct con dence intervals for the IRFs: A. bootstraing with factor estimation and B. bootstraing without factor estimation. he main feature of the Procedure A is that it includes factor estimation within each bootstra relication so that the generated con dence intervals are exected to roerly account for the uncertainty associated with factor estimation. One caution has to be rovided. When we conduct the bootstra with factor estimation, then the factor and arameter estimates always face accidental rotation since the factor models can identify only saces sanned by factors, not individual factors and arameters. Hence if the IRF is not well identi ed, then each bootstra estimate of IRF is di erently rotated and this results in senseless con dence intervals. Procedure A : Bootstraing with factor estimation 1. Estimate the model by the two-ste PC rocedure and obtain arameter estimates ^; ^; ^P and residuals ^ut and ^e t. Obtain the IRF estimate ^' i : 2. Resamle the demeaned residuals ^e t e with relacement and label it e b t. Generate the bootstraed samle Ft b by Ft b = ^(L)F t b 1 + e b t with the initial condition Fj b = ^F j for j = 1; ;. 6 Also resamle the demeaned residuals ^u t u with relacement and label it u b t. Generate the bootstraed observations X b by Xt b = ^F t b + u b t Using the bootstraed observation X b t, estimate ( ^F b ; ^ b ) and estimate the VAR equation of ^F b t to obtain the bootstraed estimate ^ b and ^R b as seci ed in section 3. his yields the bootstra IRF estimates ^' b i. 4. Obtain the re-centered statistics s ^' b i ^' i. 5. Reeat 2) - 4), R times and store s (1) ; :::; s (R). Sort the statistics and ick the ercentiles (s L ; s U ) according to the seci ed nominal level. he resulting bootstra con dence interval is then [^' i s U ; ^' i s L ]. Intuitively, the basic features that allow this rocedure to work well are the following. In ste 2, the observations of the bootstra samle X b t shares the same data generating rocess 6 At this stage, some tyes of bias correction methods can be alied such as the bootstra after bootstra discussed in Kilian (1998). See Aendix B. 7 When the model includes olicy variables Y t in VAR, let Ft b be (Ft b ; Yt b ) and ^F t be ( ^F t ; Y t ). 11

13 as the original samle X t. In ste 3, the bootstra estimate involves the same estimation methods as the original. Given these two observations, the bootstraed estimates should exhibit disersions which mimic those of the original estimates. Procedure B : Bootstraing without factor estimation Second, we consider the rocedure B. bootstraing without factor estimation. It actually needs a modi cation only in the ste three in the rocedure A and is formalized as follows. 3. Using the bootstraed observation X b t, estimate ^ b, ^ b and ^P b using F b t. his yields the bootstra IRF estimates ^' b i. he other stes are ket the same as those in Procedure A. his rocedure can be regarded as a natural and simle extension of the methods conducted in standard vector autoregression analyses where the variables forming vector autoregressions are all observable hence easier to imlement. Also the accidental rotation caused by H N does not show u in bootstra relication because no factor estimations are involved in bootstra relications so that the identi cation roblem does not a ect the con dence intervals in contrast to the revious rocedure. However, more imortantly, the features include the fact that the generated con dence intervals do not take the nite samle uncertainty associated with factor estimation into account. 5 Asymtotic results his section investigates the asymtotic validity of the roosed bootstra rocedure. o this end, we extensively use the asymtotic normality results develoed by Bai and Ng (2006). We initially assume a set of standard assumtions. Assumtion 1 he common factor F t satis es E kf t k 4 M, a nite matrix, and P 1 t=1 F tft 0! F > 0, a nonrandom ositive de nite matrix. Assumtion 2 he factor loading, ; is either deterministic, such that k i k M; or stochastic, such that k i k 4 < M < 1; where i is the ith row of. In either case, N 1 0! > 0, a nonrandom ositive de nite matrix. Assumtion 3 F t and u t satisfy the following conditions: u t are identically and indeendently distributed in t. 12

14 E(u i;t ) = 0; E ju i;t j 8 M < 1: E(u i;t; u j;t ) = ij and j ij j ij for all i; j such that N 1 P N i;j=1 i;j M. P 1=2 t=1 F tu 0 d i;t! N(0; ui ) where ui = lim P 1 t=1 (u2 i;tf t Ft) 0 > 0. Assumtion 4 F t and e t satisfy the following conditions: e t are identically and indeendently distributed in t. E(e t ) = 0 and E(e t e 0 t) =, a time invariant matrix, and E je i;t j 8 M < 1: P 1=2 t=1 F te 0 d i;t! N(0; ei ) where ei = lim P 1 t=1 (e2 i;tf t Ft) 0 > 0. Assumtion 5 he eigenvalues of the r r matrix F are distinct. Assumtions 1 and 2 allow general rocess for the factors and loadings and are the same as used elsewhere in the literature. Assumtion 3 is stronger than what is commonly seci ed in the sense that we do not allow autocorrelation and heteroskedasticity in the idiosyncratic errors. his assumtion is necessary here for the consistency of the i.i.d. bootstra estimates roosed above. It could, however, be relaxed by alying other versions of the bootstra which are robust to autocorrelation and/or heteroskedasticity. Assumtion 4 is standard in the VAR literature. Assumtion 5 guarantees the uniqueness of the limit of ^F 0 F=; discussed later. Next, we introduce a su cient condition that guarantees the identi cation of the structural model. Condition 1 Let the models (3) and (4) be estimated by the two-ste PC method with normalization of F 0 F= = I. Let ^P be the estimate of the structural relationshi by underlying restrictions. hen the following holds as N;! 1. ^P H N P = o (1); where H N = V 1 N ( ^F 0 F= )( 0 =N) with V N eigenvalues of X 0 X=(N ) in descending order. a diagonal matrix of its elements r largest 13

15 Asymtotic normality for the individual arameters out of the reduced-form model was derived in Bai and Ng (2006). Based on their results, we establish asymtotic normality for structural arameters de ned in (3) and (4). heorem 1 (Parameters for the structural model) Under Assumtions 1-5 and Condition 1, the followings hold for the two-ste PC estimators of and in (3) and (4): (^ i as N;! 1 and =N! 0 where vec(^ i ) d! N(0; i); i = P 1 F ui 1 F P 0 ; ) d! N(0; ); = ( P 0 P 1 ) ( P 10 P ): where ^ i and ^ are de ned in Lemma 2 and P HP with H de ned in Lemma A1 in aendix. Next, we resent the asymtotic distribution of the IRFs. heorem 2 (Structural IRFs) Let ^' i;h be the two-ste PC estimates of the structural IRFs of ith observation at delay h. Under Assumtions 1-5 and Condition 1, d ^'i;h ' i;h! N(0; 'i;h ); 8i uniformly in h = 0; 1; 2; as ; N! 1 and =N! 0 h =@ 6= 0 where = [ i vec( )] and and = diag( i ; ). 'i;h @ 0 ; Finally, we move on to the asymtotic validity of the bootstra rocedure. In articular, we show that the asymtotic distributions of the original estimators and that of bootstra estimators conditional on any given samle are equivalent under N;! 1 and =N! 0. 14

16 heorem 3 (Consistency of the rocedure A) Let ' i;h be the structural IRFs of ith observation at delay h and let ^' i;h and ^' b i;h be the original estimate and bootstra estimate of structural IRFs resectively. Under Assumtions 1-5 and Condition 1, d ^'i;h ' i;h! N(0; 'i;h ); and ^' b i;h ^' i;h d! N(0; 'i;h ); conditionally on the original samle, 8i and uniformly in h = 0; 1; 2; as N;! 1, and =N! 0, where 'i;h is de ned in heorem 2. he issue is now whether the bootstra rocedure has good nite samle roerties. In this vain, we conduct simle Monte Carlo simulations including an exeriment calibrated to emirical data in the next section. 6 Finite samle roerties 6.1 Monte Carlo simulations In this subsection, we rovide simulation results to assess the nite samle roerties of the roosed bootstra rocedures. For simlicity we consider a two-factor model with VAR(1), however, the results are qualitatively robust when we use several di erent yet reasonable seci cations. he observable variables x i;t are generated as x i;t = i f t + u i;t ; and the factors (f t : 2 1) evolve such that f t = f t 1 + e t ; for i = 1; :::; N and t = 1; :::; with i = [ i;1 ; i;2 ] a 1 2 vector and is a 2 2 matrix. We generate quasi-random variables e j;t (j = 1; 2) and u i;t as indeendently and identically distributed. o assess the robustness of the results to the distributional seci cations, we consider both errors following the standard normal distribution and errors following a centered chi-squared distribution with one degree of freedom. 15

17 We study the structural IRF to a unit shock to the rst factor. As for the arameter, we secify 2 3 0:4 = 4 0:2 5 ; 0:2 0:4 he individual elements of i indeendently and dentically follow the uniform distribution U[0; 1]. Since the e ect of the samle size on the results is of major interest, we comare the results of four (N; ) combinations of N = f50; 200g and = f40; 120g. he bootstra inference is conducted for 1; 000 relications and con dence intervals at the 95% nominal levels are reorted. he bias correction 8 in the sirit of Kilian (1998) is alied where the bias for ^ is estimated by another R b = 1; 000 times bootstra loo and evaluated by Bias = P i R 1 Rb b j=1 h^b j ^(H b ) j with j = 1; :::; R b. he number of relications for the Monte Carlo simulations to evaluate the coverage ratio are 1; 000 and the results are shown for 5 eriods. he coverage ratios and the median of the length of the con dence intervals are reorted. he length of the con dence intervals are normalized with the value of (N = 200 and = 120, h = 0) as unity. he results are shown in able 1. he rst observation to note is that throughout the exeriments, the bootstra rocedure shows coverage robabilities close to the corresonding nominal levels. his result is robust to the samle sizes considered. he second notable result is that samle sizes a ect the lengths of the con dence intervals, with more available data inducing tighter intervals as exected. Next we comare the results of con dence intervals constructed by bootstraing without factor estimation. his is esecially meaningful since it is often conducted by existing emirical researches. he case of smaller samle size is of interest since the factors are estimated less recisely and it should enlighten the e ects of the factor estimation uncertainty. he samle sizes are now chosen (N; ) = f(10; 120); (30; 120); (50; 120)g. As for VAR arameters, we consider the case of the diagonal elements 0:7 and 0:4; and the o -diagonal elements 0:2. Everything else is same as the baseline simulation and results of only 95% nominal level and the normal errors are reorted in able 2. he results show that the roosed bootstra rocedure including factor estimation (w/estim) rovides better erformance in nite samle than that without factor estimation (w/o estim). First for smaller samle sizes, the e ect of neglecting the factor estimation uncertainty becomes more distinct. Second and as frequently shown in emirical data, when 8 See aendix 2 for detail. 16

18 factors are more ersistent and have more variability (diagonal elements are larger) the difference of two rocedures becomes more distinct. Although the di erence is often considered minor, it is advised that the two methods can give signi cant di erence in certain cases. 6.2 Monte Carlo simulation using U.S. macroeconomic data Finally, we resent an emirical exeriment to ascertain the robustness of the roosed bootstra rocedure to actual economic data. o this end, we use 110 US macroeconomic series which are investigated by Stock and Watson (2008). he frequencies of the data are a mixture of quarterly and monthly, sanning from 1959Q1 to 2006Q4. We conducted the following treatment, as the original aer did. First, monthly data are converted into quarterly by taking a simle average over three months. Second, all series are transformed into stationary rocesses following Stock and Watson s (2008) guidelines. In addition, the data are demeaned and standardized to have unit standard deviations. he details are contained in Stock and Watson (2008). We here consider two tyes of models: case 1, IRFs to factor innovations, and case 2, IRFs to an innovation of observable variable in VAR. In this setting, we chose the order of the vector autoregression to be four and the number of factors is set at r = 2. hese choices are justi ed by the ICP2 criteria of Bai and Ng (2002) although moderate variations of the lag order and number of factors do not a ect the qualitative results. he observation equations and the VAR equations are identical to those described in the revious subsection excet now with the larger lag order. he aim is to evaluate the coverage ratios of the IRFs. Hence, we use the following calibration exeriment. 1. Estimate the model and obtain the coe cient estimates and residuals. 2. Generate quasi-observations from the calibrated model with the errors resamled from f^u t g and f^e t g with relacement. Note that, f^e t g are orthogonalized such that t = ^e t ^ 1 where ^ is the Cholesky decomosition of covariance matrix of ^e t. his makes t interretable as structural innovations. 3. Using each generated data set, construct 95% con dence intervals of the IRFs by bootstraing and see if the true (calibrated) IRF is included in the estimated interval. 4. Reeat stes 2. and 3. 1; 000 times and evaluate the coverage robabilities. 17

19 he considered IRFs are to rice (ersonal consumtion exenditures rice index, nondurable excluding food, clothing, and oil : GDP275_4), long term interest rate (interest rate, US treasury bills, 10 year : FYGM10), roduction index (industrial roduction index, manufacturing : IPS43), and unemloyment rate (unemloyment rate, all workers, 16 years & over : LHUR). For case 2, the federal funds rate (FYFF) is chosen as the olicy variable. Results for both cases 1 and 2 are rovided in ables 3 for 8 eriods. he results for both cases generally yield values very close to 95% nominal level for all four variables when using the full samle data set. herefore, the good nite samle roerties of the bootstra rocedure are con rmed by this calibrated exeriment. Finally we comare the results with the bootstraing without factor estimation in able 4. Here we use a smaller data set by choosing only aggregate series from the Stock and Watson s data set. For examle, we use industrial roduction index (total) instead of using industrial roduction index for durable, nondurable, and so on. his rocedure leaves us 47 data series, however, the basic structure of the data set remains the same and gives clearer results. 9 We consider the full time length ( = 190). It is also shown that if the bootstraing is alied without considering uncertainty associated with factor estimation as in Procedure B, the resulting con dence intervals become narrower and the coverage ratios are mostly below the nominal level. 7 Conclusions Based on recent advances in theoretical and emirical research, we further develoed FAVAR analysis in terms of commonly used residual-based bootstra inference methods and established imortant caveats. First, the uncertainty associated with factor estimation cannot be neglected in ractice although asymtotic results agree on this device. Our Monte Carlo simulation results actually show that the con dence intervals of the IRF can be much narrower and coverage ratio can be lower if this e ect is not taken into account. Second, in order to achieve the roosed right rocedure, the imortance of the identi cation of IRFs shall be reconsidered. Since the latent factor model can only identify the sace sanned by all factors, the identi cation of IRFs which are basically comrised of individual arameters in FAVAR may not be trivial. As there is no free lunch, alying the traditional factor analysis to VAR models naturally has its costs and bene ts. his aer mainly focused on the cost asects of FAVARs as 9 he results using full data set follow the same line. 18

20 oosed to most existing works which emhasize solely on the bene ts. FAVARs can, on the one hand, enjoy the ability to concisely extract information out of large data set researchers tyically face with comrehensive macroeconomic data series or individual stock or bond rice data. On the other hand simly treating estimated factors as observed variables and conducting conventional inferences should induce overly otimistic forecasts. However, these costs can be aid o in a simle and straightforward manner as far as rather rimitive settings are concerned as this aer shows. More comlicated situations including inference with uncertainty regarding number of factors should be further develoed to meet with recently growing demands for alications of factor analysis in economics and nance. However, correctly accounting for the technical exenses will surely make FAVARs more owerful and romising tools for future researches. 19

21 Aendix A : echnical derivations In aendix, we omit the subscrit N attached to H N simlicity. and denote H for notational De nition 1 (Normalized arameters) Let S and G be arameter saces on which 2 S and H 2 G are de ned. he normalized arameters f(h) : S G! Sg of is a measurable continuous maing that guarantees the consistency of the two-ste rincial comonents estimates ^ when ; N! 1. For the model (1) and (2), we articularly have (H) H 1 and (H) HH 1. Proof of Lemma 1: For this lemma, the roof for the IRF i;h is all that is necessary since other results are obtained in Bai and Ng (2006). We omit the notation of the lag oerator L here and write (L)L without a ecting the results of the roof. Since = [I ] 1, the normalized arameters will simly be obtained by lugging (H) into. By De nition 1, we have (H) = HH 1 so that [I (H)] 1 = I HH 1 1 = H [I ] 1 H 1 ; hen (H) = H 1 yields the IRF: i;h(h) = i (H) (H) = i [I ]H 1 ; which comletes the roof. Proof of Lemma 2: his is straightforward from Lemma 1, the de nition of arameters in structural model and ' i;h = i;h P by continuous maing theorem. Now I turn to roofs of heorem 1 to heorem 3. Lemma A1: Under assumtion 1-6 and the rincial comonents estimation, the normalization factor H converges to a xed invertible matrix H = V 1=2 0 1=2 in robability as ; N! 1 where V = diag(v 1 ; : : : ; v r ) and are the r largest eigenvalues and associated eigenvector matrix for F resectively. Proof of Lemma A1: See Stock and Watson (1998) theorem 1(b). 20

22 Proof of heorem 1: For reduced-form arameters, Bai and Ng (2006) derived (^i i H 1 ) d! N(0; i ); vec(^ HH 1 ) d! N(0; ); where i = H 10 1 F ui 1 F H 1 and = 1 F H 1. By Condition 1, denote the robability limit of ^P as P = HP. hen Slutsky s lemma yields (^ i vec(^ i ) d! N(0; i); ) d! N(0; ); where i = P 1 F ui 1 F P 0 and = ( P 0 P 1 ) ( P 10 P ): Proof of heorem 2: Since ' i;h is globally di erentiable function for the structural arameters, heorem 1 and the delta method yield the asymtotic normality of ^' i;h. 'i;h is given by 'i;h @ 0 ; where = diag( i ; ). Lemma A2: Let ^v 1; b :::; ^v r b and b N be the r largest eigenvalues and the associated eigenvector matrix of X b0 X b =N, and let fv 1 ; :::; v r g and be those of F. hen under Assumtion 1-6, ^v j b! v j for j = 1; ; r; and b N! unconditionally. Proof of Lemma A2: Let P be the distribution of the original samle and P be the distribution of the bootstra samle conditional on the original samle. In Bai (2003) it is shown that ^v j! vj for j = 1; ; r; and N! where f^v1 ; :::; ^v r g and N be the r largest eigenvalues and the associated eigenvector matrix of X 0 X=N. Hence it is su cient to show the equivalence of the unconditional second moments of X b and X. In articular I rove that the bootstra samle Xt b = ^F t b + u b t = (^H)(H 1 Ft b ) + u b t has the following roerties. H 0 ^0 ^H=N! ; (A.1) H 10 F b0 F b H 1 =! F ; (A.2) and V ar(u b ) = V ar(u) (A.3) 21

23 unconditionally. First 1 1 h N H0 ^0 ^H = (H 0 ^0 N = 1 N N (H0 ^0! ; i i 0 ) + h 0 + (^H ) ; 0 ) + 1 N 0 (^H since ^H = (^ H 1 )H = o (1) and it shows (A.1). Show (A.2). ) + 1 N (H0 ^0 0 )(^H ); 1 H 10 F b0 F b H 1 = 1 H 10 0 ^F ^F H 1 + o (1); = 1 h( ^F i i 0 H 10 F 0 ) + F hf 0 + (H 1 ^F F ) + o (1); = 1 F 0 F + 1 (H 10 ^F 0 F 0 )F + 1 F 0 (H 1 ^F F ); since ^F H (H 10 ^F 0 F 0 )(H 1 ^F F ) + o (1);! F ; F = H 1 ( ^F HF ) = o (1) and the rst equality follows from the facts F b t = ^ (L)e b t and ^F t = ^ (L)^e t. Since we know V ar(e b j ^P ) = V ar(^e) by the construction of e b, the law of iterated exectation immediately yields V ar(e b ) = V ar(^e). Finally, (A.3) is similarly shown and these guarantee that X 0 X=N and X b0 X b =N have the same eigenstructure as N;! 1 unconditionally. Lemma A3: Let H b be the bootstra counterart of H such that H b = V b 1 N hen, H b H has the same limit in robably as H, that is, H b H! H unconditionally. Proof of Lemma A3: It is straightforward to show kh b Hk = V ^F b 1 b0 ^F ^ 0 ^ N N H ; = V ^F b 1 b0 F 0 N N + o (1): unconditionally. By Lemma A2, we know VN b ^F 0 F= have the same limits. Consider the identity (1= N)X b0 X b ^F b = ^F b V b N ^F b0 ^F ^ 0 ^. N! V. Hence I need to show that ^F b0 F= and remultilied by 1 ( 0 =N) 1=2 F on 22

24 both sides: 0 1=2 X 1 b0 X b F N ^F b 0 = N 1=2! F 0 b ^F VN b : Exanding X b0 X b by X b = ^F b + u b, one can write 0 1=2!! F 0 F b ^0 ^ F b0 b ^F 0 + N N N = N 1=2! F 0 b ^F VN b ; with N = o (1). hen, further transform: 0 1=2! F 0 F 0 F 0 b ^F 0 + d N = N N N 1=2! F 0 b ^F VN b ; (A.4) where d N = o (1). Rewrite (A.4) as BN + d N R 1 N RN = R N V b N ;. he same logic as in the with B N = 0 1=2 F 0 F 0 1=2 N N and RN = 0 1=2 F 0 ^F b N roof of Proosition 1 in Bai (2003) alies for the rest to reach the conclusion F 0 ^F b = 0 N 1=2 N V b1=2 N ;! 1=2 V 1=2 : Here I also use the result of Lemma A2: V b N! V. Lemma A4 (Uniform delta method for bootstra): Let g : R k! R m be a measurable ma de ned and continuously di erentiable in a neighborhood of. Let S N and SN b be random vectors taking their values in the domain of g. Let N and b N be random vectors taking their values in the domain of g and both converge in robability to. If (S N N ) d! S and (S b N b N ) d! S conditionally in robability, then (g(s N ) g( N ))! d g 0 (S) and (g(sn b ) g(b N ))! d g 0 (S), conditionally in robability. Proof of Lemma A4: he roof is an extension of heorem 23.5 of van der Vaart (1998). I aly the same logic for (g(s N ) g( N )) and (g(sn b ) g(b N )). Also, I use the convergence in robability instead of almost sure, but the roof remains valid excet some minor change hence it is omitted here 23

25 Proof of heorem 3: I show that the limit distribution of the bootstra arameter estimates conditional on the original samle are same as those of the original estimates. For the original estimates, Pr lim ^0 i H 0 0 i x j P ;N!1 Pr due to lim ;N!1 ^F t ^ HH 1 x j P = Pr = Pr = Pr = Pr lim ;N!1 lim ;N!1 lim ;N!1 lim ;N!1 1 P t=1 1 P H 1 P t=1 1 H ^F t^u it x j P ; F t u it x j P ; t=1 ^F t 1^e 0 t x j P ; P H 0 x j P F t t=1 HF t = o (1) and roerties of ^u t u t and ^e t He t. On the other hand, for the bootstra estimate ^ b, Pr ^b0 i Hb 0 ^ 0 i x j ^P lim ;N!1 = Pr = Pr lim ;N!1 lim ;N!1 t=1 1 e 0 t 1 P H b ^F t^u b it x j ^P ; t=1 1 P H F t u it x j P ; where the second equality follows from Lemma A3. For the VAR arameter, I similarly have Pr lim 1 ^b H b ^H b x j ^P 1 P = Pr lim H b ^F t ;N!1 ;N!1 1^e b0 t x j ^P ; t=1 1 P = Pr lim H b ^F t ;N!1 1^e 0 t Hb 0 x j ^P ; t=1 1 P = Pr H H 0 x j P : lim ;N!1 F t t=1 herefore the bootstraed individual arameter estimates have the same limit distribution conditionally on the original samle as the original arameter estimates. Finally, aly Lemma A4 to a measurable continuously di erentiable function (). In order to do this, I need to show that ^(H b ) = [^ i (H b ); vec(^(h b ))] and (H) = [ i (H); vec((h))] converge to the same limit unconditionally. Using De nition 1, one can show ^(H b ) = (H b H), hence ^(Hb ) = (H b H)! by Lemma A3 and the continuous maing theorem. he subsequent result for the reduced-form IRFs follows from Lemma A4. Using Condition1 and Slutsky s lemma, the result for ^' i;h follows. 1 e 0 t ; 24

26 Aendix B : Finite samle bias-correction rocedure for bootstra For both of the simulation studies resented in this aer, I alied the following nite samle bias-correction rocedure, in the sirit of Kilian (1998), for the VAR arameter. he imortant di erence for our setu from Kilian (1998) is to estimate the bias by using ^ b ^(H b ), where (H) is de ned in De nition 1. his estimation should be straightforward to imlement given the asymtotic results for the two- ste PC estimates described in the main text. 1. Estimate the model and generate M bootstra relications ^ b j, j = 1; 2; ; M. hen MP aroximate the bias B = E(^ (H)) by B b = 1 (^ b M j ^(Hj b )) where Hj b is estimated by regressing ^F b t on F b t. 2. Calculate the modulus of the largest eigenvalues of the comanion matrix : I k ; I k 0 j=1 and if it is less than 1, construct the bias-corrected coe cient estimate e = ^ not, let e = ^. his will reserve the stationarity of the generated rocess. B b. If 3. Generate the bias-corrected bootstra relications for the IRFs by using ^, e, ^e t, and ^u t. 25

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