HIAS-E-72

Identifying Factor-Augmented Vector Autoregression Models
via Changes in Shock Variances
Yohei Yamamoto(a)(a) Department of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8601 Japan

August, 2018

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
2-1, Naka, Kunitachi, Tokyo 186-8601, Japan
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http://hias.ad.hit-u.ac.jp/
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Identifying Factor-Augmented Vector Autoregression
Models via Changes in Shock Variances!
Yohei Yamamotoy
Hitotsubashi University
August 6, 2018

Abstract
We propose a new method for the structural identiÖcation of a dynamic causal relationship in factor-augmented vector autoregression models based on changes in theunconditional shock variances that occur on a historical date. The proposed method
can incorporate both observed and unobserved factors in the structural vector autoregression system and it allows the contemporaneous matrix to be fully unrestricted.
We derive the asymptotic distribution of the impulse response estimator and consider a bootstrap inference method. Monte Carlo experiments show that the proposed
method is robust to the misspeciÖcation of the contemporaneous matrix unlike theexisting methods. Both the asymptotic and bootstrap methods obtain a satisfactory
coverage rate when the shock of an observed factor is studied, although the latter ismore accurate when the shock of an unobserved factor is considered. An empiricalexample based on the same data employed by Bernanke et al. (2005) provides similar point estimates and somewhat wider conÖdence intervals, thereby supporting theiridentiÖcation strategy.
JEL ClassiÖcation Numbers: C14, C22
Keywords: dynamic causal e§ect, factor-augmented vector autoregression, identiÖcation through heteroskedasticity, impulse response!
The author acknowledges Önancial support for this study under MEXT Grants-in-Aid for ScientiÖc
Research No.17H00985 and from Hitotsubashi Institute for Advanced Study, Hitotsubashi University. I
also thank participants at the 2018 Kansai Econometrics Study Group Meeting and the 5th IAAE Annual
Conference for useful comments and suggestions. All remaining errors are my own.
yHitotsubashi University, Department of Economics, 2-1 Naka, Kunitachi, Tokyo, Japan 186-8601(yohei.yamamoto@econ.hit-u.ac.jp).

1

Introduction

One of the most challenging issues in structural VAR analysis1 is that the model can only
accommodate a small number of variables in the equation system. If the model is lacking
important variables, the empirical results can su§er from a severe omitted variable bias.
One solution to address this concern is to incorporate a small number of unobserved factors
extracted from an information set spanned by a large panel of data. This approach is calledfactor-augmented VAR (FAVAR) and it was initiated by Bernanke et al. (2005), advanced instudies summarized by Stock and Watson (2016). It is well-known that the factors extractedusing the popular principal component method are only identiÖed up to a random rotation(Bai and Ng, 2013) and this often causes di¢culties when making a particular interpretationof the estimated unobserved factors, which plays a critical role when identifying a dynamic
causal relationship in the VAR system. Recently, Bai et al. (2016) addressed this concern byproposing exclusion restrictions for identifying the unobserved factors or, more speciÖcally, toÖx the random rotation matrix in the FAVAR model. However, their exclusion restrictions
assume zero contemporaneous correlations between the sets of observed and unobserved
factors, which may not be supported without an economic interpretation of the unobserved
factors. This chicken-and-egg problem demands a method for identifying a structural FAVAR
model where its validity can be investigated by a statistical procedure.2
Thus, in this study, we consider an approach for identifying a simultaneous equation
system based on changes in the unconditional variances proposed by Rigobon (2003) andRigobon and Sack (2003). Recently, this method was extended to the conventional smallscale structural VAR system by Lanne and L¸tkepohl (2008), L¸tkepohl (2013), and L¸tkepohl and Netsunajev (2017).3 The fundamental idea of this method is to increase the numberof equations in the system by considering changes in the variances4 to achieve the order con1

See Sims (1980) and the vast amount of work surveyed by Kilian and L¸tkepohl (2017).
It should also be noted that identiÖcation restrictions motivated by economic reasoning may be di¢culteven in the conventional structural VAR when its dimension becomes relatively large. For instance, Brunnermeier et al. (2017) used the structural VAR to investigate the e§ects of up to 10 orthogonal shocks on
macroeconomic variables and showed that their contemporaneous ordering is not a trivial problem when thedimension is relatively large.
3
See Section 4.5 of Stock and Watson (2016) and Chapter 14 of Kilian and L¸tkepohl (2017) for a surveyof a wider class of identiÖcation methods that use heteroskedasticity. These include discrete changes in
the unconditional variances, variances that follow Markov switching and smooth transition dynamics, andconditionally heteroskedastic variances. Lewis (2018) discusses a method that is robust to the misspeciÖcationof heteroskedasticity process.
4
Importantly, the coe¢cient parameters are typically assumed to be constant.
2

1

dition for identiÖcation. As a result, we can avoid the agnostic zero restrictions that are
imposed by particularly ordering the VAR variables or imposed on the impulse responses
based on economic meaning. Instead, we can use a statistical procedure to verify the identiÖcation conditions. See Qu and Perron (2007) and Casini and Perron (2017), for recent
developments in structural break analysis that incorporates changes in the error variances.5
More speciÖcally, in this study, we develop a new method for the structural identiÖcationof the impulse responses in FAVAR models based on the changes in shock variances that occur
on a historical date. In particular, we identify the rotated version of the contemporaneous
matrix and this problem is fundamentally the same as that addressed by Bai et al. (2016)
but we allow the contemporaneous matrix to be fully unrestricted. We derive the asymptotic
distributions of the identiÖed structural coe¢cients and impulse response estimators underp
the condition that T =N ! 0 as N and T ! 1, where T and N are the time and crosssection dimensions, respectively. We also consider a bootstrap method for constructing

the conÖdence interval, based on the seminal studies of GonÁalves and Perron (2014)6 andYamamoto (2017).
Our Monte Carlo experiments show that the impulse responses are estimated more accurately by the proposed method as the magnitude of the variance change and the sample size
increase. More importantly, the proposed method is robust to the misspeciÖcation of thecontemporaneous matrix unlike the existing methods proposed by Bai et al. (2016). Even
when it is correctly speciÖed, the mean squared error (MSE) with the proposed method iscomparable to that with the existing methods when the magnitude of the variance change
and the time dimension are large. We also Önd that both the asymptotic and bootstrap
conÖdence intervals yield satisfactory Önite sample properties when we study the responses
to the shock of an observed factor. However, the latter provides a more accurate coveragerate when the shock of an observed factor is considered.
5
Many studies considered factor models. In particular, Han and Inoue (2015) proposed a test for changes
in the factor loadings at a common time by investigating a change in the variances of factors. Chen et al.
(2014) considered the relationships between the estimated Örst factor and the remaining estimated factors.
Breitung and Eickmeier (2011) and Yamamoto and Tanaka (2015) investigated changes in the factor loadings
for an individual response variable. Because a change in the variance of factors and some type of changes in
the factor loadings at the same timing may not be identiÖed, we need to ìassumeî that the factor loadings
do not have such common breaks.
6GonÁalves and Perron (2014) allow for serial correlations in the idiosyncratic errors and show that thebootstrap does not have to replicate them. This is extended to the case where the errors in the factoraugmented model are serially correlated, as introduced by Djogbenou et al. (2015), and to the case where
the idiosyncratic errors are cross-sectionally correlated, as introduced by GonÁalves and Perron (2016).
GonÁalves et al. (2017) consider the bootstrap prediction intervals in the factor-augmented model by explicitly accounting for the uncertainty of unobservable factors and coe¢cients.

2

Finally, we investigate whether the recursive identiÖcation strategy of the well-known
study of Bernanke et al. (2005) is valid by using the same data employed in the original
study but with the proposed identiÖcation strategy. To ensure consistency with the empiricalevidence of the Great Moderation, we consider changes in the shock variances in December
1984. We Önd that our identiÖcation method could obtain similar point estimates of the impulse responses to those in the original study but with somewhat wider conÖdence intervals.
These Öndings support the validity of the identiÖcation strategy implemented by Bernanke
et al. (2005).
The remainder of this paper is organized as follows. In Section 2, we introduce the modeland assumptions. In Section 3, we propose a new identiÖcation method for FAVAR models
and derive the asymptotic distributions of the estimators. We also design a bootstrap method
for constructing the conÖdence interval in the same section. In Section 4, we examine theÖnite sample properties of the proposed estimator based on comparisons with alternative
methods proposed by Bai et al. (2016), and an empirical illustration is provided in Section
5. In Section 6, we give our conclusions, and the appendices provide the technical derivations
and details of estimators that are not explained in the main text.
Throughout this study, we use the following notation. The Euclidean norm of vector x ispddenoted by kxk. The vector-induced norm is used for matrices. The symbols ì!î and ì!î
represent convergence in probability under the probability measure P and convergence indistribution, respectively. The symbols Op (&) and op (&) denote the order of convergence under
P . We use the symbol ìA ' Bî if, for two random matrices A and B, kA ( Bk = op (1) as

N; T ! 1. Let L be the standard lag operator. The operator vec(X) transforms an m ) mmatrix X into an m2 ) 1 vector by stacking the columns, whereas vech(X) stacks only theelements on and above the main diagonal elements of a square matrix X. We denote theCholesky decomposition of a positive deÖnite matrix X by chol(X). [&] returns the integer
part.
2
2.1

Model and assumptions
Structural FAVAR models

We consider the structural factor-augmented vector autoregression (FAVAR) model of orderp:Ph"t = 4 " + pj=1 A"j h"t#j + "t ;(1)3

for t = 1; & & & T , where h"t is an r ) 1 vector of factors and A"j is the VAR coe¢cient of the jth
lag.7 The factors can be a mixture of unobserved factors denoted by an r1 ) 1 vector ft" andobserved factors denoted by an r2 ) 1 vector gt" , where h"t = [ft"0 ; gt"0 ]0 and r = r1 + r2 . The

error term "t is considered as structural shocks and is assumed to be serially uncorrelatedwith a diagonal unconditional covariance matrix E("t "0t ) = ). The constant term 4 " can beomitted for simplicity to focus on the essential components. We also observe a large numberof economic variables that are driven by the factors so an N ) 1 vector xt is generated by
xt = <" + *" ft" + +" gt" + u"t ;= <" + C " h"t + u"t ;(2)where *" and +" are N ) r1 and N ) r2 factor loading matrices of the unobserved andobserved factors, respectively. Let C " = [*" ; +" ] denote the composite factor loadings. If
we write c"i , we denote an r ) 1 vector of factor loadings of the ith response variable for
i = 1; :::; N . Similarly, A"i and B "i are r1 ) 1 and r2 ) 1 vectors of the factor loadings attachedto the unobserved and the observed factors, respectively. The error term u"t is an N ) 1
vector referred to as the idiosyncratic errors. Again, the constant term <" can be omitted forsimplicity.
The structural VAR allows contemporaneous correlations among the factors to identify
the causal e§ects of an economic shock. Thus, we deÖne an estimable reduced-form modelby plugging h"t = B #1 ht with a nonsingular r ) r matrix B in (1)B #1 ht =or equivalently
ht =

Pp

" #1
j=1 Aj B ht#j + "t ;

Pp

" #1
j=1 BAj B ht#j + B"t :To simplify the notation, we deÖne the reduced-form VAR coe¢cient of the jth lag Aj =BA"j B #1 and the reduced-form errors et = B"t and we write the reduced-form VAR model
ht =

Pp

j=1 Aj ht#j + et :(3)This is the same model as that described by Bai et al. (2016). If we simplify the notation byletting W = [H(#1) ; H(#2) ; & & & ; H(#p) ] be a T )rp matrix where H(#j) = [h1#j ; & & & ; hT #j ]0 ; and7This is a simpliÖed model of the formulation given by Stock and Watson (2016) where the number ofshocks is the same as the number of factors.

4

A = [A1 ; & & & ; Ap ]0 is an rp ) r matrix, then (3) can be written equivalently as H = W A + e.
The corresponding reduced-form factor model is

xt = *ft + +gt + ut ;(4)

= Cht + ut ;where C = C " B #1 and u"t = ut .

In this study, we develop a method for identifying the structural FAVAR model by estimating the matrix B but without imposing any exclusion restrictions on it. Our approach
follows the method for identiÖcation based on the changes in shock variances proposed by
Rigobon (2003) and Rigobon and Sack (2003) in a demand-and-supply system, which was
extended to the conventional small-scaled structural VAR models by Lanne and L¸tkepohl(2008), L¸tkepohl (2013) and L¸tkepohl and Netsunajev (2017).8 In particular, we deÖne
the structural impulse response of an observed variable xi;t+h to the structural shocks in "tat horizon h as

@xi;t+h"0= c"0i -h = ci -h B;@"0twhere -"h and -h are the structural and the reduced-form vector moving-average coe¢cients.
P
P
They are recursively deÖned by -"h = hj=1 -"h#j A"j and -h = hj=1 -h#j Aj , respectively,,i;h *

with -"0 = -0 = Ir .
2.2

Assumptions

We consider the following assumptions on the models introduced in the previous subsection.

Assumption A (VARs)(i) The structural VAR shocks "t are independent and identically distributed (i.i.d.) withmean zero, and the unconditional covariance matrix changes on a known date Tb = [GT ]where G 2 (0; 1), i.e.,
E("t "0t ) =
8

8
< )

for t = 1; :::; Tb

1

: )

2

;

for t = Tb + 1; :::; T

The proposed method considers a discrete change in the unconditional variance of errors, where changes
in the conditional variances and various types of volatility changes are studied (see chapter 14 of Kilianand L¸tkepohl (2017)). Brunnermeier et al. (2017) apply this method to measure the e§ects of up to 10
orthogonal shocks on macroeconomic variables in Önancial markets.

5

(1)

(2)

(r)

(1)

(2)

(r)

where )1 = diag[J 1 ; J 1 ; & & & ; J 1 ] and )2 = diag[J 2 ; J 2 ; & & & ; J 2 ] are diagonal matrices
with distinct positive diagonal elements.
(l)(l)(ii) The VAR variables in h"t are ordered according to the magnitudes of ratio J 1 =J 2
for l = 1; & & & r.

(iii) The r ) r contemporaneous matrix B is nonsingular.

(iv) The roots of det(Ir ( A1 y ( A2 y 2 ( & & & ( Ap y p ) = 0 lie outside the unit circle.
Assumption B (idiosyncratic errors)(i) E(uit ) = 0 and E juit j8 - M , for all (i; t):
P PN
PT PT
#1
(ii) N #1 N
i=1
k=1 jM ik j - M , where M ik = E(uit ukt ). T
s=1
t=1 jB st j - M; wherePN#1
B st = E(Ni=1 uis uit ): %%4
P%%(iii) For every (s; t), E %N #1=2 N
[u
u
(
E(u
u
)]
is it % - M:
i=1 is it&'PPd(iv) For any i, T #1=2 Tt=1 ht uit ! N (0; 1Hui ), where 1Hui * lim V ar T 1=2 Tt=1 ht uit .
Assumption C (factor model)(i) The factor loadings of the unobserved factors Ai are deterministic and *0 *=N ! 5*a Öxed positive deÖnite matrix.
(ii) The eigenvalues of the r1 ) r1 matrix 5* 5F , where 5F = p limT !1 F 0 MG F=T and

MG = IT ( G(G0 G)#1 G0 , are distinct.

(iii) "t and uis are mutually independent for all (i; s; t).
Assumption D (normalization through structural impulse responses) The val-

ues are known for the main diagonal elements of the matrix of contemporaneous structural
impulse responses of the Örst r variables.
The key assumptions of this paper are Assumptions A (i), (ii) and D. Assumptions A(i) and (ii) are indeed similar to that considered by L¸tkepohl (2013) and L¸tkepohl andNetsunajev (2017). They normalize the covariance matrix of one regime to be an identity
matrix to Öx the scale of the structural shocks, we also Öx the scale of the eigenvectors basedon Assumption D. Assumption A (iii) assumes a known break date for simplicity, but this
could be relaxed by using the standard break date estimate developed by Qu and Perron(2007).9 The standard break date fraction estimator, i.e., the break date estimator divided
pby the sample size, would have a faster rate than T so the estimation uncertainty associated9
Strictly speaking, as far as the author knows, this particular technique in the comon factor model is still
an open question.

6

with the break date does not a§ect the asymptotic inference for the least square coe¢cientestimators. Assumptions A (iv) guarantees a stable VAR system and it also ensures thatthe probability limit of F 0 MG F=T becomes a Öxed positive deÖnite matrix which we use in
Assumption C (ii). Assumption B is essentially the same as the standard literature of thefactor model, e.g., see Bai (2003) and Bai and Ng (2006), and it allows idiosyncratic errors
to have weak serial and cross-sectional dependence. Assumption C is the standard regularity
condition and it guarantees the statistical identiÖcation of the unobserved factors up to arandom rotation, as discussed in the next section. Finally, by Assumption D, we need toÖx the scale of the unobserved economic shocks where this normalization depends on the
researcherís discretion, discussed in the empirical illustration.10
3

Estimation and inference

3.1

Estimation of the reduced-form models

We use the two-step principal component method to estimate the reduced-form models (3)and (4). First, we obtain the regression residuals for xit on the observed factors gt . Then, theunobserved factors ft are estimated as the r1 principal components of the residuals. Thus,
we obtain an estimate Fe as a T ) r1 eigenvector matrix corresponding to the r1 largest

eigenvalues of MG XX 0 MG =(T N ) where MG * IT ( G(G0 G)#1 G0 , under the normalizatione * [Fe; G] and theof Fe0 Fe=T = Ir . We then construct a T ) r matrix of factor estimate H
f * [He (#1) ; & & & ; H
e (#p) ]. The factor loadings are estimatedlagged variables up to pth order W
e = X 0 H(
e H
e 0 H)
e #1 and the VAR coe¢cients are estimatedusing the least squares method C
by the feasible generalized least squares (FGLS) such that

wheree0 ) = [(Wf 0 . Ir )5
e #1 (Wf . Ir )]#1 (Wf 0 . Ir )5
e #1 vec(He 0 );
vec(A
e
e
2
3
e
5
0 &&& 0
6 1
7
6
7
e2
60 5
7
ee = 6
7;
5
6 ..
7
...
6 .
7
4
5
eT
0
5

10
This is a common problem when normalizing the structural shocks in FAVARs, e.g., see Stock andWatson (2016).

7

e t = 1 PTb
et =is a T r ) T r block diagonal matrix with 5^t e^0t for t = 1; & & & Tb and 5
t=p+1 eTb #p
PT1^t e^0t for t = Tb + 1; & & & T and e^t comprises the ordinary least squares residuals oft=Tb +1 e
T #Tb
e on W
f . It should be noted that if we denote the random rotation matrix
the regression of H
induced by the two-step principal component estimation by R, then it is known that the
ej are consistent estimators11 for c0i R#1 and RAj R#1 and theleast squares estimators e
c0i and A
following asymptotic normality results hold according to Bai and Ng (2006)12
p
p

d

T (e
ci ( R0#1 ci ) ! N (0; Vci );de ( (Ip . R0#1 )AR0 ] !
T [AN (0; VA );where the asymptotic variances Vci and VA are stated in Appendix A as Lemma A1. Hence,
the reduced-form impulse response c0i -h is also estimated up to a rotation matrix R suchthat it is a consistent estimator for c0i -h R#1 .
Remark 1 Bai et al. (2016) considers the same problem but without explicitly considering
the structural identiÖcation, i.e., the contemporaneous correlation matrix B. Hence, theirsetup is the same as our reduced-form model (3) and (4), where they aim to identify thematrix R. We do not separately identify R and B, so our problem is essentially the same astheirs. We explain their approaches in Section 3.2 and Appendix B.
3.2

IdentiÖcation of the structural parameters

We propose a method for identifying the structural coe¢cients and impulse responses. The
aim is to consistently estimate C " ; A" , and ,i;h and the asymptotic distribution results arepresented.
Algorithm:
e C,
e and eet . Let C
e1:r be the
1. Estimate the reduced-form models (3) and (4) to obtain A;eÖrst r rows of C.

0
0
0
0 p
0e
$1 0 pIn particular, 0eci ( R0$1 ci 0 ! 0 and 0A
0 ! 0 where R is a random matrix.
j ( RAj R
12
In the Proof of Theorem 1 in Appendix A, we show that2
3
Q$1
0r1 %r2
5;
R=4(G0 G)$1 G0 F
Ir2
11

where Q * V $1 (Fe0 MG F=T )(%0 %=N ) and V is a r1 ) r1 diagonal matrix where the main diagonal elements
are the r1 largest eigenvalues of MG XX 0 MG =(T N ).

8

2. Construct the sample covariance matrices for the pre- and post- break VAR residuals,
respectively, by
1 PTb
e 1,
eet ee0 = 1
Tb t=p+1 t
1 PT
e 2.
eet ee0 = 1
T ( Tb t=Tb +1 t

3. Obtain an r ) r eigenvector matrix of

e1:r 1
e 11
e #1 Ce#1
Se = C2
1:r

in descending order of their associated eigenvalues. Let the kth eigenvector divided bye =[eits kth element be e
U k . Then, 7
U1; & & & ; e
U r ] is an r ) r matrix.

4. Estimate the contemporaneous matrix B by

#1 e
e=C
e1:r
B
7:

e the structural parameters are conventionally constructed by comAfter we obtain B,
eB
e and A^"j * B
e #1 Ae:j B
e and the structuralbining with the reduced-form estimates C^ " * C^ ih * ee h B.
e The following result is presented for these
impulse response is constructed by ,c0 iparameters.

ppTheorem 1 Under Assumptions A-D, the algorithm given above yields: c^"i ! c"i , A^"j ! A"j ,p^ ih !
and ,,ih for each i and j, uniformly in h, as N; T ! 1.

A formal proof is given in Appendix A, but an intuitive explanation of this identiÖcae such that it is asymptoticallytion mechanism is given as follows. The goal is to obtain B
equivalent to the rotated version RB. Thus, in step 3, we obtain Se such thatSe =!

e1:r 1
e1
C
|{z}
|{z}

e #1
1
2
|{z}

e#1 ;
C
1:r
|{z}

CR!1 RB,1 B 0 R0 R0!1 B 0!1 ,!1 B !1 R!1 RC !1
2

#1
#1 #1
p (C1:r B))1 )2 (B C1:r ),""#1= C1:r)1 )#12 C1:r :In the last line in the equations given above, )1 )#1
2 is a diagonal matrix in descending order"by Assumptions A (i) and (ii), and C1:rhas unit diagonal elements by Assumption D, the"eigenvector of Se becomes a consistent estimate for C1:r, which is C1:r B. Then, in step 4,
#1 e
#1
e=C
e1:r
B
7 ' (RC1:r)(C1:r B) = RB:9

e in the structural parameters, we can show that the Rs are
Furthermore, by plugging B
cancelled out in the structural parameter and impulse response estimates.
We next derive the asymptotic distributions, where the following lemma is useful. We
note that the matrix C1:r is a submatrix of C but is only used for the identiÖcation of RB.
"
Thus, in order to avoid notational confusion, we denote C1:r by 8 and C1:rby 8" in thefollowing.
Lemma 1 Under Assumptions A-D, the following holds:pas N; T ! 1 underpde ( RB) !
T vec(BN (0; VB );T =N ! 0;where

9 #1 )V- (Ir . 80#1 R
9 0 ) + (70 80#1 R
9 0 . R8
9 #1 )V. (70 80#1 R
9 0 . R8
9 #1 );VB * (Ir . R8
with V. * [(R5H R0 )#1 . 51:ru ] and V- is given in Lemma A5.
e1:r is denoted by V. and it is constructed based on the
The asymptotic variance of C

existing results given by Bai (2003) and Bai and Ng (2006). The asymptotic variance ofe denoted by V- is described in Lemma A5 in Appendix A. We
the eigenvector matrix 7
now derive the asymptotic distributions of the structural parameters and impulse responses
estimators in the following theorem.

Theorem 2 Under Assumptions A-D, the algorithm given above yields: (i) for the factor
loadings:p

as N; T ! 1 and

p

d

T (^
c"i ( c"i ) ! N (0; 1ci );T =N ! 0 for any i = 1; :::; N , where0
9 0#1 c0i . Ir )Krr VB Krr
9 #1 . Ir ) + B 0 R
9 0 Vc RB;
91 ci * ( R(ci Riwhere Vci is deÖned in Lemma A1 and Krr is the r2 ) r2 commutation matrix. (ii) For theVAR coe¢cients:

as N; T ! 1 andppd

T vec(A^"j ( A"j ) ! N (0; 1Aj );T =N ! 0 for any j = 1; :::; p, where0
9 0 . B #1 R
9 #1 )VA (RB
9 .R
9 0#1 B 0#1 );1Aj * DB VB DB+ (B 0 R
j

10

with
9 #1 ) ( (B 0 A0j B 0#1 . B #1 R
9 #1 )]DB * [(Ir . B #1 Aj R(iii) For the impulse responses:pas N; T ! 1 andpd^ i;h ( ,i;h ) ! N (0; 1/ );
T (,
i;h

T =N ! 0 uniformly in h for any i = 1; :::; N , where

9 h BVc B 0 -0 R
9 0 + DA;h VA D0 + (Ir . c0 -h R
9 #1 )VB (Ir . R
9 0#1 -0 ci );1/i;h = Rh
A;h
i
h
i
with

&
'
90 . R
9 0#1 ci ) Ph#1 JA0h#1#j . R9 jR
9 #1 ;
DA;h * (B 0 R
j=0

where J = [Ir ; 0r'r(p#1) ] and A is a companion form of [A1 ; & & & ; Ap ].

Several remarks are given pertaining to this result. First, all of the asymptotic variances
given above can be consistently estimated by combining the reduced-form estimates, because
all of the entities in the expressions are the rotated versions (some of them are cancelled out
in the expressions given above). For example R0#1 c0i is consistently estimated by e
c0i and B 0 R0
0e . Second, we only derive the asymptotic distribution underis consistently estimated by Bp
T =N ! 0 in this study. GonÁalves and Perron (2014) showed that relaxing this conditionpin the form of T =N ! c where 0 - c < 1 generates asymptotic bias of the estimators andthe inference that dealing explicitly with this asymptotic bias is important. However, weleave this issue for future research. Finally, when there are more than one break in the shock
variances, utilizing all these changes yields an overidentifying situation and the proposed
method is not directly applicable. A simple solution would be to use the sample covariancesin only two regimes in step 3 at a cost of e¢ciency.
3.3

Strategies employed by Bai et al. (2016)

Bai et al. (2016) considers identiÖcation strategies for FAVAR models to provide estimatesfor coe¢cients and impulse responses that are free from random rotation. Their setup is thesame as our reduced-form models (3) and (4), where
3
2
1
1
11
12
5;
E(et e0t ) = 5e = 4
121 122for all t. The following three sets of exclusion restrictions are considered.
11

/ IRa: 111 = Ir1 , 112 = 121 = 0r1 'r2 , and N #1 *0 5#1
u * is a diagonal matrix where its
diagonal elements are distinct and arranged in descending order.

/ IRb: 111 = Ir1 , 112 = 121 = 0r1 'r2 , and *1:r = [A1 ; & & & ; Ar1 ] is a lower triangularmatrix.

/ IRc: 112 = 121 = 0r1 'r2 and *1:r = [A1 ; & & & ; Ar1 ] = Ir1 .
In the form of structural VAR models (1) and (2), the above restrictions are translated
into those on the matrix B through the relationship 5e = BB 0 . Thus, we can restrict thematrix B so 5e = BB 0 satisÖes the assumptions given above. A common feature of thesethree methods is assuming that 112 = 121 = 0r1 'r2 . Except for extreme coincidence, this isachieved by assuming that the same o§-diagonal parts of the matrix B are zeros, i.e., theunobserved and observed factors are not contemporaneously correlated. In this sense, ourmethod will complement the previously proposed methods, because it allows the matrix B
to be fully unrestricted.
It should be noted that Bai et al. (2016) considers the quasi-maximum likelihood method
instead of the principal component method for estimating the unobserved factors. However,their identiÖcation strategies are valid when the principal component factor estimate is usedas they point out, and the steps after factor extraction can be followed exactly. Hence, inthe following, we compare our method with that of Bai et al. (2016) by including their identiÖcation methods in the two-step estimation using the principal component factor estimate.
3.4

Bootstrap conÖdence interval

The seminal studies of GonÁalves and Perron (2014) and Yamamoto (2017) show that asymptotic inference of the factor-augmented models may under-evaluate the sampling errors(asymptotic bias) of the coe¢cient estimators because it is assumed that the asymptoticpresults are free from factor estimation errors under T =N ! 0. They develop bootstrap inp
ference as an alternative method that is consistent even under T =N ! c where 0 - c < 1.
We also consider the bootstrap method in a similar manner. Thus, we also assume independence13 but allow for heteroskedasticity in the idiosyncratic errors uit . The procedure is
described as follows.
13GonÁalves and Perron (2014) show that the serial correlations in the idiosyncratic errors are irrelevant
when we focus on the coe¢cients of the factor-autoregression model in which the dependent variable isobserved. However, this is not the case when some of the dependent variables are the estimated factors as
in the FAVAR model.

12

Bootstrap algorithm:e
1. Estimate the reduced-form models (3) and (4), and obtain the parameter estimate C,e B,e and the residuals eet and uA,
eit . Construct the structural impulse response estimator^ i;h by using the algorithm proposed in Section 3.2.
,b
2. Ensure that the pre- and post- break VAR residuals fe
et gTt=1
and feet gTt=Tb +1 are de-meaned, respectively, in the time direction. Resample with replacements the pre-break
5 6Tbbresiduals fe
et gTt=1
as r ) 1 vectors in an i.i.d. manner and label them ebt t=1. Construct
5 b 6T
the same process for the post-break residuals and label them et t=T +1 . Generate thebP
ej hb + eb ; for t = p + 1; :::; T:14
bootstrapped factors via hbt = p A
j=1

t#j

t

3. Demean the idiosyncratic residuals u
eit in both the time and cross-sectional directions.

For each i = 1; :::; N , if fuit gTt=1 are considered homoskedastic, then we propose the i.i.d.
5 6T
resampling of fe
uit gTt=1 to obtain ubit t=1 . If fuit gTt=1 are considered heteroskedastic,

then use the wild bootstrap ubit = u
eit Z it , where Z it 2 i:i:d:(0; 1) is an external random
5 b 6Tvariable to obtain uit t=1 . Generate the bootstrapped observations xbit = e
ci hbt + ubit
for t = 1; :::; T and i = 1; :::; N .

eb , Aeb , and B
e b , by
4. Using the bootstrap observations xbit , obtain the parameter estimate C
using the same estimation and identiÖcation methods proposed in Section 3.2, thereby^b .
obtaining the bootstrap estimate of the structural impulse response ,i;h^b \ ( ,^ i;h \,
5. Repeat steps 2 to 4 nb times and store the recentered statistic si;h * ,i;hwhere \ is an r ) 1 vector of shocks of interest.

(;=2)(1#;=2)].
6. Sort the statistics and select the 100 ;2 th and 100(1 ( ;2 )th percentiles [si;h ; si;h(1#;=2) ^(;=2)^
The resulting 100(1(])% conÖdence interval for ,i;h \ is [,i;h \ (si;h; ,i;h \ +si;h ].
The bootstrap consistency of the method underp

T =N ! c (0 - c < 1) would be

similar to that shown by Yamamoto (2017) when explicitly deriving the asymptotic bias ofthe original and bootstrap estimates for the impulse responses, but this problem is rather
trivial and it is not presented in this study.
14

The bias-correction method discussed by Kilian (1998) can be applied. The bias is estimated by taking
the average of A^!j ( R! A^j R!$1 in another bootstrap loop, where R! is a bootstrap analogue of R and it canbe constructed based on the original estimate and the bootstrap samples.

13

4Monte Carlo experiments

We investigate the Önite sample properties of the impulse response estimator by using Monte
Carlo simulations. First, we compare the accuracy of estimator obtained using our identiÖcation method with that using the three methods based on the IRa , IRb and IRc assumptions
proposed by Bai et al. (2016). To ensure that the comparisons are fair and to simplify the
implementation, we estimate the unobserved factors by the principal component method instead of the quasi-maximum likelihood method.15 We also investigate the empirical coverage
properties of the asymptotic and bootstrap conÖdence intervals when the proposed identiÖcation method is used. We use the following reduced-form models as the data generatingprocess (DGP):
ht = Aht#1 + et ;(5)c0i ht + uit ;

(6)

xit =

with ht = [ft0 ; gt0 ]0 where ft is an r1 ) 1 vector of unobserved factors and gt is an r2 ) 1 vectorof observed factors. The number of unobserved factors is r1 = 2 and that of the observed

factors is r2 = 1, so the total number of factors is r = r1 + r2 = 3. The order of the VAR
model (5) is 1 with the VAR coe¢cient matrix
2
3
0:6 0
0
6
7
6
7
A = 6 0 0:4 0 7 ;
4
5
0
0 0:2

while the VAR errors are generated by et = B"t with the structural shock "t 2 i:i:d:N (0; )t )and the matrix B is arbitrarily set at23
1:0 0:4 0:8
6767
B = 6 0:2 1:0 (0:37 ;(7)
4
5(0:6 0:4 1:0

unless stated otherwise. The unconditional shock covariance matrix )t changes in the middleof the sample period such that8
< diag(2 + d; 1 + d=2; 0:25), for t = 1; :::; [T =2]
)t =
;
: I
for t = [T =2] + 1; :::; Tr

15

As discussed in Section 3.3, this does not a§ect the qualitative results for the main conclusion.

14

where the parameter d controls the magnitude of the variance change, and we speciÖcallyconsider d = 0:5; 1, and 2 in this experiment. For the factor model (6), the factor loadings
ci are generated as an r ) 1 vector where each element is independently drawn from thestandard normal distribution. Furthermore, the idiosyncratic errors are generated by uit 2

i:i:d:N (0; 1). We consider the impulse responses of the N th variable ,N;h .16 The number ofMonte Carlo replications is 3,000.
4.1

MSE of the impulse response estimatorFirst, we examine the MSE of the impulse response estimator to one unit structural shock of
the observed factor and an unobserved factor. In particular, we compare our method (in thecolumn denoted as ìIRv î) with the methods proposed by Bai et al. (2016) (in the columns
denoted as ìIRa î, ìIRb î, and ìIRc î). In this experiment, the matrix B for IRa , IRb , and
IRc is restricted as follows. Let 1 = BB 0 where B is speciÖed by (7). Then, for IRa , IRb ,and IRc , we impose the o§-diagonal blocks 112 and 121 to be zero matrices. Further, for
IRa and IRb , we impose the upper left r1 ) r1 block 111 to be the identity matrix. This
results in
8 2
3>> I r1 0>>
4
5 for IRa and IRb
>
>
< 0 1
22
2
3
1=
;
(8)
>
>
1
0
11
>
> 4
5
for IRc
>
>
:
0 1
22

where 122 is the lower right r2 ) r2 block of 1 = BB 0 . After obtaining such 1, we construct
the matrix B by imposing a recursive assumption such that B = chol(1) for IRa , IRb ,and IRc , respectively. Hence, the true or correct B matrix can be di§erent in di§erent
identiÖcation strategies.
The DGPs are di§erent when diverse identiÖcation methods are compared, so it is difÖcult to determine whether the di§erence in MSEs is due to the DGP or the identiÖcation
method. To address this concern, we consider the following four scenarios. In scenario A, theDGPs conform with the identiÖcation assumptions, and thus the four methods are applied
to di§erent DGPs. In scenario B, the DGP uses the unrestricted B matrix (7) for every
16
The factor loadings of the Örst r response variables are used for identiÖcation and they can be di§erent
according to the identiÖcation method employed in this experiment. The remaining factor loadings are thesame even when di§erent identiÖcation methods are considered. The exception is IRa where the factor
loading matrix has a block structure (if 'u is diagonal). In this experiment the Örst factor loading of the
N th response variable is zero.

15

method, and thus the method based on IRv is regarded as a correct method, whereas IRa ,IRb , and IRc are not and the latter is a§ected by misspeciÖcation. In scenario C, the DGP
uses the matrix B based on IRa and IRb (which are the same). In scenario D, the DGP uses
the B matrix based on IRc . As shown in (8), scenario C is the case with the most restrictions
and all four methods satisfy their assumptions. Scenario D is somewhat less restrictive thanscenario C because the 111 part is unrestricted, so the methods based on the IRa and IRb
assumptions involve the misspeciÖcation, whereas the methods based on the IRc and IRv
assumptions are correctly speciÖed.
By contrast, the identiÖcation assumptions on the factor model are correctly imposedin every case as follows. When we use the method based on IRa , the Örst [0:7N ] response
variables have zero second factor loadings and the last [0:3N ] response variables have zeroÖrst factor loadings, so *0 5u *=N becomes a diagonal matrix. When we use the method
based on IRb , we put three zeros in the factor loading matrix so that the Örst r1 response
variables *1:r1 becomes a lower triangular matrix. When we use the method based on IRc ,*1:r1 is the identity matrix. When we use the method based on IRv , we follow Assumption D
and sets C1:r such that the main diagonal elements of C1:r B are unity.17 We use the impulse
response of the N th variable, so the objectives are the same regardless of the assumptions,
except for IRa where the Örst factor loading is restricted to be zero. However, that has very
little e§ect on the MSEs.
Table 1 shows the MSEs of the impulse response estimator of the time horizon h = 1
for scenarios A to D. For each scenario, the left panel presents the MSEs of the impulse
response estimator to a shock of the observed factor and the right panel shows the MSEs of
the impulse response estimator to a shock of the Örst unobserved factor. We investigatedthe MSEs for di§erent values of the horizon h, but the results were qualitatively similar, so
we only present the case where h = 1.18 In each table, we consider the sample size pairs(T; N ) = (50; 50); (50; 150); (150; 50); and (150; 150).
Throughout the four scenarios, the MSE of IRv tends to decrease as d increases because
the eigenvalues of )1 )#1
2 are more distinct and the asymptotic variance of the eigenvectors
will become smaller.19 By contrast, the MSEs of IRa , IRb , and IRc tend to increase as
17

First, we compute C1:r B and set its diagonal elements as one and get back C1:r by right-multiplying.
18
When the sample size and the break size are not large, the MSE of IRv may be driven by a very few
large errors and the results may be unstable. Hence, we report the MSEs after truncating the 1% largest and1% smallest squared errors. However, this does not a§ect qualitative results of the Monte Carlo simulation.
(k)
(j)
19
This is consistent with Lemma A5, in which the asymptotic variance is shown to have (2 1=2 ( 2 1=2 )2 as
the denominator.
B

$1

16

d becomes large because the variance of the VAR errors increases in (the Örst half of) the
sample period, thereby resulting in larger sampling errors for the impulse response estimator.
We then compare the MSEs of the four identiÖcation methods in each scenario. In
scenario A, when the shock of the observed factor is considered, the MSE of IRv is larger
than that with any other method for every sample size pair. However, it becomes morecomparable to the others, especially with IRb , under a large break d = 2 with a large time
dimension T = 150. A larger N also reduces the MSE except in a few cases, but its e§ectp
is more blurred than that of T because the estimator is T consistent. When the shock ofthe Örst unobserved factor is considered, the MSEs of IRa and IRb increase whereas thoseof IRc decreases compared with the shock of the observed factor case. The MSE of IRv ismuch smaller than those of IRa and IRb and it outweighs that of IRc when T is large.
Scenario B is the most interesting experiment because the matrix B is fully unrestricted
and it is the same for all of the identiÖcation methods. When the shock of the observed
factor is considered, the MSEs of IRa , IRb , and IRc are considerably larger than those in
scenario A, whereas the MSEs of IRv does not increase. This is because the methods basedon IRa , IRb , and IRc su§er from misspeciÖcation whereas the method based on IRv doesnot. When we consider the shock of the Örst unobserved factor, IRv still tends to give thesmallest MSE when T = 150:
In scenario C, the matrix B has the most restrictions, and thus the relative merit of IRv
is expected to be the smallest among the four scenarios. However, we Önd that IRv gives
MSEs that are comparable with those of IRa , IRb , and IRc when d and T are large and theshock of the observed factor is considered. The results remained qualitatively similar evenwhen the shock of the Örst unobserved factor is considered. Finally, the results obtained
under scenario D agree with those under scenario C even when matrix B is slightly more
restricted.
In summary, the Monte Carlo results conÖrm that the impulse responses are estimatedmore accurately using the proposed method as the magnitude of the variance change increases
and as the sample size grows. More importantly, the proposed method is robust to themisspeciÖcation of the matrix B unlike the existing methods proposed by Bai et al. (2016).
Even when the matrix B is correctly speciÖed, the MSEs with the proposed method arecomparable with those as d and T become large.

17

4.2

Coverage property of the conÖdence interval

Next, we investigate the coverage property of the conÖdence interval for the impulse responses identiÖed using the proposed method. Table 2 presents the empirical coveragerate and the median value of the length, i.e., the upper conÖdence limit minus the lower
conÖdence limit, for the 95% asymptotic conÖdence interval based on 3,000 Monte Carlo
replications. Given that the order of VAR is 1, it is su¢cient to investigate the horizon
h up to 5. The same set of sample size pairs described in the previous subsection, i.e.,(T; N ) = (50; 50); (50; 150); (150; 50), and (150; 150) are used to study the sample size e§ect.
The upper table shows the results for the impulse responses to a unit shock of the observed
factor and the lower table shows the impulse responses to a unit shock of the Örst unobserved
factor. When we consider the shock of the observed factor, the coverage rate is close to thenominal level of 95% in every case, validating Theorem 2. Importantly, when we consider
the responses to an unobserved factor shock as shown in the lower table, the asymptotic
conÖdence interval tends to fall short of the nominal coverage rate. This under-coverage
property is exacerbated when the sample size is small, especially when N is smaller than Tp(T = 150; N = 50) because the condition that T =N ! 0 is considered less appropriate
in this empirical situation. Unsurprisingly, the coverage rate deteriorates as h increases,

because the degree of nonlinearity of impulse responses increases and this undermines theasymptotic approximation. The median length decreases as d becomes larger because theasymptotic variance decreases for the same reason explained in the previous subsection. We
also observe that the median length decreases as T increases, which is consistent with thep
T rate of the estimator, whereas the e§ect of a large N is more blurred.
In the aforementioned studies of bootstrapping factor models, it is suggested that bootstrap methods can improve the coverage property. To examine this suggestion, Table 3 shows
the coverage rate and median length of the 95% bootstrap conÖdence interval, as proposed
in Section 3.3. We consider the i.i.d. bootstrap based on 499 repetitions. The upper tableagain shows the results for a unit shock of the observed factor, where the coverage rate isvery close to the nominal level of 95% and the median length is somewhat longer than theasymptotic interval, thereby reáecting the factor estimation errors. The lower table shows
the results with respect to a unit shock of the Örst unobserved factor. Clearly, the bootstrap
interval has a coverage rate that is much closer to the nominal level than the asymptotic
interval. As suggested in the seminal study of GonÁalves and Perron (2014), the bootstrap
accounts for the e§ects of factor estimation errors that are not accounted for by the asympp
totic interval under the assumption of T =N ! 0. A well-known caveat that the coverage
18

rate can deteriorate as h becomes larger even with the bootstrap method is also relevant inthis case. This could potentially be Öxed by introducing the double bootstrap bias correction
procedure proposed by Kilian (1998).
5

Empirical illustration

We apply the proposed identiÖcation method to the well-known empirical study by Bernanke
et al. (2005), who investigated the e§ect of a contractionary monetary shock on macroeconomic variables using the same FAVAR model as (3), with the federal funds rate (FFR) as
the observable factor gt and the unobservable factors ft extracted from 120 macroeconomic
data series for the United States. The aim of this study is to examine whether we can obtain impulse response estimates similar to those in the original study by using the proposed
identiÖcation method.
Thus, we Örst explain their identiÖcation strategy. Bernanke et al. (2005) assumed
a recursive structure among the unobserved and observed factors. They Örst extract theunobserved factors from a subset of ìslow-moving variablesî among the 120 macroeconomic
variables. The remaining unobserved factors are identiÖed as the residuals from the regressionof the standard principal component estimate on the slow-moving factors and gt . Clearly,
the identiÖed unobserved factors are orthogonal to the observed factor and the slow-moving
factors. Our method does not impose these restrictions but a change in the unconditional
shock variances is accounted for in December 1984. The timing of this change in the varianceis consistent with the evidence for the so called Great Moderation20 and a reasonable variationin the timing does not qualitatively a§ect our results.21 To Öx the scale of the shocks, we
use the point estimate of the reduced-form factor loading estimate for the main diagonal
elements of the contemporaneous impulse responses. Our speciÖc choice is based on thevariables that take a large value for each factor loading. Our speciÖc choices comprise theindustrial production (Önal products) for the Örst factor, the NYSE common stock priceindex (composite) for the second factor, the producer price index (Önished goods) for thethird factor, and the spread of FYGT1 from FYFF for the observed factor. As discussed
in Section 2, this choice may a§ect the empirical results and making this step more robust
20

Stock and Watson (2002) document that about 40% of 168 U.S. macroeconomic variables experienced achange of the variance during 1983 and 1985.
21
Han and Inoue (2015) pointed out that common breaks in the factor loadings can increase the numberof factors and Yamamoto (2016) investigated a consequence of this fact in a forecasting exercise. Any
criticism of the time-invariant factor loading assumption may apply to this example, so to the original studyof Bernanke et al. (2005).

19

should be addressed in future research.
Following the original study, we use the VAR lag order p = 13 and the number ofunobserved factors r = 3. We use exactly the same data employed by Bernanke et al. (2005),
which ranges from January 1959 to August 2001. A 25-basis point increase in the FFR isinvestigated, and the impulse responses and associated bootstrap conÖdence intervals are
retransformed to level. The conÖdence level is 90% and the number of bootstrap repetitions
is 1,000. We could compute the impulse responses of the 120 variables, but given space
limitations, the following six selected variables are presented: FFR, industrial production
index (total, IP), consumer price index (all item; CPI), monetary base, exchange rate yen,commodity price index up to 48 periods ahead as shown in Figure 1.
First, we Önd that the directions of the point estimates of the impulse responses are largely
consistent with those in the original study.22 This suggests that imposing the identiÖcation
assumption of Bernanke et al. (2005) has little e§ect on their main results. Second, notsurprisingly, the conÖdence intervals with our identiÖcation method are somewhat wider thanthose in the original study in most cases. We recall that the MSE with our method tends
to become larger when the DGP is restricted correctly (in Scenarios C and D) in our Monte
Carlo investigations, so this Önding also supports the identiÖcation strategy implemented by
Bernanke et al. (2005).
6

Conclusion

In this study, we develop a new structural identiÖcation method for FAVAR models. The
main idea is based on previous studies that use changes in the unconditional shock variances
on a historical date. Similar to the methods developed by Bai et al. (2016), the proposed
method can identify the rotated version of the contemporaneous matrix, but it is advantageous because it allows the contemporaneous matrix to be fully unrestricted. Our method
can utilize statistical methods to verify the identiÖcation assumptions rather than relying on
economically motivated assumptions. We derive the asymptotic distributions of the strucp
tural parameter and impulse response estimators under T =N ! 0 as N; T ! 1 andpropose a bootstrap method for constructing the conÖdence interval. Monte Carlo experi-ments suggest that the MSEs with the proposed method are larger than the existing methods
if the identifying assumptions of the latter is indeed correct. However, the proposed method
obtains much smaller MSEs when the contemporaneous matrix is indeed unrestricted. This22

We can directly compare the results with Figure II (page 408) of Bernanke et al. (2005).

20

feature becomes more distinct when the magnitude of the change in the variance is large andas T increases. Both the asymptotic and bootstrap conÖdence intervals provide a satisfactory
coverage rate when the impulse response to a shock of the observed factor is investigated.
The bootstrap interval gives more accurate coverage rate when a shock of an unobserved factor is considered, especially when N is relatively smaller than T , as suggested in the previous
studies. Finally, our empirical investigation of the same model employed by Bernanke et al.
(2005) provides similar point estimates and wider conÖdence intervals in general. These Öndings support the validity of the identiÖcation strategy used by Bernanke et al. (2005) in theircontext. Several important issues need to be addressed in future research. The asymptotic
ptheory under T =N ! c (0 - c < 1) must be developed to justify the bootstrap methods.
More importantly, the new identiÖcation strategy can potentially be applied to empirical
data given the large number of studies of structural VARs reviewed by Stock and Watson(2016) and Kilian and L¸tkepohl (2017), e.g., especially in a data-rich environment.

21

Appendix A: Proof of Theorems
Proof of Theorem 1: We rewrite the reduced-form factor model (4) as
xi = F Ai + GB i + ui ;= MG F Ai + G^T;i + ui ;where ^T;i * [(G0 G)#1 G0 F Ai + B i ]. Because MG F and G are orthogonal, by using Theorem 1of Bai and Ng (2006), the reduced-form factor loading estimator has the following property
2
32 3
Q#1 0r1 'r2A
54 i 5;eci ' 4
0r2 'r1
I r2^T;iso that2

Q#1

0r1 'r2

32

I r1

54
e
ci ' 4
0r2 'r1
I r2(G0 G)#1 G0 F
2
3
Q#1
0r1 'r2
5 ci ;= 40#1 0(G G) G F
I r2

0r1 'r2
I r2

32 3
A
5 4 i5 ;Bi

where Q * V #1 (Fe0 MG F=T )(*0 *=N ) and V is an r1 ) r1 diagonal matrix where the main
diagonal elements are the r1 largest eigenvalues of MG XX 0 MG =(T N ). Thus, we have
where

p
e*C
e1:r !
9 #1 * 8R
9 #1 ;
8
C1:r R2

Q#1

R*4(G0 G)#1 G0 F

0r1 'r2
I r2

(A.1)

3

5;

9 = p lim(R). This implies that the inverse has the probability limitand Rpe #1 !
9 #1 :
8
R8(A.2)

Because the deÖnition of the reduced-form VAR
yields E(et e00t ) = B)j B 0 for j = 1
0 errorsP001
and 2 and Lemma 2 of Yamamoto (2017) gives 0 T1j j eet ee0t ( R)j R0 0 = Op ( minfN;T), thegsample covariance matrix of the reduced-form VAR residuals for regime j has a probability
limit
p
0 90
ej !
9
1
RB)
(A.3)
jB R ;
for j = 1 and 2 because et = B"t and Assumption A (i).

22Therefore, by using (A.1), (A.2) and (A.3),
p0 90
0 9 0 #1 9 #1
e1
e 11
e #1
e #1 !
9 #1 )(RB)99
Se * 8(8R
1 B R )(RB)2 B R ) (R8 );
2 8
#1 #1
= 8B)1=2 B 8 * S;
(k)

(k)

(k)

where )1=2 is a diagonal matrix where the kth main diagonal element is J 1=2 * J 1 =J 2 fore is the eigenvectors of S,e the probability limit of 7
e is the eigenvectors
k = 1; :::; r. Because 7of
S = 8B)1=2 B #1 8#1 ;"and the diagonal elements of 8B * C1:rare normalized by Assumption D, we obtainand, by using (A.2) and (A.4),

p
e !
7
8B;

(A.4)

p
e=8
e #1 7^ !
9 #1 8B = RB:
9
B
R8

Finally, because the reduced-form estimator satisÖes eci ' R0#1 ci for i = 1; :::; N and
ej ' RAj R#1 for j = 1; :::; p,
A
and

e0e
c^i = Bci ' B 0 R0 R0#1 ci = B 0 ci = c"i ;e #1 A
ej B
e ' B #1 R#1 RAj R#1 RB = B #1 Aj B = A" :A^j = Bj

The results follow.!

In the Lemmas A1 and A2, we only consider the case of spherical idiosyncratic errors
in order to simplify the notation for the long-run variance. However, the results can beextended to the case with heteroskedasticity and standard type of serial correlations.
Lemma A1. Under Assumptions A-D, the following hold as N; T ! 1 and
(i)
p
d
T (e
ci ( R0#1 ci ) ! N (0; Vci );9 HR9 0 )#1 .
where Vci * ` 2i (R5(ii)
pe ( (Ip . R0#1 )AR0 ] ! N (0; VA );T vec[Ap

T =N ! 0:where

VA * p lim T [f(Ip . R0 )W 0 g . Ir ]#1 [(IT . R)5e (IT . R0 )][fW (Ip . R)g . Ir ]#1 :Proof of Lemma A1: (i) This is a direct result of Theorem 1 of Bai and Ng (2006).
23

(ii) First, we note thate * HR + (He ( HR);H= W AR + eR + W AR + (H ( HR);f (Ip . R#1 )AR + eR + [W ( W
f (Ip . R#1 )]AR + (He ( HR):= W

f 0 . Ir )5
e #1
fConsider the FGLS estimator. To simplify the notation, we let D * [(We (W . Ir )].
Then,e0 ) = D#1 (Wf 0 . Ir )5
e #1 vec(He 0 );
vec(Aef 0 . Ir )5
e #1 vec[R0 A0 (Ip . R0#1 )W
f 0 ];= D#1 (We

#1 f 00 0
e #1+D (W . Ir )5e vec(R e );f 0 . Ir )5
e #1 vec[R0 A0 fW 0 ( (Ip . R0#1 )W
f 0 g];+D#1 (Wef 0 . Ir )5
e #1 vec(He 0 ( R0 H);+D#1 (We= I + II + III + IV:

For I, because

f 0 ] = (Wf . Ir )vec[R0 A0 (Ip . R0#1 )];vec[R0 A0 (Ip . R0#1 )W

we obtain

I = vec[R0 A0 (Ip . R0#1 )]:Hence,

pf0 ] =
T vec[R0 A0 (Ip . R0#1 )Wp

T (II + III + IV ):Terms
III and IV are factor estimation errors and they diminish as N; T ! 1 underp
T =N ! 0. This yieldspf 0 ])AV ar( T vec[R0 A0 (Ip . R0#1 )W#1
0#1
f 0 . Ir )5
e #1
e #1
f
= p lim(T D#1 )(W
vec(R0 e0 )vec(R0 e0 )0 ]5):e [Te (W . Ir )(T D
However, becausee e ) = E[vec(R0 e0 )vec(R0 e0 )0 ];E(5we have

or

0 0
0 0 0
#1
f 0 . Ir )5
e #1
e #1 fp lim(WD;e [vec(R e )vec(R e ) =T ]5e (W . Ir ) = p lim T

We note thatpf 0 ]) = p lim(T D#1 ):AV ar( T vec[R0 A0 (Ip . R0#1 )W

D ' [f(Ip . R0 )W 0 g . Ir ][(IT . R)5e (IT . R0 )]#1 [fW (Ip . R)g . Ir ]:24

The result follows.!
Lemma A2. Under Assumptions A-D, the following holds:pde ( 8R#1 ) !
T vec(8N (0; V. );

where

9 HR
9 0 )#1 . 51:r ];V. * [(R5u23` 21 & & & ` 1r
6
7.. 7 2
6 ..
...
51:r
*
6
7 ; ` i = E(u2it ) and ` ij = E(uit ujt ).
.
.
u
4
5
` r1 & & & ` 2r

Proof of Lemma A2: This proof is obtained directly from Lemma A1 (i).!
Lemma A3. The following holds as N; T ! 1 with
p

where

p

T =N ! 0:de 11
e #1 ( R11 1#1 R#1 ) ! N (0; V3 );T vec(122
1=2

9 0#1 1#1
9 #1 . Ir )V31 (R
9 0#1 1#1
9 #1 . Ir )V31=2 * (R
2 R
2 R
9 0#1 1#1
9 #1 . R1
9 1 1#1
9 #1
9 0#1 1#1
9 #1 . R1
9 1 1#1
9 #1
+(R
2 R
2 R )V32 (R2 R
2 R );with
9 jR
9 0 . R1
9 jR
9 0 )PD ; for j = 1 and 2;V3j * 2PD (R1
PD * Dr (Dr0 Dr )#1 Dr0 ;where Dr is a duplication matrix of dimension r.
Proof of Lemma A3: First, we obtain the following result for the asymptotic distribution of the sample covariance matrix of the VAR errors by using Lemma 4 of Yamamoto(2017)pde j ( R1j R#1 ) !
T vec(1N (0; V3j );
(A.5)
where

9 jR
9 0 . R1
9 jR
9 0 )PD ;V3j * 2PD (R1(A.6)
p
9 j * R1j R0 for j = 1; 2: The delta
for j = 1; 2 as N; T ! 1 and T =N ! 0. Then, we let 1
method yields
p
d
#1 #1
e 11
e #1
T vec(12 ( R11 12 R ) ! N (0; V31=2 );25

whereV31=2 *

0
9 11
9 #1
9 11
9 #1
@vec(1@vec(1
2 )
2 )
V
3
1
0
9 1)
9 1)
@vec(1@vec(1
#1
0
9
9
9
9 #1
@vec(11 12 )@vec(11 1
2 )
+
V
3
2
9 2 )09 2) :@vec(1@vec(1(A.7)

Furthermore,
9 11
9 #1
@vec(12 )9 #1 . Ir = (R
9 0#1 1
9 #1 R
9 #1 ) . Ir ;= 122
9 1 )0@vec(1
9 11
9 #1@vec(12 )
9 1 )(1
9 #1 . 1
9 #1 );= ((Ir . 122
9@vec(12 )0
9 #1 . 1
9 11
9 #1 );= ((122
9 0#1 1#1 R
9 #1 . R1
9 1R
90R
9 0#1 1#1 R
9 #1 ):
= ((R
2

2

(A.8)

(A.9)

Plugging (A.6), (A.8), and (A.9) in (A.7) yields the result. !
pLemma A4. The following holds as N; T ! 1 with T =N ! 0:
p

where

d
#1
e ( 811 1#1
T vec(Q2 8 ) ! N (0; VQ );9 0 . 8R
9 #1 )V3 (R8
9 #1 . R
9 0#1 80 )VQ * (80#1 R
1=2
9 0 . 811=2 8#1 )V. (R8
9 #1 . 80#1 11=2 80 )+(80#1 R
9 #1 . Ir )V. (R
9 0#1 11=2 8#1 . Ir );+(80#1 11=2 R
where V31=2 and V. are given in Lemmas A3 and A2, respectively.
e 11
e #1
9
9 9 0 9 9 0 #1 =Proof of Lemma A4: We let the probability limit of 1
2 be 11=2 * R11 R (R12 R )#1
9 1 12 R
9 #1 and 8
9 * 8R
9 #1 . Then,
R1
e '
Q
=
=

91
9 1=2 89 #1
89 #1 )(R1
9 1 1#1
9 #1 9 #1(8R
2 R )(R8 )#1 #1
811 12 8 * Q:

Using Lemmas A2 and A3, the delta method yieldspe ( Q) =T vec(Qpde ( 811=2 8#1 ) ! N (0; VQ );T vec(Q26

as N; T ! 1 withp

T =N ! 0; where
VQ *@vec(Q)@vec(Q)0V3
9 1=2 )0 1=2 @vec(19 1=2 )
@vec(10@vec(Q) @vec(Q)+9 0 V. @vec(8)9 :@vec(8)(A.10)

Furthermore,
91
9 1=2 89 #1 )@vec(8@vec(Q)=
9 1=2 )0
9 1=2 )0@vec(1@vec(10#1
9 . 8)
9 = (80#1 R0 . 8R#1 );= (8(A.11)

and91
9 1=2 89 #1 )@vec(8@vec(Q)=
9 0
9 0@vec(8)@vec(8)91
9 1=2 )(8
9 0#1 . 8
9 #1 ) + (89 0#1 1
9 01=2 . Ir );= ((Ir . 8
9 0#1 . 891
9 1=2 89 #1 ) + (89 0#1 1
9 0 . Ir );
= ((8
1=2
0#1 9 0#1
0#1 0
9 #1 . Ir ):= ((8 R . 811=2 8 ) + (8 11=2 R(A.12)

Hence, plugging (A.11) and (A.12) in (A.10) yields the result.!
e =Lemma A5. Under Assumptions A-D, for the normalized
eigenvector matrix 7pe the following holds as N; T ! 1 with T =N ! 0:[e
U1; & & & ; e
U k ] of Q,
p

where

d
e ( C" ) !
T vec(7N (0; V- ) ;
1:r

2

3
V@11 & & & V@1r
6
7
.. 7
6 .
..
V- * 6 ..
.
. 7;
4
5
V@r1 & & & V@rr

with
V@kk *

Pr

V@kl * (vij is the (i; j)th element ofp

AV ar(vii )j=1;j6=k(k)

AV ar(vkl )
(k)

(j)

(J 1=2 ( J 1=2 )2(j)(J 1=2 ( J 1=2 )2c"+j c"0+j ;c"+k c"0+l ; for k 6= l;"e ( Q) and c"+j is the jth column of C1:r
T (Q
.

27

Proof of Lemma A5: To prove this lemma, we follow the derivation of the asymptotic
distribution of eigenvectors of a sample covariance in Theorem 13.5.1 of Anderson (2003).
e which is not symmetric in Önite samples and its
However, we take the eigenvectors of QÖnite sample distribution is not necessarily a Wishart distribution. However, the asymptotic
e is that obtained in Lemma A4. First, we consider the asymptotic distribudistribution of Q
e " * S. Then, we consider that of 7
e = C " E.
tion of the normalized eigenvector E of C "#1 QC
This relation holds because if we let D be the eigenvalue matrix thenSE

If we letp

= ED , C " SC "#1 (C " E) = (C " E)De7
e = 7D:e, Qp
T (S ( )1=2 ) = V and T (E ( Ir ) = W; then SE = ED implies that
8
98
9 8
98
9
1
1
1
1
I+p W = I+p W)1=2 + p D ;)1=2 + p V
T
T
T
Tso that

V = W )1=2 ( )1=2 W + D + Op (T #1=2 ):If we neglect the terms of order T #1=2 , thenV = (W )1=2 ( )1=2 W ) + D:
It is shown that wii = 0 because of the diagonal elements are known by Assumption D. Fori 6= jvij
wij = (j):(i)
J 1=2 ( J 1=2"
Because e
U k is the kth column of C1:r
E,

p

d
T (e
U k ( c"+j ) ! N

0;Pr

AV ar(vii )j=1;j6=k(k)(j)(J 1=2 ( J 1=2 )2c"+j c"0+j!

;

and the asymptotic covariance of e
U k and e
U l (k 6= l) is given by(

AV ar(vkl )
(k)

(l)

(J 1=2 ( J 1=2 )2c"+k c"0+l :The result follows.!
9 * RB;9 c9i * ci R
9 #1 ; and A9j * RA
9 jR
9 #1 . (i) The delta
Proof of Theorem 2: We let B
method yields
9 0 c9i @999 0 c9i
9
@B
c0 B
@B
@9
c0i B
1ci *
V ci i +
VB
0
0
9
9 ;
@9
ci
@9
ci
@vec(B)@vec(B)28

but

9 0 c9i
@B
9 0;
=B
@9
c0i

and
9 0 c9i
9 0)@B@vec(B
0
=
(9
c
.
I
)
r
i
9 0
9 0
@vec(B)@vec(B)0
9 0#1 c0i . Ir )Krr ;= (9ci . Ir )Krr = (Ryield the results.
(ii) The delta method yields
1Aj =9 #1 A9j B)
99 #1 A9j B)
9 0 @vec(B
9 #1 A9j B)
99 #1 A9j B)
9 0@vec(B@vec(B@vec(B
V
+
V
;
A
B
j
9 0
9
@vec(A9j )0@vec(A9j )@vec(B)@vec(B)but
9 #1 A9j B)
9@vec(B
90 . B
9 #1 );= (B09@vec(Aj )9 0 . B #1 R
9 #1 )= (B 0 R
and
9 #1 A9j B)
99 #1@vec(B
9 #1 A9j ) + (B
9 0 A90j . Ir ) @vec(B ) ;
=
(I
.
B
r
9 0
9 0
@vec(B)@vec(B)
9 #1 A9j ) ( (B
9 0 A90 . Ir )(B
9 0#1 . B
9 #1 );= (Ir . Bj
9 #1 A9j ) ( (B
9 0 A90j B
9 0#1 . B
9 #1 );= (Ir . B

9 #1 ) ( (B 0 A0j B 0#1 . B #1 R
9 #1 );= (Ir . B #1 Aj R

yield the result.
(iii) The proof is essentially the same as that given in Proposition 3.6 of L¸tkepohl (2005)
and thus it is omitted.!

29

Appendix B: Estimation under the IRa , IRb , and IRc assumptions
e under the three sets of identiÖcation assumptions given by Bai et al. (2016)
We obtain B
as follows.
e C,
e and eet . Let C
e1:r be the Örst r rows
1. Estimate the reduced-form models to obtain A;
of C.

2. Construct the sample covariance matrices of the VAR residuals for the whole sample
by
PT
1
e=
1
eet ee0 ;T ( pk ( 1 t=p+1 t
e 11 , 1
e 12 , 1
e 21 , and 1
e 22 . Let
and obtain its partitioned components 1
e 11,2 = 1
e 11 ( 1e 12 1e #1
e1
22 121 :IRa Let Va be the eigenvector matrix ofe 1=2
e 0 e #1 ee 1=2
1
11,2 (* 5u *=N )111,2 ;in descending order of their associated eigenvalues. Calculate
2
3#1
0 e #1=2 e e #1
e #1=2
Va0 1
(V
1
1
1
12 22
11,2
a 11,2e=4
5 :B
0r2 'r1
I r2
e 1=2
e0
IRb Let Vb be the same as Va and let Qb be the QR decomposition of 1
11,2 *1:r . Then,2
3#1
e #1=2e #1=2e 12 1e #1
Q0b 1(Q0b 1
1
11,2
11,2
22
e=4
5 :
B
0r2 'r1
I r2

IRc Let

2

e 1:r
*

e=4
B
0r2 'r130

e 1:r 1e 12 1
e #1
(*
22
I r2

3#1
5

:

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33

Table 1. Mean squared errors of the impulse response estimator

Scenario A: Correct DGP
Responses to a unit shock of the observed factor
d
IRa
IRb
IRc
IRv
T=50, N=50 0.5 0.068
0.126
0.116
0.425
1.0 0.067
0.132
0.114
0.278
2.0 0.066
0.128
0.113
0.262
T=50, N=150 0.5 0.065
0.125
0.104
0.274
1.0 0.063
0.122
0.104
0.264
2.0 0.068
0.132
0.112
0.260
T=150, N=50 0.5 0.021
0.041
0.034
0.054
1.0 0.022
0.040
0.034
0.050
2.0 0.022
0.045
0.036
0.051
T=150, N=50 0.5 0.022
0.043
0.035
0.054
1.0 0.021
0.043
0.035
0.054
2.0 0.022
0.043
0.034
0.048

Responses to a unit shock of the first unobserved factor
d
IRa
IRb
IRc
IRv
T=50, N=50 0.5 0.521
0.705
0.084
0.289
1.0 0.619
0.751
0.077
0.235
2.0 0.589
0.815
0.074
0.304
T=50, N=150 0.5 0.496
0.657
0.084
0.318
1.0 0.528
0.759
0.080
0.304
2.0 0.641
0.895
0.076
0.241
T=150, N=50 0.5 0.505
0.730
0.040
0.033
1.0 0.567
0.914
0.042
0.030
2.0 0.588
1.048
0.039
0.026
T=150, N=50 0.5 0.451
0.770
0.037
0.030
1.0 0.532
0.835
0.037
0.029
2.0 0.583
1.031
0.038
0.026

Scenario B: Unrestricted DGP
Responses to a unit shock of the observed factor
d
IRa
IRb
IRc
IRv
T=50, N=50 0.5 0.414
0.543
0.449
0.425
1.0 0.455
0.614
0.494
0.278
2.0 0.472
0.662
0.527
0.262
T=50, N=150 0.5 0.405
0.550
0.453
0.274
1.0 0.431
0.584
0.469
0.264
2.0 0.487
0.648
0.521
0.261
T=150, N=50 0.5 0.384
0.481
0.413
0.054
1.0 0.418
0.532
0.445
0.049
2.0 0.512
0.637
0.542
0.052
T=150, N=50 0.5 0.389
0.473
0.411
0.054
1.0 0.421
0.530
0.457
0.052
2.0 0.507
0.618
0.530
0.049

34

Responses to a unit shock of the first unobserved factor
d
IRa
IRb
IRc
IRv
T=50, N=50 0.5 0.678
0.868
0.070
0.280
1.0 0.693
0.849
0.068
0.295
2.0 0.756
1.036
0.068
0.220
T=50, N=150 0.5 0.644
0.893
0.068
0.276
1.0 0.685
0.862
0.070
0.335
2.0 0.786
0.939
0.066
0.318
T=150, N=50 0.5 0.684
1.023
0.031
0.034
1.0 0.788
1.103
0.033
0.040
2.0 0.847
1.125
0.032
0.030
T=150, N=50 0.5 0.699
0.944
0.032
0.034
1.0 0.759
1.050
0.032
0.032
2.0 0.763
1.103
0.030
0.025

Table 1. Mean squared errors of the impulse response estimator (continued)
Scenario C: DGP with the IRa and IRb assumptions
Responses to a unit shock of the observed factor
d
IRa
IRb
IRc
IRv
T=50, N=50 0.5 0.068
0.126
0.094
0.373
1.0 0.067
0.132
0.093
0.266
2.0 0.066
0.126
0.092
0.219
T=50, N=150 0.5 0.061
0.116
0.085
0.251
1.0 0.069
0.126
0.094
0.246
2.0 0.071
0.141
0.099
0.242
T=150, N=50 0.5 0.022
0.042
0.029
0.050
1.0 0.022
0.041
0.029
0.050
2.0 0.023
0.043
0.029
0.049
T=150, N=50 0.5 0.019
0.039
0.027
0.045
1.0 0.021
0.042
0.029
0.048
2.0 0.021
0.045
0.029
0.048

Responses to a unit shock of the first unobserved factor
d
IRa
IRb
IRc
IRv
T=50, N=50 0.5 0.536
0.718
0.046
0.300
1.0 0.563
0.738
0.040
0.287
2.0 0.656
0.850
0.038
0.296
T=50, N=150 0.5 0.507
0.661
0.044
0.272
1.0 0.542
0.749
0.045
0.347
2.0 0.568
0.762
0.038
0.343
T=150, N=50 0.5 0.547
0.858
0.011
0.034
1.0 0.532
0.833
0.011
0.035
2.0 0.639
1.003
0.011
0.032
T=150, N=50 0.5 0.516
0.831
0.012
0.035
1.0 0.513
0.835
0.011
0.028
2.0 0.587
1.079
0.011
0.031

Scenario D: DGP with the IRc assumption
Responses to a unit shock of the observed factor
d
IRa
IRb
IRc
IRv
T=50, N=50 0.5 0.080
0.162
0.116
0.471
1.0 0.079
0.169
0.114
0.322
2.0 0.079
0.161
0.113
0.273
T=50, N=150 0.5 0.077
0.156
0.104
0.308
1.0 0.075
0.153
0.103
0.312
2.0 0.081
0.171
0.114
0.318
T=150, N=50 0.5 0.025
0.052
0.034
0.063
1.0 0.024
0.051
0.033
0.056
2.0 0.026
0.056
0.036
0.061
T=150, N=50 0.5 0.025
0.051
0.034
0.059
1.0 0.025
0.052
0.035
0.061
2.0 0.025
0.053
0.033
0.055

35

Responses to a unit shock of the first unobserved factor
d
IRa
IRb
IRc
IRv
T=50, N=50 0.5 0.821
1.010
0.082
0.385
1.0 0.795
1.071
0.077
0.358
2.0 0.898
1.270
0.078
0.351
T=50, N=150 0.5 0.774
1.067
0.084
0.482
1.0 0.805
1.175
0.080
0.325
2.0 0.910
1.275
0.074
0.322
T=150, N=50 0.5 0.744
1.080
0.041
0.040
1.0 0.933
1.299
0.042
0.035
2.0 1.025
1.455
0.041
0.035
T=150, N=50 0.5 0.777
1.076
0.038
0.038
1.0 0.837
1.308
0.041
0.038
2.0 0.924
1.438
0.040
0.036

Table 2. Coverage rate and median length of the asymptotic conÖdence interval
Responses to a shock of the observed factordT=50, N=50
0.5
1.0
2.0
T=50, N=150 0.5
1.0
2.0
T=150, N=50 0.5
1.0
2.0
T=150, N=150 0.5
1.0
2.0

Coverage rate
h=1
2
97.7
94.6
98.5
95.1
98.6
96.1
97.5
95.2
98.4
95.6
98.5
95.8
97.3
94.9
97.6
95.7
97.5
95.2
97.9
94.6
97.9
95.1
97.4
94.3

3
94.6
94.8
96.2
95.2
95.5
95.3
94.8
95.5
95.4
95.0
94.9
94.5

4
93.5
93.8
94.6
93.8
93.0
94.2
94.1
94.9
94.8
94.2
94.7
94.4

5
91.5
91.8
92.7
92.2
91.9
92.5
93.0
94.0
93.7
93.2
93.7
93.3

Median length
h=1
2
2.63
1.69
2.54
1.66
2.46
1.66
2.56
1.68
2.54
1.69
2.44
1.68
1.17
0.81
1.15
0.82
1.15
0.83
1.18
0.81
1.17
0.82
1.14
0.82

3
1.06
1.08
1.09
1.07
1.09
1.10
0.54
0.55
0.58
0.55
0.56
0.56

4
0.70
0.71
0.72
0.69
0.72
0.72
0.36
0.36
0.39
0.36
0.37
0.38

5
0.45
0.45
0.47
0.45
0.47
0.48
0.23
0.24
0.26
0.24
0.24
0.25

3
0.77
0.75
0.72
0.78
0.74
0.71
0.39
0.39
0.37
0.39
0.40
0.37

4
0.51
0.51
0.48
0.52
0.51
0.48
0.26
0.27
0.25
0.27
0.27
0.25

5
0.33
0.33
0.32
0.35
0.33
0.32
0.17
0.18
0.17
0.17
0.18
0.17

Responses to a shock of the Örst unobserved factordT=50, N=50
0.5
1.0
2.0
T=50, N=150 0.5
1.0
2.0
T=150, N=50 0.5
1.0
2.0
T=150, N=150 0.5
1.0
2.0

Coverage rate
h=1
2
90.3
94.0
89.7
94.5
89.1
93.5
90.7
94.9
89.2
93.4
89.0
93.4
91.7
94.5
91.5
93.9
90.4
94.1
91.9
94.2
92.5
95.1
90.8
94.1

3
93.4
93.5
92.0
94.4
92.4
91.9
94.6
93.8
94.1
94.1
94.6
94.5

4
90.3
90.6
88.9
91.9
90.2
90.3
92.9
91.5
91.3
91.2
91.7
92.1

36

5
87.3
87.0
86.3
89.1
87.6
87.5
89.6
89.4
87.7
89.2
89.0
89.3

Median length
h=1
2
1.51
1.12
1.45
1.08
1.35
1.04
1.52
1.12
1.43
1.07
1.33
1.03
0.77
0.55
0.76
0.56
0.69
0.53
0.79
0.56
0.76
0.55
0.69
0.52

Table 3. Coverage rate and median length of the bootstrap conÖdence interval
Responses to a unit shock of the observed factordT=50, N=50
0.5
1.0
2.0
T=50, N=150 0.5
1.0
2.0
T=150, N=50 0.5
1.0
2.0
T=150, N=150 0.5
1.0
2.0

Coverage rate
h=1
2
98.3
98.7
97.9
98.1
97.6
98.4
97.9
98.8
97.9
98.5
97.7
98.3
96.1
96.2
96.5
96.7
96.2
95.8
96.7
96.1
96.3
96.1
95.6
95.4

3
96.6
96.0
96.1
96.5
97.1
96.8
94.3
94.5
94.1
94.4
94.1
92.8

4
96.1
95.3
95.6
96.4
96.2
96.2
92.7
93.6
93.0
93.4
93.0
92.1

5
94.6
93.9
93.7
95.3
94.4
94.7
90.3
91.4
91.0
91.7
91.6
90.4

Median length
h=1
2
2.80
1.96
2.59
1.89
2.38
1.78
2.77
1.93
2.59
1.88
2.36
1.81
1.10
0.83
1.06
0.83
1.03
0.82
1.09
0.83
1.07
0.83
1.02
0.81

3
1.12
1.09
1.07
1.14
1.13
1.07
0.53
0.52
0.54
0.53
0.54
0.52

4
0.72
0.69
0.67
0.73
0.71
0.67
0.33
0.33
0.34
0.34
0.34
0.33

5
0.47
0.45
0.42
0.47
0.46
0.44
0.21
0.21
0.22
0.22
0.22
0.21

3
1.01
1.02
0.97
1.03
0.99
0.99
0.43
0.44
0.43
0.43
0.43
0.42

4
0.63
0.64
0.60
0.66
0.62
0.62
0.29
0.29
0.29
0.30
0.30
0.28

5
0.40
0.40
0.39
0.43
0.41
0.41
0.19
0.20
0.19
0.20
0.20
0.19

Responses to a unit shock of the Örst unobserved factordT=50, N=50
0.5
1.0
2.0
T=50, N=150 0.5
1.0
2.0
T=150, N=50 0.5
1.0
2.0
T=150, N=150 0.5
1.0
2.0

Coverage rate
h=1
2
98.6
98.3
98.5
98.3
98.3
97.5
98.3
98.3
98.5
97.7
98.6
97.8
98.0
97.7
98.9
97.2
98.3
97.4
98.1
97.3
98.8
97.6
98.5
97.9

3
95.2
94.7
93.6
95.2
94.3
94.5
95.0
95.1
95.3
94.4
95.2
96.0

4
93.0
91.6
90.6
93.3
91.8
92.1
92.6
92.2
92.9
91.9
92.8
93.4

37

5
89.2
89.3
87.5
91.0
88.8
89.7
89.9
90.0
89.7
89.7
90.3
90.5

Median length
h=1
2
2.91
1.73
2.98
1.75
2.82
1.68
3.00
1.73
2.93
1.72
2.91
1.69
0.99
0.64
1.03
0.66
0.99
0.64
1.02
0.65
1.02
0.66
0.98
0.64

Figure 1. Point estimates and 90% bootstrap conÖdence intervals using thesame data employed by Bernanke et al. (2005)

38