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Consistent selection rules for
the number of dynamic factors
in approximate factor models
Jorg Breitung∗
University of BonnUta Pigorsch†
University of Mannheim
This version: January 2009
Abstract
Determining the number of dynamic factors in large panel data sets is animportant issue for many economic applications. In this paper we developtwo selection procedures that allow to consistently estimate the numberof dynamic factors in a dynamic factor model. The procedures are basedon a canonical correlation analysis of the static factors obtained from aprincipal component analysis. Compared to other selection procedures,our approach has the advantage that it does not assume a finite order au-toregressive representation of the common factors, and that it is invariantto a rotation of the factor space. Monte Carlo simulations show that theproposed selection rules outperform the existing ones based on principalcomponents. The new selection procedures are also applied to the U.S.macroeconomic data panel used in Stock and Watson (2005).
∗Address: University of Bonn, Institute of Econometrics, Adenauerallee 24-42, 53113 Bonn,Germany, phone: +49 (0)228 73 9201, fax: +49 (0)228 73 9189, email: [email protected].†Address: University of Mannheim, Department of Economics, L7, 3-5, 68131 Mannheim,
Germany, phone: +49 (0)621 181 1945, fax: +49 (0)621 181 1931, email: [email protected].
1 Introduction
In many economic applications it is very appealing to represent a large number
of series by a small number of latent factors. In macroeconomics, for example,
factor models have been used in the business cycle analysis (see e.g. Forni and
Reichlin, 1998 and Giannone, Reichlin, and Sala, 2006) and in the identification of
common macroeconomic or policy shocks (see e.g. Favero, Marcellino, and Neglia,
2005, Forni, Lippi, and Reichlin, 2008 and Stock and Watson, 2005). Recently, it
has also been shown that forecasts based on a few number of so–called diffusion
indices, that summarize a huge number of candidate predictor variables, obtain
smaller forecast errors relative to alternative techniques based on ARIMA models
or small structural economic models (see e.g. Angelini, Henry, and Mestre, 2001,
Brisson, Campbell, and Galbraith, 2003, Eickmeier and Ziegler, 2008, Marcellino,
Stock, and Watson, 2003, and Stock and Watson, 1999, 2002a,b). Knowing the
correct number of common factors is crucial for all of these applications. Bai and
Ng (2002) propose information criteria that can be used to consistently select the
number of common factors in a static factor model with both N and T converging
to infinity. The shortcoming of such a factor model, however, is the assumption
of static factors. In view of the common fluctuations of many economic series, it
seems to be more realistic to specify dynamic, e.g. autoregressive, factors that
explain the comovement of the observed series. A leading example of such a
dynamic factor model is considered in Stock and Watson (2002a). There, a large
number of series is represented by a few number of dynamic factors whose lags
may also enter the factor representation. Applying the information criteria of
Bai and Ng (2002) yields a consistent estimate of the number of common factors
in this so–called static representation of the factors.1
An important drawback of this approach, however, is that it cannot disen-
tangle the dynamic factors from their lags and, thus, the estimated number of
factors is equal to the sum of the number of dynamic factors and their lags in
the factor representation. Forni, Hallin, Lippi, and Reichlin (2000), therefore,
suggested informal criteria for determining the number of dynamic factors. Their
approach is based on the computation of the variance explained by the common
components estimated from the Stock-Watson procedure. The procedure is also
applied in Favero et al. (2005) and Giannone, Reichlin, and Sala (2002).
1Pesaran (2006) suggested a methodology that only assumes an upper bound for the numberof common factors. Thus, this approach sidesteps the problem of determining the number ofcommon factors.
1
A weakly consistent selection rule for the number of dynamic (or “primitive”)
factors (k) is suggested by Bai and Ng (2007). Their empirical procedure is based
on the fact that fitting a vector autoregressive (VAR) model to the vector of r > k
principal components yields an asymptotically singular covariance matrix of the
residuals. Therefore, the r − k smallest eigenvalues of the residual covariance
matrix converge to zero as N and T tend to infinity. Based on the eigenvalues of
the residual covariance matrix, Bai and Ng (2007) derive two statistics and corre-
sponding asymptotic bounds that allow for a consistent selection of the number of
dynamic factors. A related test procedure is suggested by Amengual and Watson
(2007) and Stock and Watson (2005). Their empirical procedure is based on the
innovations of the vector of time series. Since the factor representation of the
innovations involve k common components (instead of r factors of the original
time series), they suggest to apply the information criteria suggested by Bai and
Ng (2002) for selecting the number of static factors of these innovations, which
corresponds to the number of dynamic factors of the observed data.
In this paper two alternative specification criteria for the number of dynamic
factors are proposed. Our approach is based on a canonical correlation analysis
(CCA) between the current and past values of the r “static” (or reduced form)
factors. The first selection procedure is based on the fact that lagged values of
common factors are perfectly predictable conditional on the past of the factors
and, therefore, the associated eigenvalues of the CCA tend to unity as the sample
size tends to infinity, whereas the remaining eigenvalues converge to values smaller
than one. The second selection procedure exploits the fact that if a sufficient
number of lags of the dynamic factors enter the static representation of the factors,
then there exist k linear combinations that cannot be predicted based on the past
of the factors and, therefore, the respective eigenvalues converge to zero. To test
the hypothesis that k eigenvalues are equal to zero, we propose a likelihood ratio
statistic.
The CCA procedure has several advantages. First, it can also be applied to
the approximate (instead of a strict) factor model. Second, our first selection
procedure is not based on a finite order VAR representation of the dynamic
factors. Therefore, in contrast to the approach of Bai and Ng (2007) our selection
procedure can also be applied if the common factors are generated by moving
average processes. Third, the selection procedures based on the CCA are invariant
to any rotation of the factor space. Finally, our Monte Carlo simulations suggest
that the CCA criteria are less sensitive to nuisance parameters and generally
2
outperform alternative selection procedures.
The rest of the paper is organized as follows. In Section 2 we introduce the
factor model and review the existing selection procedures for the number of dy-
namic factors. Section 3 proposes two selection procedures based on canonical
correlations. In Section 4 we compare the finite sample properties of the alter-
native selection methods and apply the new procedures to the dataset of Stock
and Watson (2005). Section 5 concludes.
2 Determining the number of dynamic factors
Following Stock and Watson (1999, 2002a,b) we consider the (restricted) dynamic
factor model of the form2
xt = A0ft + · · ·+ Amft−m + ut (1)
≡ AFt + ut , (2)
where xt and ut are N -dimensional column vectors, ft is k×1 vector with k � N ,
A = [A0, . . . , Am] and Ft = [f ′t , . . . , f′t−m]′. For some part of our analysis we
assume that the vector of common dynamic factors possesses a VAR(p) represen-
tation
ft = Γ1ft−1 + · · ·+ Γpft−p + εt , (3)
where Γ1, . . . ,Γp and Σ = E(εtε′t) are k × k matrices. However, for our first
selection criterion we do not assume a finite order VAR representation and just
assume that T−1∑T
t=1 ftf′t
p→ Σf is a positive definite matrix.
It is important to note that A need not have full column rank. For example,
a subset of the factors may not enter with all lags. In this case the respective
columns of A are zero. Let r ≤ (m + 1)k be the rank of the matrix A. Then
there exists a N × r matrix A such that
AFt = AFt
where Ft = RFt and R is a nonsingular r × (m + 1)k matrix. Ft is called the
vector of static factors.
2Note that the model is a restricted version of the general dynamic factor model developedin Forni et al. (2000) and Forni and Lippi (2001) since we assume that the factors enter withfinite order lag polynomials. An identification procedure for the number of dynamic factors ina more general dynamic framework is proposed in Hallin and Liska (2007).
3
We assume that the idiosyncratic error ut is weakly correlated across series
and time, whereas the common factors ft, . . . , ft−p give rise to a strong correla-
tion among the series.3 To identify the number of dynamic factors (or primitive
factors) ft from the static factors Ft, Favero et al. (2005) and Forni et al. (2000)
inspect the fraction of variance explained by the common components. An im-
portant problem of this criterion is that the choice of the cut-off point is rather
arbitrary. Therefore, Amengual and Watson (2007), Bai and Ng (2007) and Stock
and Watson (2005) suggest consistent selection rules for the number of dynamic
factors.
In the following we briefly review these approaches. For the ease of exposition
we assume for the moment that A has full column rank such that A = A and Ft =
Ft. Furthermore, assume that m ≥ p so that the vector Ft can be represented by
a VAR(1) system given by
Ft = CFt−1 + ηt , (4)
where
C =
Γ1 Γ2 · · · Γm−1 ΓmIq 0 · · · 0 00 Iq 0 0...
. . ....
0 0 Iq 0
,ηt = [ε′t, 0]′ and Γj = 0 for j > p. As shown by Bai and Ng (2002) and Stock and
Watson (2002a), the Principal Components (PC) estimator Ft is a consistent es-
timator for HFt, where H is an appropriately chosen rotation matrix. Therefore,
if Ft is replaced by Ft the VAR becomes
Ft = C∗Ft−1 + et . (5)
In this representation C∗ = HCH−1 and, for N → ∞ and T → ∞, the asymp-
totic covariance matrix of et is Σe = HΣηH′, where Ση = E(ηtη
′t). Obviously, Σe
is of rank k since rk(Ση) = k.
Thus, the number of dynamic factors can be obtained by determining the rank
of Ση (or Σe). Bai and Ng (2007), for example, propose a selection rule based
on the ordered eigenvalues c1 ≥ · · · ≥ cN of the estimated residual covariance
matrix
Σe = T−1
T∑t=2
FtF′t − T−1
T∑t=2
FtF′t−1
(T∑t=2
Ft−1F′t−1
)−1 T∑t=2
Ft−1F′t .
3These properties are formalized in the assumption given in e.g. Bai and Ng (2002) andStock and Watson (2002a).
4
They suggest two test statistics based on the eigenvalues of Σe:
D1,k∗ =ck∗+1√∑r
j=1 c2j
(6)
D2,k∗ =
(∑rj=k∗+1 c
2j∑r
j=1 c2j
)1/2
. (7)
Since Σep→ Σe as N and T tend to infinity, it follows that Dj,k∗
p→ 0 for k∗ ≥ k
and j = 1, 2. Specifically, Bai and Ng (2007) show that the estimates of the
number of dynamic factors
kj = min{k : Dj,` < m∗/min(N2/5, T 2/5)} (8)
are (weakly) consistent in the sense that kjp→ k as N, T →∞. Based on Monte
Carlo simulations, Bai and Ng (2007) recommend the value m∗ = 1.
An alternative selection procedure is suggested by Amengual and Watson
(2007) and Stock and Watson (2005). They assume a factor model, where the
idiosyncratic components possess the autoregressive representation
αi(L)uit = νit i = 1, . . . , N, (9)
with αi(L) = 1−α1L−· · ·−αqiLqi and qi is a variable specific lag order. Consider
the vector of innovations:
ξt = xt − E(xt|xt−1, xt−2, . . .)
= A0εt + νt (10)
where νt = [ν1t, . . . , νNt]′. It follows that the number of common factors εt in (10)
is identical to the number of dynamic factors k. Therefore, the selection criteria
suggested by Bai and Ng (2002) can be used to determine k from the factor model
(10). To obtain an estimator of ξt, Amengual and Watson (2007) and Stock and
Watson (2005) suggest a two-step procedure. First, the r static factors in Ft are
estimated from xt by using the usual PC estimator. In a second step, an estimate
of the element ξit is obtained as the residual of a regression of xit on lags of Ft and
xit.4 The number of dynamic factors are estimated by applying the information
criteria suggested by Bai and Ng (2002) to the estimated innovation vector ξt.
4Alternatively, a regression of αi(L)xit on lags of Ft = αi(L)Ft can be performed, where αiis obtained from an autoregression of the estimated idiosyncratic components.
5
Jacobs and Otter (2008) suggest a test procedure for determining the number
of factors in a strict factor model (with small N) under the assumption that the
dynamic factors ft have a finite order moving average representation of order q
and the idiosyncratic components are white noise. In the first step, the lag order
q is selected by testing the hypothesis that all eigenvalues from a CCA of xt on
xt−q−1 are equal to zero. In a second step, a LR test is applied to find out the
number of nonzero eigenvalues from a CCA of xt on xt−q. If all factors have a
MA(q) representation, then the second CCA tends to indicate k nonzero eigenval-
ues, which corresponds to the estimated number of dynamic factors. An obvious
disadvantage of this test procedure is that it is based on a number of restrictive
assumptions (e.g. that the MA orders of all common factors are identical and
that the idiosyncratic components are white noise) that are difficult to verify in
empirical applications. In the next section alternative selection procedures are
suggested that are applicable to the dynamic factor model considered above.
3 Selection rules based on canonical correlations
An important drawback of the selection procedure suggested by Bai and Ng
(2007) is that the statistics depend on the scaling of the factor estimates Ft in
the sense that a transformation of the factors such as Ft = QFt affects the value
of the test statistics D1,k∗ and D2,k∗ . The test statistic of Bai and Ng (2007) is
obtained from the eigenvalue problem |c∗Ir−QΣeQ′| = 0. Obviously, the resulting
eigenvalues c∗j , j = 1, . . . , r depend on the rotation matrix Q. Dividing the
eigenvalues by the sum of eigenvalues does not solve the problem in general. This
is an undesirable property as the normalization of the factor space is arbitrary.5
In what follows we suggest a test statistic that is invariant to a transformation
of the form Ft = QFt. Consider first the case m = 1. In this case linear
combinations of ft and ft−1 may enter Ft. Our selection procedure is based on
the generalized eigenvalues from
|λ∗j S00 − Σe| = 0,
or, equivalently,
|λjS00 − S01S−111 S
′01| = 0 (11)
5For example, the test statistic is different if the estimated factors are computed as ft = V ′rxt(as in Bai and Ng, 2007), where Vr is the matrix of r eigenvectors associated with the r largesteigenvalues of Σ, or whether it is normalized as T−1
∑Nt=1 ftf
′t = Ir (as in Bai and Ng, 2002).
6
where λj = 1− λ∗j and
Sij = T−1
T∑t=2
Ft−iF′t−j.
An important advantage of using the eigenvalue problem (11) is that the eigenval-
ues are invariant to a rotation of the system. Therefore, selection criteria based
on the eigenvalue λκ or the sum of eigenvalues λκ + · · ·+ λr are scale invariant.
For the more general case with some lag order m ≥ 1 the following theo-
rem considers the asymptotic properties of the eigenvalues λ1, . . . , λr from the
generalized eigenvalue problem
|λjS00 − S01S−111 S
′01| = 0, (12)
where
S00 =T∑
t=m+1
FtF′t , S01 =
T∑t=m+1
FtG′t−1 , S11 =
T∑t=m+1
Gt−1G′t−1
and Gt−1 = [F ′t−1, . . . , F′t−m]′ is the vector of m lags.
Theorem: Let Ft denote the PC estimator of the vector of static components Ft
in (1) and assume that the assumptions of Bai and Ng (2002) are fulfilled such
that T−1∑T
t=1 ||Ft −HFt||2 = Op(C−2NT ), where CNT = min(
√N,√T ).
(i) If ft is a k-dimensional vector of dynamic factors, then
(1− λj) = Op(C−2NT ) for j = 1, . . . , r − k
(ii) There exists a constant M > 0 such that as N →∞ and T →∞
P (1− λj > M)→ 1 for j = r − k + 1, . . . , r.
(iii) If ft has a VAR(p) representation as in (3) with p ≤ m and rk(A) = (m+1)k
then, as T →∞ and N →∞,
T
r∑j=r−k+1
λjd→ χ2(k2)
Proof: See appendix.
Remark A: The results (i) and (ii) are derived under the fairly weak conditions
of an approximate factor model (e.g. Bai, 2003, Bai and Ng, 2002 and Stock and
7
Watson, 2002a). They do not require restrictive assumptions on the dynamic
process generating ft with the exception that T−1∑T
t=1 ftf′t
p→ Σf , where Σf is
a positive definite matrix. This assumption rules out that (some of) the factors
are I(1) but allows for a wide range of stationary processes. For practical appli-
cations, the choice of the lag order m is an important issue, as it ensures that
all lags in the factor model are contained in the vector Gt−1. So far, there does
not exist a statistical criterion to choose the lag length m. In practice it seems
reasonable to try out various values of m to find out the minimum lag length
necessary to obtain stable results.
Remark B: The additional assumptions of a finite order VAR representation
of the factors with p ≤ m and rk(A) = (m + 1)k are required for part (iii)
of the theorem. These assumptions ensure that p lags of the dynamic factors
are contained in Ft−1. If these additional assumptions are satisfied, a powerful
selection procedure can be constructed (see below), since the contrast between
the eigenvalues are maximal, i.e. r − k eigenvalues converge to unity, whereas
the remaining k eigenvalues converge to zero. Unfortunately, in practice it is not
known whether these conditions are fulfilled. Therefore, a reasonable selection
strategy is to apply selection procedures based on (ii) and (iii). If both procedures
find the same number of dynamic components one can be confident that this
number is a good choice.
Remark C: The term (1− λj) for j = 1, . . . , r − k is stochastically bounded by
a complicated function of N and T . Following Bai (2003) and Bai and Ng (2002)
the limiting properties are expressed by using the minimum function CNT . For
practical applications the use of this function suffers from an important drawback.
The term CNT is identical for fairly small sample sizes (e.g. N = 50, T = 50) and
for very large samples (such as N = 1000 and T = 50), although the estimation
error is considerably smaller in the latter case. In order to take into account the
relative sizes of the two sampling dimensions we employ an alternative rate CNT
with the property
CNTCNT
→ κ with 0 < κ <∞.
Specifically, our preferred selection criterion employs
C−2NT =
a
N+a
T.
This function has the property that a ≤ C2NT/C
2NT ≤ 2a and, therefore, the lim-
iting result presented in part (i) of the theorem can be alternatively represented
8
as
(1− λj) = Op(C−2NT ) for j = 1, . . . , r − k. (13)
Based on the theorem we can construct two selection rules for the number of
dynamic factors. Using the modified rate CNT of Remark C the first selection
criterion is constructed as
ξ(k∗) = C2−δNT
r−k∗∑j=1
(1− λj) , (14)
where 0 < δ < 2. The theorem implies that under the hypothesis that k∗ is
equal to the correct number of factors k, then the largest r − k∗ eigenvalues
λ1, · · · , λr−k∗ tend to unity and, thus, ξ(k∗) tends to zero. Under the assumption
that k∗ < k, the eigenvalue λr−k∗ has a probability limit smaller than one and,
therefore, the statistic ξ(k∗) tends to infinity. Define ξ(r) = ∞. The number of
dynamic factors can be estimated consistently by the largest number k∗ in the
sequence k∗ = r − 1, r − 2, . . . , 0, where the statistic ξ(k∗) is larger than some
fixed threshold level τ . Thus,
k = max{k∗ : ξ(k∗) > τ}. (15)
In our Monte Carlo simulations we found that a = 20, δ = 0.5 and τ = 1 generally
performs well.
If the conditions of part (iii) of the Theorem are satisfied, the test statistic
LR(k∗) = Tr∑
j=r−k∗+1
λj (16)
can be used to test the null hypothesis H0 : k∗ = k. This test statistic is approx-
imatively equivalent to the likelihood ratio statistic for the hypothesis that the
k∗ smallest eigenvalues of the canonical correlation analysis between Ft and Gt−1
are zero (cf. Tiao and Tsay, 1989). Accordingly, the number of dynamic factors
can be selected by testing the sequence of hypotheses k∗ = r − 1, r − 2, . . . , 1.
For example, if the test rejects for k∗ ≥ 3 but cannot reject for k∗ < 3, then the
maintained number of dynamic factors is k = 2. In other words, the estimated
number of dynamic factors is the largest value of k∗, which does not lead to a re-
jection of the hypothesis. Since it is well known that a selection procedure based
9
on tests with a fixed significance level is not consistent as the error probability
for selecting a smaller number of dynamic factors does not vanish, we construct
an information criterion as
IC(k∗) =r∑
j=r−k∗+1
λj + k∗c(T )
T(17)
where c(T ) is a penalty function. Using part (ii) of the Theorem it is not difficult
to show that the IC(k) is a weakly consistent selection criterion whenever c(T )→∞ and c(T )/T → 0 as T →∞. Consider the difference
IC(k∗)− IC(k∗ + 1) = λr−k∗ − c(T )/T.
It follows that k∗+1 is selected if T λr−k∗ > c(T ). If k ≥ k∗ then T λr−k∗ is Op(1),
whereas T λr−k∗ is Op(T ) if k < k∗. Therefore, a consistent selection rule requires
c(T )→∞ and c(T )/T → 0 as T →∞.
4 Finite sample properties
In the following we analyze the finite sample properties of the alternative selec-
tion procedures and present an empirical application of the new methods to the
dataset used in Stock and Watson (2005).
4.1 Simulation study
To investigate the performance of the proposed procedures to select the number
of dynamic factors, some Monte Carlo experiments were conducted. The data
are generated based on the model
xt = A0ft + A1ft−1 + ut , (18)
where the components of the k-dimensional vector ft are independent Gaussian
AR(1) processes, i.e. fit = γifit−1 + εit for i = 1, . . . , k with εiti.i.d.∼ N(0, 1− γ2
i ).
The elements of the N × k matrices A0 and A1 are i.i.d. standard normal.
The idiosyncratic errors are generated as uti.i.d.∼ N(0, 4ψIN) with ψ controlling
the signal–to–noise ratio. We also tried out alternative models but the general
conclusions remain the same. All results are based on 10,000 replications of the
model.
To investigate the ability of the selection procedures to identify the correct
number of dynamic factors k, we simulate data according to the above model with
10
k = 2 and the following two sets of autocorrelation coefficients: γ1 = γ2 = 0.5
and γ1 = 0.2, γ2 = 0.8. We further set ψ = 2.5, 1, and 0.5, corresponding to low,
medium and high signal–to–noise–ratios.
The performance of the newly proposed procedures is compared to the se-
lection criteria of Amengual and Watson (2007), Stock and Watson (2005) and
Bai and Ng (2007). Note that for the ease of exposition we only report results
for the criteria that have been shown to perform best in previous Monte Carlo
simulations. We therefore consider the information criterion of Amengual and
Watson (2007) and Stock and Watson (2005)
SWP2 = lnV (k∗) + k∗N + T
NTln min{N, T} (19)
with V (kk) denoting the sum of squared residuals (divided by NT ) of the factor
model for the innovation vector ξt as given in (10), if k∗ factors are estimated.
This resembles the ICP2 criterion suggested by Bai and Ng (2002) for the selection
of the number of static factors from the dataset. We further report results for
the selection rule of Bai and Ng (2007) based on the D1 statistics, see equations
(8) and (6). Following Bai and Ng (2007) we set m∗ = 1.
To concentrate on the properties of the various selection procedures, we as-
sume that the number of reduced form factors r is known. In practice, the
information criteria suggested by Bai and Ng (2002) can be used to obtain a
consistent estimate of r. We further base the construction of the D1 statistic on
a VAR(1) model for the factors and treat the number of lags of Ft and xit in the
regressions of the selection procedure of Amengual and Watson (2007) and Stock
and Watson (2005) as given. The computation of the new selection rules (14)
and (16) is based on one lag in the generalized eigenvalue problem, i.e. m = 1.
For the selection criterion ξ(k∗) we further set a = 20, δ = 0.5 and τ = 1, as
these values have been found to perform well in a wide range of Monte Carlo
simulations.
Tables 1 and 2 present the performance of different selection criteria for the
two different sets of the autoregressive parameter values. Reported are the fre-
quencies (in percent) of choosing the correct number of dynamic factors. The
results show that our selection procedures based on canonical correlations out-
perform the existing ones especially in small sample sizes. Moreover, these new
procedures find out the correct number of dynamic factors with a probability
converging to one and to 95% for the ξ(k∗) criterion and the LR(k∗) test, respec-
tively. In the tables we further present the success rates of the various criteria
11
for different values of ψ. Interestingly, the new selection criteria are quite robust,
while the selection procedures based on principal components seem to be very
sensitive to the signal–to–noise ratio. In particular, for low signal–to–noise ratios
these procedures, especially the SWP2 criterion, have difficulties in determining
the correct number of factors. This becomes even more pronounced in the pres-
ence of more persistent factors (see Table 2, which presents the success rates for
the parameter set γ1 = 0.2 and γ2 = 0.8). The performance of the new selection
procedures instead seem to be also robust to changes in the factor persistence.
As mentioned previously, the LR test is only a powerful selection criteria
if rk(A) = (m + 1)k. As this assumption is difficult to verify in practice, we
also investigate the finite sample performance of the procedure in the case of the
violation of this assumption. In particular, we consider the above factor model
with k = 2 and A1 having zero second column, which implies that r = 3. Table 3
presents the corresponding success rates of the different selection procedures for
γ1 = γ2 = 0.5. As expected the LR(k∗) test is not able to identify the correct
number of dynamic factors, while the previous conclusions for the alternative
criteria also hold for this data generation process. As the selection procedures
based on canonical correlations are both very powerful if rk(A) = (m + 1)k,
we suggest for empirical applications to compute both criteria. If both come to
similar conclusions, then this indicates that the assumption is not violated.
4.2 Empirical application
In the following we apply our selection procedures to a large U.S. macroeco-
nomic dataset constructed by Stock and Watson (2005)6. The data comprises
132 monthly macroeconomic time series ranging from January 1960 to December
2003, i.e. T = 528. For our empirical application we perform the same data
transformations as in Stock and Watson (2005). The dataset has been widely
used to test for the number of static and dynamic factors driving the U.S. econ-
omy. Bai and Ng (2007) and Stock and Watson (2005), for example, select seven
static factors based on the information criteria of Bai and Ng (2002). Given these
seven factors, their selection procedures indicate four and seven dynamic factors,
respectively.
Using the ICP2 information criteria of Bai and Ng (2002) we also find seven
6The dataset is kindly provided at Mark Watson’s homepage: http://www.princeton.edu/~mwatson/.
12
static factors to which we apply our selection procedures. Table 4 shows the
statistic ξ(k∗) and the LR(k∗) test statistics using m = 1 in the generalized
eigenvalue problem (12). The first selection criterion ξ(k∗) selects k = 4 dynamic
factors, while the LR(k∗) test indicates one dynamic factor. The observation
that both criteria choose a different number of dynamic factors suggests that the
rank conditions of part (iii) of the theorem are not fulfilled and we thus rely in
the following on the ξ(k∗) criterion.
For the accuracy of the ξ(k∗) criteria, we need to ensure that all lags in the
factor model are also contained in the vector Gt−1. We therefore also consider
larger values of m. The results for m ranging from 2 to 4 are presented in Table
5. Interestingly, the selected number of dynamic factor remains at k = 4 for all
lags, which indicates that a lag order of one is sufficient. We therefore conclude
that there are four dynamic factors that drive the U.S. economy - a result that
is consistent with the findings of Bai and Ng (2007).
5 Conclusion
In this paper two identification procedures are suggested that allow to estimate
the number of dynamic factors (or “structural factors”) from a set of static factors
(or “reduced form factors”) that are obtained from a principal component analysis
of a large number of observed series, see Stock and Watson (2002a,b). While the
procedures of Bai and Ng (2007), Amengual and Watson (2007) and Stock and
Watson (2005) use the method of principal components to determine the number
of dynamic factors, our approach is based on a canonical correlation analysis
(CCA) of the static factors. An important advantage of our approach is that
it is invariant to the normalization of the factor space and does not assume a
finite order VAR model for the vector of common factors. Furthermore, our
Monte Carlo simulations indicate that the new criteria generally outperform the
alternative selection criteria and are much more reliable especially in the presence
of low signal–to–noise ratios and/or persistent factors. Applying the new criteria
to the dataset used in Stock and Watson (2005) yields that seven static factors
can be represented by four dynamic factors and three lags which corroborates the
earlier findings of Bai and Ng (2007).
13
Appendix A: Proof of the Theorem
We first state the following lemma:
Lemma A.1: Under Assumptions A–F of Bai (2003) it holds for any fixed j
that
(a) T−1
T∑t=j+1
(Ft − Ft)F ′t−j = Op(C−2NT ), T−1
T∑t=j+1
(Ft − Ft)F ′t−j = Op(C−2NT )
(b) T−1
T∑t=j+1
FtF′t−j = T−1
T∑t=j+1
FtF′t−j +Op(C
−2NT )
Proof: The proof follows closely the proof of Lemma B.2 and Lemma B.3 of
Bai (2003) and can be found in Breitung and Tenhofen (2008, Lemma A.1). �
To proof part (i) of the theorem, let F = [Fm+1, . . . , FT ]′, F = [Fm+1, . . . , FT ]′,
G = [Gm, . . . , GT−1]′, G = [Gm, . . . , GT−1]
′ and Gt−1 = [Ft−1, . . . , Ft−m]′. From
Lemma A.1 it follows that
T−1F ′F = T−1F ′F +Op(C−2NT ), T−1F ′G = T−1F ′G+Op(C
−2NT ),
T−1G′G = T−1G′G+Op(C−2NT )
and, therefore, λj = λ0j + Op(C
−2NT ), where λ0
j denotes the eigenvalue from the
eigenvalue problem
|λ0jF′F − F ′G(G′G)−1G′F | = 0. (20)
The eigenvalues of can be written as
λ0j =
v′jF′G(G′G)−1G′Fvj
v′jF′Fvj
.
If Ft contains r − k lagged values up to the maximal lag m that is used to
construct Gt−1 then there exist r − k linear independent combinations of the
form v′jFt = w′jGt−1 where wj is some mr× 1 vector. Therefore, the r− k largest
eigenvalues λ01, . . . , λ
0r−k are equal to one. This implies that for j = 1, . . . , r − k
we have (1− λj) = Op(C−2NT ).
To proof part (ii) we first note that if there exist r dynamic factors, then
the covariance matrix of the projection residual ηt = Ft − E(Ft|Gt−1) is positive
definite. Hence,
1− λ0j =
v′jF′Fvj − v′jF ′G(G′G)−1G′Fvj
v′jF′Fvj
14
and by the weak law of large numbers there exists a constant M such that
limT→∞ P (1 − λ0j > M) = 1 for j = r − k + 1, . . . , r. Finally, if N → ∞
and T → ∞, then P (1 − λj > M − Op(C−2NT )) will converge to unity as well for
j = r − k + 1, . . . , r.
Under the conditions stated in part (iii) of the theorem, all lags of the VAR
representation of Ft enter Gt−1 and, thus, there exist r − k vectors vj, j =
r − k + 1, . . . , k such that v′jFt = w′jut is white noise. The following lemma
presents the limiting distribution of the sum of eigenvalues in this situation.
Lemma A.2. Let yt = [z′t, y′2t]′ and xt = [z′t, x
′2t]′, where x2t and y2t are m × 1
and zt is a n × 1 vector. Furthermore E(xt) = E(yt) = 0 and E(xtx′t) = Σx,
E(yty′t) = Σy. Assume that there exist m linear combinations [w′1yt, . . . , w
′myt]
′ ≡W ′yt with E(W ′yt|xt) = 0 and V ar(W ′yt|xt) = Σm, where Σm is positive definite.
Then, as T →∞
Tn+m∑i=n+1
µi + op(1)d→ χ2(m2),
where µi denotes the eigenvalues (in descending order) of
|µSyy − SyxS−1xx S
′xy| = 0 , (21)
and
Sab = T−1
T∑t=1
atb′t, a, b ∈ {x, y}.
Proof: Let x2t (y2t) denote the projection residuals of x2t (y2t) on zt, and
xt =
[ztx2t
], yt =
[zty2t
].
Accordingly we define
Sxx =1
T
T∑t=1
xtx′t =
[Szz 00 Sx2x2
]
Syy =1
T
T∑t=1
yty′t =
[Szz 00 Sy2y2
]
Sxy =1
T
T∑t=1
xty′t =
[Szz 00 Sx2y2
].
15
Since zt enters both xt and yt, the first n eigenvalues are unity and the corre-
sponding eigenvectors are [v1, . . . , vn] = V = [In, 0]′. The remaining matrix of m
eigenvectors can be normalized such that WT = [wT1, . . . , wTm]′ = [0, Im]′. The
eigenvalues µn+1, . . . , µn+m can be written as
µn+j = w′TjS−1/2yy S ′xyS
−1xx SxyS
−1/2yy wTj
= e′jS−1/2y2y2
S ′x2y2S−1x2x2
Sx2y2S−1/2y2y2
ej,
where ej is the j’th column of Im and
n+m∑i=n+1
µi = tr{S−1/2y2y2
S ′x2y2S−1x2x2
Sx2y2S−1/2y2y2
}.
Using Sy2y2p→ Σy2y2 , Sx2x2
p→ Σx2x2 and√Tvec(Σ
−1/2y2y2
S ′x2y2Σ−1/2x2x2
)d→ N(0, Im2)
it follows that T∑n+m
j=n+1 µj has an asymptotic χ2 limiting distribution with m2
degrees of freedom. �
Under the conditions of part (iii) of the theorem we can find rotations Q1Ft =
[z′t, x′t]′ and Q2Gt−1 = [z′t, y
′t]′ that fulfill the conditions of Lemma 2. Therefore,
the sum of the k smallest eigenvalues have a χ2 limiting distribution with k2
degrees of freedom. Let[z∗tx∗t
]= QFt and
[z∗ty∗t
]= QGt−1
and denote by xt and yt the respective residuals from a projection on z∗t . Then,
by using the results in part (i) of the theorem
T−1/2
T∑t=m+1
x∗t y′∗t = T−1/2
T∑t=m+1
xty′t +Op(
√T/C2
NT )
T−1
T∑t=m+1
x∗t x′∗t = T−1
T∑t=m+1
xtx′t +Op(1/C
2NT )
T−1
T∑t=m+1
y∗t y′∗t = T−1
T∑t=m+1
yty′t +Op(1/C
2NT )
and, therefore,
Tr∑
j=r−k+1
λj = T
r∑r−k+1
µj +Op(√T/C2
NT )
where the eigenvalues µj are defined as in Lemma A.2 with m = k. It follows
that T∑r
j=r−k+1 λj has a χ2 distribution with k2 degrees of freedom.
16
References
Amengual, D. and Watson, M. (2007), “Consistent Estimation of the Number of
Dynamic Factors in a Large N and T Panel,” Journal of Business & Economic
Statistics, 25, 91–96.
Angelini, E., Henry, J., and Mestre, R. (2001), “Diffusion Index-based Inflation
Forecasts for the Euro Area,” ECB Working Paper 61.
Bai, J. (2003), “Inferential Theory for Factor Models of Large Dimensions,”
Econometrica, 71, 135–172.
Bai, J. and Ng, S. (2002), “Determining the Number of Factors in Approximate
Factor Models,” Econometrica, 70, 191–221.
— (2007), “Determining the Number of Primitive Shocks in Factor Models,”
Journal of Business & Economic Statistics, 25, 52–60.
Breitung, J. and Tenhofen, J. (2008), “GLS Estimation of Dynamic Factor Mod-
els,” Working paper, http://www.ect.uni-bonn.de/mitarbeiter/breitung/pc-
gls.pdf.
Brisson, M., Campbell, B., and Galbraith, J. (2003), “Forecasting Some Low-
predictability Time Series Using Diffusion Indices,” Journal of Forecasting, 22,
515–531.
Eickmeier, S. and Ziegler, C. (2008), “How Successful Are Dynamic Factor Models
at Forecasting Output and Inflation? A Meta-Analytic Approach,” Journal of
Forecasting, 27, 237–265.
Favero, C., Marcellino, M., and Neglia, F. (2005), “Principal Components at
Work: The Empirical Analysis of Monetary Policy with Large Data Sets,”
Journal of Applied Econometrics, 20, 603–620.
Forni, M., Hallin, M., Lippi, M., and Reichlin, L. (2000), “The Generalized Dy-
namic Factor Model: Identification and Estimation,” The Review of Economics
and Statistics, 82, 540–554.
Forni, M. and Lippi, M. (2001), “The Generalized Factor Model: Representation
Theory,” Econometric Theory, 17, 1113–1141.
17
Forni, M., Lippi, M., and Reichlin, L. (2008), “Opening the Black Box: Structural
Factor Models versus Structural VARs,” Econometric Theory, forthcoming.
Forni, M. and Reichlin, L. (1998), “Lets Get Real: A Factor-Analytical Approach
to Disaggregated Business Cycle Dynamics,” Review of Economic Studies, 65,
453–473.
Giannone, D., Reichlin, L., and Sala, L. (2002), “Tracking Greenspan: Systematic
and Unsystematic Monetary Policy Revisited,” CEPR Discussion Paper 3550.
— (2006), “VARs, Factor Models and the Empirical Validation of Equilibrium
Business Cycle Models,” Journal of Econometrics, 132, 257–279.
Hallin, M. and Liska, R. (2007), “Determining the Number of Dynamic Factors
in the General Dynamic Factor Model,” Journal of the American Statistical
Association, 102, 603–617.
Jacobs, J. and Otter, P. (2008), “Determining the Number of Factors and Lag Or-
der in Dynamic Factor Models: A Minimum Entropy Approach,” Econometric
Reviews, 27, 398–427.
Marcellino, M., Stock, J., and Watson, M. (2003), “Macroeconomic Forecasting
in the Euro Area: Country Specific versus Area-Wide Information,” European
Economic Review, 47, 1–18.
Pesaran, M. H. (2006), “Estimation and Inference in Large Heterogeneous Panels
with a Multifactor Error Structure,” Econometrica, 74, 967–1012.
Stock, J. and Watson, M. (1999), “Forecasting Inflation,” Journal of Monetary
Economics, 44, 293–335.
— (2002a), “Forecasting Using Principal Components from a Large Number of
Predictors,” Journal of the American Statistical Association, 97, 1167–79.
— (2002b), “Macroeconomic Forecasting Using Diffusion Indexes,” Journal of
Business & Economic Statistics, 20, 147–162.
— (2005), “Implications of Dynamic Factor Models for VAR Analysis,” NBER
Working Paper 11467.
Tiao, G. and Tsay, R. (1989), “Model Specification in Multivariate Time Series,”
Journal of the Royal Statistical Society, Series B, 51, 157–213.
18
Table 1: Rates of success of model selection criteria for r = 4, k = 2 andγ1 = γ2 = 0.5
SWP2 D1 ξ LR SWP2 D1 ξ LRψ = 2.5
N T = 50 T = 10020 0.0021 0.0081 0.5345 0.3708 0.0024 0.0057 0.2034 0.833850 0.0107 0.0007 0.5336 0.8697 0.0085 0.0155 0.3078 0.9464100 0.0096 0.3104 0.9125 0.9399 0.0537 0.1190 0.8569 0.9497150 0.0367 0.7794 0.9903 0.9436 0.3292 0.7072 0.9908 0.9469200 0.0648 0.9540 0.9987 0.9382 0.6074 0.9704 0.9999 0.9435N T = 150 T = 20020 0.0026 0.0076 0.1262 0.9339 0.0036 0.0105 0.0997 0.942350 0.0314 0.0174 0.1840 0.9456 0.0622 0.0220 0.1074 0.9506100 0.3353 0.1649 0.7423 0.9479 0.6269 0.1894 0.6025 0.9501150 0.6603 0.5049 0.9791 0.9454 0.9377 0.5918 0.9591 0.9443200 0.9327 0.9406 0.9995 0.9433 0.9900 0.8653 0.9991 0.9482
ψ = 1N T = 50 T = 10020 0.2772 0.3531 0.9863 0.8658 0.4644 0.4586 0.9818 0.951050 0.4876 0.6840 0.9990 0.9402 0.9167 0.8390 1.0000 0.9455100 0.9091 0.9963 1.0000 0.9371 0.9982 0.9955 1.0000 0.9470150 0.9789 0.9999 1.0000 0.9410 1.0000 1.0000 1.0000 0.9476200 0.9907 1.0000 1.0000 0.9389 1.0000 1.0000 1.0000 0.9426N T = 150 T = 20020 0.5367 0.5029 0.9833 0.9509 0.9345 0.9174 0.9846 0.945950 0.9832 0.8879 1.0000 0.9472 1.0000 0.9997 1.0000 0.9487100 1.0000 0.9991 1.0000 0.9460 1.0000 1.0000 1.0000 0.9478150 1.0000 1.0000 1.0000 0.9440 1.0000 1.0000 1.0000 0.9426200 1.0000 1.0000 1.0000 0.9440 1.0000 1.0000 1.0000 0.9486
ψ = 0.5N T = 50 T = 10020 0.7948 0.8430 0.8816 0.9388 0.9028 0.8950 1.0000 0.947750 0.9832 0.9899 0.9992 0.9377 1.0000 0.9990 1.0000 0.9452100 0.9999 1.0000 1.0000 0.9341 1.0000 1.0000 1.0000 0.9464150 1.0000 1.0000 1.0000 0.9402 1.0000 1.0000 1.0000 0.9466200 1.0000 1.0000 1.0000 0.9373 1.0000 1.0000 1.0000 0.9430N T = 150 T = 20020 0.9216 0.9127 1.0000 0.9469 0.9461 0.9234 1.0000 0.945950 1.0000 0.9991 1.0000 0.9469 1.0000 0.9997 1.0000 0.9488100 1.0000 1.0000 1.0000 0.9457 1.0000 1.0000 1.0000 0.9481150 1.0000 1.0000 1.0000 0.9446 1.0000 1.0000 1.0000 0.9425200 1.0000 1.0000 1.0000 0.9437 1.0000 1.0000 1.0000 0.9468
19
Table 2: Rates of success of model selection criteria for r = 4, k = 2 andγ1 = 0.2, γ2 = 0.8
SWP2 D1 ξ LR SWP2 D1 ξ LRψ = 2.5
N T = 50 T = 10020 0.0005 0.0118 0.6295 0.3687 0.0003 0.0049 0.4347 0.851150 0.0000 0.0029 0.4348 0.7852 0.0000 0.0035 0.2959 0.9485100 0.0000 0.1224 0.6504 0.9196 0.0001 0.0300 0.5567 0.9475150 0.0000 0.4455 0.8192 0.9334 0.0003 0.3868 0.8326 0.9451200 0.0000 0.7207 0.9102 0.9419 0.0025 0.8306 0.9515 0.9485N T = 150 T = 20020 0.0004 0.0051 0.3933 0.9372 0.0003 0.0072 0.3854 0.946450 0.0002 0.0070 0.2370 0.9484 0.0003 0.0072 0.1941 0.9478100 0.0005 0.0603 0.5297 0.9452 0.0031 0.0929 0.4883 0.9477150 0.0008 0.2468 0.8328 0.9470 0.0137 0.3587 0.8144 0.9502200 0.0126 0.7882 0.9590 0.9457 0.0323 0.6887 0.9574 0.9467
ψ = 1N T = 50 T = 10020 0.1064 0.2935 0.9747 0.8043 0.1527 0.3793 0.9810 0.944150 0.0659 0.4725 0.9784 0.9410 0.3071 0.7301 0.9939 0.9453100 0.2950 0.9780 0.9982 0.9432 0.5732 0.9808 1.0000 0.9443150 0.4498 0.9993 0.9995 0.9370 0.8849 1.0000 1.0000 0.9438200 0.5500 1.0000 1.0000 0.9404 0.9645 1.0000 1.0000 0.9476N T = 150 T = 20020 0.1892 0.4440 0.9894 0.9505 0.2026 0.4698 0.9927 0.946750 0.4946 0.8166 0.9983 0.9473 0.6212 0.8563 0.9990 0.9481100 0.8975 0.9944 1.0000 0.9451 0.9742 0.9978 1.0000 0.9482150 0.9800 0.9999 1.0000 0.9451 0.9992 1.0000 1.0000 0.9444200 0.9981 1.0000 1.0000 0.9453 0.9999 1.0000 1.0000 0.9474
ψ = 0.5N T = 50 T = 10020 0.5597 0.7894 0.9375 0.9241 0.7324 0.8589 0.9996 0.945050 0.7240 0.9726 0.9997 0.9394 0.9681 0.9956 1.0000 0.9458100 0.9596 0.9999 1.0000 0.9395 0.9992 1.0000 1.0000 0.9460150 0.9596 0.9999 1.0000 0.9333 0.9992 1.0000 1.0000 0.9408200 0.9959 1.0000 1.0000 0.9380 1.0000 1.0000 1.0000 0.9453N T = 150 T = 20020 0.7937 0.8842 1.0000 0.9488 0.8202 0.8912 1.0000 0.946150 0.9935 0.9967 1.0000 0.9489 0.9983 0.9988 1.0000 0.9449100 1.0000 1.0000 1.0000 0.9432 1.0000 1.0000 1.0000 0.9483150 1.0000 1.0000 1.0000 0.9452 1.0000 1.0000 1.0000 0.9440200 1.0000 1.0000 1.0000 0.9446 1.0000 1.0000 1.0000 0.9469
20
Table 3: Rates of success of model selection criteria for r = 3, k = 2 andγ1 = γ2 = 0.5
SWP2 D1 ξ LR SWP2 D1 ξ LRψ = 2.5
N T = 50 T = 10020 0.0009 0.0357 0.9396 0.8660 0.0014 0.0264 0.9988 0.541650 0.0001 0.0729 0.9943 0.6179 0.0075 0.0741 0.9986 0.1727100 0.0096 0.5861 0.9949 0.4696 0.0516 0.3151 0.9999 0.0721150 0.0311 0.9088 0.9962 0.4042 0.3085 0.8579 1.0000 0.0523200 0.0550 0.9856 0.9952 0.3751 0.5999 0.9910 1.0000 0.0375N T = 150 T = 20020 0.0030 0.0268 0.9999 0.3068 0.0021 0.0235 1.0000 0.161650 0.0298 0.0806 0.9990 0.0317 0.0598 0.0839 0.9998 0.0049100 0.3278 0.3718 1.0000 0.0075 0.6280 0.4123 1.0000 0.0004150 0.6676 0.7267 1.0000 0.0038 0.9357 0.7802 1.0000 0.0003200 0.9322 0.9767 1.0000 0.0036 0.9907 0.9480 1.0000 0.0000
ψ = 1N T = 50 T = 10020 0.2607 0.5445 0.3914 0.6147 0.4303 0.6249 0.7579 0.192350 0.4577 0.8509 0.8707 0.4213 0.9172 0.9358 0.9990 0.0600100 0.9068 0.9989 0.9499 0.3531 0.9977 0.9990 1.0000 0.0364150 0.9763 1.0000 0.9624 0.3273 1.0000 1.0000 1.0000 0.0309200 0.9903 1.0000 0.9715 0.3228 1.0000 1.0000 1.0000 0.0249N T = 150 T = 20020 0.5161 0.6702 0.8870 0.0581 0.5640 0.6824 0.9402 0.013250 0.9811 0.9544 1.0000 0.0044 0.9944 0.9638 1.0000 0.0004100 1.0000 0.9997 1.0000 0.0019 1.0000 1.0000 1.0000 0.0000150 1.0000 1.0000 1.0000 0.0021 1.0000 1.0000 1.0000 0.0000200 1.0000 1.0000 1.0000 0.0011 1.0000 1.0000 1.0000 0.0000
ψ = 0.5N T = 50 T = 10020 0.8072 0.9279 0.0502 0.4641 0.9370 0.9566 0.1733 0.090850 0.9847 0.9977 0.6433 0.3536 1.0000 0.9997 0.9878 0.0372100 0.9998 1.0000 0.8827 0.3187 1.0000 1.0000 0.9999 0.0272150 1.0000 1.0000 0.9316 0.3050 1.0000 1.0000 1.0000 0.0262200 1.0000 1.0000 0.9505 0.3064 1.0000 1.0000 1.0000 0.0216N T = 150 T = 20020 0.9607 0.9631 0.2989 0.0142 0.9675 0.9693 0.3726 0.001550 1.0000 0.9997 0.9997 0.0018 1.0000 0.9998 1.0000 0.0001100 1.0000 1.0000 1.0000 0.0009 1.0000 1.0000 1.0000 0.0000150 1.0000 1.0000 1.0000 0.0018 1.0000 1.0000 1.0000 0.0000200 1.0000 1.0000 1.0000 0.0008 1.0000 1.0000 1.0000 0.0000
21
Table 4: Selection criteria for the number of dynamic factors in the U.S. economy
k∗ λ ξ(k∗) LR(k∗) crit. val.1 0.0422 3.0625 1.5• 3.84152 0.1706 2.1048 23.7 9.48773 0.2237 1.2754 113.5 16.91904 0.7577 0.4991• 231.1 26.29625 0.8596 0.2567 629.6 37.65256 0.8837 0.1163 1081.8 50.9985
Reported are the selection criteria ξ(k∗) and LR(k∗) for differ-ent values of k∗ using m = 1. Note, that the critical value ofthe ξ(k∗) statistic is C−2+δ
NT =0.2873. The first column reportsthe eigenvalues of (12) with m = 1, the last columns reportsthe critical values of the LR–test. • indicates those criteriathat select the number of dynamic factors.
Table 5: Selection criteria ξ(k∗) based on different values of m
k∗ m=2 m=3 m=40 3.7484 3.6428 3.52211 2.8051 2.7212 2.61372 1.8827 1.8203 1.74503 1.1119 1.0731 1.05064 0.4218• 0.3980• 0.3883•
5 0.2244 0.2151 0.20956 0.1047 0.0981 0.0942
Reported are the selection criteria ξ(k∗) for differentvalues of k∗ and different lag lengths m. The criticalvalue of the ξ(k∗) statistic is C−2+δ
NT =0.2873. • indicatesthose criteria that select the number of dynamic factors.
22
