Forecasting Using Principal Components
From a Large Number of Predictors
James
H.
STOCK
and
Mark
W. WATSON
This article considers forecasting a single time series when there are many predictors
(N)
and time series observations (T). When the
data follow an approximate factor model, the predictors can be summarized by a small number of indexes, which we estimate using
principal components. Feasible forecasts are shown to be asymptotically efficient in the sense that the difference between the feasible
forecasts and the infeasible forecasts constructed using the actual values of the factors converges in probability to
0
as both
N
and T
grow large. The estimated factors are shown to be consistent, even in the presence of time variation in the factor model.
KEY WORDS:
Factor models; Forecasting; Principal components.
1.
INTRODUCTION
classic factor analysis model. In our macroeconomic forecast-
This article considers forecasting one series using a large
number of predictor series. In macroeconomic forecasting, for
example, the number of candidate predictor series (N) can be
very large, often larger than the number of time series obser-
vations (T) available for model fitting. This high-dimensional
problem is simplified by modeling the covariability of the
series in terms of a relatively few number of unobserved latent
factors. Forecasting can then be carried out in a two-step pro-
cess. First, a time series of the factors is estimated from the
predictors; second, the relationship between the variable to be
forecast and the factors is estimated by a linear regression. If
the number of predictors is large, then precise estimates of the
latent factors can be constructed using simple methods even
under fairly general assumptions about the cross-sectional and
temporal dependence in the variables. We estimate the factors
using principal components, and show that these estimates are
consistent in an approximate factor model with idiosyncratic
errors that are serially and cross-sectionally correlated.
To be specific, let y, be the scalar time series variable to be
forecast and let
Xi be a N-dimensional multiple time series
of candidate predictors. It is assumed that (Xi, y,,,) admit a
factor model representation with
r
common latent factors F,,
X,
=
AF,
+
e,
(1)
and
Yr+h
=
PkFr
+PLwt
+
Et+h
(2)
where
e,
is a N
x
1 vector idiosyncratic disturbances,
h
is the
forecast horizon, w, is a
m
x
1 vector of observed variables
(e.g., lags of y,), that together with
F,
are useful for forecasting
y,+,, and
st+,
is the resulting forecast error. Data are available
for {y,, X,, w,}:,, and the goal is to forecast y,+,.
If the idiosyncratic disturbances e, in (1) were cross-
sectionally independent and temporally iid, then (1) is the
James
H.
Stock is Professor, Kennedy School of Government, Harvard
University, Cambridge, MA 02138, and the National Bureau of Economic
Research (E-mail:
james-stock@harvard.edu).
Mark W. Watson is Profes-
sor, Department of Economics and Woodrow Wilson School, Princeton
University, Princeton, NJ 08540, and the National Bureau of Economic
Research (E-mail: mwatson@princeton.edu). The results in this article origi-
nally appeared in the paper titled "Diffusion Indexes" (NBER Working Paper
6702, August 1998). The authors thank the associate editor and referees,
Jushan Bai, Michael Boldin, Frank Diebold, Gregory Chow, Andrew Harvey,
Lucrezia Reichlin, Ken Wallis, and Charles Whiteman for helpful discussions
and/or comments, and Lewis Chan, Piotr Eliasz, and Alexei Onatski for skilled
research assistance. This research was supported in part by National Science
Foundation grants SBR-9409629 and SBR-9730489.
ing application, these assumptions are unlikely to be satisfied,
and so we allow the error terms to be both serially corre-
lated and (weakly) cross-sectionally correlated. In this sense,
(1) is a serially correlated version of the approximate factor
model introduced by Chamberlain and Rothschild (1983) for
the study of asset prices. To construct forecasts of
y,,,, we
form principal components of {X,}T=, to serve as estimates of
the factors. These estimated factors, together with w,, are then
used in (2) to estimate the regression coefficients. The fore-
cast is constructed as
j,,,
=
&PT
+F,w,, where
p,,
p,,
and
pT
are the estimated coefficients and factors.
This article makes three contributions. First, under general
conditions on the errors discussed in Section 2, we show that
the principal components of
Xi are consistent estimators of
the true latent factors (subject to a normalization discussed
in Sec. 2). Consistency requires that both N and T
+
co,
although there are no restrictions on their relative rates of
increase. Second, we show that the feasible forecast, j,,,,
constructed from the estimated factors together with the esti-
mated coefficients converge to the infeasible forecast that
would be obtained if the factors and coefficients were known.
Again, this result holds as N,
T
+
co.
Thus the feasible fore-
cast is first-order asymptotically efficient. Finally, motivated
by the problem of temporal instability in macroeconomic fore-
casting models, we study the robustness of the consistency
results to time variation in the factor model. We show that
these results continue to hold when the temporal instability
is small (as suggested by empirical work in macroeconomics)
and weakly cross-sectionally dependent, in a sense that is
made precise in Section
3.
This article is related to a large literature on factor anal-
ysis and a much smaller literature on forecasting. The liter-
ature on principal components and classical factor models is
large and well known
(Lawley and Maxwell 1971). Sargent
and Sims (1977) and Geweke (1977) extended the classical
factor model to dynamic models, and several researchers have
applied versions of their dynamic factor model. In most appli-
cations of the classic factor model and its dynamic general-
ization, the dimension of
X
is small, and so the question of
O
2002 American Statistical Association
Journal of the American Statistical Association
December 2002, Vol. 97, No. 460, Theory and Methods
DO1 10.1 198101 621450238861 8960
1168
consistent estimation of the factors is not relevant. However,
several authors have noted that with large N, consistent esti-
mation is possible. Connor and Korajczyk (1986, 1988, 1993)
discussed the problem in a static model and argue that the fac-
tors can be consistently estimated by principle components as
N
-t
co even if the errors terms are weakly cross-sectionally
correlated. Forni and Reichlin (1996, 1998) and Forni, Hallin,
Lippi, and Reichlin (1999) discussed consistent (N, T
+
co)
estimation of factors in a dynamic version of the approximate
model. Finally, in a prediction problem similar to the one con-
sidered here, Ding and Hwang (1999) analyzed the properties
of forecasts constructed from principal components in a set-
ting with large N and T. Their analysis is conducted under
the assumption that error process {e,, is cross-sectionally
and temporally iid, an assumption that is inappropriate for
economic models and when interest focuses on multiperiod
forecasts. We highlight the differences between our results and
those of others later in the article.
The article is organized as follows. Section 2 presents the
model in more detail, discusses the assumptions, and presents
the main consistency results. Section 3 generalizes the model
to allow temporal instability in the factor model. Section 4
examines the finite-sample performance of these methods in a
Monte
Carlo study, and Section
5
discusses an application to
macroeconomic forecasting.
2.
THE MODEL AND ESTIMATION
2.1
Assumptions
As described in Section 1, we focus on a forecasting sit-
uation in which N and T are both large. This motivates our
asymptotic results requiring that N, T
+
co jointly or, equiv-
alently, that N
=
N(T) with lim,,, N(T)
+
co. No restric-
tions on the relative rates of N and T are required.
The assumptions about the model are grouped into assump-
tions about the factors and factor loading, assumptions about
the errors in the (I), and assumptions about the regressors and
errors in (2).
Assumption
Fl
(Factors and Factor Loading).
a.
(A'AIN)
-+
I,.
b.
E(F,FI1)
=
C,,,
where
ZFF
is a diagonal matrix with
elements
a,,
>
a,
>
0
for i
<
j.
C.
Ihi,,,l
5
h
<
M.
d. T-I
C,
F,F;
i;
XFF.
Assumption F1 serves to identify the factors. The nonsin-
gular limiting values of (A'AlN) and
ZFF
imply that each of
the factors provides a nonnegligible contribution to the aver-
age variance of
x,,,
where
x,,
is the ith element of
X,
and
the average is taken over both i and t. Moreover, because
AF,
=
ARR-'F, for any nonsingular matrix R, a normaliza-
tion is required to uniquely define the factors. Said differ-
ently, the model with factor loadings AR and factors R-IF,
is observationally equivalent to the model with factor load-
ings
A
and factors
F,.
Assumption Fl(a) restricts R to be
orthonormal, and this together with Assumption Fl(b) restricts
R to be a diagonal matrix with diagonal elements of
*I.
This
identifies the factors up to a change of sign. Equivalently,
Assumption Fl provides this normalization (asymptotically)
Journal
of
the American Statistical Association, December
2002
by associating A with the ordered orthonormal eigenvectors of
(NT)-I xT=, AF,F:A' and {F,}~=, with the principal compo-
nents of {AF,}:=,. The diagonal elements of
ZFF
correspond
to the limiting eigenvalues of (NT)-I
cT=,
AF,F,'A1. For con-
venience, these eigenvalues are assumed to be distinct. If they
were not distinct, then the factors could only be consistently
estimated up to an orthonormal transformation.
Assumption
Fl(b) allows the factors to be serially corre-
lated, although it does rule out stochastic trends and other pro-
cesses with nonconstant uncondititional second moments. The
assumption also allows lags of the factors to enter the equa-
tions for
x,,
and y,+,
.
A leading example of this occurs in the
dynamic factor model
and
Yt+h
=
Ps(L)'fr
+
P:.wI
+
&i+h, (4)
where A,(L) and Pf(L) are lag polynomials is nonnegative
powers of the lag operator L. If the lag polynomials have
finite order
q,
then (3)-(4) can be rewritten as (1)-(2) with
F,
=
(f(
fk,
.
.
.
f(-,)', and Assumption Fl(b) will be satisfied
if the
f,
process is covariance stationary.
In the classical model, the errors or "uniquenesses" are
assumed to be iid and normally distributed. This assumption is
clearly inappropriate in the macroeconomic forecasting appli-
cation, because the variables are serially correlated, and many
or the variables
(e.g., alternative measures of the money sup-
ply) may be cross-correlated even after the aggregate factors
are controlled for. We therefore modify the classic assump-
tions to accommodate these complications.
Assumption
MI (Moments of the Errors e,)
Let e,, denote the zth element of e,; then
b. E(e,telr)
=
rll,
r,
limN+,
SUP,
N-'
CLl
C,N_l
IriJ
l
1
<
"9
and
C. 1lmN-m SUP,
5
N-I
EL1 Ey=1
lcov(e,sei,, e,seJ,)l
<
CQ.
Assumption Ml(a) allows for serial correlation in the
el,
processes. As in the approximate factor model of Chamber-
lain and Rothschild (1983) and Connor and Korajczyk (1986,
1993), Assumption Ml(b) allows (e,,} to be weakly corre-
lated across series. Forni et al. (1999) also allowed for serial
correlation and cross-correlation with assumptions similar to
Ml(a)-(b). Normality is not assumed, but Ml(c) limits the
size of fourth moments.
It is assumed that the forecasting equation
(2)
is well
behaved in the sense that if {F,} were observed, then ordi-
nary least squares (OLS) would provide a consistent estimator
of the regression coefficients. The specific assumption is as
follows.
Assumption
YI
(Forecasting Equation).
Let
z,
=
(F: w:)'
and
p=
(pk
PL,)'.
Then the following hold:
a.
E(z,zi)
=
XI
=
['FF
xu,io
a positive definite matrix.
xw~
'~~'1
1169
Stock and Watson: Forecasting From Many Predictors
Assumptions Yl(a)-(c) are a standard set of conditions that
imply consistency of OLS from the regression of
y,+,
onto
(Fi wi). Here
F,
is not observed, and the additional assump-
tions are useful for showing consistency of the OLS regres-
sion coefficients in the regression of y,,, onto
(e
w:) and the
resulting forecast of y,+,.
2.2
Estimation
In "small-N" dynamic factor models, forecasts are gener-
ally constructed using a three-step process (see, e.g., Stock
and Watson 1989). First, parametric models are postulated for
the joint stochastic process {y,,,, X,, w,, e,}, and the sample
data {y,,,, X,, w,}:_;~ are used to estimate the parameters of
this process, typically using a Gaussian Maximum likelihood
estimator (MLE). Next, these estimated parameters are used
in signal extraction algorithms to estimate the unknown value
of F,. Finally, the forecast of y,,, is constructed using this
estimated value of the factor and the estimated parameters.
When N is large, this process requires estimating many param-
eters using iterative nonlinear methods, which can be compu-
tationally prohibitive. We therefore take a different approach
and estimate the dynamic factors nonparametrically using the
method of principal components.
Consider the nonlinear least squares objective function,
written as a function of hypothetical values of the factors
(F)
-
and factor loadings (A), where
F=
(4F2.
.
.
FT)' and
Xi
is the
ith row of
X.
Let
FI
and
;i
denote the minimizers of
v(F,
X).
After concentrating out
F,
minimizing (5) is equivalent to
maximizing ~~[X'X'XX]
=
where X is subject to A'A/N
I,,
the T
x
N data matrix with tth row Xi and tr(.) denotes the
matrix trace. This is the classical principal components prob-
lem, which is solved by setting
;i
equal to the eigenvectors of
X'X corresponding to its r largest eigenvalues. The resulting
principal components estimator of
F
is then
Computation of
F
requires the eigenvectors of the N
x
N
matrix X'X; when
N
>
T, a computationally simp@ approach
uses the T
x
T matrix XX'. By concentrating ou@, minimiz-
ing (5) is equivalent to maximizing ~~[F(xx')F], subject to
FFIT
=
I,
which yields the estimator, say
?,
which is the
matrix of the first r eigenvectors of XX'. The column spaces
-
of
F^
and
F
are equivalent, and so for forecasting purposes
they can be used interchangeably, depending on computational
convenience.
2.3 Consistent Estimation of Factors
and Forecasts
(1)
and
(2)
The
first result presented in this section shows that the prin-
cipal component estimator is pointwise (for any date t) con-
sistent and has limiting mean squared error (MSE) over all t
that converges in probability to 0. Because Assumption
F1
does not identify the sign of the factors, the theorem is stated
in terms of sign-adjusted estimators.
Theorem
1.
Let Si denote a variable with value of
f
1, let
N, T
+
co,
and suppose that F1 and
MI
hold. Suppose that
k
factors are estimated, where
k
may be
5
or
>
r, the true
number of factors. Then Si can be chosen so that the following
hold:
P
a. For
i
=
1,2,
. . .
,
r,
T-'
CL,(S~E,
-
-+
0.
-
P
b. For i= 1,2,..
.
,r, SiFl,-+
Fir.
c. Fori=r+l,
...,
k,
T-'C~~~$O.
The details of the proof are provided in the Appendix;
here we offer only a few remarks to provide some insight
into problem and the need for the assumptions given in
the preceding section. The estimation problem would be
considerably simplified if it happened that
A were known,
because then
F,
could be estimated by the least squares
regression of {xit)El onto {Ai)E1. Consistency of the result-
A
ing estimator would then be studied by analyzing
F,
-
F,
=
(A'A/N)-' (N-'
El
hie,,). Because N
-+
oo,
(AfA/N)
-+
I,
[by Fl(a)], and N-'
xi
hiei,
5
0 [by Ml(a) and Fl(c)], the
consistency of would follow directly. Alternatively, if
F
were known, then
A,
could be estimated by regression {xit}~=,
onto {F,}:,
,
and consistency would be studied analyzing
(T-'
C,
F,F:)-'T-'
Er
Freir, as T
+
oo
in a similar fashion.
Because both
F
and
A
are unknown, both N and T
+
co
are
needed, and as it turns out, the proof is more complicated than
these two simple steps suggest. The strategy that we have used
is to show that the first r eigenvectors of
(NT)-'X'X behave
like the first r eigenvectors of (NT)-'A'F'FA (Assumption
M1 is critical in this regard), and then show that these eigen-
vectors can be used to construct a consistent estimator of
F
(Assumption F1 is critical in this regard).
The next result shows that the feasible forecast (constructed
using the estimated factors and estimated parameters) con-
verges to the optimal infeasible forecast and thus is asymptoti-
cally efficient. In addition, it shows that the feasible regression
coefficient estimators are consistent.
The result assumes that the forecasting equation is estimated
using the
k
=
r factors. This is with little loss of generality,
because there are several methods for consistently estimating
the number of factors. For example, using analysis similar to
that in Theorem 1, Bai and Ng (2001) constructed estimators
of
r
based on penalized versions of the minimized value of
(5), and in an earlier version of this article (Stock and Watson
1998a), we developed a consistent estimator of r based on the
fit of the forecasting equation
(2).
Theorem
2.
Suppose that Y1 and the conditions of The-
orem
1
hold. Let
@,
and
fi,
denote the OLS estimates of
@,
and
@,
from the regression of {y,,,}~. onto
{c,
w,}Y_;h.
Then the following conditions hold:
a. (fi>F^,
+
fiU,wT>
-
(P>FT +@mwT)
+
P
0.
b.
p,,
-
@,
0
and Si (defined in Theorem
1)
can be cho-
sen so that SipiF
-
Pi,
0
for i
=
I,
. . .
,
r.
1170
The theorem follows directly from Theorem 1 together with
Assumption Y1. The details of the proof are given in the
Appendix.
3.
TIME-VARYING FACTOR LOADINGS
In practice, when macroeconomic forecasts are constructed
using many variables over a long period, some degree of tem-
poral instability is inevitable. In this section we model this
instability as stochastic drift in the factor loadings, and show
that if this drift is not too large and not too dependent across
series (in a sense made precise later), then the results of Theo-
rems 1 and 2 continue to hold. Thus the principal components
estimator and forecast are robust to small and idiosynchratic
shifts in the factor loadings.
Specifically, replace the time-invariant factor model
(1) with
and
'it
=
'it-1
+gi~lir
@I
for
i
=
1,
.
.
.
,
N and t
=
1,
. . .
,
T, where g,, is a scalar and
Lit
is an r
x
1 vector of random variables. This formulation
implies that factor loadings for the ith variable shift by an
amount, giTli,, in time period t. The assumptions given in this
section limit this time variation in two ways. First, the scalar
giT is assumed to be small [g,,
--
Op(T-I)] which is consistent
with the empirical literature in macroeconomics that estimates
the degree of temporal instability in macroeconomic relations
(Stock and Watson 1996,
1998b). This nesting implies that
means that
A,, -Aio
-
O,(T-'/~). Second,
lit
is assumed to
have weak cross-sectional dependence. That is, whereas some
of the
x
variables may undergo related shifts in a given period,
wholesale shifts involving a large number of the x's are ruled
out. Presumably such wholesale shifts could be better repre-
sented by shifts in the factors rather than in the factor load-
ings. In any event, this section shows that when these assump-
tions hold (along with technical assumptions given later), then
the instability does not affect the consistency of the principal
components estimator of
F,.
To motivate the additional assumptions used in this section,
rewrite
(7)
as
where air
=
eit
+
(A,,
-
Aio)
F,
=
eit
+
giT C:=,
lL!sF,.
This equa-
tion has the same general form of the time-invariant factor
model studied in the last section, with
A,
and a,, in (9) replac-
ing
A,
and e,, in the time-invariant model. This section intro-
duces two new sets of assumptions that imply that
Aio
and a,,
in (9) satisfy the assumptions concerning hi and e,, from the
preceding section. This means that the conclusions of Theo-
rems 1 and 2 will continue to hold for the time-varying factor
model of this section.
The
first new assumption is as follows.
Assumption F2.
a.
g,, is independent of F,, ej,, and
lj,
for all i, j, and t
and supi,j.k.m T[E(lgrTgjTgkTgmT1)1141
<
<
co
i,
j,
k,
and m.
Journal of the American Statistical Association, December 2002
b. The initial values of the values loadings satisfy
N-I
CiA:,Aio
=
AbA,,/N
4
I,
and sup,,,
IAij,,\
<
A,
where
Aij,,
is the jth element of
A,.
As discussed earlier, Assumption F2(a) makes the amount
of time variation small. Assumption F2(b) means that the ini-
tial value of the factor loadings satisfy the same assumptions
as the time-invariant factor loading of the preceding section.
The next assumption limits the dependence in
if,. This
assumption is written in fairly general form, allowing for some
dependence in the random variables in the model.
Assumption
M2.
Let
l,,,,
denote the mth element of
lit.
Then the following hold:
This assumption essentially repeats Assumption M1 for
the components of the composite error term a,, in (9).
To interpret the assumption, consider the leading case in
which the various components
{E,},
{F,}, {ei,}, and {lit}
are independent and have mean
0.
Then, assuming that
the
F,
have finite fourth moments, and given the assump-
tions made in the last section, Assumption M2 is satis-
fied if (a) limT+m
T-s
Cu=l-s su~i,m IE(lis,/lis+u,m)l
<
~23;
(b) lim~,m N-'
Ci
Cj
s'JPi.s,u IE(lis,/lju.m)I
<
and (c)
lim~+mN-'
Ci
Cj
SUP(ikl;=,.
I
lit2.iZ'
ljtj,
m.
{tkl;=,
~~~(lit,,i~
l,,,
lA)l
<
co,
which are the analogs of the assumptions in M1
applied to the
[
error terms.
These two new assumptions yield the main result of this
section, which follows.
Theorem
3.
Given Fl(b), Fl(d), F2, MI, and M2, the
results of Theorems 1 and 2 continue to hold.
The proof is given in the Appendix.
4.
MONTE CARL0 ANALYSIS
In this section we study some of the finite-sample proper-
ties of the principal components estimator and forecast using
a small Monte Carlo experiment. The framework used in the
preceding two sections was quite rich, allowing for distributed
lags of potentially serially correlated factors to enter the x and
y
equations, error terms that were conditionally heteroscedas-
tic and serially and cross-correlated, and factor loadings that
1171
Stock and Watson: Forecasting From Many Predictors
evolved through time. The design used here incorporates all
of these features, and the data are generated according to
and
a:
=
So +S,U:~, +Slv;,-l,
(15)
where
i
=
1,
. . .
,
N,
t
=
1,
. .
.
,
T,
f,
and
A:,,
are
J
x
1, and
the variables {Jijt}, {uj,}, and {77il} are mutually independent
iid N(0, 1) random variables. Equation (10) is dynamic factor
model with q lags of J factors that, as shown in Section 2, can
be represented as the static factor model (1) with r
=
J(l
+
q)
factors. From (12), the factors evolve as a vector autoregres-
sive [VAR(l)] model with common scalar autoregressive (AR)
parameter
a.
From (13), the error terms in the factor equation
are serially correlated, with an AR(1) coefficient of
a,
and
cross-correlated, [with spatial moving average [MA(1)] coeffi-
cient
b].
The innovations
wit
are conditionally heteroscedastic
and follow a GARCH(1, 1) process with parameters So, S,,
and 6, [see (14) and (15)l. Finally, from (1 I), the factor load-
ings evolve as random walk, with innovation standard devia-
tion proportional to
c.
The scalar variable to be forecast is generated as
where
L
is an J
x
1 vector of 1s and
E,+,
-
iid N(0, 1) and is
independent of the other errors in (10)-(15).
The other design details are as follows. The initial factor
loading matrix, A,, was chosen as a function of RZ, the frac-
tion of the variance of xio explained by
Fo.
The value of R?
was chosen as an iid random variable equal to 0 with proba-
bility
T
and drawn from a uniform distribution on
[.
1, .8] with
probability 1
-
T.
A nonzero value of
.rr
allows for the inclu-
sion of x's unrelated to the factors. Given this value of R;, the
initial factor loading was computed as
Aijo
=
A*
(R;) hijo, where
A*(R;) is a scalar and
hijo
-
iid N(O,1) and independent of
{qi,, lij, u,}. The initial values of the factors were drawn from
their stationary distribution. The parameter So was chosen so
that the unconditional variance of
vi,
was unity.
Principal components were used to estimate k factors, as
discussed in Section 2.2. These k estimated factors were
used to estimate
r
(the true number of factors) using meth-
ods described later, and the coefficients
P
in the forecasting
regression (2) were estimated by the OLS coefficients
6
in the
A
regression of y,+, onto F,,,
j
=
1,
. . .
,
i,
t
=
1,
.
. .
,
T
-
1,
where
F
is the estimated ntm_ber of factors. The out-of-sample
forecast is
jT+,/,
=
c:=, PjFjT. For comparison purposes, the
-
infeasible out-of-sample forecast
jT+,/,
=
PIFT
was also com-
puted, where
p
is
the OLS estimator obtained from regress-
ing y,,, onto F,,
t
=
1,
. . .
,
T
-
1. The free parameters in
the Monte Carlo experiment are N, T,
?,
q, k, T,
a,
b,
c,
S,,
and 6,.
The results are summarized by two statistics. The first statis-
tic is a trace R2 of the multivariate regression of
FI
onto F,
where
2
denotes the expectation estimated by averaging the
relevant statistic over the Monte Carlo repetitions and
PF
=
P
F(F'F)-IF'. According to Theorem 1, R;,
-+
1.
The second statistic measures how close the forecast based
on the estimated factors is to the infeasible forecast based on
the true factors,
P
Because
jT+llT
-
jT+,/,
-+
0 when k
=
r
from Theorem 1,
S;,
j,
should be close to 1 for appropriately large N and T.
S;,?
is computed for several choices of
i.
First, as a benchmark,
results are shown for
i
=
r. Second,
F
is formed using three
of the information criteria suggested by Bai and Ng (2001).
These criteria have the form ICp(k)
=
ln(c)
+
kg(T, N),
where
f;,
is the minimized value of the objective function (5)
for a model with k factors and gj(T, N) is a penalty function.
Three of the penalty functions suggested by Bai and Ng are
used:
and
where
C;,
=
min(N, T), resulting in criteria labeled IC,,,
IC,,, and ICp3. The minimizers of these criteria yield a con-
sistent estimator of r, and interest here focuses on their rela-
tive small-sample accuracy. Finally, results are shown with
i
computed using the conventional Akaike information criterion
(AIC) and Bayes information criterion (BIC) applied to the
forecasting equation (2).
The results are summarized in Table 1. Panel A presents
results for the static factor model with iid errors and factors
and with large N and
T(N, T
>
100). Panel B gives corre-
sponding results for small values of N and T(N, T
1
50).
Panel C adds irrelevant xi,'s to the model (T
>
0). Panel
D
extends the model to idiosyncratic errors that are serially
correlated, cross-correlated, conditionally heteroscedastic, or
some combination of these. Panel
E
considers the dynamic
factor model with serially correlated factors and/or lags of the
factors entering'
X,.
Finally, panel F gives time-varying factor
loadings.
First, consider the results for the static factor model shown
in panel A. The values of R;, exceed .85 except when many
redundant factors are estimated. The smallest value of
R;,~
is
.69,
which obtains when N and T are relatively small
(N
=
T
=
100) and there are 10 redundant factors (r
=
5 and
