Wo r k i n g Pa p e r S e ri e s
n o 9 9 8 / J a n ua Ry 2 0 0 9
InfInIte-dImensIonal
VARS AND FACTOR MODELS
by Alexander Chudik
and M. Hashem Pesaran
WO R K I N G PA P E R S E R I E S
N O 9 9 8 / J A N U A RY 20 0 9
INFINITE-DIMENSIONAL VARS
AND FACTOR MODELS 1
by Alexander Chudik 2
and M. Hashem Pesaran 3
In 2009 all ECB
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This paper can be downloaded without charge from
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1 We are grateful to Elisa Tosetti for helpful comments and suggestions. This version has also benefited from comments by the seminar participants
at University of California, San Diego, University of Southern California, Columbia, Leicester, McGill, Princeton, Pennsylvania, and Stanford.
We would also like to acknowledge Takashi Yamagata for carrying out the estimation of the results reported in Section 6. The views
expressed in this paper do not necessarily reflect the views of the European Central Bank or the Eurosystem.
2 University of Cambridge, Faculty of Economics, Austin Robinson Building, Sidgwick Avenue, Cambridge, CB3 9DD, United Kingdom
and European Central Bank, Kaiserstrasse 29, 60311 Frankfurt am Main, Germany; e-mail: Alexander.Chudik@ecb.europa.eu;
http://www.econ.cam.ac.uk/phd/ac474/
3 University of Cambridge, CIMF and USC; Faculty of Economics, Austin Robinson Building, Sidgwick Avenue, Cambridge,
CB3 9DD, United Kingdom; e-mail: mhp1@econ.cam.ac.uk; http://www.econ.cam.ac.uk/faculty/pesaran/
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ISSN 1725-2806 (online)
CONTENTS
Abstract
4
Non-technical summary
5
1 Introduction
7
2 Infinite-dimensional vector
autoregressive models
10
3 Cross sectional dependence in stationary
IVAR models
3.1 Contemporaneous dependence: spatial or
network dependence
3.2 IVAR models with strong cross
sectional dependence
12
16
17
4 Estimation of a stationary IVAR
21
5 Monte Carlo experiments: small
sample properties of CALS estimator
5.1 Monte Carlo design
5.2 Monte Carlo results
28
28
32
6 An empirical application: a spatiotemporal
model of house prices in the U.S.
33
7 Concluding remarks
34
Figures and tables
35
References
39
Appendix
42
European Central Bank Working Paper Series
61
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Working Paper Series No 998
January 2009
3
Abstract
This paper introduces a novel approach for dealing with the ‘curse of dimensionality’in the
case of large linear dynamic systems. Restrictions on the coe¢ cients of an unrestricted VAR
are proposed that are binding only in a limit as the number of endogenous variables tends to
in…nity. It is shown that under such restrictions, an in…nite-dimensional VAR (or IVAR) can
be arbitrarily well characterized by a large number of …nite-dimensional models in the spirit
of the global VAR model proposed in Pesaran et al. (JBES, 2004). The paper also considers
IVAR models with dominant individual units and shows that this will lead to a dynamic factor
model with the dominant unit acting as the factor. The problems of estimation and inference in
a stationary IVAR with unknown number of unobserved common factors are also investigated.
A cross section augmented least squares estimator is proposed and its asymptotic distribution
is derived. Satisfactory small sample properties are documented by Monte Carlo experiments.
Keywords: Large N and T Panels, Weak and Strong Cross Section Dependence, VAR, Global
VAR, Factor Models
JEL Classi…cation: C10, C33, C51
4
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Working Paper Series No 998
January 2009
Non-technical summary
Vector autoregressive models (VARs) provide a ‡exible framework for the analysis of complex
dynamics and interactions that exist between variables in the national and global economy. However, the application of the approach in practice is often limited to a handful of variables which
could lead to misleading inference if important variables are omitted merely to accommodate the
VAR modelling strategy. Number of parameters to be estimated grows at the quadratic rate with
the number of variables, which is limited by the size of typical data sets to no more than 5 to 7.
In many empirical applications, this is not satisfactory.
The objective of this paper is to analyze large linear dynamic systems of endogenously determined variables. In particular, we study VAR models where both the number of variables (N )
and the number of time periods (T ) tend to in…nity. In this case, parameters of the VAR model
can no longer be consistently estimated unless suitable restrictions are imposed to overcome the
dimensionality problem. Two di¤erent approaches have been suggested in the literature to deal
with this ‘curse of dimensionality’: (i) shrinkage of the parameter space and (ii) shrinkage of the
data. This paper proposes a novel way to deal with the curse of dimensionality by shrinking part
of the parameter space in the limit as the number of endogenous variables (N ) tends to in…nity.
An important example would be a VAR model where each unit is related to a small number
of neighbors and a large number of non-neighbors. The neighborhood e¤ects are …xed and do
not change with N , but the coe¢ cients corresponding to the remaining units are small, of order
O N
1
. Another model of interest arises when in addition to the neighborhood e¤ects, there is
also a …xed number of dominant units that have non-negligible e¤ects on all other units. In the case
where the VAR contains neighborhood e¤ects our speci…cation would converge to a spatiotemporal
as N ! 1. Finally, when the VAR includes dominant units the limiting outcome will be a dynamic
factor models. Such VAR models of growing dimension (N ! 1) are referred in the paper as the
in…nite-dimensional VARs, or IVARs.
The analysis of the paper also formally establishes the conditions under which the Global VAR
(GVAR) approach proposed by Pesaran, Schuermann and Weiner (JBES, 2004) is applicable. In
particular, the IVAR featuring all macroeconomic variables could be arbitrarily well approximated
by a set of …nite-dimensional small-scale models that can be consistently estimated separately in
ECB
Working Paper Series No 998
January 2009
5
the spirit of the GVAR.
Besides the development of an econometric approach for the analysis of groups that belong
to a large interrelated system, the second main contribution of the paper is in considering the
problems of the estimation and inference in stationary IVAR models with known as well as an
unknown number of unobserved common factors. A simple cross sectional augmented least-squares
estimator is proposed and its asymptotic distribution derived. Satisfactory small sample properties
are documented by Monte Carlo experiments. As an illustration of the proposed approach we follow
the recent empirical analysis of real house prices across the 49 U.S. States by Holly, Pesaran and
Yamagata (2008) and show statistically signi…cant dynamic spill over e¤ects of real house prices
across the neighboring states.
6
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January 2009
1
Introduction
Vector autoregressive models (VARs) provide a ‡exible framework for the analysis of complex
dynamics and interactions that exist between economic variables. The traditional VAR modelling
strategy postulates that the number of variables, denoted as N , is …xed and the time dimension,
denoted as T , tends to in…nity. The number of parameters to be estimated grows at the quadratic
rate with N and consequently the application of the approach in practice is often limited (by the
size of typical datasets) to a handful of variables.
The objective of this paper is to analyze VAR models where both N and T tend to in…nity. In
this case, parameters of the VAR model can no longer be consistently estimated unless suitable
restrictions are imposed to overcome the dimensionality problem. Two di¤erent approaches have
been suggested in the literature to deal with this ‘curse of dimensionality’: (i) shrinkage of the
parameter space and (ii) shrinkage of the data. Spatial and/or spatiotemporal literature shrinks
the parameter space by using the concept of spatial weights matrix, which links individual units
with the rest of the system. Alternatively, one could use techniques whereby prior distributions are
imposed on the parameters to be estimated. Bayesian VAR (BVAR) proposed by Doan, Litterman
and Sims (1984), for example, use what has become known as ‘Minnesota’ priors to shrink the
parameters space.1 In most applications, BVARs have been applied to relatively small systems2
(e.g. Leeper, Sims, and Zha, 1996, considered 13- and 18-variable BVAR), with the focus mainly
on forecasting.3
The second approach to mitigating the curse of dimensionality is to shrink the data, along the
lines of index models. Geweke (1977) and Sargent and Sims (1977) introduced dynamic factor
models, which have more recently been generalized to allow for weak cross sectional dependence
by Forni and Lippi (2001) and Forni et al. (2000, 2004). Empirical evidence suggests that few
dynamic factors are needed to explain the co-movement of macroeconomic variables: Stock and
Watson (1999, 2002), Giannoni, Reichlin and Sala (2005) conclude that only few, perhaps two,
factors explain much of the predictable variations, while Bai and Ng (2007) estimate four factors
and Stock and Watson (2005) estimate as much as seven factors. This has led to the development
1
Other types of priors have also been considered in the literature. See, for example, Del Negro and Schorfheide
(2004) for a recent reference.
2
A few exceptions include Giacomini and White (2006) and De Mol, Giannone and Reichlin (2006).
3
Bayesian VARs are known to produce better forecasts than unrestricted VARs and, in many situations, ARIMA
or structural models. See Litterman (1986) and Canova (1995) for further references.
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January 2009
7
of factor-augmented VAR (FAVAR) models by Bernanke, Boivin, and Eliasz (2005) and Stock and
Watson (2005), among others.
This paper proposes a novel way to deal with the curse of dimensionality by shrinking part of
the parameter space in the limit as the number of endogenous variables (N ) tends to in…nity. An
important example would be a VAR model where each unit is related to a small number of neighbors
and a large number of non-neighbors. The neighborhood e¤ects are …xed and do not change with
N , but the coe¢ cients corresponding to the remaining units are small, of order O N
1
. Another
model of interest arises when in addition to the neighborhood e¤ects, there is also a …xed number
of dominant units that have non-negligible e¤ects on all other units. This set-up naturally arises
in the context of global macroeconomic modelling. When all economies are small and open, using
a multicountry DSGE model Chudik (2008) shows that the coe¢ cients of the foreign variables in
the rational expectations solution are all of order O N
1
. In the case where the VAR contains
neighborhood e¤ects our speci…cation would converge to a spatiotemporal as N ! 1. Finally,
when the VAR includes dominant units the limiting outcome will be a dynamic factor models.
Such VAR models will be referred as the in…nite-dimensional VARs, or IVARs.
The analysis of the paper also provides a link between data and parameter shrinkage approaches
to mitigating the curse of dimensionality. By imposing limiting restrictions on some of the parameters of the VAR we e¤ectively end up with a data shrinkage. We apply the concept of strong
and weak Cross Section (CS) dependence (introduced by Pesaran and Tosetti, 2007) in the context
of IVARs and show that only strong CS dependence can be ‘transmitted’through O N
1
coe¢ -
cients. This …nding links our analysis to the factor models by showing that dominant unit becomes
(in the limit) a dynamic common factor for the remaining units in a large system of endogenously
determined variables. Static factor models are also obtained as a special case of IVAR. Last but
not least, this paper formally establishes the conditions under which the Global VAR (GVAR)
approach proposed by Pesaran, Schuermann and Weiner (2004) is applicable.4 In particular, the
IVAR featuring all macroeconomic variables could be arbitrarily well approximated by a set of
…nite-dimensional small-scale models that can be consistently estimated separately in the spirit of
4
GVAR model has been used to analyse credit risk in Pesaran, Schuermann, Treutler and Weiner (2006) and
Pesaran, Schuerman and Treutler (2007). Extended and updated version of the GVAR by Dees, di Mauro, Pesaran
and Smith (2007), which treats Euro area as a single economic area, was used by Pesaran, Smith and Smith (2007)
to evaluate UK entry into the Euro. Global dominance of the US economy in a GVAR model is explicitly considered
in Chudik (2007). Further developments of a global modelling approach are provided in Pesaran and Smith (2006).
Garratt, Lee, Pesaran and Shin (2006) provide a textbook treatment of GVAR.
8
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January 2009
the GVAR.
Besides the development of an econometric approach for the analysis of groups that belong to a
large interrelated system, the second main contribution of the paper is in considering the problems of
the estimation and inference in stationary IVAR models with known as well as an unknown number
of unobserved common factors. Our set-up extends the analysis of Pesaran (2006) to dynamic
models where all variables are determined endogenously. A simple cross sectional augmented leastsquares estimator (or CALS for short) is proposed and its asymptotic distribution derived. Small
sample properties of the proposed estimator are investigated through Monte Carlo experiments. As
an illustration of the proposed approach we follow the recent empirical analysis of real house prices
across the 49 U.S. States by Holly, Pesaran and Yamagata (2008) and show statistically signi…cant
dynamic spillover e¤ects of real house prices across the neighboring states.
The remainder of this paper is organized as follows. Section 2 outlines IVAR model, introduces
limiting restrictions, and provides few examples, which link IVAR with the literature. Section 3
investigates cross section dependence in IVAR models where key asymptotic results are provided.
Section 4 focusses on estimation of a stationary IVAR. Section 5 presents Monte Carlo evidence
and a spatiotemporal model of the US house prices is presented in Section 6. The …nal section
o¤ers some concluding remarks. Proofs are provided in the Appendix.
are the eigenvalues of A 2 Mn n , where
Pn
max
Mn n is the space of real-valued n n matrices. kAkc
i=1 jaij j denotes the maximum
1 j n
Pn
max j=1 jaij j is the absolute row-sum matrix
absolute column sum matrix norm of A, kAkr
1 i n
p
max fj i (A)jg is the spectral
norm of A.5 kAk = % (A0 A) is the spectral norm of A; % (A)
A brief word on notation: j
1 (A)j
:::
j
n (A)j
1 i n
radius of
A.6
Row i of A is denoted by
a0i
and the column i is denoted as ai . All vectors are column
vectors. Row i of A with the ith element replaced by 0 is denoted as a0 i . Row i of A 2 Mn
the element i and the element 1 replaced by 0 is a0
1; i
= (0; ai2 ; :::; ai;i
n
with
1 ; 0; ai;i+1 ; :::; ai;iN ).
The
matrix constructed from A by replacing its …rst column with a column vector of zeros is denoted
_
by A
1.
j
Joint asymptotics in N; T ! 1 are represented by N; T ! 1. an = O(bn ) states the
deterministic sequence an is at most of order bn . xn = Op (yn ) states random variable xn is at most
of order yn in probability. N is the set of natural numbers, and Z is the set of integers. We use K
5
The maximum absolute column sum matrix norm and the maximum absolute row sum matrix norm are sometimes
denoted in the literature as k k1 and k k1 ,p
respectively.p
6
Note that if x is a vector, then kxk = % (x0 x) = x0 x corresponds to the Euclidean length of vector x.
ECB
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9
and
to denote large and small positive constants that do not vary with i; t; N or T . Convergence
d
p
q:m:
in distribution and convergence in probability is denoted by ! and !, respectively. Symbol !
represents convergence in quadratic mean.
2
In…nite-Dimensional Vector Autoregressive Models
Suppose there are N cross section units indexed by i 2 S
f1; ::; N g
N. Depending on empirical
application, units could be households, …rms, regions, countries, or macroeconomic indicators in a
given economy. Let xit denote the realization of a random variable belonging to the cross section
unit i in period t, and assume that xt = (x1t ; :::; xN t )0 is generated according to the following
stationary structural VAR model
A0 xt = A1 xt
1
+ A2 "t ,
(1)
where one lag is assumed for the simplicity of exposition, A0 , A1 and A2 are N
unknown coe¢ cients, and innovations collected into N
N matrices of
1 vector "t = ("1t ; :::; "N t )0 are IID (0; IN ).
The model (1), for example, arises as the rational expectations solution of a multi-country DSGE
model. (See, for example, Pesaran and Smith, 2006). Assuming matrix A0 is invertible, the reduced
form of structural model (1) is:
xt =
xt
1
+ ut ,
(2)
where the vector reduced-form errors ut is given by the following ‘spatial’model
ut = R"t ,
(3)
= A0 1 A1 , and R = A0 1 A2 . Focus of this paper is on a sequence of reduced-form models (2) of
growing dimension (N ! 1), where the elements of
and R (and hence the variance matrix of ut )
depend on N . But to simplify the notations we do not show this dependence explicitly, although it
will be understood throughout that all the parameters and the dimension of the random variables xt
and ut vary with N , unless otherwise stated. The sequence of models (2) and (3) with dim(xt ) = N
growing will be referred to as the in…nite-dimensional VAR(1) model.
To allow for neighborhood e¤ects it is convenient to decompose
10
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January 2009
into two components: a
sparse N
N matrix matrix,
a,
with …xed elements (that do not vary with N ) which captures
b,
the neighborhood e¤ects, and a complement matrix,
so that
=
a
+
b.
An example of
a
B 11
B
B
B 12
B
B
B 0
=B
B
B 0
B
B
B
B
@
0
is given by
a
0
characterizing the remaining interactions,
12
0
0
0
22
23
0
0
32
33
34
0
43
0
0
44
..
1
.
0
..
.
..
.
N 1;N
N 1;N
NN
C
C
C
C
C
C
C
C,
C
C
C
C
C
C
A
(4)
where the nonzero elements are …xed coe¢ cients that do not change with N . This represents
an ‘approximate line’ model where each unit, except the …rst and the last unit has one left and
1 ),
in particular
for any i; j 2 f1; ::; N g and any N 2 N. Equation for unit i 2 f2; ::; N
1g can be
one right neighbor. In contrast the individual elements of
bij
<
K
N
b
are of order O(N
written as
xit =
i;i 1 xi 1;t 1
+
ii xi;t 1
+
i;i+1 xi+1;t 1
+
0
bi xt 1
Next section shows that under weak CS dependence of errors fuit g,
considers problem of estimation of the individual-speci…c parameters
+ uit .
q:m:
0
bi xt 1 !
i;i 1 ;
ii ;
(5)
0, and Section 4
i;i+1
. We refer
to this model as a two-neighbor IVAR model which we use later for illustrative purposes as well as
in the Monte Carlo experiments.
The above decomposition of matrix
is a pivotal example of limiting restrictions developed in
this paper. More generally, we have
b;
=D S+
where as before, the individual elements of matrix
0
B
B
B
B
D=B
B
B
@
0
1
0
0
..
.
0
2
0
0
b
(6)
are (uniformly) of order O N
0
..
.
1
,
1
C
C
C
C
C,
C
0 C
A
(7)
0
N
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11
1 dimensional vector containing the unknown coe¢ cients to be estimated for unit
P
N matrix partitioned as
i 2 f1; ::; N g, hi is bounded in N , h = N
i=1 hi , and S is a known h
i
is an hi
S = (S1 ; S2 ; :::; SN )0 , with Si being the N
hi selection matrix, which de…nes the neighbors for unit
i as in the example above. S could also be related to a spatial weights matrix as in the following
example.
Example 1 Consider the following spatiotemporal model
where Sx and Su are N
xt =
x Sx xt 1
ut =
u Su ut
+ ut ,
(8)
+ "t ,
(9)
N spatial weights matrices. Spatiotemporal model (8)-(9) is a special case
of the model (2)-(3) by setting
u Su )
R = (I
1
,
i
=
x
for i 2 f1; ::; N g , S = Sx , and
b
= 0.
Remark 1 Note, however, that not all types of structural models (1) have reduced forms that satisfy
the restrictions given by (6). For example, consider the spatiotemporal model:
xt = Sx xt + xt
Assuming matrix (I
(2) with
=
(I
x Sx )
x Sx )
and unknown parameters
x
1
+ "t .
is invertible, the reduced form of spatiotemporal model (10) is model
1
and R = (I
x Sx )
1
. For the known spatial weights matrix Sx
and , the reduced form coe¢ cient matrix
in equation (6), where the matrix S is assumed to be known, f i g and
elements of
3
b
(10)
are uniformly O N
1
cannot be decomposed as
b
are unknown, and the
.
Cross Sectional Dependence in Stationary IVAR Models
Here we investigate the correlation pattern of fxit g, over time, t, and across the cross section units,
i. Unlike the time index t which is de…ned over an ordered integer set, the cross section index, i,
refers to an individual unit of an unordered population distributed over space or more generally over
networks. To avoid having to order the cross section units we make use of the concepts of weak
12
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January 2009
and strong cross section dependence recently developed in Pesaran and Tosetti (2007, hereafter
PT). A process fxit g is said to be cross sectionally weakly dependent (CWD) with respect to a
pre-determined information set, It
1,
if for all weight vectors, wt = (w1t ; :::; wN t )0 , satisfying the
‘granularity’conditions7
kwt k = O N
wjt
kwt k
= O N
1
2
,
1
2
(11)
for any j
N,
(12)
we have
lim V ar wt0
N !1
1 xt
j It
1
= 0, for all t 2 T .
Since we will be dealing with stationary processes in what follows we con…ne our analysis to time
invariant weight vectors, w, and information sets, I = ;. Accordingly, we adopt the following
concept.
De…nition 1 Stationary process fxit ; i 2 S; t 2 T ; N 2 Ng, generated by the IVAR model (2), is
said to be cross sectionally weakly dependent (CWD), if for any sequence of non-random vectors of
weights w satisfying the granularity conditions (11)-(12),
lim V ar (xwt j I) = lim V ar (xwt ) = 0,
N !1
(13)
N !1
where xwt = w0 xt . fxit g is said to be cross sectionally strongly dependent (CSD) if there exists a
sequence of weights vectors w satisfying (11)-(12) and a constant K such that
lim V ar (xwt )
N !1
K > 0.
(14)
Necessary condition for covariance stationarity for …xed N is that all eigenvalues of
of the unit circle. For a …xed N , and assuming that maxi j
`
de…ned by T r
`
`0
1=2
lie inside
i(
)j < 1, the Euclidean norm of
P
`u
! 0 exponentially in `; and the process xt = 1
t ` will
`=0
be absolute summable, in the sense that the sum of absolute values of the elements of
7
`,
for
Condition (12) is understood as
wjt
kwt k
K
p , for any j 2 f1; ::; N g and any N 2 N,
N
where constant K < 1 does not depend on N nor on j.
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January 2009
13
` = 0; 1; ::: converge. Observe that as N ! 1, V ar (xit ) need not necessarily be bounded in N if
maxi j
i(
)j < 1
. For example, consider the IVAR(1) model with
0
B
B
B
B
B
=B
B
B
B
B
@
'
0
0
'
0
0
..
.
0
'
..
.
0
..
1
C
C
0 C
C
C
0 C
C,
C
C
0 C
A
'
.
0
and assume that var (uit ) is uniformly bounded away from zero as N ! 1. It is clear all eigenvalues
of
are inside the unit circle if and only if j'j < 1, regardless the value of the neighboring coe¢ cient
. Yet the variance of xN t increases in N without bounds at an exponential rate for j j > 1
j'j.8
Therefore, a stronger condition than stationarity is required to rule out variances of xit exploding
as N ! 1. This is set out in the following assumptions.
ASSUMPTION 1 Individual elements of double index process of errors fuit ; i 2 S; t 2 T g are
random variables de…ned on the probability space ( ; F; P ). ut is independently distributed of ut0 ,
for any t 6= t0 2 T . For each t 2 T , ut has mean and variance,
E (ut ) = 0, E ut u0t =
where
is an N
2
ii
i 2 S and
= V ar (uit ) is the i-th diagonal element of covariance matrix
,
k k = O N1
V ar fxN t g =
where
14
k`
=
1
(k 1)!
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Working Paper Series No 998
January 2009
kQ2
j=0
(` + k
1
.
(15)
,
(16)
> 0 is an arbitrarily small positive constant.
It can be shown that
< K < 1 for any
and CWD ut )
k k<1
8
2
ii
N symmetric, nonnegative de…nite matrix, such that 0 <
ASSUMPTION 2 (Coe¢ cients matrix
where
,
N
X
2(N
j)
j=1
j) for k > 1 and
1
X
`=0
1`
= 1.
2
N
j+1;` '
2`
,
Remark 2 Assumption 1 and equation (16) of Assumption 2 imply fuit g is CWD.
Remark 3 Condition (15) of Assumption 2 is a su¢ cient condition for covariance stationarity
and also delivers bounded variance of xit , as N ! 1. Note that Assumption 2 also rules out
cases where strong cross sectional dependence arises due to a particular unit (or units) since both
p
p
k kc < N k k = O
N and k kc cannot diverge to in…nity at the rate N .
Proposition 1 Consider model (2) and suppose that Assumptions 1 and 2 hold. Then for any
arbitrary sequence of …xed weights w satisfying condition (11), and for any t 2 T ,
lim V ar (xwt ) = 0.
(17)
N !1
Proposition 1 has several interesting implications. Suppose that unit i has a …xed number
of neighbors, j = 1; 2; ::; p, for which coe¢ cients
ij
= O (1), while the in‡uence of each of the
remaining units on the unit i through coe¢ cient matrix
following decomposition of the ith row of matrix
0
ai
j=1
0
bi :
11=2
2 A
bij
=O N
Remark 4 Obvious examples of the decomposition of
=(
i1 ; :::;
i;i 1 ; 0;
hood model where
with
i;i 1 ;
ii
and
0
ai
i;i+1 ; :::;
iN ),
= (0; :::; 0;
i;i+1
into a possibly sparse vector
S be a non-empty index set. For any i 2 K,
0
N
X
k bi k = @
0
bi
0
i,
, denoted as
and the remaining coe¢ cients collected into vector
ASSUMPTION 3 Let K
is small. In particular, consider the
where
i;i 1 ;
ii ;
ii
i
1
2
i
=
ai
+
bi ,
where
.
is when
(18)
0
ai
= (0; :::; 0;
ii ; 0; :::; 0),
and
does not depend on N , and the left-right neigbour-
i;i+1 ; 0:::; 0)
and
0
bi
=(
i1 ; ::: i;i 2 ; 0; 0; 0;
i;i+2 :::;
,
iN )
being …xed parameters that do not vary with N .
Remark 5 As we shall see in Section 4, for estimation and inference the following slightly stronger
condition on the row norm of
bi
will be needed.
k
bi kr
=O N
Remark 6 Assumption 3 implies that for i 2 K,
1
PN
j=1
.
bij
k
bi kc
= O (1). Therefore, it is
possible for the dependence of each individual unit on the rest of the units in the system to be large
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15
even if
ai
= 0. However, as we shall see below, in the case where fxit g is a CWD process, vector
does not play a role in the model for the ith cross section unit as N ! 1.
bi
Corollary 1 Consider model (2) and suppose Assumptions 1-3 hold. Then,
lim V ar xit
N !1
Observe that if
ii
0
ai xt 1
uit = 0, for i 2 K.
is the only nonzero element of
ai ,
(19)
then the regression model for unit i
completely de-couples from the rest of the system as N ! 1, in the sense that
lim V ar (xit
N !1
ii xi;t 1
uit ) = 0.
The above corollary in e¤ect states that in econometric modelling of xit one can ignore the
e¤ects of those cross section units that have zero entries in
ai
as N becomes large, so long as xt
is a CWD process.9
3.1
Contemporaneous Dependence: Spatial or Network Dependence
An important form of cross section dependence is contemporaneous dependence across space. The
spatial dependence, pioneered by Whittle (1954), models cross section correlations by means of
spatial processes that relate each cross section unit to its neighbor(s). Spatial autoregressive and
spatial error component models are examples of such processes. (Cli¤ and Ord, 1973, Anselin,
1988, and Kelejian and Robinson, 1995). However, it is not necessary that proximity is measured
in terms of physical space. Other measures such as economic (Conley, 1999, Pesaran, Schuermann
and Weiner, 2004), or social distance (Conley and Topa, 2002) could also be employed. All these
are examples of dependence across nodes in a physical (real) or logical (virtual) networks. In the
case of the IVAR model, de…ned by (2) and (3), such contemporaneous dependence can be modelled
through an N
9
4.
N network topology matrix R.10,11 For example, in the case of a …rst order spatial
j
Appropriate rates for N ,T ! 1 needed for inference about the nonzero parameters in
10
ai
are established Section
A network topography is usually represented by graphs whose nodes are identi…ed with the cross section units,
with the pairwise relations captured by the arcs in the graph.
11
It is also possible to allow for time variations in the network matrix to capture changes in the network structure
over time. However, this will not be pursued here.
16
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moving average model, R would take the form
0
RSM A
where
s
B
B
B
B
B
B
B
= IN + s B
B
B
B
B
B
B
@
0
1
0
0
:::
0
0
1=2
0
1=2
0
:::
0
0
0
..
.
1=2
..
.
0
..
.
1=2 : : :
..
..
.
.
0
..
.
0
..
.
0
0
0
0
: : : 1=2 0
0
0
0
0
:::
0
1
1
0 C
C
0 C
C
C
C
0 C
C
.. C ;
. C
C
C
1=2 C
C
A
0
is the spatial moving average coe¢ cient.
The contemporaneous nature of dependence across i 2 S is fully captured by R. As shown in
PT the contemporaneous dependence across i 2 S will be weak if the maximum absolute column
and row sum matrix norm of R are bounded, namely if kRkc kRkr < K < 1. It turns out that
all spatial models proposed in the literature are in fact examples of weak cross section dependence.
More general network dependence such as the ‘star’ network provides an example of strong contemporaneous dependence that we shall consider below. The form of R for a typical star network
is given by
0
RStar
where
N
X
1
0
B
B
B r21 1
B
B
=B
B r31 0
B .
..
B .
B .
.
@
rN 1 0
0 0
..
.
1
C
C
0 0 C
C
C
0 0 C
C;
C
C
1 0 C
A
0 1
rj1 = O(N ).
j=2
The IVAR model when combined with ut = R"t yields an in…nite-dimensional spatiotemporal
model. The model can also be viewed more generally as a ‘dynamic network’, with R and
capturing the static and dynamic forms of inter-connections that might exist in the network.
3.2
IVAR Models with Strong Cross Sectional Dependence
Strong dependence in IVAR model could arise as a result of CSD errors fuit g, or could be due to
dominant patterns in the coe¢ cients of
, or both. Strong cross section dependence could also
arise in the case of residual common factor models where the weighted averages of factor loadings
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17
do not converge to zero.12 Section 4 considers estimation and inference in the case of stationary
CSD IVAR models with unobserved common factors and/or deterministic trends. An example of
a stationary IVAR model where the column corresponding to unit i = 1 in matrices
, and R is
dominant is provided below.
The following assumption postulates that for any i, coe¢ cient vector
into a sparse vector
0;
i2 ; :::;
i;i 1 ; 0;
ai
= (
i;i+1 ; :::;
ASSUMPTION 4 Let
of
0
ii ; 0; :::; 0)
i1 ; 0; :::; 0;
iN
=
0
and a vector
i
can be decomposed
=
1; i
1i ; :::;
0
N i)
bi
where
1; i
=
.
PN
i=1
0
i ei
0
1 e1
=
+ _
1
where
i
=(
is the ith column
; ei is an N 1 selection vector for unit i, with the ith element of ei being one and the remaining
elements zero. Denote by _
the matrix constructed from
by replacing its …rst column with a
PN
vector of zeros, and note that _ 1 = i=2 i e0i . Suppose as N ! 1
1
1
r
= O (1) .
(20)
Further, suppose that
1; i r
=O N
1
uniformly for all i 2 N,
(21)
namely there exists a constant K such that
1; i r
<
K
for any i 2 S and any N 2 N.
N
ASSUMPTION 5 (Stationarity) k kr <
< 1 for any N 2 N.
ASSUMPTION 6 The N 1 vector of errors ut is generated by the ‘spatial’model (3). E (ut u0t ) =
P
0
th
0
0
_
= RR0 is time invariant, where R = N
i=1 ri ei = r1 e1 + R 1 , and ri = (r1i ; :::; rN i ) is the i
column of matrix R. Suppose as N ! 1
2
_
R
1
= O N1
kr1 kr = O (1) ,
12
18
See Pesaran and Toesetti (2007, Theorem 16).
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,
(22)
(23)
and
lim kr
N !1
where
is an arbitrarily small constant, r
1; i k
1; i
= 0 for any i 2 N,
= (0; ri1 ; :::; ri;i
(24)
0
1 ; 0; ri;i+1 ; :::; riN ) ,
and rij denotes
the (i; j) element of matrix R.
Remark 7 Assumptions 4 and 6 imply matrix
has one dominant column and matrix R has at
least one dominant column, but the absolute column sum for only one column could rise with N at
the rate N . Part (21) of Assumption 4 allows the equation for unit i 6= 1 to de-couple from the
equations for units j 6= 1, for any j 6= i, as N ! 1.
Remark 8 Using the maximum absolute column/row sum matrix norms rather than eigenvalues
in principle allows us to make a distinction between cases where dominant e¤ ects are due to a
particular unit (or a few units), and when there is a pervasive unobserved factor that makes all
column/row sums unbounded. Eigenvalues of the covariance matrix
will be unbounded in both
cases and it will not be possible from the knowledge of the rate of the expansion of the eigenvalues
of
,
and/or R to known which one of the two cases are in fact applicable.
Remark 9 As it will become clear momentarily, conditional on x1t and its lagged values, process
fxit g become cross sectionally weakly dependent. We shall therefore refer to unit i = 1 as the
dominant unit.
Remark 10 It follows that under Assumptions 5 and 6 the IVAR model speci…ed by (2) and (3)
is stationary for any N , and the variance of xit will be uniformly bounded.
Proposition 2 Under Assumptions 4-6 and as N ! 1, equation for the dominant unit i = 1 in
the IVAR model de…ned by (2) and (3) reduces to
x1t
where # (L; e1 ) =
condition (11),
P1
`=0
e01
`r
1
q:m:
# (L; e1 ) "1t ! 0,
(25)
L` . Furthermore, for any …xed sequence of weights w satisfying
xwt
q:m:
# (L; w) "1t ! 0.
(26)
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19
The model for unit i = 1 can be approximated by an AR(p1 ) process, which does not depend
on the realizations from the remaining units as N ! 1. Let the lag polynomial
a (L; p1 )
be an approximation of #
1
#
1
(L; e1 )
(27)
(L; e1 ). Then equation for unit i = 1 can be written as
a (L; p1 ) x1t
"1t .
(28)
The following proposition presents mean square error convergence results for the remaining cross
section units.
Proposition 3 Consider system (2), let Assumptions 4-6 hold and suppose that the lag polynomial
# (L; e1 ) de…ned in Proposition 2 is invertible. Then as N ! 1, equations for cross section unit
i 6= 1 in the IVAR model de…ned by (2) and (3) reduce to
(1
ii L) xit
q:m:
i (L) x1t
rii "it ! 0; for i = 2; 3; :::
(29)
where
i (L)
=
i1 L
+ ri1 + # L;
1; i
L #
1
(L; e1 ) ;
and
# L;
1; i
=
1
X
0
1; i
`
r1 L` , for i > 1.
`=0
Remark 11 Exclusion of the current value of x1t from (29) is justi…ed only if ri1 = 0. But even
in this case xit will depend on lagged values of x1t .
Remark 12 Cross section unit 1 becomes (in the limit) a dynamic common factor for the remaining units in the IVAR model. Note that setting x1t = ft , (29) can be written as13
(1
ii L) xit
Remark 13 Conditional on fx1t ; x1;t
13
rii "it +
1 ; x1;t 2 ; ::::g,
i (L) ft ,
for i > 1.
(30)
the process fxit g for i > 1 is CWD.
x1t could be equivalently approximated by cross sectional weighted averages of xt and its lags, namely
xwt ; xw;t 1 ; :::.
20
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Remark 14 For
1
= 0 and
i
= 0, we obtain from (29) the following static factor model as a
special case
(1
ii L) xit
rii "it +
ri1
r11
ft , for i > 1,
(31)
where ft = x1t .
We now turn our attention to the problems of estimation and inference in IVAR models. In
what follows we consider the relatively simple case where there are no dominant units, but allow
for the possibility of unobserved common factors. The analysis of IVAR models featuring both
unobserved common factors, ft , and
matrices with unbounded maximum absolute column sum
matrix norms is provided in a supplement, which is available from the authors on request.
4
Estimation of a Stationary IVAR
Assume xt = (x1t ; :::; xN t )0 is generated according to the following factor-augmented IVAR(1):
(L) (xt
f t ) = ut ,
(32)
for t = 1; 2; :::; T , where the vector of errors ut is generated by spatial model (3), namely ut = R"t ,
(L) = IN
L,
is N
N dimensional matrix of unknown coe¢ cients,
1 dimensional vector of …xed e¤ects, ft is m
N
factors (m is …xed but otherwise unknown),
factor loadings with its i-th row denoted as
=(
0,
i
=(
1 ; :::;
0
N)
is
1 dimensional vector of unobserved common
1;
2 ; :::;
0
N)
is N
m dimensional matrix of
and "t = ("1t ; "2t :::; "N t )0 is the vector of error terms
assumed to be independently distributed of ft0 8t; t0 2 f1; ::; T g.
Without major di¢ culties, one could also add observed common factors and/or additional deterministic terms to the equations in (32), but in what follows we abstract from these for expositional
simplicity. System (32) models deviations of endogenous variables from common factors in a VAR.
Alternatively, one could introduce common factors directly in the residuals. This extension is
pursued in Pesaran and Chudik (2008), who focus on estimation of IVARs with dominant units.14
De…ne the following vector of weighted averages xW t = W0 xt , where W = (w1 ; w2 ; :::; wN )0
and fwj gN
j=1 are mw
1 dimensional vectors. Subscripts denoting the number of groups are again
14
This extension is not straightforward as it introduces in…nite-lag polynomials in the corresponding auxiliary
cross-section augmented regressions for the individual units.
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21
omitted where not necessary, in order to keep the notations simple. Matrix W does not correspond
to any spatial weights matrix. It is any arbitrary matrix of pre-determined weights satisfying the
following granularity conditions15
kWk = O N
kwj k
kWk
= O N
1
2
,
1
2
(33)
for any j
N.
(34)
We consider the problem of estimating the parameters of equation i 2 N in a non-nested
sequence of models (32) as both N and T tend to in…nity, where
=D S+
b.
can be decomposed as
See (6). As an important example we consider the two-neighbor IVAR model
de…ned by (5). In the case of this model the vector of unknown coe¢ cients of interest for the ith
equation is on the ith row of D, de…ned by (7) namely
i
hi = 3, and the corresponding N
1 ; ei ; ei+1 )
3 matrix Si = (ei
=
i;i 1 ;
ii ;
i;i+1
0
for i 2
= f1; N g, with
in S = (S1 ; S2 ; :::; SN )0 ; which
selects the unit i and the left and the right neighbors of unit i.16 In what follows we set
and note that it reduces to (xi
0
1;t ; xit ; xi+1;t )
it
= S0i xt ;
in the case of the two-neighbor IVAR model.
We suppose that the following assumptions hold for any N 2 N and i 2 f1; ::; N g, unless
otherwise stated.
ASSUMPTION 7 (General limiting restrictions) The ith row of matrix
i
= Si
i
+
bi ,
can be decomposed as,
(35)
where
k
Si is predetermined and known N
bi kr
=
max
j2f1;::;N g
bij
<
K
,
N
(36)
hi dimensional matrix, kSi kc < K, and hi < K. The unknown
coe¢ cients and the …xed e¤ ects are bounded, namely k i k < K and j i j < K. For any i 2 N, there
exists constant N0 2 N such that the vector of unknown coe¢ cients
15
do not change with N > N0 .
Condition (34) is understood as
kwj k
kWk
K
p , for any j 2 f1; ::; N g and any N 2 N,
N
where constant K < 1 does not depend on N nor on j.
16
The …rst and the last unit has only one neighbor.
22
i
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ASSUMPTION 8 (Stationarity) k k <
< 1.
ASSUMPTION 9 (Weakly dependent errors with …nite fourth moments) Innovations f"jt gN
j=1
are identically and independently distributed with mean 0, unit variances and …nite fourth moments.
Furthermore, matrix R has bounded row and column matrix norms.
ASSUMPTION 10 (Available observations) Available observations are x0 ; x1 ; :::; xT with the
P
` R" ( `) +
+ f 0 .17
starting values x0 = 1
`=0
ASSUMPTION 11 (Common factors) Unobserved common factors f1t ; :::; fmt follow stationary
MA(1) processes:
fst =
where polynomials
s (L)
=
P1
`=0
s (L) "f st ;
`
s` L
for s = 1; ::; m,
(37)
are absolute summable, "f st
IID 0;
2
"f s
, and the
fourth moments of "f st are bounded, E "4f st < 1. "f st is independently distributed of "t0 for any
t; t0 2 T , and any s 2 f1; ::; mg. Polynomials
s (L)
and variances
2
"f s
, for s 2 f1; ::; mg, do not
change with N and the covariance matrix E (ft ft0 ) is positive de…nite.
ASSUMPTION 12 (Bounded factor loadings) k i k < K.
Remark 15 (Eigenvalues of
) Assumption 8 implies polynomial
(L) is invertible (for any
N 2 N) and
%( ) <
< 1.
(38)
This is in line with the …rst part of Assumption 2 and is therefore su¢ cient for stationarity of xt for
any N 2 N. Also, as noted in Section 3, this assumption rules out explosive variance of individual
p
N k k, Assumption 8 rules
elements of the vector xt as N ! 1. Furthermore, since k kc
out cases where k kc diverges to in…nity at the rate N . Hence the dominance of a particular unit
or units due to the coe¢ cient matrix
is also ruled out by this assumption.
Remark 16 The spectral norm of covariance matrix E (ut u0t ) =
sumption 9, since kRR0 k
kRR0 kr
is bounded in N under As-
kRkr kRkc .18 ut is therefore a cross sectionally weakly
dependent process, which, as shown in Pesaran and Tosetti (2007), includes all commonly used
spatial processes in the literature.
17
We use notation " ( `) instead of " ` in order to avoid possible confusion with the notation used in previous
sections. p
18
kAkc kAkr for any matrix A.
kAk
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23
(L) and then by W0 yields
Multiplying system (32) by the inverse of polynomial
xW t =
where xW t = W0 xt ,
W
= W0 ,
W
= W0 ,
W
t
=
+
Wt
1
X
+
W ft
W t,
= W0
`
t,
(39)
and
ut ` .
(40)
`=0
Under Assumption 9, fut g is weakly cross sectionally dependent and therefore
kV ar (
1
X
W t )k =
`
W0
0`
W ,
`=0
kWk2 k k
= O N
1
,
1
X
`
2
,
`=0
(41)
where kWk2 = O N 1 by condition (33), k k = O (1) by Assumption 9 (see Remark 16) and
P1
P1
1
`
`
2
and the
`=0
`=0 k k = O (1) under Assumption 8. This implies W t = Op N
unobserved common factors can be approximated as
provided that the matrix
W
1
0
W
W
0
W
W
0
W
(xW t
W)
= ft + Op N
1
2
,
(42)
is nonsingular. It can be inferred that the full column rank of
is important for the estimation of unit-speci…c coe¢ cients. Pesaran (2006) shows that the full
column rank is, however, not necessary if the object of interest is a panel estimation of the common
mean of the individual coe¢ cients as opposed to the consistency of individual-speci…c estimates.
Using system (32), equation for unit i can be written as:
xit
i
0
i ft
=
0 0
i Si (xt 1
ft
1)
+
i;t 1
+ uit ,
(43)
where
it
24
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January 2009
=
0
ib t
= Op N
1
2
,
(44)
since the vector
ib
0
i ft
satis…es condition (33) under Assumption 7. It follows from equation (39) that
0
ia
0
W
0
i
where bi1 =
ft
1
= b0i1 xW t + b0i2 xW;t
1
W
0
W
(bi1 + bi2 )0
1
0
W
0 0
i Si
and bi2 =
b0i1
W
1
W
0
W.
Wt
b0i2
W;t 1 ,
(45)
Substituting equation (45)
into equation (43) yields
xit = ci +
where ci =
(bi1 + bi2 )0
0
ia
i
0 0
i Si xt 1
qit =
i;t 1
W,
+ b0i1 xW t + b0i2 xW;t
1
+ uit + qit ,
(46)
1
2
(47)
and
b0i1
Wt
b0i2
W;t 1
= Op N
.
Consider the following auxiliary regression based on the equation (46):
0
xit = git
where
it
= uit + qit ,
regressors git = 1;
estimator of
i
0
0
0 0
i ; bi1 ; bi2 is ki
0
0
0
0
i;t 1 ; xW t ; xW;t 1 , and ki
i
= ci ;
:
T
X
bi =
t=1
We denote the estimator of coe¢ cients
i
i
+
it ,
(48)
1 vector of coe¢ cients associated to the vector of
= 1 + hi + 2mw . Let b i be the least squares (LS)
0
git git
!
1 T
X
git xit .
(49)
t=1
given by the corresponding elements of the vector b i
as the cross section augmented least squares estimator (or CALS for short), denoted as bi;CALS .
Asymptotic properties of b i (and bi;CALS in the case where the number of unobserved common
factors is unknown) are the objective of this analysis as N and T tend to in…nity.
First we consider the case where the number of unobserved common factors equals to the
dimension of xW t (m = mw ), and make the following additional assumption.
ASSUMPTION 13 (Identi…cation of
N
N0 and for any i 2 f1; ::; N g, T
and CiN1 = O (1).
1
i)
PT
There exists T0 and N0 such that for all T
0
t=1 git git
1
T0 ;
0 ) is positive de…nite,
exists, CiN = E (git git
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25
Remark 17 Assumption 13 implies
W
is a square, full rank matrix and, therefore, the number
of unobserved common factors is equal the number of columns of the weight matrix W (m = mw ).
In cases where m < mw , full augmentation of individual models by (cross sectional) averages is not
necessary.
Theorem 1 Let xt be generated by model (32), Assumptions 7-13 hold, and W is any arbitrary
(pre-determined) matrix of weights satisfying conditions (33)-(34) and Assumption 13. Then as
j
N; T ! 1 (in no particular order), the estimator b i de…ned in equation (49) has the following
properties.
a)
bi
b) If in addition T =N ! {, with 0
p
2
ii;N
! 0:
{ < 1,
T
ii;N
where
p
i
1
2
CiN
(b i
i)
D
! N (0; Ik1 ) ,
(50)
1
2
= V ar (uit ) = E (e0i RR0 ei ), and CiN
is square root of positive de…nite matrix
0 ). Also
CiN = E (git git
c)
CiN
where
p
b iN !
C
0, and
p
ii;N
bii;N ! 0;
T
T
X
1X 2
0
b iN = 1
git git
; b2ii;N =
u
bit ,
C
T
T
t=1
and u
bit = xit
0 b .
git
i
(51)
t=1
Remark 18 Suppose that in addition to the assumptions of Theorem 1, the limits of CiN1 and
2
ii;N
, as N ! 1; exist and are given by Ci11 , and
p
19
T (b i
i)
D
2
ii;1 ;
! N 0;
respectively.19 Then (50) yields
1
2
ii;1 Ci;1
.
(52)
Su¢ cient condition for limN !1 CiN to exist is the existence of the following
with Assumptions
P limits0 (together
`
7-12): limN !1 S0i , limN !1 S0i , limN !1 W0 , limN !1 W0 , and limN !1 1
RR0 0` Si .
`=0 Si
26
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Consider now the case where the number of unobserved common factors is unknown, but it is
known that mw
m. Since the auxiliary regression (48) is augmented possibly by a larger number
of cross section averages than the number of unobserved common factors, we have potential problem
of multicollinearity (as N ! 1). But this observation has no bearings on estimates of
i
so long as
the space spanned by the unobserved common factors including a constant and the space spanned
by the vector (1; x0W t )0 are the same as N ! 1. This is the case when
W
has full column rank.
Using partition regression formula, the cross sectionally augmented least squares (CALS) estimator
1 dimensional vector
of hi
i
in the auxiliary regression (48) is
where xi = (xi1 ; :::; xiT )0 , Zi =
r 2 f1; ::; hi g ; MH = IT
1
bi;CALS = Z0 MH Zi
i
i1 (
1) ;
i2 (
H (H0 H)+ H0 , H =
1) ; :::;
Z0i MH xi ,
ihi
( 1) ,
(53)
ir
; XW ; XW ( 1) ,
( 1) =
is T
ir0 ; :::; i;r;T 1
0
for
1 dimensional vector of
ones, XW = (xW 1 ; :::; xW mw ), XW ( 1) = [xW 1 ( 1) ; :::; xW mw ( 1)], xW s = (xW s1 ; :::; xW sT )0
and xW s ( 1) = (xW s0 ; :::; xW s;T
S0i
t
=
it
S0i f t
0
1)
for s 2 f1; ::; mw g. De…ne for future reference vector vit =
S0i , and the following matrices.
Q = [ ; F; F ( 1)] ,
and
0
(54)
0
W
B 1
B
0
=B
A
0
W
(2m+1) (2mw +1) B
@
0 0m mw
0
W
0m
mw
0
W
1
C
C
C,
C
A
(55)
where F = (f1 ; :::; fm ), F ( 1) = [f1 ( 1) ; :::; fm ( 1)], fr = (fr1 ; :::; frT )0 and fr ( 1) = (fr0 ; :::; fr;T
0
1)
for r 2 f1; ::; mg.
For this more general case we replace Assumption 13 with the following (and suppress the
subscript N to simplify the notations)
ASSUMPTION 14 (Identi…cation of
i)
There exists T0 and N0 such that for all T
1
N
vi
exists,
N0 and for any i 2 f1; ::; N g, T 1 Z0i MH Zi
P1 0 `
0 )=
= E (vit vit
RR0 0` Si is positive de…nite and
`=0 Si
W
1
vi
T0 ;
is full column rank matrix,
= O (1).
ECB
Working Paper Series No 998
January 2009
27
Theorem 2 Let xt be generated by model (32), Assumptions 7-12, and 14 hold, and W is any
arbitrary (pre-determined) matrix of weights satisfying conditions (33)-(34). Then if in addition
j
{ < 1, the asymptotic distribution of bi;CALS de…ned by
N; T ! 1 such that T =N ! {, with 0
(53) is given by.
p
T
ii
where
2
ii
= V ar (uit ),
vi
1
2
vi
D
bi;CALS
! N (0; Ihi ) ,
i
0 ) and v = S0
= E (vit vit
it
i
t
=
P1
0
`=0 Si
(56)
`u
t `.
Remark 19 As before, we also have
p
where
vi;1
= limN !1
vi ,
T bi;CALS
and
2
ii;1
D
i
! N 0;
2
ii;1
1
vi;1
;
2.
ii
= limN !1
Extension of the analysis to a IVAR(p) model is straightforward and it is relegated to a Supplement available from the authors upon request.
5
Monte Carlo Experiments: Small Sample Properties of CALS
Estimator
5.1
Monte Carlo Design
In this section we report some evidence on the small sample properties of the CALS estimator in the
presence of unobserved common factors and weak error cross section dependence and compare the
results with standard least squares estimators. The focus of our analysis will be on the estimation
of the individual-speci…c parameters in an IVAR model that also allows for other interactions that
are of order O(N
1 ).
The data generating process (DGP) is given by
xt
ft =
(xt
1
ft
1)
+ ut ,
where ft is the only unobserved common factor considered (m = 1), and
N
(57)
=(
1 ; :::;
0
N)
is the
1 vector of factor loadings.
We consider two sets of factor loadings to distinguish the case of weak and strong cross section
dependence. Under the former we set
28
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Working Paper Series No 998
January 2009
= 0; and under the latter we generate the factor loadings
i;
for i = 1; 2; :::; N , from a stationary spatial process in order to show that our estimators are
invariant to the cross section dependence of the factor loadings. The following bilateral Spatial
Autoregressive Model (SAR) is considered.
=
i
where
IIDN 0;
i
2
1
=
2b ;
2
i 1
+
a
i+1
+
i,
(58)
. As established by Whittle (1954), the unilateral SAR(2) scheme
i
with
a
2
=
1 i 1
q
= b2 and b = 1
+
+
2 i 2
i,
(59)
a2 =a , generates the same autocorrelations as
1
the bilateral SAR(1) scheme (58). The factor loadings are generated using the unilateral scheme
(59) with 50 burn-in data points (i =
a = 0:4,
2
= 1, and choose
49; :::; 0) and the initializations
51
=
50
= 0. We set
such that V ar ( i ) = 1.20 The common factors are generated
according to the AR(1) process
ft =
with
f
f ft 1
+
f t,
2
f
IIDN 0; 1
ft
,
= 0:9.
In line with the theoretical analysis the autoregressive parameters are decomposed as
b,
where
a
a+
capture own and neighborhood e¤ects as in
0
a
20
=
0
0
1
B '1
B
B
0
B 2 '2
2
B
B
B 0
3 '3
3
=B
B
B 0
0
4 '4
B
B
..
B
.
B
@
0
0
0
0
0
0
..
.
..
.
N 1
'N
N
The variance of factor loadings is given by
2
=
1+
1
2
1
2
1
C
C
C
C
C
C
C
C;
C
C
C
C
C
C
A
2
2
1
.
2
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Working Paper Series No 998
January 2009
29
and the remaining elements of
, de…ned by
=
bij
8
>
<
i ! ij
>
:
0
are generated as
for j 2
= fi
1; i; i + 1g
for j 2 fi
1; i; i + 1g
, where
& ij
IIDU ( 0:1; 0:2) and ! ij = PN
i
j=1 & ij
IIDU (0; 1). This ensures that
with & ij
b,
bij
= Op (N
1 ),
,
(60)
and limN !1 E
bij
= 0, for all i and
j.
With
a
as speci…ed above, each unit i, except the …rst and the last, has two neighbors: the
‘left’neighbor i
1 and the ‘right’neighbor i + 1. The DGP for the ith unit can now be written as
x1t = '1 x1;t
xit = 'i xi;t
1
1
xN t = 'N xN;t
+
+
1
+
+
0
b1 xt 1
i (xi 1;t 1
+ xi+1;t
1 x2;t 1
N xN 1;t 1
+
+
1)
0
1
1 ft
0
bi xt 1
+
0
b;N xt 1
+
ft
1
+
0
i
i ft
0
N
N ft
To ensure the DGP is stationary we generate 'i
+ u1t ,
ft
1
ft
1
+ uit ; i 2 f2; ::; N
1g ,
+ uN t .
IIDU (0:4; 0:6), and
i
IIDU ( 0:1; 0:1)
for i 6= 2. We choose to focus on the equation for unit i = 2 in all experiments and we set '2 = 0:5
and
2
= 0:1. This yields k kr
0:9, and together with
f
< 1 it is ensured that the DGP is
stationary and the variance of xit is bounded in N . The cross section averages, xwt ; are constructed
P
as simple averages, xt = N 1 N
j=1 xit .
The N -dimensional vector of error terms, ut ; is generated using the following SAR model:
u1t = au u2t + "1t ,
uit =
au
(ui
2
uN t = au uN
1;t
1;t
+ ui+1;t ) + "it , i 2 f2; ::; N
1g
+ "N t ,
for t = 1; 2; ::; T . We set au = 0:4 which ensures that the errors are cross sectionally weakly
dependent, and draw "it , the ith element of "t , as IIDN 0;
30
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Working Paper Series No 998
January 2009
2
"
. We set
2
"
= N=tr (Ru R0u ) so that
on average V ar(uit ) is 1, where Ru = (I
0
B 0
B
B 1
B 2
B
B
B 0
S=B
B
B
B
B
B
B
@
0
au S)
1
; and the spatial weights matrix S is
1
0
0
0
1
2
0
1
2
0
..
.
1
2
0
..
.
..
.
1
2
0
0
1
1
0 C
C
0 C
C
C
C
0 C
C.
C
C
C
C
1 C
2 C
A
0
(61)
In order to minimize the e¤ects of the initial values, the …rst 50 observations are dropped.
N 2 f25; 50; 75; 100; 200g and T 2 f25; 50; 75; 100; 200g. For each N , all parameters were set at the
beginning of the experiments and 2000 replications were carried out by generating new innovations
"it ,
ft
and
i.
The focus of the experiments is to evaluate the small sample properties of the CALS estimator of
the own coe¢ cient '2 = 0:5 and the neighboring coe¢ cient
regression for estimating coe¢ cients f
2;
2g
2
= 0:1. The cross-section augmented
in the case of the second cross section unit is given
by (similar results are also obtained for other cross section units)
x2t = c2 +
2 (x1;t 1
+ x3;t
1)
+ '2 x2;t
1
+
2;0 xt
+
2;1 xt 1
+
2t .
(62)
We also report results of the Least Squares (LS) estimator computed using the above regression
but without augmentation with cross-section averages. The corresponding CALS estimator and
b 2;LS (own coe¢ cient), or b 2;CALS and
non-augmented LS estimator are denoted by '
b 2;CALS and '
b
2;LS
(neighboring coe¢ cient), respectively.
To summarize, we carry out two di¤erent sets of experiments, one set without the unobserved
common factor (
= 0), and the other with unobserved common factor (
6= 0). There are
many sources of interdependence between individual units: spatial dependence of innovations fuit g,
spatiotemporal interactions due to coe¢ cient matrices
with
a
and
b,
and …nally in the latter case
6= 0 the cross section dependence also arises via the unobserved common factor ft and
cross-sectionally dependent factor loadings. Additional intermediate cases are also considered, the
results of which are available in a Supplement from the authors on request.21
21
Supplement presents experiments with all combination of zero or nonzero coe¢ cient matrix
b,
zero or nonzero
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Working Paper Series No 998
January 2009
31
5.2
Monte Carlo Results
Tables 1-2 give the bias ( 100) and RMSE ( 100) of CALS and LS estimators as well as size and
power of tests at the 5% nominal level. Results for the estimated own coe¢ cient, '
b 2;CALS and
'
b 2;LS , are reported in Table 1. The top panel of this table presents the results for the experiments
with an unobserved common factor (
6= 0). In this case, fxit g is CSD and the standard LS
estimator without augmentation with cross section averages is not consistent. The bias of '
b 2;LS is
indeed quite substantial for all values of N and T and the tests based on '
b 2;LS are grossly oversized.
CALS, on the other hand, performs well for T
100 and all values of N . For smaller values of T ,
there is a negative bias, and the test based on '
b 2;CALS is slightly oversized. This is the familiar time
series bias where even in the absence of cross section dependence the LS estimator of autoregressive
coe¢ cients will be biased in small T samples.
Moving on to the experiments without a common factor (given at the bottom half of the table),
we observe that the LS estimator slightly outperforms the CALS estimator. In the absence of
common factors, fxit g is weakly cross sectionally dependent and therefore the augmentation with
cross section averages is (asymptotically) redundant. Note that the LS estimator is not e¢ cient
because the residuals are cross sectionally dependent. Augmentation by cross-section averages
helps to reduce part of this dependence. Nevertheless, the reported RMSE of '
b 2;CALS does not
outperform the RMSE of '
b 2;LS .
The estimation results for the neighboring coe¢ cient,
2;
are presented in Table 2. These are
qualitatively similar to the ones reported in Table 1. Cross section augmentation is clearly needed
when common factors are present. But in the absence of such common e¤ects, the presence of weak
cross section dependence, whether through the dynamics or error processes, does not pose any
di¢ culty for the least squares estimates so long as N is su¢ ciently large. Finally, not surprisingly,
the estimates are subject to the small T bias irrespective of the size of N or the degree of cross
section dependence.
Figure 1 plots the power of the CALS estimator of the own coe¢ cient, '
b 2;CALS , (top chart)
and the neighboring coe¢ cient, b 2;CALS , (bottom chart) for N = 200 and two di¤erent values
of T 2 f100; 200g. These charts provide a graphical representation of the results reported in
Tables 1-2, and suggest signi…cant improvement in power as T increases for a number of di¤erent
factor loadings
32
ECB
Working Paper Series No 998
January 2009
, and low or high cross section dependence of errors (au = 0:4 or au = 0:8).
alternatives.
6
An Empirical Application: a spatiotemporal model of house
prices in the U.S.
In a recent study Holly, Pesaran and Yamagata (2008, HPY) consider the relation between real
house prices, pit ; and real per capita personal disposable income yit (both in logs) in a panel of
49 US States over 29 years (1975-2003), where i = 1; 2; :::; 49 and t = 1; 2; :::; T . Controlling
for heterogeneity and cross section dependence, they show that pit and yit are cointegrated with
coe¢ cients (1; 1), and provide estimates of the following panel error correction model:
pit = ci + ! i (pi;t
1
yi;t
1)
+
pi;t
1i
1
+
2i
yit +
it .
(63)
To take account of unobserved common factors HPY augmented (63) with cross section averages
and obtained common correlated e¤ects mean group and pooled estimates (denoted as CCEMG
and CCEP) of f! i ;
1i ; 2i g
which we reproduce in the left panel of Table 3. HPY then showed that
the residuals from these regressions, ^ it , display a signi…cant degree of spatial dependence. Here
we exploit the theoretical results of the present paper and consider the possibility that dynamic
neighborhood e¤ects are partly responsible for the residual spatial dependence reported in HPY.
To this end we considered an extended version of (63) where the lagged spatial variable psi;t 1 =
PN
j=1 sij pj;t 1 is also included amongst the regressors, with sij being the (i; j) element of a spatial
weight matrix, S, namely
pit = ci + ! i (pi;t
1
yi;t
1)
+
1i
pi;t
1
+
i
psi;t
1
+
2i
yit +
it .
(64)
Here we consider a simple contiguity matrix sij = 1 when the states i and j share a border and
zero otherwise, with sii = 0. Possible strong cross section dependence is again controlled for by
augmentation of the extended regression equation with cross section averages. Estimation results
are reported in the right panel of Table 3. The dynamic spatial e¤ects are found to be highly
signi…cant, irrespective of the estimation method, increasing R2 of the price equation by 6-9%.
The dynamics of past price changes are now distributed between own and neighborhood e¤ects
ECB
Working Paper Series No 998
January 2009
33
giving rise to much richer dynamics and spill over e¤ects. It is also interesting that the inclusion
of the spatiotemporal variable
psi;t
1
coe¢ cient of the real income variable,
7
in the model has had little impact on the estimates of the
2i .
Concluding Remarks
This paper has proposed restrictions on the coe¢ cients of in…nite-dimensional VAR (IVAR) that
bind only in the limit as the number of cross section units (or variables in the VAR) tends to
in…nity to circumvent the curse of dimensionality. The proposed framework relates to the various
approaches considered in the literature. For example when modelling individual households or …rms,
aggregate variables, such as market returns or regional/national income, are treated as exogenous.
This is intuitive as the impact of a …rm or household on the aggregate economy is small, of the
order O N
1
. This paper formalizes this idea in a spatio-dynamic context.
It was established that, under certain conditions on the order of magnitudes of the coe¢ cients
in a large dynamic system, and in the absence of common factors, equations for individual units
decouple as N ! 1 and can be estimated separately. In the presence of a dominant economic agent
or unobserved common factors, individual-speci…c VAR models can still be estimated separately
if conditioned upon observed and unobserved common factors. Unobserved common factors can
be approximated by cross sectional averages, following the idea originally introduced by Pesaran
(2006).
The paper shows that the GVAR approach can be motivated as an approximation to an IVAR
featuring all macroeconomic variables. This is true for stationary models as well as for systems
with integrated variables of order one. Asymptotic distribution of the cross sectionally augmented
least-squares (CALS) estimator of the parameters of the unit-speci…c models was established both
in the case where the number of unobserved common factors is known and in the case where it is
unknown but …xed. Small sample properties of the proposed CALS estimator were investigated
through Monte Carlo simulations, and an empirical application was provided as an illustration of
the proposed approach.
Topics for future research could include estimation and inference in the case of IVAR models
with dominant individual units, analysis of large dynamic networks with and without dominant
nodes, and a closer examination of the relationships between IVAR and dynamic factor models.
34
ECB
Working Paper Series No 998
January 2009
Figure 1: Power Curves for the CALS t-tests of Own Coe¢ cient, '2 (the upper chart) and the Neighboring
Coe¢ cient,
2
(the lower chart), in the Case of Experiments with
6= 0.
ECB
Working Paper Series No 998
January 2009
35
36
ECB
Working Paper Series No 998
January 2009
21.15
21.70
20.40
20.80
21.30
7.20
9.45
8.80
9.35
10.10
6.95
6.85
7.15
6.40
7.35
7.05
7.65
7.95
7.10
8.35
13.20
13.80
13.85
12.65
13.50
13.30
14.35
14.00
13.20
15.45
8.40
9.10
8.35
8.75
8.95
9.35
10.90
10.20
10.55
10.55
6.47
6.56
6.80
6.83
6.80
6.67
6.74
6.56
6.61
6.62
6.43
6.69
6.57
6.63
6.67
6.85
6.35
7.00
7.15
6.10
6.65
6.00
6.80
5.95
5.20
7.60
7.95
8.10
8.70
7.55
27.70
30.30
27.50
28.40
28.75
4.80
5.80
6.90
5.65
5.65
5.35
5.75
6.25
5.25
5.00
7.00
6.15
6.80
6.50
6.70
33.25
32.35
32.05
32.80
32.70
4.85
6.40
5.25
6.00
6.30
5.45
6.85
5.20
6.30
6.30
5.80
5.30
6.00
6.55
6.25
45.55
42.65
43.60
44.25
42.85
Size (5% level, H0 : '2 = 0:50)
25
50
75
100
200
17.63
17.24
17.55
17.63
17.18
100)
200
18.20
19.45
19.75
18.90
19.30
15.80
17.25
17.35
16.70
17.30
23.80
24.85
26.05
25.85
27.00
16.85
16.00
16.45
14.90
15.30
19.50
20.20
21.05
18.35
21.80
19.25
18.85
19.70
17.75
20.80
22.90
26.90
26.35
26.80
26.40
19.80
20.85
18.45
18.05
20.95
21.00
23.85
22.95
24.65
24.90
21.70
22.60
22.90
24.25
24.15
27.10
28.10
28.00
28.40
30.00
23.55
26.80
24.60
23.55
26.35
23.85
25.95
28.65
27.85
25.65
25.00
25.45
27.65
26.05
25.10
31.20
33.50
33.25
31.50
33.85
29.20
29.50
31.10
30.35
29.00
39.50
39.90
42.50
42.80
41.90
42.60
41.20
42.25
43.40
41.00
41.00
45.15
46.55
47.80
47.80
42.00
39.20
41.85
43.65
41.45
Power (5% level, H1 : '2 = 0:60)
25
50
75
100
200
2
i
according to stationary spatial autoregressive processes. Please refer to Section 5 for detailed description of Monte Carlo design.
+
xit = 'i xi;t
1
Notes: '2 = 0:5,
= 0:1, a = au = 0:4, and V ar ( i ) = 1. The DGP is given by 2-neighbor IVAR model (57) where the equation for unit i 2 f2; ::; N 1g is
0
(xi 1;t 1 + xi+1;t 1 ) + 0bi xt 1 + i ft
2 is computed using
i ft 1 + uit . The CALS estimator of the own coe¢ cient '2 and the neighboring coe¢ cient
the following auxiliary regression, x2t = c2 + 2 (x1;t 1 + x3;t 1 ) + '2 x2;t 1 + 2;0 xt + 2;1 xt 1 + 2t . Estimators '
b 2;LS and b 2 are computed from the auxiliary regressions
not augmented with cross section averages. Unobserved common factor ft is generated as stationary AR(1) process, and factor loadings and innovations fuit g are generated
Bias ( 100)
Root Mean Square Errors (
(N,T)
25
25
50
50
75
75
100
100
200
Experiments with nonzero factor loadings
LS estimator not augmented with cross section averages, '
b 2;LS
25
-4.34
25.78
18.82
5.46
17.48
7.34
17.49
8.76
11.02
50
-3.04
25.14
18.83
4.39
18.37
8.21
17.45
8.61
10.65
75
-2.94
25.20
18.50
4.58
17.55
6.99
17.47
8.36
10.59
100
-2.67
24.19
18.27
4.89
17.65
7.60
17.74
8.94
10.74
200
-3.71
24.59
18.90
4.11
17.98
7.79
17.62
8.84
10.26
b 2;CALS
CALS estimator '
25 -18.43
29.04
15.52
-7.63
12.14
-5.22
-3.38
9.86
-0.72
50 -19.65
29.40
16.73
-9.58
12.44
-5.69
10.13
-4.77
-1.76
75 -19.76
29.83
16.56
-9.36
12.41
-5.82
10.35
-4.41
-2.30
100 -20.19
29.94
16.82
-9.31
12.60
-6.00
10.18
-4.29
-2.30
200 -20.96 -10.17 -6.27 -4.85
30.33
17.17
12.27
10.41
-2.37
Experiments with zero factor loadings
LS estimator not augmented with cross section averages,b
'2;LS
25 -13.10
24.20
14.80
-6.17
11.45
-3.89
9.35
-2.60
-1.59
50 -13.32
24.79
14.98
-6.16
11.51
-4.22
9.45
-2.76
-1.35
75 -12.74
24.15
14.97
-6.19
11.60
-4.14
9.97
-3.50
-1.37
100 -12.36
23.69
14.63
-5.65
11.87
-4.41
9.46
-3.15
-1.72
200 -13.30
24.49
15.03
-6.42
11.43
-4.54
9.47
-2.79
-1.46
b 2;CALS
CALS estimator '
25 -14.43
25.84
15.23
-6.22
11.36
-3.58
9.22
-2.21
-0.86
50 -15.38
27.13
15.61
-6.87
11.77
-4.54
9.48
-2.81
-1.10
75 -15.03
26.69
15.72
-6.93
11.95
-4.48
10.14
-3.64
-1.34
100 -14.90
26.49
15.40
-6.51
12.15
-4.78
9.70
-3.48
-1.75
200 -15.67
26.94
15.92
-7.51
11.91
-5.14
9.75
-3.23
-1.61
Table 1: MC results for the own coe¢ cient '2 .
ECB
Working Paper Series No 998
January 2009
37
7.67
7.14
7.03
6.99
7.21
7.47
7.22
6.31
6.69
6.45
7.02
6.33
6.55
6.67
6.76
1.18
0.88
0.87
0.71
1.14
1.17
0.89
0.52
0.55
0.72
0.78
0.43
0.64
0.58
0.46
18.01
16.96
17.27
17.00
18.06
22.64
21.44
21.46
21.54
21.37
10.04
10.12
10.16
10.14
10.56
10.69
10.56
10.37
10.74
10.55
16.86
16.83
16.11
16.33
17.61
7.98
7.89
8.01
7.94
7.99
8.20
8.39
8.25
8.23
8.30
15.10
15.24
14.87
15.07
15.20
6.58
6.81
6.82
6.77
6.75
6.99
6.79
6.82
6.72
6.88
14.62
14.83
13.71
14.15
14.18
2;LS
4.66
4.80
4.67
4.72
4.65
4.93
4.73
4.79
4.60
4.57
13.27
13.19
13.48
13.20
13.26
0.77
0.67
0.98
0.52
1.11
0.51
0.41
0.60
0.53
0.48
0.39
0.33
0.54
0.68
0.40
0.09
0.36
0.09
0.30
0.14
2.15
1.92
1.15
2.07
1.31
1.02
0.91
1.05
0.58
1.17
0.66
0.47
0.64
0.58
0.54
0.58
0.43
0.61
0.82
0.43
0.23
0.46
0.13
0.38
0.14
CALS estimator b 2;CALS
2.04
1.74
1.22
1.94
1.38
17.64
16.88
16.48
17.40
16.80
16.10
15.76
15.49
16.19
15.63
10.47
10.48
10.43
10.42
10.90
8.29
8.08
8.12
8.18
8.13
6.70
6.92
6.98
6.84
6.87
4.63
4.84
4.69
4.77
4.68
LS estimator not augmented with cross section averages, b 2;LS
with zero factor loadings
1.70
1.52
1.35
1.26
1.22
See the notes to Table 1.
25
50
75
100
200
25
50
75
100
200
7.88
8.06
7.62
7.35
8.28
CALS estimator b 2;CALS
9.51
8.04
8.49
7.75
8.52
25 2.31
50 2.20
75 2.71
100 1.82
200 2.55
Experiments
25
50
75
100
200
Bias ( 100)
Root Mean Square Errors ( 100)
(N,T)
25
25
50
50
75 100 200
75
100
200
Experiments with nonzero factor loadings
LS estimator not augmented with cross section averages, b
2.
8.25
7.50
7.00
8.00
7.10
8.20
7.05
6.80
7.25
7.15
9.10
8.15
8.05
8.10
9.00
19.70
18.60
17.40
18.25
18.85
6.35
6.70
6.60
6.15
7.40
5.55
6.00
6.10
6.00
6.45
6.75
6.85
6.75
6.90
6.45
27.30
27.45
26.60
27.05
29.50
5.85
5.90
5.40
5.80
5.40
6.15
5.65
5.30
5.00
5.10
5.75
7.40
6.05
6.75
6.50
36.35
35.60
36.00
35.85
36.75
5.00
5.90
5.60
5.60
5.95
5.60
5.75
5.55
5.60
6.05
6.70
5.95
5.70
5.45
6.00
43.50
44.10
41.85
42.30
41.00
4.70
5.75
5.30
5.50
4.85
5.20
5.90
5.00
5.25
5.05
7.30
6.05
6.25
5.25
5.55
60.30
59.25
58.25
59.15
60.40
Size (5% level, H0 : 2 = 0:10)
25
50
75
100
200
Table 2: MC results for the neighboring coe¢ cient
11.80
10.00
11.45
12.20
11.70
11.55
10.75
11.10
12.25
11.30
12.65
11.60
11.45
11.00
11.95
16.70
15.85
17.30
17.50
16.50
17.45
16.30
16.20
17.65
16.85
17.40
17.25
17.45
18.45
18.45
15.75
16.00
15.60
17.00
17.80
23.15
24.80
22.20
24.45
25.30
23.55
23.30
22.25
23.70
23.05
23.10
23.90
22.85
24.05
24.05
21.85
24.60
23.35
24.20
21.95
28.65
29.95
30.25
30.20
29.65
28.20
29.95
30.20
27.50
30.05
30.05
31.80
30.70
29.20
30.65
28.10
29.55
30.70
30.50
29.90
33.05
34.45
34.40
34.00
34.90
56.35
54.30
56.20
53.50
57.35
58.05
55.95
57.15
54.70
56.65
52.80
57.35
53.85
53.90
55.70
43.10
44.80
47.00
45.35
44.10
Power (5% level, H1 : 2 = 0:20)
25
50
75
100
200
Table 3: Alternative Average Estimates of the Error Correction Models for House
Prices Across 49 U.S. States over the Period 1975-2003
yi;t
1
Regressions augmented with
without dynamic spatial e¤ects
dynamic spatial e¤ects
MG
pit
pi;t
Holly et al. (2008) regressions
1
0:105
(0:008)
pi;t
1
yit
psi;t
0:524
(0:030)
0:500
1
R2
Average Cross Correlation
CCEMG
0:183
(0:016)
0:449
(0:038)
0:277
CCEP
0:171
(0:015)
0:518
(0:065)
0:227
(0:040)
(0:059)
(0:063)
-
-
-
0:54
0:70
0:284
0:005
MG
0:095
(0:009)
0:296
(0:060)
0:497
(0:040)
0:331
CCEMG
0:154
(0:018)
0:188
(0:049)
0:284
(0:059)
0:350
CCEP
0:152
(0:018)
0:272
(0:082)
0:201
(0:088)
0:431
(0:066)
(0:085)
(0:105)
0:66
0:60
0:79
0:72
0:016
0:267
0:012
0:016
Coe¢ cients ( ^ )
Notes: MG stands for Mean Group estimates. CCEMG and CCEP signify the Common Correlated E¤ects Mean Group and
Pooled estimates de…ned in Pesaran (2006). Standard errors are in parentheses. ^ denotes the average pair-wise correlation of
the residuals from the cross-section augmented regressions across the 49 U.S. States.
38
ECB
Working Paper Series No 998
January 2009
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ECB
Working Paper Series No 998
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41
Appendix
A
Lemmas and Proofs
Proof of Proposition 1. For any N 2 N, the variance of xt is
=V ar (xt ) =
1
X
0`
`
,
(65)
`=0
and, k k is under Assumptions 1-2 bounded by
k k
k k
1
X
`=0
k k2` = O N 1
.
(66)
It follows that for any arbitrary nonrandom vector of weights satisfying granularity condition (11),
V ar w0 xt
where % ( ) = k k = O N 1
O N
= w0 w
% ( ) w0 w
, and w0 w = kwk2 = O N
,
(67)
by condition (11). This implies kV ar (w0 xt )k =
1
and limN !1 kV ar (w0 xt )k = 0.
Proof of Corollary 1.
Assumption 3 implies that for i 2 K, vector
bi
satis…es condition (11). It follows from
Proposition 1 that
0
bi xt
lim V ar
N !1
= 0 for i 2 K.
(68)
System (2) implies
0
a xt 1
xit
uit =
0
b xt 1 ;
for any i 2 S and any N 2 N:
(69)
Taking variance of (69) and using (68) now yields (19).
Proof of Proposition 2. Solving (2) backwards yields
xt =
1
X
`
R "t
`,
(70)
`=0
_
where R = r1 e01 + R
1.
Hence
1
X
x1t
e01
`
r1 "1;t
`
1
X
=
`=0
e01
`
_
R
"t
1
`.
(71)
`=0
Under Assumptions 4-6,
V ar
1
X
`=0
e01
`
_
R
1
"t
`
!
=
1
X
e01
`
_
R
_0
1R 1
_
e01 R
_0
1 R 1 e1
+
1
X
`=1
42
ECB
Working Paper Series No 998
January 2009
0`
e1 ,
`=0
e01
`
_
R
_0
1R 1
0`
e1 .
(72)
But
_
e01 R
lim
N !1
under Assumption 6. Set a0`
e01
`
and let a`;
_0
1 R 1 e1
1
2
1k
= lim kr
N !1
(73)
=0
= (0; a`2 ; :::; a`N )0 . Note that under Assumptions 4-5:
`
ka` kc
ka`;
,
(74)
a`1
=
O (1) ,
1 kr
=
O N
(75)
1
,
(76)
for ` = 0; 1; 2; :::. Result (74) follows from Assumption 5 by taking the maximum absolute row-sum matrix norm of
a0` = e01
` 22
.
Results (75)-(76) follow by induction directly from Assumptions 4-5. Using (74)-(76), we have
1
X
e01
`
_
R
_0
1R 1
0`
1
X
=
e1
`=1
_
a0` R
_0
1 R 1 a`
,
`=1
1
X
a2`1 r0 1 r
1
+
`=1
kr
1
X
a0`;
_
_0
1 R 1 R 1 a`; 1
,
`=1
2
1k
1
X
2`
2
_
+ R
1
`=1
1
X
`=1
ka`;
1 kr
ka`;
(77)
1 kc
2
P
2`
_ 1 = O N1
where as before limN !1 kr 1 k2 = 0 under Assumption 6, 1
= O (1) by Assumption 5, R
`=1
P1
P
2
1
by properties (74)-(76). It follows that
by Assumption 6, and 1
`=1 ka`; 1 k
`=1 ka`; 1 kr ka`; 1 kc = O N
hP
i
P1 0 `
1
0
0`
0
`
_ 1R
_ 1
_ 1 "t ` = 0, we have
R
R
limN !1
e1 = 0. Noting that E
`=1 e1
`=0 e1
1
X
e01
`
_
R
1
q:m:
"t
`
`=0
! 0, as N ! 1.
(78)
This completes the proof of equation (25). To prove (26), we write
1
X
xwt
w0
`
r1 "1;t
`
`=0
Since the vectors w0
`
=
1
X
w0
`
_
R
1
"t
`.
(79)
`=0
have the same properties as vectors fa` g in equations (74)-(76), it follows that (using the
same arguments as above),
1
X
w0
`
_
R
1
"t
q:m:
`
`=0
! 0, as N ! 1.
(80)
This completes the proof of equation (26).
Proof of Proposition 3.
xit
The vector r
1; i
0
ii xi;t 1
1; i xt 1
ka` kc
e01
ri1 "1t
rii "it = r0
1; i "t
(81)
satis…es equation (24) of Assumption 6, V ar ("t ) = IN , and E ("t ) = 0, which implies
r0
22
i1 x1;t 1
`
r
ke01 kr k k`r
`
1; i "t
q:m:
! 0:
(82)
.
ECB
Working Paper Series No 998
January 2009
43
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44
ECB
Working Paper Series No 998
January 2009
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Working Paper Series No 998
January 2009
45
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46
ECB
Working Paper Series No 998
January 2009
8 and 9 that & b;n has following properties2 9
h
E
E
Nt
cN t
j Ft
n
i2
& b;0 = O (1) , and & b;n ! 0 as n ! 1.
(95)
is therefore bounded by & n = & a;n + & b;n . Equations (94) and (95) establish
& 0 = O (1) , & n ! 0 as n ! 1.
By Liapunov’s inequality, E jE (
Nt
j Ft
q
E [E (
n )j
that the two-dimensional array f
1
1
,
N t ; Ft gt= 1
N =1
thermore, (96) establishes array f
N t =cN t g
0
j Ft
2
n )]
(Davidson, 1994, Theorem 9.23). It follows
is L1 -mixingale with respect to a constant array fcN t g. Fur-
is uniformly bounded in L2 norm. This implies uniform integrability.3 0
Since also equations (91) and (92) hold, array f
P N
L1
which implies Tt=1
N t ! 0, i.e.:
T
1 X
T t=1
Nt
(96)
0
t p'
1
1
N t ; Ft gt= 1
N =1
t q
E
0
satis…es conditions of a mixingale weak law,3 1
0
t p'
L
!1 0,
t q
j
as N; T ! 1 at any rate. Convergence in L1 norm implies convergence in probability. This completes the proof of
p
1
result (86). Under the condition k k = O N 2 , result (88) follows from result (86) by noting that
N
= O (1).
j
Lemma 2 Let xt be generated by model (32), Assumptions 7-12 hold, and N; T ! 1 at any rate. Then for
and ' with growing dimension N
any p; q 2 f0; 1g, and for any sequences of non-random vectors
1 such that
k kc = O (1) and k'kc = O (1), we have
T
1 X
T t=1
29
=
=
max
n2f1;::;N g
max
n2f1;::;N g
max
n2f1;::;N g
max
N
X
s=1
N
X
s=1
N
X
s=1
n2f1;::;N g
RR0
31
xt
p
E
Matrix B is symmetric by construction. Therefore kBk
kBkr
30
0
2
r
k
0
xt
p
p
(97)
! 0,
p
kBkr kBkc = kBkr , where
ns k
max
i2f1;::;N g
max
i2f1;::;N g
N X
N
X
s=1 `0 =0
N X
N
X
j=1 `=0
jri` rj` rs` rn` j
N
N
X
X
j=1
`=0
!
jrs`0 rn`0 j
kRk2r kRk2c = O (1)
jri` rj` j
max
N
X
`0 =0
i2f1;::;N g
!
jrs`0 rn`0 j
N X
N
X
j=1 `=0
!
jri` rj` j
Su¢ cient condition for uniform integrability is L1+" uniform boundedness for any " > 0.
Davidson (1994, Theorem 19.11).
ECB
Working Paper Series No 998
January 2009
47
and
T
1 X
T t=1
0
xt
0
p ' xt q
0
E
p
0
p ' xt q
xt
(98)
! 0.
Furthermore, for k k = O (1) and k'kc = O (1) we have
T
1 X
T t=1
where
t
0
0
t p'
p
ft
(99)
! 0,
q
is de…ned in equation (40).
Proof. Let TN = T (N ) be any non-decreasing integer-valued function of N such that limN !1 TN = 1. Consider
the following two-dimensional array f
1
1
,
N t ; Ft gt= 1
N =1
Nt
1
TN
=
0
where fFt g denotes an increasing sequence of -…elds (Ft
time t and
Nt
independence of ft and
t0
1
TN
ft
q,
Ft ) such that Ft includes all information available at
1
for any t; t0 2 Z, we have
where
& n = sup
N 2N
k k2 = O (1), k k <
(
1
1 N =1
fcN t g1
t=
be two-dimensional array
for all t 2 Z and N 2 N. Using submultiplicative property of matrix norm, and
E
q
0
t p'
is measurable with respect to Ft for any N 2 N. Let
of constants and set cN t =
ft
de…ned by
2
(
2
E
Nt
j Ft
cN t
2 maxf0;n pg
k k k kk k
`=0
by Assumption 8, and k k
is covariance stationary and k'0
0
1
X
n
)
2`
k k E
& n,
n
0
E '
ft
q
j Ft
2
n
o
)
.
p
k kc k kr = O (1) by Assumption 9. Furthermore, since
'k = O (1) (by condition k'kc = O (1) and Assumption 12), we have
E
n
E '0 f t
q
j Ft
2
n
o
= O (1) .
It follows that & n has following properties
& 0 = O (1) and & n ! 0 as n ! 1.
Array f
N t =cN t g
is thus uniformly bounded in L2 norm. This proves uniform integrability of array f
Furthermore, using Liapunov’s inequality, two-dimensional array f
1
1
N t ; FN t gt= 1
N =1
N t =cN t g.
is L1 -mixingale with respect
to constant array fcN t g. Noting that equations (91) and (92) hold, it follows that the array f N t ; Ft g satis…es
P N
L1
conditions of a mixingale weak law,3 2 which implies Tt=1
N t ! 0. Convergence in L1 norm implies convergence in
probability. This completes the proof of result (99).
Assumption 11 implies that sequence
0
(as well as '0 ) is deterministic and bounded. Vector of endogenous
variables xt can be written as
xt =
32
48
Davidson (1994, Theorem 19.11)
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Working Paper Series No 998
January 2009
+ ft +
t.
Process ft is independent of
t.
j
Suppose N; T ! 1 at any rate. Processes f
in mean by Lemma 1 since k k
0
t pg
and f
0
0
t p'
t qg
are ergodic
k kc = O (1). Furthermore,
T
1 X
T t=1
0
p
ft
0
E (ft ) ! 0,
q
0
E ft ft0
and
T
1 X
T t=1
since ft is covariance stationary m
0
f t '0 f t
0
q
p
' ! 0,
1 dimensional process with absolute summable autocovariances (ft is ergodic in
mean as well as in variance), and
0
0
0
0
'
'
2
=
O (1) ,
=
O (1) ,
by Assumption 12, conditionk kc = O (1) and condition k'kc = O (1). Sum of bounded deterministic process and
independent processes ergodic in mean is a process that is ergodic in mean as well. This completes the proof.
j
Lemma 3 Let xt be generated by model (32), Assumptions 7-12 hold and N; T ! 1 at any rate. Then for any
p; q 2 f0; 1g, for any sequence of non-random matrices of weights W of growing dimension N
mw satisfying
conditions (33)-(34), and for any r 2 f1; ::; mw g,
p
N
T
p
N
T
p
N
T
p
N
T
where the process
t
T
X
p
W0
t p
W0
t p xW;t q
W0
t p xi;t q
t=1
T
X
(100)
! 0,
p
t=1
T
X
t=1
T
X
t=1
(101)
! 0,
p
(102)
! 0,
p
(103)
git qit ! 0,
is de…ned in equation (40), vector git = 1;
0
0
0
0
i;t 1 ; xW t ; xW;t 1
and qit is de…ned equation (47).
Proof. Let wr for r 2 f1; ::; mw g denote the rth column vector of matrix W. Noting that
granularity condition (33), result
p
T
N X 0
wr
T t=1
p
t p
p
N wr
= O (1) by
(104)
!0
follows directly from Lemma 1, equation (87). This completes the proof of result (100).
Let ' be any sequence of non-random N
We have
p
T
N X 0
wr
T t=1
1 dimensional vectors of growing dimension such that k'kc = O (1).
0
t p ' xt q
=
p
T
N X 0
wr
T t=1
0
t p'
(
+ ft
q
+
t q) .
(105)
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Working Paper Series No 998
January 2009
49
Since
p
N wr = O (1) for any r 2 f1; ::; mw g by condition (33), we can use Lemma 1, result (88), which implies
p
T
N X 0
wr
T t=1
0
t p'
E wr0
t q
0
t p'
p
t q
(106)
! 0.
Sequence f'0 g is deterministic and bounded in N , and therefore it follows from Lemma 1, result (87), that
p
T
N X 0
wr
T t=1
p
0
t p'
(107)
! 0.
Similarly, Lemma 2 equation (99) implies
p
T
N X 0
wr
T t=1
0
t p'
p
ft
q
(108)
! 0.
Results (106), (107) and (108) establish
p
T
N X 0
wr
T t=1
p
0
t p ' xt q !
(109)
0.
Result (101) follows from equation (109) by setting '= wl for any l 2 f1; ::; mw g. Result (102) follows from equation
1 dimensional selection vector for the ith element.
(109) by setting ' = ei where ei is N
Finally, the result (103) directly follows from results (100)-(102). This completes the proof.
j
Lemma 4 Let xt be generated by model (32), Assumptions 7-12 hold, and N; T ! 1 at any rate. Then for any
sequence of non-random matrices of weights W of growing dimension N
T
1 X
0
git git
T t=1
0
) and vector git = 1;
where matrix Ci = E (git git
mw satisfying conditions (33)-(34),
p
(110)
Ci ! 0,
0
0
0
0
i;t 1 ; xW t ; xW;t 1 .
Proof. Result (110) directly follows from Lemmas 1, 2 and 3.
j
Lemma 5 Let xt be generated by model (32), Assumptions 7-12 hold, and N; T ! 1 at any rate. Then for any
sequence of non-random matrices of weights W of growing dimension N
for any …xed p
0,
T
1 X 0
W
T t=1
where the process
mw satisfying conditions (33)-(34), and
t
t p uit
p
(111)
! 0,
is de…ned in equation (40). If in addition T =N ! {, with 0
T
1 X 0
p
W
T t=1
t p uit
p
{ < 1,
(112)
! 0.
Proof. Let TN = T (N ) be any non-decreasing integer-valued function of N such that limN !1 TN = 1 and
limN !1 TN =N = { < 1, where {
0 is not necessarily nonzero. De…ne
N it
50
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Working Paper Series No 998
January 2009
1
= p
TN
W0
t p uit
E W0
t p uit
,
(113)
where the subscript N is used to emphasize the number of cross section units.3 3 Let fFt g denote an increasing
sequence of
-…elds (Ft
Ft ) with
1
i 2 N, the vector array f
N it
measurable with respect of Ft . First it is established that for any …xed
1
1
N it =cN t ; Ft gt= 1
N =i
1
N TN
is uniformly integrable, where cN t = p
. For p > 0, we can
write
0
N it N it
2
cN t
E
=
N
=
E
2
ii
N
"
1
X
W
0
`
` p uit
ut
`=0
1
X
W0
0`
`
!
1
X
W
0
`
ut
` p uit
`=0
!0 #
,
W ,
`=0
N
1
kWk2 k k
O (1) ,
=
where kWk2 = O N
2
ii
1
X
`
2
,
`=0
by condition (33), k k = O (1) by Assumption 9, and
8. For p = 0, we have
E
0
N it N it
2
cN t
=
is N
and f
2
kWk k
ii k
+
2
2
N ii
`
ut
kWk k k
O (1) ,
N it =cN t g
N it =cN t g.
E jE (
W
0
` uit
` 2
`=0
!
1
X
= O (1) by Assumption
,
`
2
+O N
1
`=1
!
,
N symmetric matrix with the element (n; s) equal to E (uit uit unt ust ). Therefore for p
the two-dimensional vector array f
f
1
X
`=1
=
ii
W ut uit +
N V ar
N
where as before
0
P1
N it
1
1
N it ; FN t gt= 1
N =i
j Ft
n )j
=
8
<
:
0,
is uniformly bounded in L2 norm. This proves uniform integrability of
for any n > 0 and any …xed p
0
mw cN t O (1)
for n = 0 and any …xed p
0
,
(114)
0
is L1 -mixingale with respect to constant array fcN t g.3 4 Note that
lim
N !1
TN
X
cN t = lim
N !1
t=1
TN
X
t=1
1
p
= lim
N !1
N TN
r
p
TN
= { < 1,
N
and
lim
N !1
TN
X
t=1
c2N t = lim
N !1
TN
X
t=1
1
1
= lim
= 0.
N !1 N
TN N
Therefore for each …xed i 2 N, each of the mw two-dimensional arrays given by the elements of vector array
33
Note thatW and t p change with N , but as before we ommit subscript N here to keep the notation simple.
The last equality in equation (114) takes advatage of Liapunov’s inequality. mw is mw 1 dimensional vector
of ones.
34
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Working Paper Series No 998
January 2009
51
f
1
1
N it ; Ft gt= 1
N =i
satis…es conditions of a mixingale weak law3 5 , which implies
TN
1 X 0
p
W
TN t=1
p
TN E W0
t p uit
t p uit
L
!1 0.
But
p
TN E W0
t p uit
=
c
p
TN E W0 ut uit
=
c
p
TN O
1
N
! 0,
since limN !1 TN =N = { < 1. Convergence in L1 norm implies convergence in probability. This completes the
proof of result (112).
Result (111) is established in a very similar fashion. De…ne new vector array qN it = p1
TN
N it
where
N it
is array
de…ned in (113) and i 2 N is …xed. Let TN = T (N ) be any non-decreasing integer-valued function of N such that such
n p
o1
1
TN qN it =cN t ; Ft t= 1
is uniformly
that limN !1 TN = 1. Notice that for any …xed i 2 N, vector array
N =i
integrable because
f
1
N it =cN t ; Ft gt=
1
1 N =i
is uniformly integrable. Furthermore,
p1 cN t
mixingale with respect to the constant array
since f
TN
1
1
N it ; Ft gt= 1
N =i
fqN it ; Ft g1
t=
1
1 N =i
is L1 -
is L1 mixingale with respect
to the constant array fcN t g. Note that
lim
N !1
TN
X
t=1
TN
X
1
1
1
p
p = lim p = 0,
cN t = lim
N !1
N
!1
TN
T
N
N
t=1 N
and
lim
N !1
TN
X
t=1
1
p
cN t
TN
2
= lim
N !1
Therefore for any …xed i 2 N, a mixingale weak law
TN
X
t=1
36
TN
X
2
1
p
TN N
t=1
= lim
N !1
1
= 0.
TN N
implies
L
qN it !1 0 as N ! 1.
(115)
Since also
E W0
t p uit
=O N
1
,
it follows
T
1 X 0
W
T t=1
t p uit
L
!1 0,
j
as N; T ! 1 at any rate. Convergence in L1 norm implies convergence in probability. This completes the proof of
result (111).
j
Lemma 6 Let xt be generated by model (32), Assumptions 7-12 hold and N; T ! 1 such that T =N ! {, with
{ < 1. Then for any sequence of non-random matrices of weights W of growing dimension N
0
conditions (33)-(34), we have,
35
36
52
See Davidson (1994, Theorem 19.11).
See Davidson (1994, Theorem 19.11).
ECB
Working Paper Series No 998
January 2009
mw satisfying
a) under Assumption 13,
1
Ci
T
1 X
D
p
eit uit ! N (0; Iki ) ,
g
T t=1
1
2
ii
0
eit = 1;
eit
where Ci = E (e
git g
) and g
0
0
W ; ft 1
0
0
i;t 1 ; ft
0
0
W
(116)
.
b) under Assumption 14,
1
p
ii T
where matrix
vi
1
2
vi
T
X
1 uit
t=1
P1
0
= E (vit vit
) and vector vit = S0i
Proof. Let a be any ki
vi;t
`
`=0
D
! N (0; Ihi ) ,
ut
(117)
`.
1 dimensional vector such that kak = 1 and de…ne
Nt
1
= p
TN
1
2
a0 Ci
ii
eit uit ,
g
where TN = T (N ) is any non-decreasing integer-valued function of N such that limN !1 TN = 1 and limN !1 TN =N =
{ < 1, where 0
a0 Ci
1
2
{ < 1. Array f
N t ; Ft g
is a stationary martingale di¤erence array.3 7 Lemmas 1 and 2 imply
eit is ergodic in variance, in particular
g
TN
1 X 0
a Ci
TN t=1
1
2
0
eit g
eit
g
Ci
1
2
p
a ! 1.
eit and uit are independent and the fourth moments of uit are …nite. Therefore a0 Ci
g
and
TN
X
t=1
Furthermore, E
ii
1 0
a Ci
1=2
eit uit
g
p
2
Nt !
1
2
eit uit is ergodic in variance
g
(118)
1.
4
= O (1) and therefore
lim
N !1
TN
X
4
Nt
E
= 0.
t=1
Using Liapunov’s theorem (Davidson, 1994, Theorem 23.11), Lindeberg condition3 8 holds, which in turn implies
max j
1 t TN
Ntj
p
! 0 as N ! 1.
(119)
Results (118), (119) and the martingale di¤erence array central limit theorem (Davidson, 1994, Theorem 24.3)
establish
TN
X
t=1
Since equation (120) holds for any ki
Nt
1
= p
TN
a0 Ci
ii
1
2
TN
X
t=1
D
eit uit ! N (0; Iki )
g
(120)
1 dimensional vector a such that kak = 1, result (116) directly follows from
equation (120) and Davidson (1994, Theorem 25.6).
Result (117) can be established in the same way as the result (116), but this time we set
37
Ft .
38
As before, let fFt g denote an increasing sequence of
-…elds (Ft
1
Ft ) with
Nt
Nt
1
a0
TN ii
= p
1
2
vi
vi;t
1 uit ,
measurable with respect of
See Davidson (1994, Condition 23.17).
ECB
Working Paper Series No 998
January 2009
53
where a is any hi
1 dimensional vector such that kak = 1.
j
Lemma 7 Let xt be generated by model (32), and suppose Assumptions 7-12 hold and N; T ! 1 at any rate. Then
for any arbitrary matrix of weights W satisfying conditions (33)-(34), and for any p; q 2 f0; 1g :
T
1 X
T t=1
T
1 X
T t=1
T
1 X
T t=1
T
1 X
T t=1
W;t p
0
W;t p ft q
W;t p
,
,
(122)
1
p
N
= op
0
W;t q
(121)
1
p
N
= op
0
W;t q
i;t p
1
p
N
= op
,
1
p
N
= op
(123)
,
(124)
0
iQ
= op (1) .
T
(125)
Furthermore,
H0 Q
Q0 Q
= A0
+ op
T
T
1
p
N
,
(126)
Z0i H
Z0 Q
= i A + op
T
T
1
p
N
,
(127)
H0 H
Q0 Q
= A0
A + op
T
T
1
p
N
,
(128)
H0 ui
Q0 ui
= A0
+ op
T
T
1
p
N
,
(129)
where
T
vit = S0i
P1
`=0
`
ut
`,
i
hi
= (vi0 ; vi1 ; :::; vi;T
0
1)
,
(130)
matrices H and Zi are de…ned below equation (53), and matrices Q, F and A are de…ned in
equations (54)-(55).
Proof. Result (121) follows directly from equation (87) of Lemma 1 since the spectral norm of any column vector
of the matrix W is O N
of
W;t
1
2
. Result (122) follows from result (121) by noting that ft is independently distributed
P
p
and all elements of the variance matrix of ft are …nite. Furthermore, since (by Lemma 1) T1 Tt=1 vit ! 0,
equation (125) follows. Results (123) and (124) follows directly from equation (88) of Lemma 1 by noting that
p
NE
i;t p
0
W;t q
=O
1
p
N
W;t p
0
W;t q
=O
1
p
N
(131)
as well as3 9
p
NE
.
In order to prove equations (126)-(129), …rst note that the row t of the matrix H
39
(132)
QA is 0;
0
W t;
0
W;t 1
.
Results (131) and (132) are straightforward to establish by taking the row norm and by noting that the granularity
conditions (33)-(34) imply kWkr = O N 1 .
54
ECB
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January 2009
Using results (121)-(124), we have
(H
QA)0 Q
T
Z0i (H
QA)
T
H0 (H QA)
T
(H
QA)0 (H
T
QA)
=
=
=
=
1
T
20
T 6B
X
6B
6B
4@
t=1
2
C
C
C
A
Wt
W;t 1
2
T 6
1 X6
6
T t=1 6
4
1
0
i;t 1
0
0
B
B
B
@
Wt
W;t 1
0
1
0
T 6
B
xW t
1 X6
6@
AB
B
6
T t=1 4 x
@
W;t 1
20
T 6B
1 X6
6B
B
T t=1 6
4@
ft0 ,
1,
0
Wt
W;t 1
CB
CB
CB
A@
Wt
W;t 1
1
10 3
C7
C7
= op
C7
A7
5
10
0
ft0
0
Wt
W;t 1
3
7
7
7 = op
5
1
p
N
10 3
C7
C7
= op
C7
A7
5
10 3
C7
C7
= op
C7
A7
5
1
p
N
,
,
(133)
(134)
1
p
N
,
(135)
1
p
N
,
(136)
Equations (133)-(134) establish results (126) and (127). Note that
H0 H
T
=
=
=
H0 (H QA)
H0 (QA)
+
,
T
T
0
(H QA)0 Q
H (H QA)
Q0 Q
+
A + A0
A,
T
T
T
0
1
QQ
,
A+op p
A0
T
N
where the last equality uses equations (133) and (135). This completes the proof of result (128).
Equation (115) (see proof or Lemma 5) implies
T
1 X
T t=1
W;t p uit
E(
W;t p uit )
p
NE (
j
as N; T ! 1 at any rate. Result (129) follows by noting that
p
! 0,
W;t p uit )
=O N
1
2
. This completes the
proof.
j
Lemma 8 Let xt be generated by model (32), suppose Assumptions 7-12, 14 hold, and N; T ! 1 at any rate. Then
for any arbitrary matrix of weights W satisfying conditions (33)-(34) and Assumption 14, we have
Q0 Q p
!
T
QQ
Q
,
(137)
is nonsingular, and
0
i
T
p
i
vi
! 0,
(138)
ECB
Working Paper Series No 998
January 2009
55
where
Q
f
(`) = E (ft ft0
` ),
vi
0
1
0
B
B
=B 0
@
0
1
0
f
(0)
f
f
(1)
f
C
C
(1) C ,
A
(0)
= E (vi vi0 ), matrix Q is de…ned in equation (54), and matrix
Proof. Assumption 11 implies matrix
Q
i
= (vi0 ; vi1 ; :::; vi;T
0
1) .
is nonsingular. Result (137) directly follows from the ergodicity properties
of the covariance stationary time-series process ft .
j
Consider now asymptotics N; T ! 1 at any rate. Lemma 1 implies that hi 1 dimensional vector vit = S0i
P
p
is ergodic in variance, in particular T1 Tt=1 S0i t 0t Si E (S0i t 0t Si ) ! 0.4 0 This completes the proof.
t
j
Lemma 9 Let xt be generated by model (32), suppose Assumptions 7-12 and 14 hold, and N; T ! 1 at any rate.
Then for any arbitrary matrix of weights W satisfying conditions (33)-(34) and Assumption 14, we have
Z0i MH Zi
T
Z0i MQ Zi
T
Z0i MH ui
p
T
vi
1
p
N
,
(139)
p
=
!
T
N
op
0
i MQ ui
p
T
=
is de…ned in Assumption 14, MH = IT
(140)
! 0,
r
vi
Z0i MH Q
p
T
where
Z0i MQ Zi
+ op
T
=
,
(141)
r
+ op
T
N
!
,
(142)
+
H (H0 H) H0 , matrices H and Zi are de…ned below equation
(53), matrices Q and F are de…ned in equation (54), and matrix
i
= (vi0 ; vi1 ; :::; vi;T
0
1) .
Proof.
Z0i MH Zi
Z0 Zi
= i
T
T
Z0i H
T
H0 H
T
+
H 0 Zi
.
T
(143)
Results (127)-(128) of Lemma 7 imply
Z0i H
T
H0 H
T
+
H 0 Zi
Q0 Q
Z0 Q
= i A A0
A
T
T
T
+
A0
Q0 Zi
+ op
T
p1
N
.
(144)
Using de…nition of the Moore-Penrose inverse, it follows
A0
Multiply equation (145) by
Q0 Q
T
Q0 Q
A
T
1
A0
(AA0 )
1
Q0 Q
A
T
A0
Q0 Q
A
T
A0
=
Q0 Q
A .
T
A from the left and by A0 (AA0 )
A A0
40
+
Q0 Q
A
T
+
A0 =
Q0 Q
T
1
Q0 Q
T
(145)
1
from the right to obtain4 1
1
.
(146)
kSi kc = O (1) by Assumption 7.
Note that plimT !1 T1 Q0 Q is nonsingular by Lemma 8, equation (137). AA0 is nonsingular, since matrix A has
full row-rank by Assumption 14.
41
56
ECB
Working Paper Series No 998
January 2009
Equations (146) and (144) imply
Z0i H
T
H0 H
T
+
H 0 Zi
Z0 Q
= i
T
T
1
Q0 Q
T
Q0 Zi
+ op
T
p1
N
.
(147)
Result (139) follows from equations (147) and (143).
System (32) implies
Zi =
0
i Si
0
i Si
+ F ( 1)
+
i.
(148)
Since Q = [ ; F; F ( 1)], it follows
Z0i MQ Zi
=
T
0
i MQ
i
T
0
i
=
i
T
0
iQ
+
Q0 Q
T
T
1
Q0
T
i
.
(149)
Using equations (125), (137) and (138), result (140) follows directly from (149).
Results (126)-(128) of Lemma 7 imply
Z0i H
T
+
H0 H
T
+
H0 Q
Q0 Q
Z0 Q
= i A A0
A
T
T
T
A0
Q0 Q
+ op
T
p1
N
.
(150)
Substituting equation (146), it follows
Z0i H
T
+
H0 H
T
H0 Q
Z0 Q
= i
T
T
1
Q0 Q
T
Equation (151) implies
r
Z0i MH Q
Z0 MQ Q
p
= ip
+ op
T
T
T
N
!
Q0 Q
+ op
T
= op
r
T
N
p1
N
!
.
(151)
.
This completes the proof of result (141).
Results (127)-(129) of Lemma 7 imply
Z0i H
T
H0 H
T
+
+
Q0 Q
H0 ui
Z0 Q
= i A A0
A
T
T
T
A0
Q0 ui
+ op
T
p1
N
.
Substituting equation (146), it follows
Z0i H
T
Noting that MQ (
0
i Si
+F
0
i Si )
H0 H
T
+
H0 Q
Z0 Q
= i
T
T
Q0 Q
T
1
Q0 ui
+ op
T
p1
N
.
(152)
= 0 since Q = [ ; F; F ( 1)], equations (152) and (148) imply
Z0i MH ui
p
T
0
=
Zi MQ ui
p
+ op
T
0
=
i MQ ui
p
T
+ op
!
T
,
N
r !
T
.
N
r
This completes the proof.
j
Lemma 10 Let xt be generated by model (32), and suppose Assumptions 7-12 and 14 hold, and N; T ! 1 at any
ECB
Working Paper Series No 998
January 2009
57
rate. Then for any arbitrary matrix of weights W satisfying conditions (33)-(34) and Assumption 14, we have
Z0i MH i ( 1)
T
Z0i MH ui
p
T
=
1
p
N
op
,
0
ui
pi
+ op
T
=
where matrices MH ,and Zi are de…ned below equation (53),
i;0 ; :::;
i;T
1
0
(153)
r
T
N
!
+ op (1) ,
(154)
= (vi0 ; vi1 ; :::; vi;T
i
1)
and vector
i
( 1) =
.
Proof.
Z0i
H0
k
ib kr
=O N
1
( 1)
T
i
=
i ( 1)
T
=
"
T
1 X
xi;t 1
T t=1
20
T
1 X 4@ xW t
T t=1
x
W;t 1
p
T
=
=
Q0 u i
p
T
1
X
`
ut
` 1
`=0
1
A
0
ib
1
X
!0 #
`
ut
,
` 1
`=0
!0 3
5.
by Assumption 7, therefore result (153) directly follows from equations (134) and (135).
0
i MQ ui
where
0
ib
0
0
ui
Q Q0 Q
pi
+ i
T
T
T
0
u
i
pi
+ op (1) ,
T
= Op (1), plimT !1 T1 Q0 Q is nonsingular by Lemma 8, and
1
Q0 ui
p ,
T
(155)
0
iQ
T
= op (1) by Lemma 7, equation (125).
Substituting (155) into equation (142) implies result (154). This completes the proof.
Proof of Theorem 1.
a) Substituting for xit in equation (49) yields
bi
j
i
=
T
1 X
0
git git
T t=1
!
1
T
T
1 X
1 X
git qit +
git uit
T t=1
T t=1
!
.
(156)
With N; T ! 1 in any order, Lemma 5 yields4 2
T
1 X
p
git uit ! 0.
T t=1
(157)
Also using Lemmas 3 and 4 we have
T
1 X
p
git qit ! 0,
T t=1
and
42 1
T
PT
T
1 X
0
git git
T t=1
p
58
ECB
Working Paper Series No 998
January 2009
(159)
CiN ! 0,
p
xj;t 1 uit ! 0 since xjt is ergodic in mean by Lemma 2 and uit is independent of xj;t
P
p
f1; ::; N g and any N 2 N. Furthermore, using similar arguments, T1 Tt=1 ft uit ! 0.
t=1
(158)
1
for any j 2
respectively. Assumption 13 postulates that the matrix CiN is invertible for any N
N0 and
CiN1
is
bounded in N . It follows from equation (159) that
T
1 X
0
git git
T t=1
!
1
p
CiN1 ! 0.
(160)
p
Result b i
i
! 0 directly follows from equations (157), (158) and (160).
b) Multiplying equation (156) by
p
T (b i
p
T yields
i)
T
1 X
0
git git
T t=1
=
!
1
T
T
1 X
1 X
p
git qit + p
git uit
T t=1
T t=1
!
.
(161)
j
With N; T ! 1 such that T =N ! { < 1, Lemma 3 can be used to show that
T
1 X
p
p
git qit ! 0.
T t=1
(162)
Since CiN1 = O (1), equations (160) and (162) now yield
T
1 X
0
git git
T t=1
!
1
T
1 X
p
p
git qit ! 0.
T t=1
(163)
Lemma 5 establishes
T
1 X
p
T t=1
W;t p uit
p
! 0 for p 2 f0; 1g .
(164)
It follows from equation (164) that
eit = 1;
where g
0
0
i;t 1 ; ft
0
0
W ; ft 1
0
0
W
T
1 X
p
(git
T t=1
p
eit ) uit ! 0,
g
(165)
. Lemma 6 establishes that
T
1
1 X
D
eit uit ! N (0; Iki ) ,
g
CiN2 p
ii;N
T t=1
1
(166)
Equations (160), (163), (165) and (166) imply result (50).
c) Lemma 4 establishes
equal to u
bit = uit
where
1
T
1
T
PT
t=1
0
git
(b i
i ),
T
T
1 X 2
1 X 2
u
bit =
uit
T t=1
T t=1
PT
t=1
u2it
2
ii;N
p
0
git git
CiN ! 0. The estimated residuals from auxiliary regression (48) are
which implies
is established in Lemma 4, and
1
T
T
1 X
git uit + (b i
T t=1
p
p
! 0, b i
0
i)
2 (b i
i
PT
t=1
0
i)
T
1 X
0
git git
T t=1
! 0 is established in part (a) of this proof,
p
1
T
!
(b i
PT
t=1
i) ,
0
git git
(167)
p
CiN ! 0
git uit ! 0 is established in equation (157). This completes the proof.
ECB
Working Paper Series No 998
January 2009
59
Proof of Theorem 2. Vector xi can be written, using system (32), as
xi =
where
i
( 1) =
i0 ; :::;
i;T
0 0
i Si
i
1
0
+ Zi
i
+F
Si
i
+
i
( 1) + ui ,
(168)
. Substituting equation (168) into the partition least squares formula (53) and noting
that by Lemma 9,
Z0i MH Q
p
= op
T
it follows
p
T bi
0
F ( 1)
i
i
=
1
Z0i MH Zi
T
"
r
T
N
!
,
Z0i MH (ui +
p
T
(169)
i
( 1))
+ op
r
T
N
!#
.
(170)
Lemma 9 also establishes that
Z0i MH Zi
T
where
vi
p
vi
j
! 0, as N; T ! 1 at any rate,
(171)
0
= E (vit vit
) is nonsingular by Assumption 14.
j
Consider now asymptotics N; T ! 1 such that T =N ! { < 1. Lemma 10 establishes
Z0i MH i ( 1) p
p
! 0,
T
and
0
Z0i MH ui
ui
p
= pi
+ op
T
T
where
i
= (vi0 ; :::; vi;T
0
1) .
1
2
vi
T
X
t=1
The desired result (56) now follows from (170)-(174).
60
T
N
!
+ op (1) ,
(173)
Also from Lemma 6
1
p
ii T
ECB
Working Paper Series No 998
January 2009
r
(172)
vi;t
1 uit
D
! N (0; Ihi ) .
(174)
European Central Bank Working Paper Series
For a complete list of Working Papers published by the ECB, please visit the ECB’s website
(http://www.ecb.europa.eu).
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62
ECB
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January 2009
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997 “Financing obstacles and growth: an analysis for euro area non-financial corporations” by C. Coluzzi, A. Ferrando
and C. Martinez-Carrascal, January 2009.
998 “Infinite-dimensional VARs and factor models” by A. Chudik and M. H. Pesaran, January 2009.
ECB
Working Paper Series No 998
January 2009
63
Wo r k i n g Pa p e r S e r i e s
N o 1 0 0 1 / J a n ua ry 2 0 0 9
Identifying
the elasticity
of substitution
with biased
technical change
by Miguel A. León-Ledesma1,
Peter McAdam
and Alpo Willman