Exponent of Cross-sectional Dependence:
Estimation and Inference
Natalia Bailey? , George Kapetanios† and M.Hashem Pesaran‡
and University of Southern California
March 2012
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Outline
Introduction - Motivation
Weak and Strong Cross Section Dependence
Exponent of Cross-sectional Dependence (α)
Estimation and Inference
Small Sample Results using Monte Carlo Experiments
Empirical Application
Summary and Conclusions
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Introduction
Over the past decade there has been a resurgence of interest
in the analysis of cross-sectional dependence applied to
households, …rms, markets, regional and national economies.
Researchers in many …elds have turned to network theory,
spatial and factor models to obtain a better understanding of
the extent and nature of such cross dependencies.
There are many issues to be considered: how to test for the
presence of cross-sectional dependence, how to measure the
degree of cross-sectional dependence, how to model
cross-sectional dependence, and how to carry out
counterfactual exercises under alternative network formations
or market inter-connections.
In this paper we focus on measures of cross-sectional
dependence and how such measures are related to the
behaviour of cross-sectional averages or aggregates.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
A popular and convenient framework for investigating
cross-sectional dependence is the factor model.
The methodology has been applied widely in areas such as:
Finance
APT - Ross (1976), Hubermann (1982), Chamberlain (1983)
and Ingersoll (1984).
Portfolio choice problem, Markowitz (1959), Sharpe (1963),
Brandt (2004) and Fan, Fan and Lv (2008).
Contagion - Forbes and Rigobon (2002), Corsetti et al.
(2005), Acemoglu et al. (2010).
Macroeconomics
Forecasting - Stock and Watson (2002).
Monetary policy evaluation - Bernanke and Boivin (2003),
Favero et al.(2005).
Construction of composite indicators - Altissimo et al.
(2007),Giannone and Matheson (2006).
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
The Multi-Factor Model
Consider the multi-factor model
xit = ai + β0i ft + uit , for i = 1, 2, ..., N; t = 1, 2, ..., T ,
where ft is an m 1 vector of factors which are unobserved. We
focus on the behaviour of the loadings, since:
1
0
τN = Op @
In general we could consider,
N
∑ βi
j =1
2
A
N
∑ E ( β i ` ) = O (N α ` ),
i =1
where α` is referred to as the "cross-sectional exponent" of the `τh
factor. We assume that 1/2 < α` 1 and predominantly consider
the case where α1 = α > α2 > ... > αm .
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
To identify and estimate α = max` (α` ) it is also interesting to
note the following decomposition of the variance of the
cross-section averages of xit over i = 1, ..., N,
1
N2
Var (x̄t ) =
N
1
N
N
∑ σx2 + N 2 ∑ ∑
i
i =1
σij =
i =1 j =1,i 6=j
2
σ̄xN
+ τN ,
N
(1)
∑N
i =1 xit . (Weighted averages can also be used).
It is then easily seen that
where x̄t =
1
N
τN = O N 2α
2
, for 1/2 < α
1.
Note that even if xit are cross-sectionally independent
Var (x̄t ) = O (N 1 ) which shows that α can only be identi…ed
if α > 1/2.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Strong and Weak Cross-sectional Dependence
The large factor literature assumes that all factors are strong:
α` = 1, for all ` = 1, 2, ..., m.
See, for example, the contributions of Bai and Ng, and Lippi
and his collaborators.
At the other extreme the literature on spatial econometrics
assumes that all factors are weak:
α`
1/2, for all ` = 1, 2, ..., m.
See, for example, Pesaran and Tosetti (2011), Azomahou
(2008) and Fingleton (1999).
Onatski (2009) and Kapetanios and Marcellino (2010)
consider the intermediate case of semi-strong/weak factors
(Chudik, Pesaran and Tosetti, 2011, Econometrics J).
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Implications of Weak Cross-sectional Dependence for
Inference
Incorrectly assuming that factors are strong can lead to misleading
inference.
Even if factor loadings are known the variance of the least
squares estimator of ft ` , say fˆt ` depends on α` and standard
asymptotics will apply only if α` = 1. Also fˆt ` need not be
consistent if α` < 1.
The Bai and Ng (2002) procedure for selecting the number of
factors will no longer be valid if factors are weak.
When factors are weak the average pair-wise correlation
coe¢ cient across the cross section units tends to zero as
N ! ∞. This will not be the case if the factors are strong.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Portfolio Diversi…cation and Network Formation
The extent to which portfolio diversi…cation can be achieved
critically depends on α = max` (α` ).
Portfolio risk (measured by the variance of the portfolio
return) is diversi…able if α < 1 and is non-diversi…able if
α = 1. Di¤erent values of α < 1 represent di¤erent degrees of
portfolio diversi…cation.
Similar considerations also arise in models of networks and
contagions. The star network is an example of a strong factor
case - Chudik and Pesaran (2011 JoE), Pesaran and Chudik
(2012 forthcoming Econometrics Review ).
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Methodological Aims of this Paper
For a given set of observations, xit for i = 1, 2, ..., N, and
t = 1, 2, ..., T , this paper investigates the possibility of
estimating α = max` (α` ), when the factors are unobserved
but N and T are su¢ ciently large.
We shall also consider the conditions under which inference
can be carried out on α.
We derive the asymptotic distribution of a bias-corrected
variance estimator of α.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
A Single Factor Model
Our starting point is the single factor model (m = 1) which we
write as
xit = ai + β i ft + uit ,
for i = 1, 2, ..., N and t = 1, 2, ..., T
Assumption 1: (Factor Loadings) The loadings are given by
β i = vi for i = 1, 2, ..., [N α ] ,
β i = cρ
i [N α ]+1
α
(2)
α
, for i = [N ] + 1, [N ] + 2, ..., N,
[N α ]
where 1/2 < α 1 and fvi gi =1 is an i.i.d. sequence of random
variables with mean µv 6= 0, and variance σv2 < ∞.
Note that the speci…cation of β i for i > [N α ] need not be strictly
as in (2). Any sequence of loadings, for which
∑N
i =[N α ]+1 β i = Op (1) is acceptable.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
A Single Factor Model
Assumption 2: (Idiosyncratic components) For each i, uit follows
a linear stationary process given by
!
∞
∞
uit =
∑
∑ ψil
s= ∞
l =0
where νit s iid (0, σν2i ), i = ...,
σi2
∞
=
∑
s= ∞
ξ is νs ,t
,
1, 0, 1, ..., t = 0, 1, ...
ξ is2
!
∞
∑
∞
l =0
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
ψis2
s =0
sup ∑ l ζ jψil j < ∞ and sup
i
l
i
∞
∑
s= ∞
!
σν2i ,
js jζ jξ is j < ∞.
Exponent of Cross-sectional Dependence: Estimation and Inferenc
A Single Factor Model
Assumption 3: (common components) ft is distributed
independently of uit 0 8i, t,t 0 (ft for m > 1):
∞
ft =
∑ ψf ,j νft j ,
j =0
∞
∑ jζ
ψf ,j < ∞.
j =0
νft s iid (0, Σ2νf ).
It is required that all innovations in Assumptions 2 and 3 have
uniformly …nite κ-th order moments (κ > 4).
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Estimation - the Basic Idea
Consider the cross sectional averages of the observables given by
0 x /N, where τ is an N
x̄t = τ N
1 vector of ones. Therefore
t
N
Var (x̄t ) = N
=N
2 0
τ N Cov (xt )τ N
2 0
τN
σf2 Σ β + Σu τ N +σf2
0 E ( β)
τN
N
2
.
Given Assumption 1, we have that ∑N
i =1 β i =
!!
α
[N α ]
ρ
1 ρ(N [N ])
1
α
vi + α
= v̄N + O N
[N ]
1 ρ
[N α ] i∑
[N ]
=1
where v̄N =
i > [N α ].
1
[N α ]
(3)
α
[N α ]
[N α ]
∑i =1 vi is Op (1) and implicitly, we set vi = 0, for
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Estimation - the Basic Idea
Hence,
N
1 0
τ N E ( β)
= µv N α
1
+ O (N
1
).
Also,
N
2 0
0
τ N Σ β τ N =N 2 τ 1N
Σ β(1) τ 1N
Nα
2
λmax Σ β ,
where τ 1N is an [N α ] 1 vector of ones and Σ β(1) is the upper
[N α ] [N α ] sub-matrix of Σ β . Using the above results in (3) we
now have
Var (x̄t )
Nα
2
σf2 λmax Σ β + N
where
cN =
1
cN + σf2 µ2v N 2α
0 Σ τ
τN
u N
< K < ∞.
N
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
2
,
(4)
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Estimation - the Basic Idea
Or, if 1
α > 1/2, we have
Var (x̄t ) = κ 2 N 2α
2
+N
1
cN + O ( N α
2
),
(5)
by assuming λmax Σ β < K < ∞, which is straightforwardly
satis…ed, and κ 2 = σf2 µ2v .
Depending on how σf2 is normalized, di¤erent estimators of α
can be envisaged. Since, by assumption, µv 6= 0, a natural
normalization would be σf2 = 1/µ2v or κ 2 = 1.
Then, a simple manipulation of (5) gives
2( α
1) ln(N ) t ln(σx̄2 ) + ln 1
or
α t 1+
1 ln(σx̄2 )
2 ln(N )
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
N
1c
σx̄2
N
t ln(σx̄2 )
cN
.
2 [N ln(N )] σx̄2
N
1c
σx̄2
N
,
(6)
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Estimation - the Basic Idea
When α 1/2, the second term of the RHS of (5), that
arises from the contribution of the idiosyncratic components
will be at least as important as the contribution of the weak
factors, and, in consequence, α cannot be identi…ed.
Assuming α > 1/2, it can be estimated from (6), using a
consistent estimator of Var (x̄t ) = σx̄2 , given by
σ̂x̄2 =
1
T
T
∑ (x̄t
x̄ )2 ,
(7)
t =1
∑Tt=1 x̄t .
A simple and consistent estimator of α is given by
where x̄ = T
1
α̂ = 1 +
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
1 ln(σ̂x̄2 )
.
2 ln(N )
(8)
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Estimation - the Basic Idea
In the case of exact factor models where Σu is a diagonal
matrix, the third term in (6) can be estimated by
ĉN = N
1
N
∑ σ̂j2 = σ̄cN2 ,
j =1
where σj2 is the j th diagonal term of Σu and σ̂j2 is its estimator.
Hence, an improved estimator of α which can be used for
inference is given by
α̃ = 1 +
1 ln(σ̂x̄2 )
2 ln(N )
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
2
σ̄c
N
.
2 [N ln(N )] σ̂x̄2
(9)
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Estimator based on Cross Sectional Averages
Asymptotic Distribution of b
α
N
σ2
De…ne σ̄N2 = ∑i =N1 i and sf2 = T1 ∑Tt=1 ft T1 ∑Tt=1 ft
Let Assumptions 1-3 hold with m = 1. Then,
q
min(N ακ , T ) ln(N ) (b
α
ακ
ω=
lim
N ,T !∞
σ̄N2
N 2α 1 v̄N2 sf2
ακ )
ακN = α +
2
. Then,
!d N (0, ω ) ,
ln µ2v σf2
,
ln (N )
min(N α , T )
min(N α , T ) 4σv2
Vf 2 +
,
T
Nα
µ2v
∞
Vf 2 = Var f˜t2 + 2 ∑ Cov f˜t2 , f˜t2
i
,
i =1
where f˜t = (ft
µf )/σf , µf = E (ft ), and σf2 = E (ft
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
µf )2 .
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Normalisation
b
α is subject to two sources of small sample bias.
The …rst source relates to the term ln µ2v σf2 / ln (N ) in ακ .
This term cannot be removed through bias correction but only
through normalisation of κ 2 to unity.
The second source of bias is the term
unobserved.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
N 2α
σ̄N2
1 v̄ 2 s 2
N f
which is
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Further Asymptotic Results
N 2α
σ̄N2
1 v̄ 2 s 2
N f
can be consistently estimated by
1
2
σ̄c
N = N
N
∑ σ̂i2 ,
i =1
σ̂i2 =
1
T
2
σ̄c
N
N σ̂x̄2
where
T
∑ ûit2 ,
t =1
ûit = xit δ̂i x̃t , x̃t = x̄tσ̂x̄ x̄ , and δ̂i denotes the OLS estimator of
the regression of xit on x̃t . This suggests the following bias
corrected estimator
e
α=b
α
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
2
σ̄c
N
.
2 ln(N )N σ̂x̄2
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Further Asymptotic Results
Asymptotic Distribution of e
α
Theorem
Let Assumptions 1-3 hold with m = 1. Then, as long as either
T 1/2
! 0 or α > 4/7,
N 4α 2
q
α
min(N ακ , T ) ln(N ) (e
ακ ) !d N (0, ω )
(10)
Again, this estimator is only …rst order accurate since it only holds
under the speci…ed assumptions concerning α and the relative rate
of growth of N and T .
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Further Asymptotic Results
A Second-order Bias-adjusted Estimator
A second order bias correction amounts to estimating
2
σ̄c
N
σ̂x̄2 N ln (N )
1+
2
σ̄c
N
N σ̂x̄2
N 2α
σ̄N2
1 v̄ 2 s 2
N f
by
.
The new bias corrected estimator is now given by
!
c2
2
σ̄c
σ̄
N
α̌ = b
α
1 + N2 .
2σ̂x̄2 N ln(N )
N σ̂x̄
Theorem
Let assumptions 1-2 hold with m = 1. Then,
q
min(N ακ , T ) ln(N ) (α̌ ακ ) !d N (0, ω ) .
We consider e
α even though α̌ provides a more comprehensive bias
correction because small sample evidence suggests that e
α may
outperform α̌ for values of α > 4/7.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Further Asymptotic Results
Estimating ω
min(N α̇ , T )
4 min(N α̇ , T ) \
σv2
V̂f 2 +
,
T
N α̇
µ2v
where α̇ = α̂, α̃, α̌, a consistent estimator of Vf 2 is given by
ω̇ =
K
V̂f 2 (m ) =
and zt = (x̃t
∑
s= K
1
s
m+1
x̃ )2 , z̃t = zt
N 2 α̇
γ̂z (s ), γ̂z (s ) =
z̄, and z̄ = T
2
N α̇
∑
(s )
v̂i
1
N α̇
1
N α̇
∑Tt=s +1 z̃t z̃t
T
s
,
∑Tt=1 zt . Also
!2
(s )
∑ v̂i
\
i =1
i =1
σv2
=
,
2
α̇
µv
N
1
where v̂i denotes the OLS estimator of the regression of xit on x̄t
(s )
and v̂i denote the sequence of this estimator, sorted according to
the absolute values of the estimates, in descending order.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Discussion of the Condition κ 2 = 1
It is well known that in a single factor model, the factor
loadings are known only up to a scalar constant.
In the literature, identi…cation of factor loadings is achieved
by setting σf2 = 1.
However, for development of the feasible asymptotic for
inference on α we need to set σf2 equal to 1/µ2v which is only
possible if µv 6= 0. In practice this is unlikely to be restrictive
since otherwise x̄t !p 0 for all t as N ! ∞, irrespective of
whether the factor is weak or strong.
But if the condition µv 6= 0 is accepted, the condition
σf2 = 1/µ2v can be viewed as an scaling condition which has
limited impact on the cross section variance-covariance matrix
of the observations.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Extensions
Multi-factor model speci…cation: Asymptotic results are
derived for three cases α = α1 = ... = αm , and α > 1/2
For α > α2 + 1/4, or α2 < 3α/4, T b = N, b > 4 (α 1 α )
2
and α2 α3 ... αm
For α > α2 α3 ... αm coupled with two additional
conditions
Other loadings setups: We assume that loadings are given
by
β ik = N α 1 vik , 0 < α 1
(11)
where fvik gN
i =1 is an i.i.d. sequence of random variables with
mean µvk 6= 0, and variance σv2k < ∞. It is easy to see that
again
τN = O (N 2α ).
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Joint Estimation
α may be estimable even if κ 2 is set to an arbitrary constant.
We can employ the following quadratic form in the sampling
errors to estimate α and κ 2
∑
Q (α, κ 2 ) =
σ̂x̄2n
n
1
κ2
ĉn
2
+
n [N α ]
∑
σ̂x̄2n
n
1
ĉn
n
2
N 2α κ 2
2
,
n >[N α ]
where ĉn =
1
nT
∑nj=1 ∑Tt=1 û(2j )t ,
(s )
û(j )t = x(j )t x̄(j ) v̂j (x̄Nt x̄NT ), x̄nt = n 1 ∑ni=1 x(i )t ,
x̄nT = T 1 ∑Tt=1 x̄nt and x(j )t is the unit associated with the
(s )
estimated loading v̂j .
We prove consistency of this joint estimation and note that it
is possible only under Assumption 1.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Monte Carlo Simulation Study: Basic Design
We consider the following two-factor model
xit = β 1i f1t + β 2i f2t + uit ,
for i = 1, ..., N and t = 1, ..., T .
The factors are generated as
q
fjt = ρj fj ,t 1 + 1
(12)
ρ2j ζ jt , j = 1, 2,
with fj , 50 = 0 for t = 49, 48, ..., 0, 1, ..., T .
Two sets of experiments are considered: a single factor setup
(experiments A-C) and a two-factor setup (experiment D).
In experiments A-C the shocks are generated as
ζ jt s IIDN (0, 1), uit s IIDN (0, σi2 ),
σi2 s IID Chi-squared (2), i = 1, 2, ..., N.
In experiment D we set uit s IIDN (0, 1).
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Factor Loadings
In experiments A-C we set β 2i = 0. For β 1i we assume the
following generating mechanism:
β 1i = v1i , for i = 1, 2, ..., [N a ]
i [N a ]
β 1i = ρl
, for i = [N a ] + 1, [N a ] + 1, ..., N
and
v1i s IIDU (µv1
0.5, µv1 + 0.5) and ρl = 0.5.
We set σf21 = 1/µ2v1 , as suggested by the theory.
Next, we consider experiment D where β 2i 6= 0 which is
generated as
β 2i = v2i , for i = 1, 2, ..., [N a2 ]
i [N a 2 ]
β 2i = ρl
, for i = [N a2 ] + 1, [N a2 ] + 1, ..., N
where v2i s IIDU (µv2 0.5, µv2 + 0.5) and ρl = 0.5.
p
For α2 = α we set σf1 = σf2 = 1 and µv1 = µv2 = 0.5.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
The Experiments
The following experiments were conducted:
Experiments A-C: One factor model (β 2i = 0):
A: Basic design: ρj = 0.0, uit s IIDN (0, σi2 ).
B: Temporal dependence on fjt : ρj = 0.5, uit s IIDN (0, σi2 ).
C: Spatial dependence on uit : ρj = 0.0, uit = Rε it , where
ε it =
σε2
q
σε2 ηit ;
ηit s IIDN (0, 1),
0
= N/tr (RR )
for t = 49, 48, ..., 0, 1, ..., T , and R = (IN
set d = 0.2 and
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
dS)
1.
We
Exponent of Cross-sectional Dependence: Estimation and Inferenc
The Experiments (continued)
0
B
B
B
S=B
B
@
0 1 0
1/2 0 1/2
..
..
..
.
.
.
0
0
1
0
1
C
C
.. C
, S is N
. C
C
A
1/2
0
N.
Experiment D: Two-factor model (β 2i 6= 0):
D: Basic design: ρj = 0.0, uit s IIDN (0, 1) and α2 = α.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
The Estimators and Test Statistics Reported
We investigate the small sample properties of our initial
estimator of α, α̂, and its biased-adjusted counterparts α̃ and
α̌.
For α̂, α̃ and α̌ we show bias and RMSE results (all scaled by
100).
Additionally, for α̃ and α̌ we report size and power results.
For power we consider two alternatives, namely
α1 = α0 + 0.05 (power+) and α1 = α0 0.05 (power-),
where α0 is the selected null value of α.
We consider values of α = 0.5, 0.55, ..., 0.95, 1.00, and the
sample sizes N, T = 100, 200, 500, 1000. The number of
replications was set to 2, 000.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Newey West Corrected Estimator of V̂f 2
In the case of serially correlated ft , the Newey West corrected
variance estimator of ft is computed as
V̂f 2 (q ) = σ̂z2 /(1
ρ̂1
ρ̂2
...
ρ̂q )2 ,
after having run an AR(q) regression on
z̃t = zt
x̃t =
1
N
∑N
i =1 x it
,
σ̂x̄
x̃ = T
z̄, where zt = (x̃t
1
∑Tt=1 x̃t and z̄ = T
x̃ )2 ,
1
∑Tt=1 zt .
σ̂z is the regression s.e. and ρ̂i is the i th estimated AR
coe¢ cient, and q is set to T 1/3 .
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Table: Exp A: Bias and RMSE ( 100) Results: β 2i = 0, f1t
s IIDN (0, 1), uit s IIDN (0, σi2 ), v1i s IIDU (0.5, 1.5)
α
0.70
0.75
0.80
α̂
α̃
α̌
α̂
α̃
α̌
3.51
0.99
0.38
2.31
0.51
0.16
2.39
0.63
0.33
1.46
0.31
0.17
1.64
0.45
0.32
0.90
0.19
0.13
α̂
α̃
α̌
α̂
α̃
α̌
3.97
2.48
2.51
2.77
1.90
1.95
3.01
2.21
2.24
2.11
1.73
1.75
2.43
2.04
2.05
1.75
1.62
1.63
N/T
100
200
100
200
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Bias
0.85
100
0.90
0.95
1.00
1.13
0.35
0.29
0.54
0.10
0.08
RMSE
0.76
0.25
0.22
0.32
0.06
0.05
0.49
0.16
0.15
0.18
0.02
0.02
0.14
-0.07
-0.08
0.00
-0.09
-0.09
2.10
1.92
1.93
1.56
1.54
1.54
1.89
1.83
1.84
1.48
1.48
1.49
1.78
1.77
1.77
1.44
1.45
1.45
1.70
1.72
1.73
1.42
1.43
1.43
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Table: Exp A: Size and Power ( 100) Results: β 2i = 0, f1t
s IIDN (0, 1), uit s IIDN (0, σi2 ), v1i s IIDU (0.5, 1.5)
0.70
α
0.75
0.80
N/T
100
200
0.85
0.90
0.95
1.00
100
α̃
Size
Power+
Power-
11.10
52.75
84.70
8.25
61.30
83.60
6.40
66.50
84.20
6.10
70.45
85.40
5.65
75.00
85.65
5.75
78.35
86.35
7.05
84.50
84.50
α̌
Size
Power+
Power-
11.90
61.65
75.60
9.50
65.55
78.35
7.00
68.20
81.25
6.20
71.05
84.25
5.75
75.40
85.35
5.80
78.50
86.15
7.10
84.50
84.50
α̃
Size
Power+
Power-
9.15
76.10
91.50
7.55
82.30
92.20
6.10
86.40
93.15
5.70
90.00
93.15
5.00
92.10
93.85
5.30
93.65
94.10
6.45
93.25
93.25
α̌
Size
Power+
Power-
10.50
80.20
87.45
7.70
83.60
90.45
6.50
87.10
92.45
5.90
90.15
93.05
5.05
92.10
93.65
5.30
93.65
94.10
6.45
93.25
93.25
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Table: Exp B: Bias
q and RMSE ( 100) Results: β 2i = 0,
f1t = ρ1 f1,t 1 + 1 ρ21 ζ 1t , ρ1 = 0.5, ζ 1t s IIDN (0, 1), uit
s IIDN (0, σi2 ), v1i s IIDU (0.5, 1.5)
α
0.70
0.75
0.80
α̂
α̃
α̌
α̂
α̃
α̌
3.25
0.66
0.01
2.10
0.25
-0.12
2.10
0.28
-0.04
1.24
0.06
-0.10
1.32
0.09
-0.06
0.69
-0.05
-0.11
α̂
α̃
α̌
α̂
α̃
α̌
3.83
2.56
2.71
2.73
2.11
2.24
2.93
2.39
2.49
2.16
1.99
2.05
2.44
2.28
2.34
1.90
1.91
1.93
N/T
100
200
100
200
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Bias
0.85
100
0.77
-0.04
-0.11
0.34
-0.11
-0.14
RMSE
2.19
2.21
2.23
1.80
1.86
1.87
0.90
0.95
1.00
0.38
-0.15
-0.17
0.11
-0.16
-0.17
0.12
-0.23
-0.24
-0.04
-0.20
-0.20
-0.25
-0.48
-0.48
-0.21
-0.31
-0.31
2.09
2.16
2.17
1.76
1.82
1.82
2.05
2.12
2.13
1.75
1.79
1.79
2.05
2.14
2.14
1.74
1.78
1.78
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Table: Exp B: Size
qand Power ( 100) Results: β 2i = 0,
f1t = ρ1 f1,t 1 + 1 ρ21 ζ 1t , ρ1 = 0.5, ζ 1t s IIDN (0, 1), uit
s IIDN (0, σi2 ), v1i s IIDU (0.5, 1.5)
α
0.70
0.75
0.80
0.85
100
0.90
0.95
1.00
N/T
100
200
α̃
Size
Power+
Power-
11.90
55.15
76.20
9.75
60.75
71.80
9.25
64.95
70.70
8.35
67.20
71.05
8.90
69.70
70.20
10.25
72.25
70.95
12.15
67.15
67.15
α̌
Size
Power+
Power-
13.85
61.95
66.10
11.60
64.65
66.70
9.90
66.65
67.95
9.00
67.85
69.45
9.20
69.85
69.75
10.45
72.35
70.70
12.15
67.15
67.15
α̃
Size
Power+
Power-
11.70
72.75
84.15
10.65
76.50
83.40
10.35
79.30
83.55
10.90
81.00
83.25
11.10
82.80
83.45
11.10
83.90
83.40
11.80
81.75
81.75
α̌
Size
Power+
Power-
14.15
76.90
78.90
11.55
78.20
80.85
11.00
79.85
82.50
11.25
81.10
82.70
11.20
82.85
83.15
11.10
83.90
83.40
11.80
81.70
81.70
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Table: Exp C: Bias and RMSE ( 100) Results: β 2i = 0, f1t
s IIDN (0, 1), spatial structure on uit - d = 0.2, v1i s IIDU (0.5, 1.5)
α
0.70
0.75
0.80
α̂
α̃
α̌
α̂
α̃
α̌
2.84
1.49
1.31
1.72
0.75
0.65
1.95
1.03
0.95
1.07
0.46
0.42
1.35
0.73
0.70
0.62
0.25
0.24
α̂
α̃
α̌
α̂
α̃
α̌
3.39
2.55
2.50
2.28
1.82
1.81
2.67
2.23
2.22
1.82
1.64
1.64
2.23
2.02
2.02
1.57
1.52
1.53
N/T
100
200
100
200
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Bias
0.85
100
0.90
0.95
1.00
0.92
0.53
0.51
0.34
0.12
0.12
RMSE
0.63
0.37
0.36
0.18
0.04
0.04
0.41
0.24
0.24
0.07
-0.01
-0.01
0.07
-0.04
-0.04
-0.09
-0.14
-0.14
1.98
1.89
1.89
1.46
1.45
1.45
1.83
1.79
1.80
1.39
1.40
1.40
1.75
1.74
1.74
1.37
1.38
1.38
1.68
1.69
1.69
1.35
1.37
1.37
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Table: Exp C: Size and Power ( 100) Results: β 2i = 0, f1t
s IIDN (0, 1), spatial structure on uit - d = 0.2, v1i s IIDU (0.5, 1.5)
0.70
α
0.75
0.80
N/T
100
200
0.85
0.90
0.95
1.00
100
α̃
Size
Power+
Power-
10.40
44.10
91.40
7.10
54.50
89.90
6.00
62.65
88.60
5.20
69.50
88.70
5.45
73.75
88.05
5.25
77.55
87.75
6.95
85.60
85.60
α̌
Size
Power+
Power-
10.35
47.20
89.60
6.95
56.40
89.00
6.05
63.25
88.30
5.25
69.80
88.40
5.50
73.85
87.95
5.25
77.60
87.75
7.00
85.60
85.60
α̃
Size
Power+
Power-
8.05
74.35
94.50
5.95
81.15
94.60
5.50
86.70
94.75
5.25
90.65
94.65
4.95
93.35
95.35
5.35
94.30
95.50
6.05
94.80
94.80
α̌
Size
Power+
Power-
8.15
75.40
93.90
5.80
81.60
94.35
5.50
87.05
94.60
5.30
90.70
94.65
4.95
93.35
95.35
5.35
94.30
95.50
6.05
94.80
94.80
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Table: Exp D: Bias and RMSE ( 100) Results: β 2i 6= 0, α2 = α, fjt and
uit s IIDN (0, 1), vji s IIDU (0.5, 1.5), j = 1, 2
α
0.70
0.75
0.80
α̂
α̃
α̌
α̂
α̃
α̌
2.48
1.06
0.87
1.65
0.67
0.56
1.84
0.89
0.80
1.03
0.41
0.37
1.35
0.72
0.68
0.68
0.30
0.28
α̂
α̃
α̌
α̂
α̃
α̌
3.09
2.33
2.31
2.24
1.80
1.80
2.57
2.15
2.14
1.82
1.65
1.65
2.22
2.00
2.00
1.62
1.56
1.56
N/T
100
200
100
200
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Bias
0.85
100
0.90
0.95
1.00
0.85
0.43
0.41
0.44
0.21
0.21
RMSE
0.58
0.31
0.30
0.31
0.17
0.17
0.40
0.22
0.22
0.17
0.09
0.09
-0.06
-0.18
-0.18
-0.04
-0.09
-0.09
1.93
1.85
1.85
1.51
1.50
1.50
1.79
1.77
1.77
1.46
1.46
1.46
1.71
1.71
1.71
1.42
1.42
1.42
1.65
1.68
1.68
1.39
1.40
1.40
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Table: Exp D: Size and Power ( 100) Results: β 2i 6= 0, α2 = α, fjt and
uit s IIDN (0, 1), vji s IIDU (0.5, 1.5), j = 1, 2
0.70
α
0.75
0.80
N/T
100
200
0.85
0.90
0.95
1.00
100
α̃
Size
Power+
Power-
6.90
51.95
88.00
6.40
57.25
88.75
5.75
63.45
89.00
4.60
70.85
87.50
5.20
75.90
88.25
5.40
79.20
88.30
7.00
84.00
84.00
α̌
Size
Power+
Power-
6.90
54.65
85.60
6.40
59.35
87.70
5.65
63.90
88.65
4.60
70.90
87.40
5.30
75.95
88.15
5.40
79.20
88.25
7.00
84.00
84.00
α̃
Size
Power+
Power-
6.80
75.00
94.70
5.60
81.30
94.40
5.30
85.40
94.75
4.80
88.15
94.60
5.10
91.05
95.15
5.00
92.45
95.05
5.80
94.65
94.65
α̌
Size
Power+
Power-
7.45
76.20
93.50
5.70
81.85
93.95
5.55
85.70
94.60
4.95
88.25
94.60
5.05
91.05
95.15
5.00
92.45
95.05
5.80
94.65
94.65
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Application to Cross-country Macroeconomic Data Sets
We estimate the cross-sectional exponent, α, for a number of
macroeconomic variables used in the GVAR framework of Dees
et al (2007). The data cover the period 1979Q2-2009Q4.
Results are supportive of the notion that …nancial variables are
more strongly correlated as compared to the real variables.
The value of α = 1 typically assumed in the empirical factor
literature might be exaggerating the importance of the
common factors for modelling cross-sectional dependence.
Real GDP growth, q/q
In‡ation, q/q
Real equity price chg, q/q
Short-term interest rates
Long-term interest rates
N
33
33
26
32
18
T
122
123
122
123
123
α̃0.025
0.691
0.778
0.797
0.831
0.864
α̃
0.754
0.851
0.881
0.907
0.968
α̃0.975
0.818
0.924
0.966
0.983
1.072
*95% level con…dence bands; The observations were standardized as xit = (yit
of each time series, and si is the corresponding standard deviation.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
α̌0.025
0.689
0.777
0.796
0.831
0.864
α̌
0.752
0.850
0.881
0.907
0.968
α̌0.975
0.816
0.924
0.966
0.983
1.072
yi )/si , where yi is the sample mean
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Application to Within-country Macroeconomic Data Sets
We also estimate the cross sectional exponent, α, for two
within-country macroeconomic data sets used in the empirical
factor model literature - e.g. Stock and Watson (2002).
The UK and US quarterly datasets are used in Eklund,
Kapetanios and Price (2010).
For the US data set the point estimate (α̃) of α is slightly
larger than the estimate obtained for the UK.
Once again there evidence of a common factor dependence is
not as strong as it is assumed in the literature.
US
1960Q2-2008Q3
N=95, T=194
α̃0.025
α̃
α̃0.975
0.689
0.739
0.788
α̌0.025
α̌
α̌0.975
0.689
0.738
0.788
UK
1977Q1-2008Q2
N=94, T=126
α̃0.025
α̃
α̃0.975
0.636
0.715
0.793
α̌0.025
α̌
α̌0.975
0.635
0.713
0.792
*95% level con…dence bands. The observations were standardized as xit = (yit
of each time series, and si is the corresponding standard deviation.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
yi )/si , where yi is the sample mean
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Application to Stock Returns
We investigate the degree of interconnectivity of stock returns
in the US and how that ‡uctuates over time depending on the
health of prevailing …nancial conditions.
This a¤ects the extent to which …nancial risk can be
diversi…ed.
We consider the distinct monthly compositions of the S&P500
index from September 1989 to September 2011 in order to
assess the resulting exponent of cross section dependence, α.
Our analysis is based on a rolling window sample scheme.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Methodology
We work with stock returns de…ned as
rit = 100
(Pit
Pit
Pit
1
1)
+
DYit
,
12
for i = 1, ..., Nj and t = 1, ..., Tr ,
where Pit and DYit are the price and dividend yield of stock i
at time t.
j denotes the sub-sample of stock returns used.
r denotes the size of the rolling window.
We take excess stock returns, de…ned as
rite = rit
rft ,
for i = 1, ..., Nj and t = 1, ..., Tr ,
where rft denotes the risk-free rate in percent in month t.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Excess Returns
We extract the residuals (eite ) from regression
rite = γi + ε it ,
for i = 1, ..., Nj and t = 1, ..., Tr .
We impose the following standardisation
ẽite =
eite
where ēie = Tr
ēie
si
1
,
for i = 1, ..., Nj and t = 1, ..., Tr ,
h
T
T
∑t =r 1 eite and si = ∑t =r 1 (eite
ēie )2 /Tr
i1/2
.
We apply the estimator of cross-sectional dependence to the
e )0 ,
vector ~
ee = (~
ee1 ,~
ee2 , ....,~
eeN j )0 , where ~
eei = (ẽie1 , ẽie2 , ...., ẽiT
r
and obtain the bias-adjusted α̃.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Determination of Nj and Tr
We consider baskets of stocks extracted from each monthly
composition of Standard & Poor’s (S&P) 500 index ranging
from end-September 1989 to end-September 2011 with a
5-year and 10-year history.
Nj , on average, amount to 476 stocks per month in the 5-year
case and 439 stocks per month in the 10-year case.
The constructed monthly portfolios of securities are
considered reliable approximations of the overall index at each
point in time.
We set Tr = 10yrs or Tr = 5yrs and …nally construct
monthly sub-samples of dimension Nj Tr .
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
WCD test
We …rst verify that the interdependence between securities is
strong enough for α to be estimable (recall that α is
identi…able when α > 1/2).
For this purpose we use the test of Weak Cross-Sectional
Dependence (WCD) introduced by Pesaran (2012):
CDNT =
where
b̄
ρN =
TN (N
2
1)
1/2
b̄
ρN ,
(13)
N 1 N
2
∑ ρ̂ij ,
N (N 1) i∑
=1 j =i +1
∑Tt=1 ẽite ẽjte .
Note that ρ̄N = O (N 2α 2 ).
The null hypothesis of the test is H0 : α < 1/2 and the
critical value is 1.96.
and ρ̂ij = T
1
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
WCD statistics for S&P500 excess stock returns - 10yr and
5yr rolling windows
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
α̃t associated with S&P500 excess stock returns - 10yr and
5yr rolling windows
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
α̃t estimates and average pairwise correlations
The patterns observed in the estimates of α are in line with
changes in the degree of correlations in equity markets.
It is generally believed that correlations of returns in equity
markets rise at times of …nancial crises.
To this end we compare the estimates of α to average
pair-wise correlation coe¢ cients of excess returns, ρ̂N . This is
de…ned as before by
N 1
ρ̂N = (2/N (N
1))
N
∑ ∑
ρ̂ij ,
i =1 j =i +1
where ρ̂ij is the correlation of excess returns on i and j
securities.
ρ̄N estimates are calculated for the securities included in S&P
500 index, using 10-year and 5-year rolling windows.
Our estimates of α closely follow the rolling estimates of ρ̄N .
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Average pair-wise correlations of excess returns for
securities in the S&P 500 index and the associated α̃t
estimate computed using 10-year and 5-year rolling samples
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Direct (α̃t ) and Indirect (α̂d ) estimates of α
Further, we compare our estimates of α with estimates
obtained using excess returns on market portfolio as a
measure of the unobserved factor.
A direct estimate of α is given by α̂d = ln(M̂ )/ ln(N ), where
M̂ denotes the estimated number of non-zero betas, and N is
the number of securities under consideration (M = [N α ]).
M̂ can be consistently estimated (as N and T ! ∞) by the
number of t-tests of β i = 0 in the CAPM regressions
rit
rft = ai + β i (rmt
rft ) + uit , for i = 1, 2, ..., N,
that end up in rejection of the null hypothesis at 1%
signi…cance level, where rmt is the value-weighted return on all
NYSE, AMEX, and NASDAQ stocks - Pesaran and Yamagata
(2012).
The indirect ( α̃) and direct (α̂d ) estimates of α tend to move
together closely.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Direct (α̂d ) and indirect (α̃) estimates of cross-sectional
exponent of the market factor based on 10-year and 5-year
rolling samples
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Residuals from CAPM
Another setup considers the extent of cross-sectional
dependence in residuals from the CAPM regressions.
We analyse individual excess stock returns once again.
For the market as a whole, we use the excess returns de…ned
as:
rte,mkt = rtmkt rft , for t = 1, ..., Tr .
We then extract the residuals from regression
rite = γi + δi rte,mkt + ηit ,
for i = 1, ..., Nj and t = 1, ..., Tr .
We …nally standardise the residuals as shown before.
Again we …rst check the strength of interdependence of the
CAPM residuals by use of the WCD test on these residuals.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
WCD statistics for residuals from CAPM regressions using
5-yr / 10-yr rolling samples
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Estimates of cross-sectional exponent of residuals (α̃u )
from CAPM regressions using 5-yr / 10-yr rolling samples
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Concluding Remarks
Cross-sectional dependence has received considerable
attention over the past few years.
In many applications it is important to know whether the
cross dependence is weak, strong or somewhere in between.
We introduce a summary statistic which we call
cross-sectional exponent that quanti…es the degree of
cross-sectional dependence present in the panel data, xit .
We prove consistency and (under certain conditions) we show
that the proposed bias-corrected estimator is asymptotically
normal under a wide range of circumstances and for di¤erent
N and T sample sizes.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc
Concluding Remarks (continued)
Small sample properties of the proposed estimator and test
are investigated by Monte Carlo studies and shown to be
satisfactory.
We apply our measure to three widely analysed classes of data
sets. In all cases, we …nd that the results of the empirical
analysis accord with prior intuition:
In the case of cross country applications we obtain larger
estimates for the cross-sectional exponent of equity returns as
compared to those estimated for cross country output growths
and in‡ation.
For individual securities in S&P 500 index, the estimates of
cross-sectional exponents are systematically high but not equal
to unity, a widely maintained assumption in the theoretical
multi-factor literature.
Natalia Bailey, George Kapetanios and M.Hashem Pesaran
Exponent of Cross-sectional Dependence: Estimation and Inferenc