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Journal of Business & Economic Statistics

ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20

Dynamic Factor Models for Multivariate Count
Data: An Application to Stock-Market Trading
Activity
Robert C. Jung, Roman Liesenfeld & Jean-François Richard
To cite this article: Robert C. Jung, Roman Liesenfeld & Jean-François Richard (2011) Dynamic
Factor Models for Multivariate Count Data: An Application to Stock-Market Trading Activity,
Journal of Business & Economic Statistics, 29:1, 73-85, DOI: 10.1198/jbes.2009.08212
To link to this article: http://dx.doi.org/10.1198/jbes.2009.08212

Published online: 01 Jan 2012.

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Date: 11 January 2016, At: 22:58

Dynamic Factor Models for Multivariate Count
Data: An Application to Stock-Market
Trading Activity
Robert C. J UNG
Department of Economics, Universität Erfurt, 99089 Erfurt, Germany (robert.jung@uni-erfurt.de)

Roman L IESENFELD
Department of Economics, Christian-Albrechts-Universität zu Kiel, 24118 Kiel, Germany
(liesenfeld@stat-econ.uni-kiel.de)

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:58 11 January 2016

Jean-François R ICHARD
Department of Economics, University of Pittsburgh, Pittsburgh, PA 15260 (fantin@pitt.edu)
We propose a dynamic factor model for the analysis of multivariate time series count data. Our model
allows for idiosyncratic as well as common serially correlated latent factors in order to account for potentially complex dynamic interdependence between series of counts. The model is estimated under alternative count distributions (Poisson and negative binomial). Maximum likelihood estimation requires
high-dimensional numerical integration in order to marginalize the joint distribution with respect to the
unobserved dynamic factors. We rely upon the Monte Carlo integration procedure known as efficient importance sampling, which produces fast and numerically accurate estimates of the likelihood function.
The model is applied to time series data consisting of numbers of trades in 5-min intervals for five New
York Stock Exchange (NYSE) stocks from two industrial sectors. The estimated model provides a good
parsimonious representation of the contemporaneous correlation across the individual stocks and their
serial correlation. It also provides strong evidence of a common factor, which we interpret as reflecting
market-wide news.
KEY WORDS: Dynamic latent variables; Importance sampling; Mixture of distribution models; Poisson distribution; Simulated Maximum Likelihood.

1. INTRODUCTION

models introduced by McKenzie (1986) and Al-Osh and Alzaid
(1987) and is used to study spillover effects in the stock trading
activities of two individual stocks. Both models are amenable
to standard methods of estimation, but are parameter rich. In
order to analyze comovements in the number of trades for five

individual stocks, Heinen and Rengifo (2007) also adopted an
observation-driven approach extending the univariate autoregressive conditional Poisson model of Heinen (2003). A copula approach is used to represent contemporaneous correlations among time series counts. Since efficient joint maximumlikelihood (ML) estimation is not feasible, the authors rely upon
a consistent, though less efficient, two-stage ML approach for
separate estimation of the parameters of the marginal distributions and of those of the copula. Other multivariate count models rely upon panel data techniques, with an emphasis on unobserved heterogeneity in the individual series. See Winkelmann
(2008) for a recent survey.
In the present article we adopt a parameter-driven approach
and propose a new flexible, parsimonious, and easy-to-interpret
dynamic factor model for multivariate count series. It builds
upon and generalizes earlier models by Jørgensen et al. (1999)
and Wedel, Bockenholt, and Wagner (2003). The former model
includes only a single dynamic common factor and no dynamic

Modeling dispersion and serial correlation for univariate
count series received much attention over recent years. Existing approaches can be broadly classified as either observation
or parameter driven. The monographs of Kedem and Fokianos
(2002) and McKenzie (2003) provided excellent overviews.
Multivariate dynamic models for count data remain few. As
discussed by Cameron and Trivedi (1998, section 8.1), this
might be explained by the fact that classical inference in multivariate count data models has proven to be analytically as well
as computationally demanding. Pioneering multivariate applications are found in Jørgensen et al. (1999), Held, Hohle, and

Hofmann (2005), Heinen and Rengifo (2007), and Quoreshi
(2008). The specification proposed by Jørgensen et al. (1999)
belongs to the class of parameter-driven models. It consists of a
multivariate Poisson state–space model with a common factor
that can be analyzed by a Kalman filter. It is used to assess the
impact of air pollution on daily emergency admission counts in
a hospital for four sickness categories. Held, Hohle, and Hofmann (2005) and Quoreshi (2008) proposed observation-driven
models that are based upon a vector autoregressive moving average (VARMA) structure. The model of Held, Hohle, and Hofmann imposed a simple vector autoregressive (VAR) structure
for the conditional means and is applied to infectious disease
surveillance counts from a measle epidemic. The specification
of Quoreshi represented a multivariate extension of the univariate integer-valued autoregressive moving average (INARMA)

© 2011 American Statistical Association
Journal of Business & Economic Statistics
January 2011, Vol. 29, No. 1
DOI: 10.1198/jbes.2009.08212
73

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:58 11 January 2016

74

idiosyncratic components. The latter model is a static multivariate Poisson factor model for cross-sectional analyses. Our
model allows for serially correlated common as well as idiosyncratic factors driving the conditional means of the count distributions. Therefore, it can account for nontrivial contemporaneous as well as temporal interactions across count series. It can
also accommodate different distributional assumptions for the
conditional distribution of the counts given the factors. This can
be critical since the commonly used Poisson distribution has an
index of dispersion equal to 1 (the latter being defined as the
ratio between the variance and the mean). However, count data
often exhibit strong over-dispersion (index significantly larger
than 1), which cannot be fully captured by a conditional Poisson distribution even if marginalization of the conditional mean
can generate by itself an over-dispersed unconditional distribution. Hence, it might be important to allow for conditional distributions that can accommodate over-dispersion, such as the
negative binomial (hereafter Negbin).
The dynamic latent factors enter our model nonlinearly.
Thus, likelihood evaluation requires high-dimensional numerical integration, for which we use the efficient importance
sampling (EIS) procedure developed by Richard and Zhang
(2007). EIS is a generic, flexible, and easy-to-implement Monte
Carlo (MC) integration procedure specifically designed to maximize numerical accuracy. It also facilitates exploring alternative model specifications that typically require only minor modifications of a baseline EIS implementation. Last but not least,
EIS can be used to compute filtered estimates of the latent factors themselves. Several diagnostic test statistics are based upon
such estimates.

We apply our model to a multivariate time series consisting
of numbers of trades in 5-min intervals for five stocks traded
at the New York Stock Exchange (NYSE). We implicitly adopt
the information flow interpretation associated with the mixtureof-distribution model of Tauchen and Pitts (1983). See also Andersen (1996) and Liesenfeld (2001). In this context, numbers
of trades are directly influenced by the arrival of new information, whether specific to a single stock (idiosyncratic factor), to
an industry (sector factor), or to the market (market factor). Because news arrivals also drive the volatility of asset returns, our
specification for the number of trades is intimately related to the
dynamic latent factor models for the volatility of multivariate
return series proposed by Diebold and Nerlove (1989), Aguilar
and West (2000), and Chib, Nardari, and Shephard (2006).
Alternative multivariate approaches used to analyze the stock
market trading activity are based on the duration between successive trades or the trading intensity defined as the instantaneous rate of trade occurrence. Spierdijk, Nijman, and vanSoest (2002), for example, proposed a bivariate duration model
for two stocks, which combines a univariate model for the
“pooled” durations between two successive trades in any of the
two stocks with a Probit model that determines which of the
two stocks is traded. However, the application of this model to
more than two stocks becomes intractable. A multivariate approach based on the intensity that is applicable to a larger number of stocks and which is closely related to ours is proposed
by Bauwens and Hautsch (2006). They considered a model for
stock-price intensities that allows for observation-driven stockspecific intensity components and for a latent dynamic common

Journal of Business & Economic Statistics, January 2011

component that jointly drives the intensity of all stocks. Since
their model is nonlinear in the dynamic latent common factor
they also use EIS integration to obtain ML-estimates.
The article is organized as follows. The multivariate dynamic
factor model is introduced in Section 2. Section 3 discusses
ML estimation and filtering based upon EIS. The application
to NYSE data is presented in Section 4. Section 5 concludes.
2. DYNAMIC FACTOR MODEL FOR MULTIVARIATE
COUNT DATA
The econometric model we propose consists of a dynamic
extension of the static multivariate Poisson factor model introduced by Wedel, Bockenholt, and Wagner (2003). Consider a
J-dimensional vector of counts yt = (yt1 , . . . , ytJ )′ recorded at
time t (t = 1, . . . , T). Dynamics will be introduced at the level
of the latent factors. Accordingly, counts are assumed to be conditionally independently distributed with Poisson distributions
y

p(ytj | θtj ) =

exp(−θtj )θtjtj
ytj !

t = 1, . . . , T, j = 1, . . . , J,

,

(1)
whose means θtj are latent random variables. We assume the
existence of a link function b(·), whereby the mean vector
θ t = (θt1 , . . . , θtJ )′ can be expressed as a linear function of a
P-dimensional vector of latent random factors ft , say
b(θ t ) = μ + Ŵft ,

(2)

where μ denotes a vector of fixed intercepts and Ŵ a (J ×P) matrix of factor loadings. The P latent factors in ft are assumed to
be independent of each other. A log-link function b(θ t ) = ln(θ t )
is convenient since it implies positivity of θ t without parametric
restrictions on (μ, Ŵ) and is the only one considered here.

In the context of our NYSE application considering the joint
behavior of the number of trades for different stocks, we allow
for a single common market factor λt , S < J industry-specific
factors τ t = (τt1 , . . . , τtS )′ , and J stock-specific factors ωt =
(ωt1 , . . . , ωtJ )′ . Hence, ft is partitioned into ft = (λt , τ ′t , ω′t )′
and P = J + S + 1. The matrix of factor loadings is partitioned
conformably with ft into Ŵ = (Ŵ λ , Ŵ τ , Ŵ ω ), where Ŵ λ = (γjλ )
is a J-dimensional vector, Ŵ τ = (γjτs ) a J × S matrix with zero
entries for any firm j that does not belong to sector s, and
ω
Ŵ ω = diag(γj j ) a (J × J) diagonal matrix. Thus, the log-mean
function for stock j, belonging to industry s is given by
τs

ω

ln θtj = μj + γjλ λt + γj j τtsj + γj j ωtj ,

(3)

where the index sj denotes the industry of firm j.
In order to account for possible serial and cross-correlation
in the counts, we assume that the factors follow independent
Gaussian AR(1) processes, say
λt | λt−1 ∼ N(κ λ + δ λ λt−1 , [ν λ ]2 ),

(4)

τs 2

(5)

τs

τs

τts | τt−1s ∼ N(κ + δ τt−1s , [ν ] ),
ωtj | ωt−1j ∼ N(κ ωj + δ ωj ωt−1j , [ν ωj ]2 ).

(6)

|δ λ | < 1,

To ensure stationarity of the factors, it is assumed that
|δ τs | < 1, and |δ ωj | < 1. Other distributional and dynamic specifications for the factors are easily accommodated. Under an

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Jung, Liesenfeld, and Richard: Dynamic Factor Models for Multivariate Count Data

identity link b(·), for example, Gamma or log-normal transition
distributions will provide suitable factor specifications (see Jørgensen et al. 1999 and Jung and Liesenfeld 2001). Also, higher
order AR(p) processes can be accommodated.
The model as specified is unidentified. Identification for the
static case with iid factors is discussed in Wedel, Bockenholt,
and Wagner (2003) and can be extended to the dynamic model
introduced here. We impose the restrictions that κ λ = κ τs =
κ ωj = 0 for s = 1, . . . , S and j = 1, . . . , J in order to identify
ω
the μj ’s [see Equations (4)–(6)]. Furthermore, we set γj j = 1
for j = 1, . . . , J, γjτs = 1 for one arbitrarily selected stock j in
industry s for s = 1, . . . , S, and γjλ = 1 for one arbitrarily selected stock j [see Equation (3)]. This eliminates indeterminacies in the factor scales.
Under the assumed Poisson distribution, whose dispersion
index equals 1, over-dispersion of the counts can only originate from the unconditional variances of the factors, which
themselves critically depend on the persistence parameters
(δ λ , δ τs , δ ωj ). In order to relax this close relationship between
over-dispersion and persistence, we will substitute the more
flexible Negbin distribution for the Poisson. It is given by

1/σj2
Ŵ(ytj + 1/σj2 )
1
p(ytj | θtj ) =
Ŵ(1/σj2 )Ŵ(ytj + 1) 1 + σj2 θtj

ytj
θtj
, (7)
×
θtj + 1/σj2
where Ŵ(·) denotes the Gamma function. Its mean and variance are given by θtj and θtj (1 + σj2 θtj ), respectively. Its overdispersion is a monotone increasing function of σj > 0 and the
Poisson distribution in Equation (1) obtains as the limit for
σj → 0.

75

ft−1 as defined by Equations (1) to (6). The initial condition
f0 is assumed to be the known constant f0 = E(ft ) = 0. Since
ft is serially dependent and enters the model nonlinearly the
likelihood integral in Equation (8) is not amenable to standard
integration procedures.
Following Richard and Zhang (2007), the EIS MC integration for the likelihood in Equation (8) is based upon a sequence
of auxiliary importance sampling densities for ft given ft−1 of
the form
kt (ft , ft−1 ; at )
, with
mt (ft | ft−1 ; at ) =
χt (ft−1 ; at )
(9)

χt (ft−1 ; at ) = kt (ft , ft−1 ; at ) dft

for t = 1, . . . , T, where {kt (ft , ft−1 ; at ), at ∈ At }Tt=1 denotes a
(preselected) class of auxiliary parametric density kernels with
known analytical integrating factors in ft given ft−1 denoted
by χt (ft−1 ; at ). The likelihood integral in Equation (8) is then
rewritten as


 
T 
ϕt (ft , ft−1 ; ψ)χt+1 (ft ; at+1 )
L(ψ) = χ1 (f0 ; a1 ) · · ·
kt (ft , ft−1 ; at )
t=1

×

T


mt (ft | ft−1 ; at ) dfT · · · df1 ,

with χT+1 (·) ≡ 1. For any value of {at } the corresponding importance sampling MC estimate of the likelihood L(ψ) is given
by
L¯ N (ψ) = χ1 (f0 ; a1 )
×


(i) (i)
(i)
N T 
1   ϕt (˜ft , ˜ft−1 ; ψ)χt+1 (˜ft ; at+1 )
,
(i) (i)
N
kt (˜ft , ˜f ; at )
i=1 t=1

3. EIS BASED INFERENCE
In order to estimate the model we use ML based upon EIS.
An alternative will be to use GMM based on unconditional moments that can be computed in closed form. However, it is well
known that for models with dynamic latent factors a likelihoodbased approach has typically better finite-samples properties
than GMM—see, e.g., Andersen, Chung, and Sørensen (1999).
3.1 EIS
The evaluation of the likelihood function for the model described by Equations (1) to (6) requires integrating the joint
density of counts and factors with respect to the T · P latent
factor variables (in our application T · P equals 36,600). For the
likelihood evaluation, counts are kept fixed at their observed
values and are, therefore, omitted from notation except for the
fact that densities need to be time indexed to reflect their dependence on the data. The likelihood integral to be evaluated is of
the form:

 
T
ϕt (ft , ft−1 ; ψ) dfT · · · df1 ,
(8)
L(ψ) = · · ·
t=1

where ψ regroups the parameters of the model and ϕt denotes
the product of the time t densities for yt given ft and for ft given

(10)

t=1

(11)

t−1

(i)

where {{˜ft }Tt=1 }N
i=1 denotes N independent trajectories drawn
from the sequence of importance sampling densities {mt (ft |
ft−1 ; at )}Tt=1 . EIS aims at selecting values of the auxiliary parameters {at } that minimize the MC sampling variance of the
MC estimate in Equation (11). This requires minimizing period
by period the variance of the ratios ϕt · χt+1 /kt as functions of
ft and ft−1 with respect to the mt -distributions. As shown in
Richard and Zhang (2007), an MC-EIS approximate solution
of this minimization problem obtains from the following backward recursive sequence of auxiliary least squares (LS) problems:
(ˆct , aˆ t ) = arg

min

ct ∈R,at ∈At

N


 (i) (i)

ln ϕt ˜ft , ˜ft−1 ; ψ
i=1

2



˜(i)
ˆ t+1 − ct − ln kt ˜f(i)
,
· χt+1 ˜f(i)
t , ft−1 ; at
t ;a

t = T, T − 1, . . . , 1,
T
{˜f(i)
t }t=1

(12)

denotes a trajectory drawn from an initial sewhere
(0)
quence of auxiliary samplers {mt (ft | ft−1 ; at )}Tt=1 with i =
1, . . . , N (iid). As initial samplers we use the Gaussian densities associated with second-order Taylor-series approximations
(TSA) to ϕt χt+1 as described below. As detailed in Richard
and Zhang (2007), the sequence of EIS auxiliary regressions
in Equation (12) is iterated by replacing the initial auxiliary

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76

Journal of Business & Economic Statistics, January 2011

samplers by the previous stage importance samplers. For such
iterations to converge to a fixed-point solution for {ˆat }, characterizing the final EIS sampling densities {mt (ft | ft−1 ; aˆ t )}, it
(i)
is critical that all ith trajectories {˜ft }Tt=1 be obtained by transforming a single set of common random numbers (CRNs), say
(i)
{u˜ t }Tt=1 . CRNs are N(0, 1) for Gaussian EIS samplers.
The MC-EIS evaluation of L(ψ) for a given value of ψ
then obtains by substituting the fixed-point values of {ˆat } for
{at } in Equation (11). A generalization of Equations (4)–(6) to
higher order AR(p) processes only requires replacing in Equations (8)–(12) ft−1 by Ft−1 = (f′t−1 , . . . , f′t−p )′ and f0 by F0 (see
Richard and Zhang 2007).
Note that {ˆat } is an implicit function of ψ. Therefore, maximal numerical efficiency requires complete reruns of the EIS
algorithm for any new value of ψ. The (re)use of CRNs across
those reruns ensures continuity of the MC likelihood evaluation
L¯ N (ψ) with respect to ψ, which is critical for likelihood maximization. Finally, it should be mentioned that EIS-density kernels kt within the exponential family of distributions are linear
in the auxiliary parameters at under their natural parameterization as well as closed under multiplication. As detailed below,
these two properties considerably simplify the application of
EIS to our model.
3.2 Implementation of EIS for the Poisson Factor Model
The implementation of the sequential EIS procedure outlined
in Section 3.1 for the model defined by Equations (1) to (6)
proceeds as follows. Under the log-link function, the Poisson
density in Equation (1) is rewritten as
p(ytj | φtj ) =

exp(ytj φtj − eφtj )
,
ytj !

(13)

with φtj = ln θtj . Equations (2) and (3) are rewritten in matrix
form as
φ t = μ + Ŵft ,

(14)

φ ′t

with
= (φt1 , . . . , φtJ ). According to Equations (4)–(6) the
density of the conditional Gaussian distribution for ft given ft−1
has the form

term in the product ϕt χt+1 to be approximated by kt via the
EIS auxiliary regressions in Equation (12) is the product of the
J densities p(ytj | φtj ). Thus, it suffices to construct Gaussian
approximates in φtj to the latter densities in order to produce
a Gaussian kernel kt for (ft , ft−1 ). The corresponding kernel kt
consists of the product:
kt (ft , ft−1 ; at ) = p(ft | ft−1 ; ψ)ζt (ft ; at )χt+1 (ft ; at+1 ),

of J univariate Gaussian kernels in
where ζt denotes the product
φtj designed to approximate Jj=1 p(ytj | φtj ). It is parameterized
as


1 ′
′
(18)
ζt (ft ; at ) = exp − (φ t Bt φ t − 2φ t ct ) ,
2
where Bt = diag(btj ) denotes a J × J positive-definite diagonal
matrix and ct = (ctj ) a J-dimensional vector. The EIS auxiliary parameter at is defined as a′t = (vech(Bt )′ , c′t ). Note that
the factors ln p(ft | ft−1 ; ψ) and ln χt+1 appear on both sides of
the auxiliary EIS regression in Equation (12) and cancel out. It
follows that this regression simplifies into J independent linear
˜ (i) ˜ (i) 2 N
LS regressions of {ln p(ytj | φ˜ tj(i) )}N
i=1 on {(φtj , [φtj ] )}i=1 with
intercepts.
Since the density kernels kt in Equation (17) depend on
the one-period ahead integrating constants χt+1 , they are obtained back-recursively as described next. Since χt+1 is itself
a Gaussian kernel in ft , we introduce the following auxiliary
parameterization:


1
χt+1 (ft ; at+1 ) = exp − (f′t Pt+1 ft − 2f′t qt+1 + rt+1 ) , (19)
2
where (Pt+1 , qt+1 , rt+1 ) denote appropriate functions of the
EIS auxiliary parameter at+1 resulting from the backward recursion as given below. Since χT+1 ≡ 1, the “initial values” are
PT+1 = 0, qT+1 = 0, and rT+1 = 0. Next, we combine together
Equations (15), (18), and (19) according to Equation (17) to
obtain
kt (ft , ft−1 ; at )
= (2π)−P/2 |H−1 |−1/2

1
× exp − [ft − (dt + Gt ft−1 )]′ Mt [ft − (dt + Gt ft−1 )]
2

p(ft | ft−1 ; ψ)
= (2π)−P/2 |H−1 |−1/2


1
′
× exp − (ft − ft−1 ) H(ft − ft−1 ) ,
2

(15)

where  and H are both diagonal and H denotes the inverse of
the covariance matrix of ft given ft−1 .
In order to apply sequential EIS to this model, we first note
that the factor ϕt (ft , ft−1 ; ψ) in the likelihood integral in Equations (8) and (10) is given by
 J


p(ytj | φtj ) ,
(16)
ϕt (ft , ft−1 ; ψ) = p(ft | ft−1 ; ψ) ·
j=1

where φtj is a linear function in ft . Next, note that if the selected
kernel kt (ft , ft−1 ; at ) in Equation (9) is Gaussian in both ft and
ft−1 , then its integrating constant χt (ft−1 ; at ) is itself Gaussian
in ft−1 . By recursion this implies that the sole non-Gaussian

(17)

with

+ (μ′ Bt μ − 2μ′ ct + f′t−1 ′ Hft−1 + rt+1 )


− (dt + Gt ft−1 )′ Mt (dt + Gt ft−1 ) ,

(20)

Mt = Ŵ ′ Bt Ŵ + H + Pt+1 ,
′
dt = M−1
t [qt+1 + Ŵ (ct − Bt μ)],

Gt =

(21)

M−1
t H.

It immediately follows from Equation (20) that the Gaussian
EIS sampler for ft | ft−1 is given by
mt (ft | ft−1 ; at ) ∼ N(dt + Gt ft−1 , M−1
t ).

(22)

Jung, Liesenfeld, and Richard: Dynamic Factor Models for Multivariate Count Data

The integrating constant χt (ft−1 ; at ) obtains by regrouping all
remaining factors in Equation (20) and is, therefore, of the form
introduced in Equation (19) together with
Pt = ′ H − G′t Mt Gt ,
qt = G′t Mt dt ,
rt = μ′ Bt μ − 2μ′ ct + rt+1 − d′t Mt dt

(23)

+ ln |H−1 | − ln |Mt−1 |.

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:58 11 January 2016

In summary, the computation of the EIS estimate of the likelihood requires the following simple steps:
Step (i). Generate N independent P-dimensional trajecto(i)
ries {{˜ft }Tt=1 }N
i=1 from a sequence of initial samplers {m(ft |
(0)
ft−1 , at )}. This sequence obtains by using
 for ln ζt in Equation (18) a second-order TSA in φtj to ln Jj=1 p(ytj | φtj ) around
E(φtj ) = μj . The resulting TSA values of the auxiliary para-

(0) ′
(0) ′
′
meters (a(0)
t ) = (vech(Bt ) , (ct ) ) are substituted for at in
Equations (21)–(23) in order to construct the initial EIS samplers.
Step (ii). Transform the simulated ft -trajectories from the
previous sampler into the corresponding N independent Jdimensional φ t -trajectories according to φ t = μ + Ŵft . The latter are used to solve back recursively the LS-problems in Equation (12). This requires running the following J independent
linear auxiliary regressions for each period t:


1
(i)
2
(i) 
(i)
(i)
ln p yt1 | φ˜ t1 = constant − bt1 φ˜ t1 + ct1 φ˜ t1 + ςt1 ,
2
i = 1, . . . , N,

(24)

..
.


1

2
ln p ytJ | φ˜ tJ(i) = constant − btJ φ˜ tJ(i) + ctJ φ˜ tJ(i) + ςtJ(i) ,
2
i = 1, . . . , N,

(25)

(i)

where ςtj denotes the regression error term of regression (t, j).
Step (iii). Use the LS estimates of the auxiliary parameters
ˆ t = diag(bˆ tj ) and cˆ t = (ˆctj ) obtained in step (ii), to construct
B
back-recursively the sequence of EIS-sampling densities {m(ft |
ft−1 , aˆ t )} as given by Equation (22) together with the recursions
in Equations (21) and (23).
Step (iv). Generate N independent trajectories from the sequence of EIS samplers constructed in step (iii) and use them
either to repeat steps (ii) and (iii) or (at convergence) to compute the EIS-MC estimate of the likelihood according to Equation (11).
If we replace the Poisson density in Equation (1) by the
Negbin density in Equation (7), we only need to modify accordingly the dependent variables in the auxiliary EIS regressions, a trivial adjustment altogether. For a generalization of
Equations (4)–(6) to higher order AR(p) processes we need
to modify accordingly the transition density in Equation (15)
and to adjust χt+1 in Equation (19) to be a Gaussian kernel in
Ft = (f′t , . . . , f′t−p+1 )′ .

77

3.3 Filtering and Diagnostics
In state–space applications such as the one analyzed here, interest lies also in the filtered values of the latent states (factors)
whether for diagnostic checking or forecasting. Filtering entails
the computation of the conditional expectation of some function
h(ft ) (including ft itself) given the information available up to
time t − 1 denoted by Yt−1 . This expectation takes the form of
a ratio of integrals
E[h(ft ) | Yt−1 ; ψ]


= · · · h(ft ) · p(ft | ft−1 ; ψ)
·

t−1


ϕτ (fτ , fτ −1 ; ψ) dft · · · df1

τ =1


 

···

 
t−1



ϕτ (fτ , fτ −1 ; ψ) dft−1 · · · df1 ,

τ =1

(26)

with p(ft | ft−1 ; ψ) and ϕτ (fτ , fτ −1 ; ψ) as defined by Equations
(15) and (16), respectively. These integrals are functionally similar to the likelihood integral in Equation (8) and can, in turn,
be accurately approximated by EIS (see Liesenfeld and Richard
2003).
Filtering enables us to compute the standardized Pearson
residuals defined as
ztj = [ytj − E(ytj | Yt−1 )]/ Var(ytj | Yt−1 )1/2 .

(27)

These residuals are critical components of a variety of diagnostic statistics since they should have zero mean, unit variance,
and be serially uncorrelated if the model is correctly specified.
Under the Poisson model, the relevant conditional moments of
ytj are given by E(ytj | Yt−1 ) = exp{μj } · E(exp{γ ′j ft } | Yt−1 )
and Var(ytj | Yt−1 ) = exp{μj } · E(exp{γ ′j ft } | Yt−1 ) + exp{2μj } ·
Var(exp{γ ′j ft } | Yt−1 ), where γ ′j denotes the jth row of the loading matrix Ŵ.
In order to check the distributional assumptions of the factor model we use the “randomized” version of the probability integral transform (PIT) proposed by Liesenfeld, Nolte, and
Pohlmeier (2006). The PIT values of a random variable consist
of its predictive cdf evaluated at the observed values. If the predictive cdf is correctly specified and continuous, the PIT values
follow a standard uniform distribution (see Rosenblatt 1958).
The randomized PIT approach accounts for the fact that the
PIT values for discrete predictive cdf’s are no longer uniform.
Let {yotj } denote the actual count observations. The application
of randomized PIT to the J count series is based on simulated
random draws u˜ tj (t = 1, . . . , T; j = 1, . . . , J) from uniform dis(l) (u)
tributions on the intervals [ctj , ctj ],


u˜ tj ∼ U ctj(l) , c(u)
tj , with
(u)

ctj = P(ytj ≤ yotj | Yt−1 ),

(28)

(l)

ctj = P(ytj ≤ yotj − 1 | Yt−1 ).
If the model is correctly specified, u˜ tj is a serially independent
random variable following a uniform distribution on the interval [0, 1]. Using the inverse of a standardized Gaussian cdf, the

78

Journal of Business & Economic Statistics, January 2011

Table 2. Sample correlation matrix for the number of trades

variable u˜ tj can be mapped into a N(0, 1)-distribution:
z∗tj = −1 (˜utj ),

(29)

such that the resulting normalized PIT residuals z∗tj should fol(u)
low a standard normal distribution. The probability ctj in
Equation (28) obtains by setting the function h(ft ) in Equayo
tion (26) equal to ytjtj =0 p(ytj | θtj ), where p(ytj | θtj ) denotes
the conditional density of ytj given θtj = exp{μj + γ ′j ft }. The
(l)
probability ctj obtains analogously.

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4. APPLICATION TO STOCK–MARKET ACTIVITY
The dynamic factor model for counts introduced in Section 2
is used to analyze the dynamics and correlations of stock market activity measured by the number of trades aggregated over
time intervals with a fixed length. Note that time aggregation of
the irregularly spaced trading events entails a loss of information on the dynamics of the trading process, which is increasing with the interval size. This raises the question of the “optimal” length of the aggregation interval. Following Heinen and
Rengifo (2007) and Quoreshi (2008), we use an interval length
of 5 min—a choice that is regarded as a good compromise between keeping the information loss low and avoiding too many
zeros for intervals which are too small.
4.1 The Data

WPP

EDE

NU

WR

1
0.238
0.256
0.206
0.168

1
0.256
0.174
0.183

1
0.283
0.184

1
0.216

1

of trades Q10 and Q20 including 10 and 20 lags, respectively, indicate strong serial correlation.
It is well known that intra-day trading activity has a distinctive U-shape pattern (see, e.g., Admati and Pfleiderer 1988). In
order to capture it, we introduce a Fourier series for the intercept in the log-mean function in Equation (3). Specifically, μj
is replaced by a cyclical term μtj defined as
μtj = μj + α ′ xt ,

(30)

with α ′ = (α1 , . . . , α4 ) and x′t = (cos(2πt/75), sin(2πt/75),
cos(4πt/75), sin(4πt/75)). EIS trivially accommodates such
extension. Note that specification in Equation (30) imposes a
common diurnal pattern for all stocks, thereby preserving parsimony for the multivariate model. This restriction is justified
by the finding that the individual patterns obtained from estimating univariate specifications (not presented here) are quite
similar across the five stocks.
4.2 Estimation Results

The dynamic factor model is applied to the number of trades
in 5-min intervals between 9:45 a.m. and 4:00 p.m. for J = 5
stocks from two industry sectors traded at the NYSE: P.H. Glatfelter Company (GLT), Wausau Paper Corporation (WPP), Empire District Electric Company (EDE), Northeast Utilities (NU),
and Westar Energy, Inc. (WR). The first two belong to the industry subsector paper and the last three to the industry subsector conventional electricity. Data are taken from the Trades and
Quotes (TAQ) dataset, provided by the NYSE. The time period
covered is the first quarter of 2005 (January 3, 2005–March 31,
2005) with 61 trading days. As there are 75 5-min intervals per
day, the sample size is T = 4575. See the top panel of Figure 2
(in Section 4.2.1) for time series plots of the number of trades.
Descriptive statistics are provided in Table 1 and Table 2 reports
the sample correlations across the five stocks, all of which are
positive. The empirical distribution of the number of trades is
clearly over-dispersed. The Ljung–Box statistics for the number
Table 1. Descriptive statistics for the number of trades

Mean
Median
Standard dev.
Minimum
Maximum
Q10
Q20

GLT
WPP
EDE
NU
WR

GLT

GLT

WPP

EDE

NU

WR

5.80
5
4.09
0
54
2086
2549

7.90
7
5.85
0
43
4057
5026

3.47
3
3.11
0
25
1575
1908

10.41
9
5.87
0
48
3452
3927

9.61
9
5.91
0
59
4150
5942

NOTE: The number of observations per stock is T = 4575. The Ljung–Box statistics
Q10 and Q20 include 10 and 20 lags.

4.2.1 Poisson Factor Model. For the Poisson factor model
with a single common market factor λt , two sector-specific factors τtsj and five idiosyncratic factors ωtj the log-mean function
for stock j is given by
τs

ln θtj = μtj + γjλ λt + γj j τtsj + ωtj ,

(31)

together with Equations (4)–(6) and (30). We set sj = 1 for j =
{1, 2} (GLT, WPP) and sj = 2 for j = {3, 4, 5} (EDE, NU, WR)
and use the restrictions γ1λ = γ1τ1 = γ3τ2 = 1 for normalization.
Joint ML-EIS estimates based upon N = 50 trajectories
are found in Table 3. ML-EIS estimation requires approximately 100 Broyden–Fletcher–Goldfarb–Shanno (BFGS) iterations and takes of the order of 200 min on a Core 2 Duo Intel 2.7 GHz processor using GAUSS on Windows XP. MC numerical standard deviations of the ML-EIS parameter estimates
used as measures of numerical precision are obtained from
20 iid ML-EIS estimations conducted under different CRN
seeds (see Richard and Zhang 2007 for details). They indicate
that the parameter estimates are numerically very accurate. The
fact that such high accuracy obtains with as little as N = 50
trajectories indicates that the likelihood integrands in Equation
(10) are very well-behaved functions of the 36,600 latent factor
variables and are accurately approximated by the EIS-sampler
(using 73,200 auxiliary parameters). In particular, the R2 ’s of
the EIS auxiliary LS-problems in Equation (12) are typically
larger than 0.99.
All parameter estimates are reasonable and apart from α4 significant at the 1% significance level. The estimates of the factor loadings γjλ , γjτ1 , and γjτ2 suggest that the trading activity
of all stocks load significantly on the common market factor

Jung, Liesenfeld, and Richard: Dynamic Factor Models for Multivariate Count Data

79

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Table 3. ML-EIS for the dynamic Poisson factor model for the TAQ data
GLT

WPP

EDE

NU

WR

μj

1.611
(0.026)
[0.0037]

1.871
(0.017)
[0.0018]

1.032
(0.037)
[0.0030]

2.247
(0.012)
[0.0005]

2.113
(0.038)
[0.0058]

γjλ

1.000

0.980
(0.060)
[0.0138]

2.351
(0.105)
[0.0128]

0.783
(0.040)
[0.0085]

0.826
(0.053)
[0.0107]

γj 1

τ

1.000

0.322
(0.052)
[0.0181]

0.000

0.000

0.000

γj 2

τ

0.000

0.000

1.000

−0.412
(0.134)
[0.0132]

−2.122
(0.237)
[0.0116]

δ ωj

0.952
(0.009)
[0.0011]

0.700
(0.017)
[0.0017]

0.971
(0.007)
[0.0009]

0.730
(0.018)
[0.0026]

0.978
(0.004)
[0.0005]

ν ωj

0.093
(0.010)
[0.0011]

0.392
(0.009)
[0.0012]

0.077
(0.013)
[0.0017]

0.236
(0.008)
[0.0012]

0.070
(0.008)
[0.0009]

δλ

νλ

δ τ1

ν τ1

δ τ2

ν τ2

0.327
(0.022)
[0.0018]

0.206
(0.007)
[0.0011]

0.245
(0.033)
[0.0033]

0.329
(0.009)
[0.0022]

0.277
(0.035)
[0.0034]

0.139
(0.013)
[0.0007]

α1

α2

α3

α4

Log-likelihood

0.269
(0.007)
[0.0008]

−0.052
(0.009)
[0.0010]

0.038
(0.009)
[0.0005]

−0.010
(0.006)
[0.0005]

−61,402.7

Factor param.

Seasonal param.

[2.276]

NOTE: The reported numbers are the ML-EIS estimates for the parameters. Asymptotic standard errors obtained from a numerical
approximation to the Hessian are in parentheses and MC (numerical) standard deviations obtained from 20 ML-EIS estimations
conducted under different sets of CRNs are in brackets. ML-EIS estimates are based on a MC sample size of N = 50 and three EIS
iterations.

λt and on their corresponding industry factor τt1 or τt2 . The
market factor appears to be dominated by EDE and the industry factors by GLT and WR, respectively. All factor loadings
are positive except for the NU and WR loadings on the factor of the electric-service industry τt2 . Their negative signs, together with the positive EDE loading on τt2 , are indicative of a
“substitution effect” in the trading activities of this sector. The
estimates of the parameters for the market factor (δ λ , ν λ ) and
the two industry-specific factors (δ τ1 , ν τ1 , δ τ2 , ν τ2 ) reveal that
they exhibit substantial variation and a slight, yet significant,
persistence. The estimates of the parameters characterizing the
idiosyncratic factors (δ ωj , ν ωj ) indicate substantially more persistence than for the common factors. Hence, the idiosyncratic
factors appear to capture the persistent movements of the trading process, while the common factors account for the more
transitory variation. In light of the information-flow interpretation associated with the mixture-of-distribution approach à la
Tauchen and Pitts (1983), this suggests that market participants
process market and industry-specific news significantly faster
than firm-specific news. Figure 1 shows the estimated diurnal
seasonal effects for the number of trades. They exhibit the welldocumented U-shape pattern.

The parameter estimates in Table 3 can be used to compute
the implied estimates of the means and covariances of the unconditional distribution for the number of trades, to be compared with their sample counterparts. In the presence of deterministic diurnal effects, the unconditional means and variances
for the trades of stock j are given by
E(ytj ) = ET (exp{μtj + 0.5γ ′j  f γ j }),

(32)

Var(ytj ) = VarT (exp{μtj + 0.5γ ′j  f γ j })

+ ET exp{2μtj + γ ′j  f γ j }[exp{γ ′j  f γ j } − 1]

(33)
+ exp{μtj + 0.5γ ′j  f γ j } ,

where γ j represents the vector of the factor loadings for stock j
and  f the stationary covariance matrix of the vector of factors ft . The notation ET and VarT indicate sample moments
computed with respect to the deterministic variation of the diurnal seasonal effects. The corresponding covariance between
trading of stock j and stock k is obtained from the cross moments

E(ytj ytk ) = ET exp{μtj + μtk

+ 0.5(γ j + γ k )′  f (γ j + γ k )} ,
j = k. (34)

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80

Figure 1. Estimated diurnal seasonal effects for the number
of trades obtained under the Poisson factor model given by
exp{0.269 cos(2π t/75) − 0.052 sin(2π t/75) + 0.038 cos(4π t/75) −
0.010 sin(4π t/75)}.

It can be shown that the resulting covariance Cov(ytj , ytk ) is positive (negative) as long as γ ′j  f γ k is larger (smaller) than zero.
The estimates of the unconditional mean and covariance matrix
of yt are given by
⎛
⎞
5.80
⎜ 7.93 ⎟
⎟
ˆ t) = ⎜
E(y
and
⎜ 3.47 ⎟
⎝
⎠
10.41
9.51
⎞
⎛
17.28
⎟
⎜ 6.01 38.71
⎟
 t) = ⎜
Var(y
4.41 9.88
⎟,
⎜ 3.28
⎠
⎝
4.79
6.48 4.53 33.26
4.49
6.08 3.01 9.12 38.59
respectively. The corresponding sample moments are
⎞
⎛
5.80
⎜ 7.90 ⎟
⎟
⎜
and
y¯ = ⎜ 3.47 ⎟
⎠
⎝
10.41
9.61
⎛
⎞
16.75
⎜ 5.68 34.17
⎟
⎜
⎟
 y¯ = ⎜ 3.26
4.65 9.65
⎟,
⎝
⎠
4.95
5.97 5.17 34.41
4.09
6.32 3.37 7.47 34.89

respectively. The close match between those two sets of moments indicates that the common factor model provides a good
parsimonious representation of the contemporaneous correlation across the five stocks.
In order to assess the relative importance of the market factor,
the industry factors, the idiosyncratic factors, and the diurnal
component we computed their relative contribution to the overall variance of the log-link function for the individual stocks
ln θtj . The implied estimates of those contributions are reported
in the upper panel of Table 5 (in Section 4.2.2). The fraction
of variation explained by the market factor varies between 11%

Journal of Business & Economic Statistics, January 2011

(WPP) and 62% (EDE) while that of the industry-specific factors range between 2% (NU) and 39% (GLT). Hence, the common factors (market and industry), considered together, explain
a significant fraction of the variation in the ln θtj s. However,
these results also suggest that the industry factors might actually capture additional firm-specific variations of GLT and WR
rather than genuine industry-specific effects.
The upper panel of Table 6 (in Section 4.2.2) summarizes the
results of diagnostic checks on the Pearson residuals ztj and the
normalized PIT residuals z∗tj . Both sample means and standard
deviations of the residuals ztj exceed their respective benchmark
values of 0 and 1 under a correctly specified model. This indicates that there is more variation and over-dispersion in the
data than the model accounts for. This is confirmed by the inspection of the time-series plots of the ztj -residuals (see middle
row of Figure 2). The Jarque–Bera statistic JB given in Table 6
indicates a strong rejection of normality for the z∗tj ’s suggesting
that the model also has difficulties accounting for the skewness
and tail behavior of the observed counts. This is corroborated
by the quantile–quantile plots of the z∗tj ’s provided in Figure 3,
which also confirm that the variance predicted by the model is
too small to explain the variation in the data. Finally, the Ljung–
Box statistics for the residuals ztj in Table 6 indicate that the
model does not fully capture the dynamic behavior of the trading activity even though it dramatically reduces the corresponding statistics for the raw data as given in Table 1.
4.2.2 Negbin Factor Model. In order to better account for
over-dispersion and to allow for greater flexibility in capturing
the serial correlation, we replace the conditional Poisson density in Equation (1) by the Negbin density in Equation (7). As
explained in Section 3.2, this substitution only requires minor
modifications of the baseline EIS algorithm.
The results of the ML-EIS estimation of the Negbin factor
model are reported in Table 4. The substitution of the Negbin
for the Poisson distribution increases the value of the maximized likelihood function by 193, indicative of a much better
fit. Moreover, the additional σ -parameters measuring the deviation from the Poisson distribution are all statistically significantly larger than zero. Nevertheless, the estimates of the intercept parameters (μj ), the factor loadings (γjλ , γjτ1 , γjτ2 ), and
of the seasonal parameters (αi ) obtained under the Poisson and
the Negbin specification are typically quite similar, indicating
a fairly robust factor structure. This conclusion is supported by
the comparison of the factors’ contribution to the overall variance of the stock-specific link functions obtained under both
distributions (see Table 5). Note, however, that under the Negbin distribution the factor loading of NU and WR on the industry factor τt2 reported in Table 4 is no longer significant.
Furthermore, noticeable differences are observed for the parameters characterizing the idiosyncratic factors for WPP and NU
as well as the industry factor τt2 . For those factors, the move to
the Negbin model has substantially increased the δ-parameters
and decreased the ν-parameters, implying smoother evolution
over time and greater persistence. These differences also suggest that a significant part of the variation in trading activities
of the corresponding stocks, which was attributed to persistent shocks in the factor processes under the Poisson model, is
now interpreted as transitory and attributed to conditional overdispersion (σj > 0) under the Negbin model.

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Jung, Liesenfeld, and Richard: Dynamic Factor Models for Multivariate Count Data

81

Figure 2. Time series of the number of trades in 5-minute intervals ytj (top row); Pearson residuals ztj obtained from the Poisson factor model
(middle row); Pearson residuals ztj obtained from the Negbin factor model (bottom row).

The lower panel of Table 6 summarizes the properties of the
Pearson residuals ztj and the normalized PIT residuals z∗tj obtained from the Negbin factor model. The mean and standard
deviation of the ztj ’s suggest that, in contrast to the Poisson
model, the Negbin specification accounts for most of the observed over-dispersion, except possibly for GLT (with a standard deviation of 1.14). However, closer inspection of the GLTresiduals indicates that this large standard deviation is essentially due to a single outlier (see bottom panel of Figure 2). The
Jarque–Bera test still rejects normality for the GLT, NU, and
WR normalized residuals, while the WPP and EDE normalized
residuals pass this test. The corresponding quantile–quantile
plots provided in Figure 4 reveal that those rejections are due
to deviations from normality in the right tail of the distribution
generating significant skewness and excess kurtosis. This suggests replacing the Negbin by a distribution allowing for thicker
right tails. Potential candidates to be explored in future research
are Consul’s (1989) generalized Poisson distribution (see Joe
and Zhu 2005) or seminonparametric count distributions based
upon series expansions as proposed by Guo and Trivedi (2002).
Alternatively, one might consider replacing in Equations (4)–
(6) the conditional Gaussian distribution for the latent factors
by a fat-tailed distribution like the Student-t. However, assuming a Student-t distribution for the persistent factor processes

will also imply temporal clustering of extreme events as well as
an excess of zero observations, both of which are not observed
for our count data.
The Ljung–Box statistic for the ztj residuals reported in Table 6 show that the Negbin model successfully accounts for the
serial correlation in all count series expect possibly for WPP.
In order to more completely capture the dynamics one can generalize Equations (4)–(6) to higher-order AR(p) processes. An
easier alternative, which requires less modifications of our baseline EIS algorithm, consists of replacing the industry factor for
GLT and WPP (τ1t ) by a second additional idiosyncratic AR(1)
factor for the problematic WPP stock. The Ljung–Box statistic of the Pearson residuals obtained from this modified model
(not presented here) indicates that it indeed accounts for the observed serial correlation of all stocks including WPP.
All in all, our results show that the Negbin factor model provides a good description of stock market activities. They also
provide robust evidence for a common dynamic market factor—
akin to the common factors for multivariate stock-price intensities identified by Bauwens and Hautsch (2006) and for volatility processes across asset returns found by Diebold and Nerlove
(1989), Aguilar and West (2000), and Chib, Nardari, and Shephard (2006).

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82

Journal of Business & Economic Statistics, January 2011

Figure 3. Quantile plots of the normalized PIT residuals z∗tj obtained from the Poisson factor model. The solid line plots the quantiles of
the standard normal distribution against the quantiles of the standard normal distribution and the dotted line plots the sorted z∗tj ’s against the
quantiles of the standard normal distribution.

5. CONCLUSION
We can draw three sets of conclusions from the application
we present in this article. With respect to modeling multivariate time series of counts, we illustrate that our proposed parsimonious and easy-to-interpret dynamic factor model is able to
represent nontrivial contemporaneous and temporal interdependencies across count series and provides a useful framework for
the analysis of high-dimensional time series of counts.
In regard to our application to the number of trades for five
NYSE stocks itself, we find robust evidence for a common factor reflecting market-wide news and accounting for observed
comovements in trading activities across the five individual
stocks analyzed here. While the two industry-specific factors
do capture additional variations in trading activities, they appear to represent additional firm-specific factors for two stocks
rather than genuine industry-specific factors. We also find that
the Negbin clearly dominates the Poisson distribution in terms
of accounting for observed over-dispersion and serial correlation of trading volumes.
From a numerical viewpoint, we demonstrate that EIS enables us to analyze complex factor structures in the context of
dynamic multivariate discrete models, at least as long as the dynamics of the model are specified in the form of Gaussian autoregressive factors. (Work-in-progress should allow us to relax
such restrictions, but goes beyond the objectives of the present
article.) The baseline EIS algorithm requires only minor adjustments to accommodate alternative specifications (factor structure and/or discrete distribution) thereby providing a high degree of flexibility for the analysis of complex dynamic factor

structures. Last but not least, application of the proposed factor mo