61 Generalized Linear Autoregressive Moving Average Models
these two individuals as reported earlier. The GLARMA model autoregressive coefcients
also substantially differed (φ
ˆ
22
= 0.687 ± 0.079 and φ
ˆ
23
= 0.894 ± 0.024). This also holds
when all 32 individual responses are modeled in this way.
We have also applied GLARMA modeling with the glarma package on binary and
binomial time series to study responsiveness to musical features in the complete panel
of 32 listeners. A clear conclusion from this extended analysis is that there is a strong
need to consider modeling of individual responses in order to address questions such
as: “Are there differences between musical expertise groups?”. Also clear is the need to
accommodate differing amounts of serial dependence for each series in the ensemble,
something that even current longitudinal data analysis for mixed models does not read-
ily allow. In the next section we explain two approaches to allowing for variation between
individual time series.
3.4 GLARMA Models for Multiple Independent Time Series
3.4.1 Examples
The musicology example of Section 3.3 is an example where ensembles of individual inde-
pendent responses need to be modeled together. In a public health policy setting, Bernat
et al. (2004) assessed, using a pooled cross-sectional random effects analysis, the impact of
lowering the legal allowable blood alcohol concentration (BAC) in motor vehicle drivers
from 0.10 to 0.08 on monthly counts of single vehicle night time fatalities in 17 U.S. states.
In that study, serial dependence was detected in some series but could not be modeled with
software and methods available at that time. The purpose of the remainder of this chapter
is to present methods that overcome these gaps.
Two approaches are developed here. The rst is referred to as the xed effects plus
GLARMA specication. Here, regression effects and serial dependence parameters may be
constrained across series to test various hypotheses of interest primarily about regression
effects, but also, about variation in serial dependence between series. For long longitu-
dinal data each series can be estimated and modeled individually using the GLARMA
models previoulsy discussed and combined likelihoods for constrained parameterizations
constructed on which to base inference. Here, the length of the individual time series allows
considerable exibility in serial dependence structures between individual series. In tradi-
tional longitudinal data analysis where the number of repeated measures is low, it may not
be possible to allow for individuality of this type.
The second approach is based on a random effect specication of the regression compo-
nent while allowing individual series to have different serial dependence structures and
strengths. This is not possible for traditional longitudinal data and the methods proposed
here is a substantial extension of existing methodology, which is enabled by the length of
the individual series.
Before dening models that combine xed or random effects with GLARMA serial
dependence, we mention some other recent examples along these lines. Xu et al. (2007)
present a parameter-driven specication for serial dependence with random effects on
covariates. Their approach is limited by the requirement that only autoregressive serial
dependence is covered and all series must share the same structure and parameter val-
ues. Additionally, the method has not been demonstrated, neither in simulation nor in
62 Handbook of Discrete-Valued Time Series
application, on series longer than 20 time points. Zhang et al. (2012) specify an exponential
decay autocorrelation on a latent process and use marginal estimation methods for their
combined serial dependence and random effects specication. They also assume that the
structure and strength must be the same for all series. These approaches, which force the
same dependence structure on all series, may be a legacy from the modeling of traditional
longitudinal data in which the individual series are very short and for which the ability to
detect different serial dependence properties between series is limited. It is our view that
for long longitudinal data, this restriction is articial for two reasons. First, in all exam-
ples we have encountered, there is strong evidence that the serial dependence structure
and strength varies between series; some series will have strong serial dependence, while
others will have none at all. One explanation for this is that the covariates used for the
ensemble for xed and random effect terms may not be inclusive of covariates that are
unavailable and impact individual series—in such circumstances serial dependence may
be stronger simply because it acts as a proxy for unobserved covariates. The second reason
relates to the fact that the use of random effects on regression variables should logically
extend to their use for serial dependence parameters. The models and methods presented
here provide considerably more exibility than those proposed by Xu et al. (2007) and
Zhang et al. (2012). However, being based on GLARMA models, which require regular
time spacing of observed outcomes, they cannot handle irregular spacings. For this irreg-
ularly spaces observations, parameter-driven models for the individual series would be
more appropriate.
It is our view that, more frequently, increasingly long longitudinal data will become
available and methods such as those proposed here will be become needed. For exam-
ple, studies with panels of subjects equipped with automatic data loggers measuring their
physical condition and activity levels are now quite feasible.
3.4.2 General Model for Multiple Independent GLARMA Models with Random Effects
Let Y
jt
be the observation at time t = 1, ..., n
j
on the jth series of counts, where j = 1, ..., J
and let x
j,t
be the covariates for the jth series. We also let r
jt
denote d random effect covariates
that apply to all series through coefcients represented as vectors of normally distributed
random effects U
j
∼ i.i.d N(0, (λ)), where is a d × d covariance matrix determined by
parameters λ.
In addition to assuming that the series
Y
jt
are independent across the J cases, for each j,
we also assume that, given the state process W
j,t
, Y
jt
are independent with exponential
family distribution (3.1), where
W
j,t
= x
T
j,t
β
(j)
+ r
T
jt
U
j
+ Z
jt
(3.15)
is the linear state process for the jth case series. Typically, some of the covariates appear in
both x
j,t
and r
jt
in which case corresponding components of the β
(j)
will not depend on j
and some or all of the covariates can be the same across all series.
Serial dependence is modeled with the Z
jt
assumed to satisfy (3.6) with degrees
p
(j)
, q
(j)
p
(j)
q
(j)
Z
jt
= φ
l
(j)
Z
j,t−l
+ e
j,t−l
+
θ
l
(j)
e
j,t−l
. (3.16)
l=1 l=1
63 Generalized Linear Autoregressive Moving Average Models
Let τ
(j)
=
φ
1
(j)
, ..., φ
p
(j
(
)
j)
, θ
1
(j)
, ..., θ
q
(j
(
)
j)
. We distinguish three special cases for the general
state equation (3.15).
1. Generalized linear mixed model: In this specication, the serial dependence term Z
jt
is
absent, giving the standard generalized linear mixed model for longitudinal data
discussed in Diggle et al. (2002) and Fitzmaurice et al. (2012)
W
j,t
= x
j
T
,t
β
(j)
+ r
jt
T
U
j
. (3.17)
The only source of within series correlation is in the random effects component.
Various packages (such as SAS and R) are available for model tting.
2. Fixed effects multiple GLARMA model: Random effects are not included but there is
a serial dependence in the form of an observation-driven model:
W
j,t
= x
j
T
,t
β
(j)
+ Z
jt
, (3.18)
where Z
jt
is given by (3.16).
3. Random effects multiple GLARMA model: Both random effects and serial depen-
dence are present. This is the general specication for W
jt
given by (3.15), with
Z
jt
specied by the GLARMA model (3.16).
Model Class 1 was used in Bernat et al. (2004). A previous example of Model Class 2 type
of modeling was considered in Dunsmuir et al. (2004), who investigated the commonal-
ity of regression impacts on three series of daily asthma presentation counts. Extension to
datasets with substantially more time series and more complex hypotheses on the xed
effects parameters required considerable development of the previous software. The ran-
dom effects multiple GLARMA model is developed in detail in Dunsmuir et al. (2014).
The BAC dataset will be used to illustrate tting of both models (3.18) and (3.15).
3.5 Fixed Effects Multiple GLARMA Model
3.5.1 Maximum Likelihood Estimation
In this section, we develop maximum likelihood estimation for the model with individual
series state equations specied as in (3.18). Let the log-likelihood for the jth series be as
in (3.7) and denoted by l
j
θ
(j)
where θ
(j)T
=
β
(j)T
, τ
(j)T
. The log-likelihood across all
series is
J
l(θ) =
l
j
(θ
(j)
) (3.19)
j=1
where β
T
=
β
(1)T
, ..., β
(J)T
and τ
T
=
τ
(1)T
, ..., τ
(J)T
.Let θ =
β
T
, τ
T
T
denote all
parameters.
64 Handbook of Discrete-Valued Time Series
A primary focus is on testing if the parameterization across series can be simplied
so that regression coefcients or serial dependence parameters can be constrained to be
the same. We consider only linear constraints of the form θ = Aψ where A has fewer
columns than rows and ψ denotes the lower-dimensional vector of parameters in the con-
strained model. Typically ψ
T
=
ψ
β
T
, ψ
T
τ
since we will generally not be interested in
relating the regression coefcients to the serial dependence parameters. In that case A will
be block diagonal appropriately partitioned. Denote the log-likelihood with respect to the
constrained parameters as l(ψ) = l(Aθ).
Maximization of (3.19) with respect to the constrained parameters can be over a high-
dimensional parameter space. Initial estimates of
θ
ˆ
are obtained by maximizing (3.19)
without constraints which is the same as maximizing all individual likelihoods separately
and combining the resulting l
j
β
ˆ
(j)
,
τ
ˆ
(j)
. These unconstrained estimates are used to ini-
tialize the constrained parameters via
ψ
ˆ
(0)
= (A
T
A)
−1
A
T
θ
ˆ
. Next, using the appropriate
components of
θ
ˆ
(0)
= A
ψ
ˆ
(0)
, each component log-likelihood l
j
θ
ˆ
(
(
0
j)
)
and its derivatives
are calculated using the standard GLARMA software. Finally, these are combined to get
the overall l
ψ
ˆ
(0)
. Derivatives with respect to ψ can be obtained using the identities
∂l(ψ)/∂ψ = A
T
∂l(Aψ)/∂θ and ∂
2
l(ψ)/∂ψ∂ψ
T
= A
T
∂
2
l(Aψ)/∂θ∂θ
T
A. This procedure
is then iterated to convergence using the Newton–Raphson or Fisher scoring algorithm.
Fisher scoring was found to be more stable in the initial stages. Once the derivative
∂l(ψ)/∂ψ settles down, the iterative search for the optimum can be switched to the
Newton–Raphson updates, which typically gives speedier convergence.
Similar to the single series case, the asymptotic properties of the MLEs
ψ
ˆ
have not been
established for the general model previously described. Asymptotic results for longitudi-
nal data typically let the number of cases J tend to innity and the lengths of individual
series n
j
are typically held xed. For this scenario, asymptotic theory is typically straight-
forward since it relies on large numbers of independent trajectories. For our applications,
J is often bounded and we perceive of all n
j
as tending to innity in which case asymp-
totic results rely on those for individual time series which, as previously noted, are rather
underdeveloped. We assume, however, that asymptotic results hold and perform infer-
¨
ence in the usual way. For example, the matrices of second derivatives
l(
ψ
ˆ
), computed in
the course of the Newton–Raphson or Fisher scoring maximization procedures, are used to
estimate standard errors for individual parameters. Also, to test the null hypothesis of com-
mon regression effects, we use the likelihood ratio statistic G
2
=−2
l(
ψ
ˆ
0
) − l(
ψ
ˆ
1
)
, where
ψ
ˆ
0
is the estimate obtained under the null hypothesis and
ψ
ˆ
1
the estimate obtained under
the alternative. Degrees of freedom, for the chi-squared approximate reference distribu-
tion for G
2
, are calculated in the usual way. These ideas were rst illustrated in Dunsmuir
et al. (2004) for three series of daily asthma counts which are assessed for common seasonal
patterns, day of the week, weather, and pollution effects.
3.5.2 Application of Multiple Fixed Effects GLARMA to Road Deaths
Bernat et al. (2004) assessed the impact of lowering the legal allowable BAC in motor vehicle
drivers from 0.10 to 0.08 on monthly counts of single vehicle night time fatalities in 17 U.S.
states for which at least 12 months of post intervention data were available. The study
design selected 72 consecutive months of data with 36 months prior to the decrease in
65 Generalized Linear Autoregressive Moving Average Models
allowable BAC and up to 36 months after for each of the 17 states used in the analysis.
The mixed effects model presented in Bernat et al. (2004) assumed that the observed counts
of single vehicle night time deaths, Y
j,t
in month t for state j, had, conditional on the random
effects U
j
, a Poisson distribution with log mean given by
W
jt
= β
0
+ β
1
I
1
(t) + β
2
I
2,j
(t) + β
3
x
3,j,t
+ β
4
x
4,j,t
+ O
j,t
+ r
jt
T
U
j
, (3.20)
where I
1
(t) is the indicator variable for the change in BAC from 0.1 to 0.08 coded as 0 for
t ≤ 36 and 1 for t > 36 for all states in the study, and I
2,j
(t) the indicator variable taking the
value 0 for t < T
j
and 1 for t ≥ T
j
, where T
j
is the month in which an administrative license
revocation law was enacted. This potential confounder was enacted in seven of the states
during the period of data used there. The other two regression variables are x
3,j,t
, the num-
ber of Friday and Saturday nights, and a control series x
4,j,t
, the log of other motor vehicle
deaths (adjusted for population and seasonal factors), in month t for state j. An offset term,
O
j,t
, was used to adjust for unique state population trends and seasonal factors. Finally,
U
j
∼ N(0, G) is the multivariate normal distribution with covariance matrix G for the ran-
dom effects on the selected covariates in r
jt
. Further details concerning the rationale and
denition of the model components mentioned earlier can be found in Bernat et al. (2004).
Because of the differences between the time periods over which the data were collected
from each of the 17 states and because of the control series used for each state, the assump-
tion that the 17 series are independent was considered reasonable. We also conrmed
this assumption using cross-correlation analysis of Pearson residuals from individual
GLARMA model ts to the 17 series. Bernat et al. (2004) discussed the likely impact that any
serial dependence might have on their key conclusion that there is a statistically signicant
lowering of overall average single vehicle night time fatalities associated with the lower-
ing of the legal BAC level but suitable software was not available at that time to assess this
statistically.
We now illustrate the xed effects plus GLARMA model dened in (3.18) on these data.
The rst step was to use the glarma package to t the model (3.18) to each of the indi-
vidual series using regression and offset terms as specied in (3.20). Examination of the
PIT residual plots for each series suggested overtting relative to the Poisson distribution.
We identied that the main contributor to overtting was the seasonal offset term used in
(3.20); this had been determined using standard seasonal adjustment methods available in
the PROC X-11 package in SAS for each series separately and based on only n = 72 months
of data. As an alternative, we used parametric harmonic seasonal terms to all 17 response
series with the other regressors and the population offset only. In almost all cases, these
seasonal harmonics were not signicant suggesting that use of the control series (without
seasonal adjustment) and lag 12 autoregressive terms in the GLARMA model are sufcient
for modeling seasonality. In view of this, we performed our reanalysis of these data using
the regression specication in (3.20), but dropping the seasonal adjustment in both the off-
set O
j,t
and logOMVD (so that the control series is now log other motor vehicle deaths per
100,000 population without seasonal adjustment), together with a GLARMA process for
serial dependence. We compared various combinations of lags for the autoregressive and
moving average terms in the GLARMA model to allow for both serial dependence at low
orders and at seasonal lags. Based on AIC the overall best specication was an autoregres-
sion of order p = 12 with zero coefcients at lags 1 through 11. Hence we initially adopted
this seasonal model for all series.
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