3: Generalized Linear Autoregressive Moving Average Models (2/6)

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56 Handbook of Discrete-Valued Time Series

for serial dependence. To illustrate, consider the case where no serial dependence exists

but p = q > 0 is specied. Then the likelihood iterations are unlikely to converge because

the likelihood surface will be “ridge-like” on the manifold where φ

=−θ

, an issue that is

encountered for standard ARMA model tting. Corresponding to this, the second deriva-

tive matrix D

(δ) will be singular or the state variable W

can degenerate or diverge.

Because of this possibility, it is prudent to start with low orders for p and q and avoid speci-

fying them as equal. Once stability of estimation is reached for a lower-order specication,

increasing the values of p or q could be attempted.

The likelihood ratio test that there is no serial dependence versus the alternative

that there is GLARMA-like serial dependence with p =q > 0 will not have a standard

chi-squared distribution because the parameters φ

,for j = 1, ..., p, are nuisance parame-

ters which cannot be estimated under the null hypothesis. Testing methods such as those

proposed by Hansen (1996) or Davies (1987) need to be developed for this situation. Further

details on these points can be found in Dunsmuir and Scott (2015).

3.2.3 Distribution Theory for Likelihood Estimation

The consistency and asymptotic distribution for the maximum likelihood estimate

is rig-

orously established only in a limited number of special cases. In the stationary Poisson

response case in Davis et al. (2003) where x

≡1 (intercept only) and p = 0and q = 1,

these results have been proved rigorously. Similarly, for simple models in the Bernoulli

stationary case in Streett (2000) these results hold. Simulation results are also reported in

Davis et al. (1999, 2003) for nonstationary Poisson models. Other simulations not reported

in the literature support the supposition that

has a multivariate normal distribution for

large samples for a range of regression designs and for the various response distributions

considered here.

For inference in the GLARMA model, it is assumed that the central limit theorem holds

so that

≈ N(δ,

), (3.11)

where the approximate covariance matrix is estimated by



=−D

(

)

−1



−D

(

)

−1

.In the glarma package, this distribution is used to obtain standard errors

and to construct Wald tests of the hypotheses that subsets of δ are zero. It is also assumed

that Wald tests and equivalent likelihood ratio tests will be asympotically chi-squared with

the correct degrees of freedom, results which would follow straightforwardly from (3.11)

and its proof when available. Regardless of the technical issues involved in establishing

a general central limit theorem, the earlier approximate result seems plausible since, for

these models, the log-likelihood is a sum of elements in a triangular array of martingale

differences. Conditions under which this result would likely hold include identiability

conditions as discussed earlier, conditions on the regressors similar to those used in Davis

et al. (2000, 2003) and Davis and Wu (2009), where the covariates x

are assumed to be a

realization of a stationary time series or is dened as x

= x

= f(t/n) where f(u) is a piece-

wise continuous function from u ∈[0, 1] to R

. Additional conditions on the coefcients

are also needed to ensure that Z

and hence W

do not degenerate or grow without bound.

Indeed, little is known so far about suitable conditions to ensure this.

57 Generalized Linear Autoregressive Moving Average Models

3.2.4 Convergence of GLARMA Recursions: Ergodicity and Stationarity

To date, the stationarity and ergodicity properties of the GLARMA model are only partially

understood. These properties are important to ensure that the process is capable of gener-

ating sample paths that do not degenerate to zero or do not explode as time progresses, as

well as for establishing the large sample distributional properties of parameter estimates.

Davis et al. (2003) provide results for the simplest of all models: Poisson responses specied

with p = 0, q = 1, and x

β = β. Results for simple examples of the stationary Bernoulli

case are given in Streett (2000).

For the Poisson response distribution GLARMA model, failure to scale by the variance

or standard deviation will lead to unstable Poisson means (that diverge to innity or col-

lapse to zero as an absorbing state for instance) and existence of stationary and ergodic

solutions to the recursive state equation is not assured—see Davis et al. (1999, 2003, 2005)

for details. For the binomial situation, this lack of scaling should not necessarily lead to

instability in the success probability as time evolves since the success probabilities, p

,and

observed responses, Y

, are both bounded between 0 and 1. Thus, degeneracy can only arise

if the regressors x

become unbounded. As recommended in Davis et al. (1999), temporal

trend regressors should be scaled using a factor relating to the sample size n.

Asymptotic results for various types of observation-driven models without covariates

are increasingly becoming available. Tjøstheim (2012) (also see Tjøstheim [2015; Chapter 4

in this volume]) has provided a review of the ergodic and stationarity properties for var-

ious observation-driven models—primarily in the Poisson response context—as well as

presenting large sample theory for likelihood estimation. Wang and Li (2011) discuss the

binary (BARMA) model and present some asymptotic results for that case. However, the

state equation for the BARMA model differs from that for the binary GLARMA model in

that the latter involved scaled residuals while the former uses identity residuals and is also

structurally different in its use of past observation of {Y

}. Davis and Liu (2015) present

general results for the one-parameter exponential family response distributions and a semi-

parametric observation-driven specication of the state equation. Woodard et al. (2011)

present some general results on stationarity and ergodicity for the GARMA models similar

to those available in Benjamin et al. (2003). However, because the state equation recursions

involve applying the link function to both the responses and the mean responses, none of

these results apply to the specic form of GLARMA models presented here. Also, none of

these recent results consider the case of covariates; hence they are not, as yet, applicable to

likelihood estimation for regression models for discrete outcome time series.

3.3 Application of Univariate GLARMA Models

We now illustrate the tting of GLARMA models to binomial and binary time series aris-

ing in the study of listener responses to a segment of electroacoustic music. This example

also motivates studying multiple independent time series as an ensemble (as will be dis-

cussed in the next section). The background to the analysis presented here is in Dean et al.

(2014b). Members of a panel comprising three musical expertise groups (8 electroacoustic

musicians, 8 musicians, and 16 nonmusicians) provided real-time responses to a segment

of electroacoustic music. Aims of these types of experiments are to determine the way

in which features of the music (in this case the sound intensity) impact listener response



58 Handbook of Discrete-Valued Time Series

measured in various ways (we concentrate on the “arousal” response). Dean et al. (2014b)

present a variety of standard time series methods for modeling the arousal responsive-

ness in terms of lags of the musical intensity. Questions such as: “Do the musical expertise

groups display differences between them with respect to the impact of intensity on their

group average responses?” and “Are there substantial differences between individuals

within the panel or within each musical group?” were addressed. For example, in Dean

et al. (2014b), the transfer function coefcients for the impact of changes, at lag 1, of musical

intensity on the changes in arousal were modeled using 11 lag transfer function models of

the form

∇Y

= ω

+ ω

∇X

t−k

+ α

, (3.12)

k=1

where ∇A

= A

− A

t−1

and α

was modeled by an autoregression of at most order 3.

There was evidence that variation between individuals in their responsiveness was sub-

stantial and suggested use of a cross-sectional time series analysis as in Dean et al. (2014a).

However, analysis of individual responses suggested varying levels of “stasis”; that is,

frequent and sometime prolonged periods during which their arousal response did not

change. In some cases, this led to a high level of zeros in their differenced responses, which

has the potential to impact the validity of model estimates based on traditional Gaussian

linear time series analysis. In fact, listeners varied substantially in the amount of time that

their responses are in stasis, for example, from 15% to 90%. Response distributions with

such large numbers of zeros constitute a challenge for conventional time series analysis.

To examine how robust their ndings were based on standard Gaussian linear times

series methods, Dean et al. (2014b) also considered an approach similar to that employed

in Rydberg and Shephard (2003) for decomposing transaction level stock price data into

components of change and size of change. For each listener, binary responses were dened

as D

= 1 if the change in their response from time t−1tot was positive, otherwise D

= 0.

This is one of the two components of a potential trinomial model for the change process

for which the approach of Liesenfeld et al. (2006) could be used; currently, the glarma

package does not handle trinomial responses.

In order to answer “Are there variations between the three musical groups with respect

to their group average responses to the same musical excerpt?” the binary time series

were aggregated at each time into binomial counts of 8, 8, and 16 respondents at each time

in the EA, M, and NM musical groups. The multiple GLARMA xed effects modeling

(described in the next section) was used to examine the differences between group average

responses. For this, it is assumed that the aggregated counts obey a binomial distribution.

The independence of trials assumption is not in doubt since the individuals responded

independently to the musical excerpt. However, the assumption that each individual in

the group shares the same probability of response at each time appears in doubt as we now

explain.



We consider the nonmusician group of 16 listeners. Let S

j=1

count the number

of respondents whose arousal change was positive and assume that S

∼ Bin(π

,16), where

logit(π

) = ω

+ ω

∇X

t−k

+ α

(3.13)

k=1



59 Generalized Linear Autoregressive Moving Average Models

−0.5

0.0

0.5

1.0

1.5

2.0

TF coefficient

−0.5

0.0

0.5

1.0

1.5

2.0

TF coefficient

−0.5

0.0

0.5

1.0

1.5

2.0

TF coefficient

246810

246810 246810

(a)

Lag (b) Lag (c) Lag

FIGURE 3.1

Transfer function coefcients for aggregated binomial series. (a) 16 nonmusicians (b) Listener 22 (c) Listener 23.

and α

follows a suitably specied GLARMA process. For this group, we found that (p, q) =

(3, 0) was adequate (based on minimum AIC). The tted transfer function coefcients

with individual 95% condence limits under the assumption of normality are shown in

Figure 3.1a. Most transfer functions coefcients are individually signicant. The observed

binary responses (as probabilities) along with the xed effects t and GLARMA model t is

shown in Figure 3.2a. The nonrandomized probability integral transform (PIT) residual plot

for this t is shown in Figure 3.3. Clearly the binomial assumption is not correct as the PIT

plot suggests that the binomial distribution is not providing a good prediction of the prob-

ability of small or large counts. This is not surprising since the 16 individuals in this group

show substantial variability in their “stasis” levels, which of course will impact the average

probability of a positive change in arousal. The PIT analysis suggests that aggregation of

individual binary responses in this way is not appropriate.

Consequently, we turn to analysis of two individual binary responses to illustrate the

application of GLARMA model for binary data using the individual models

logit(π

) = ω

j,0

+ ω

j,k

∇X

t−k

+ α

, (3.14)

k=1

where {α

} is a GLARMA process for the jth series. For most of the series, p = 1and q = 0

seemed appropriate; hence,we settle on this for all 32 series. We illustrate the results of such

ts on two listeners: listener 22, who had 88% stasis, and listener 23, who had 52% stasis.

With such a high proportion of zero changes in arousal responses, it is not clear that appli-

cation of standard Gaussian time series transfer function modeling would be reasonable

for these two cases. We modeled the binary response series {D

} using a binary GLARMA

model with probabilities specied as in (3.14). Figure 3.1b and c shows the estimated val-

ues of the transfer function coefcients ω

j,k

along with 95% signicance levels. There are

clear differences between the two individual responses to the same musical excerpt. The

tted values for these cases are shown in Figure 3.2b and c. There are also substantial dif-

ferences between overall level as measured by the intercept terms ( ωˆ

22,0

=−3.24 ± 0.39,

ωˆ

23,0

=−1.21 ± 0.54), which is consistent with the relative levels of stasis observed for

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60 Handbook of Discrete-Valued Time Series

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FIGURE 3.2

Binomial time series for 16 NM groups, response with GLARMA ts of response to changes in musical inten-

sity and two individual binary time series with ts. (a) Binomial Fit: 16 Nonmusician Listeners (b) Binary Fit:

Listener 22 (c) Binary Fit: Listener 23.

Binomial GLARMA model: 16 nonmusician listeners

Relative frequency

0.0

1.0

2.0

0.0 0.2 0.4 0.6 0.8 1.0

PIT

FIGURE 3.3

PIT residual plot for GLARMA model for binomial counts of positive responses in 16 nonmusician group listeners.

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