44 Handbook of Discrete-Valued Time Series
For count time series, q-dependence for some positive integer q is generally not expected
based on context, as there may be no reason for independence to occur with longer lags.
Mixed Markov/q-dependent models with small p, q are better models to consider if the
dependence is not as simple as Markov dependence.
The copula version with p = q = 1 is dened below (it does extend to p, q ≥ 1by
combining the above constructions for Markov order p and q-dependence). Let C
12
and K
12
be two different bivariate copulas. Dene
= C
−1
= K
−1
= F
−1
W
s
2|1
(
s
|W
s−1
), U
t
2|1
(
t
|W
t−1
), Y
t
Y
(U
t
),
where {
t
} is a sequence of innovation random variables and {W
s
} is a sequence of unob-
served U(0, 1) random variables.
2.6 Statistical Inference and Model Comparisons
For NB margins, numerical maximum likelihood is possible for all of the Markov models in
the preceding three sections. For GP margins, all are possible except the ones based on the
generalized thinning operators denoted as I2 and I3. With numerical maximum likelihood
for Markov models, the asymptotic likelihood inference theory is similar to that for inde-
pendent observations, using the results in Billingsley (1961). The CLS method can estimate
the mean of a stationary distribution but not additional parameters such as overdispersion.
For applications to count time series with small counts, generally low-order Markov
models are adequate. The q-dependent models which are the analogies of Gaussian MA(q)
are usually less appealing within the context of count data. The analogue of Gaussian
ARMA(1,1) has been used in the literature on count time series, but there is less experience
with such models. It would be desirable to have the ARMA(1,1) equivalent of the above
models in the three categories. The joint likelihood of Y
1
, ..., Y
n
(time series of length n)
involves either a high-dimensional sum or integral so that maximum likelihood estimation
is not feasible. However for the ARMA(1,1) analogue in the CCID class or copula approach,
the joint pmf of three consecutive observations is, respectively, a triple sum, and a triple
sum plus an integral. Therefore, likelihood inference could proceed with composite like-
lihood based on sum of log-likelihoods of subsets of three consecutive observations, as in
Davis and Yau (2011) and Ng et al. (2011). Because of space constraints, the data analysis
below will only involve Markov models.
In the remainder of this section, the three classes of Markov count time series models are
tted and discussed for one data set, and some results are briey summarized for another
data set that had previously been analyzed.
For some count time data sets with low counts, serial dependence and explanation of
trends from covariates, we consider monthly downloads to specialized software. A source
for such series is http://econpapers.repec.org/software/. Such data are con-
sidered here because the next month’s total download is plausibly based on the current
month’s total download through a generalized thinning operator. A similar data set with
daily downloads for the program TeXpert is used in Weiß (2008).
The specic series consist of the monthly downloads of Signalpak (a package for
octave, which is a Matlab clone) from September 2000 to March 2013. See Figure 2.1.
45 Markov Models for Count Time Series
1.0
30
25
0.6
20
2002 2006 2010
ACF
Signalpak downloads
15
0.2
5
0
10
−0.2
0.0 0.5 1.0 1.5
Year Lag
FIGURE 2.1
Time series plot of monthly download of signalpak; and the autocorrelation function.
This series looks close to stationary. For longer periods of time, there is no reason to expect
the series to be stationary because software can go through periods or local trends of more
or less demand. We also consider a covariate for forecasting that is a surrogate for the
popularity of the octave software; the surrogate is the total monthly abstract views of
octave codes that are at the website referred to earlier. A summary of model ts is given in
Table 2.1.
The use of the covariate marginally improves the log-likelihood and root mean square
prediction error. Otherwise the Markov order 1 models t about the same, and Markov
2 models did not add explanatory power. As might be expected, the generalized thinning
operators t a little better than binomial thinning since those operators are more intuitive
for the context of these data. For thinning operators I2 and I3,the γ parameter was set
to be the largest possible so that they lead to more conditional heteroscedasticity than the
model with binomial thinning. Experience with I2 and I3 is that for short time series, if γ
is estimated, it is usually at one of the boundaries.
Table 2.1 lists maximum likelihood estimates and corresponding standard errors for one
of the better tting models without/with the covariate. The estimates of the univariate
parameters are similar for the different models as well as the strength of serial dependence
at lag 1. Note that the addition of the covariate decreases the estimation of overdispersion
and lag 1 serial dependence.
For another count time series data set, we also comment on comparisons of ts of
models for the data set in Zhu and Joe (2006) on monthly number of claims of short-
term disability benets, made by workers in the logging industry with cut injuries, to
the B.C. Workers’ Compensation Board for the period of 10 years from 1985 to 1994.
This data set of claims is one of few data sets in the literature where “survivor” inter-
pretion of binomial thinning is plausible. There is clear seasonality from the time series
plot and autocorrelation function plot, so we use the covariates (sin(2πt/12), cos(2πt/12)).
This led to root mean square prediction errors (RMSPE) that were from 2.78 to 2.86 with
no seasonal covariates and 2.64 to 2.73 with seasonal covariates. The Markov model with
a Gaussian copula tted a little better than those based on thinning operators in terms of
46 Handbook of Discrete-Valued Time Series
TABLE 2.1
The Covariate Is One Hundredth of the Number of Downloads of Octave Programs at the Website
in the Preceding Month
regr. Dependence Order −Loglik rmspe
No covariate
NB Gaussian copula 1 424.49 4.17
GP Gaussian copula 1 424.36 4.17
NB Gaussian copula 2 422.72 4.13
NB Frank copula 1 428.65 4.24
NB Gumbel copula 1 427.35 4.24
NB re.Gumbel copula 1 424.07 4.15
NB conv-closed oper. 1 425.24 4.18
NB binomial thinning 1 426.49 4.18
NB I2 thinning 1 425.36 4.18
NB I3 thinning 1 425.83 4.18
One covariate
NB1 Gaussian copula 1 416.07 4.02
NB2 Gaussian copula 1 416.03 4.02
GP1 Gaussian copula 1 415.99 4.02
NB1 Gaussian copula 2 415.79 4.01
NB2 Gaussian copula 2 415.76 4.02
NB1 Frank copula 1 416.98 4.05
NB2 Frank copula 1 416.90 4.05
NB1 Gumbel copula 1 416.00 4.01
NB2 Gumbel copula 1 416.25 4.01
NB1 re.Gumbel copula 1 417.14 4.02
NB2 re.Gumbel copula 1 417.04 4.02
NB1 conv-closed oper. 1 415.99 4.02
NB binomial thinning 1 416.10 4.02
NB I2 thinning 1 415.63 4.02
NB I3 thinning 1 415.86 4.02
Model (Markov order 1) MLE SE
NB with Gaussian copula and no covariates β
ˆ
0
= 2.40 0.05
ξ
ˆ
=
0.89 0.27
ρˆ = 0.42 0.08
NB1 with Gaussian copula β
ˆ
0
= 1.93 0.11
β
ˆ
1
= 0.09 0.02
ξ
ˆ
=
0.48 0.18
ρˆ = 0.22 0.09
Note: For the models with covariates based on the I2 and I3 thinning operators, the formulation of Section 2.3.5
n
is used. The root mean square prediction error (rmspe) is dened as {(n − p)
−1
(y
t
− E
ˆ
(Y
t
|Y
t−1
=
t=1+p
y
t−1
, ..., Y
t−p
= y
t−p
)}
1/2
,where n is the length of the series and p is the Markov order and E
ˆ
means the
conditional expectation with parameter equal to the maximum likelihood estimate (MLE) of the model. In
the bottom, MLEs and corresponding standard errors are given for one of the better tting models without
and with the covariate.
47 Markov Models for Count Time Series
log-likelihood, but the Frank copula model with NB2 regression margins was the best in
terms of log-likelihood and RMSPE.
Similar to Table 2.1, there is a bit more variation among copula model ts than those
based on thinning operators (the latter models tend to be close to the Gaussian copula
model). An explanation is that different copula families have a wide variety of shapes of
conditional expectation and variance functions.
Finally, we indicate an advantage of count time series models with known univariate
marginal distributions. In this case, univariate and conditional probabilities can be easily
obtained and also predictive intervals with and without the previous observation(s). A pre-
dictive interval without the previous time point is a regression inference and a predictive
interval with the previous time point is an inference that combines regression and forecast-
ing. The rst type of predictive interval would not be easy to do with other classes of count
time series models that are based on conditional specications.
Acknowledgments
This work was supported by an NSERC Discovery grant.
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