141 State Space Models for Count Time Series
6.4 Forecasting
For the nonlinear state-space framework of Section 6.1 given in equations (6.2) and (6.3), the
forecast density of the next observation Y
n+1
given the current data Y
(n)
can be computed
recursively via Bayes’ theorem. We follow the development in Section 8.8.1 of Brockwell
and Davis (2002). First, using (6.2) and (6.3), the ltering and prediction densities can be
recursively obtained via
p
p
p
s
t
|y
(t)
=
y
t
p
|s
t
y
t
|y
(
s
t
t
−
|y
1)
(
t−1)
(6.33)
and
p s
t+1
|y
(t)
= p
(
s
t+1
|s
t
)
p s
t
|y
(t)
dμ
(
s
t
)
, (6.34)
where μ(·) is the dominating measure for the state density function p
(
s
t
|s
t−1
)
. Note that the
forecasting density p y
t
|y
(t−1)
is a normalizing constant that ensures the ltering density
integrates to 1, that is, p s
t
|y
(t)
dμ
(
s
t
)
=
1. The updating equation to calculate the forecast
density for the observations is then found from
p y
t+1
|y
(t)
= p y
t+1
|s
t+1
p s
t+1
|y
(t)
dμ
(
s
t+1
)
. (6.35)
In practice, of course, these recursions are not computable in closed form and one needs
to resort to Monte Carlo procedures, see Durbin and Koopman (2012). Specically, if one
can generate replicates of S
n+1
given Y
(n)
through MCMC or via importance sampling
as described in Section 6.2, then the forecasting density can be computed by averaging
p(y
n+1
|s
n+1
) over those replicates.
In the case that the count time series is modeled under the assumptions as specied in
(6.4) with S
t
given by (6.7), then one can derive a rather nice expression for E(Y
n+1
|Y
(n)
).
To see this, condition rst on Y
(n)
and S
(n+1)
, which is the same as conditioning on α
(n+1)
,
and then using (6.4), we have
E(Y
n+1
|Y
(n)
) = E E(Y
n+1
|S
n+1
, S
(n)
, Y
(n)
)|Y
n)
= E E(Y
n+1
|S
n+1
)|Y
n)
= E E(Y
n+1
|α
n+1
)|Y
n)
= E h(α
n+1
)|Y
n)
,
where h
(
α
n+1
)
=
E
(
Y
n+1
|α
n+1
)
. Since p y
(n)
|α
n+1
, α
(n)
= p y
(n)
|α
(n)
, it follows that
p α
n+1
|y
(n)
, α
(n)
= p α
n+1
|α
(n)
and hence
E Y
n+1
|Y
(n)
= E E(h(α
n+1
)|α
(n)
)|Y
n)
. (6.36)
If the {α
t
} process is Gaussian, then often E h
(
α
n+1
)
|α
(n)
can be expressed as an explicit
function of α
(n)
. Hence, to compute the conditional expectation in (6.36), it is enough
142 Handbook of Discrete-Valued Time Series
to generate a large number of replicates α
˜
(n)
, ..., α
˜
(n)
computed from the conditional
1
N
distribution of α
(n)
given Y
(n)
and then approximate the conditional expectation by
N
(n)
E(Y
n+1
|Y
(n)
) ≈
i=1
E h(α
n+1
)|α
˜
i
.
N
The same ideas can be applied for predicting lead times further into the future.
Example: Suppose Y
t
given the state S
t
is Poisson (e
s
t
)andthat S
t
is the linear regression
model with Gaussian AR(p) noise as given in (6.7). In this case,
h(α
n+1
) = E(Y
n+1
|α
n+1
) = exp x
n
T
+1
β + α
n+1
.
Since {α
t
} is a stationary Gaussian time series with zero mean,
α
n+1
|α
(n)
∼ N γ
T
Vα
(n)
, γ(0) − γ
T
Vγ
n
, (6.37)
n n
where γ(h) = cov(α(0), α(h)) is the autocovariance function for {α
t
},
n
is the covariance
matrix for α
(n)
and γ = cov(α(n + 1), α
(n)
) is a 1 × n covariance vector. Using the {α
t
}
process in (6.37), we have
1
E(e
α
n+1
|α
(n)
) = exp γ
T
Vα
(n)
+
γ(0) − γ
T
Vγ
n
n n
2
and hence
1
E(Y
n+1
|Y
(n)
) = exp x
T
n+1
β +
γ(0) − γ
n
T
Vγ
n
E exp γ
T
n
Vα
(n)
|Y
(n)
. (6.38)
2
In order to compute the righthand side of this equation, one needs to integrate
exp γ
T
Vα
(n)
relative to the conditional density α
(n)
|Y
(n)
, which can be obtained using
n
some of the same methods described in Section 6.2.
The conditional variance var(Y
n+1
|Y
(n)
) can be computed using a similar development.
First note that
var(Y
n+1
|Y
(n)
) = E var(Y
n+1
|α
n+1
, α
(n)
, Y
(n)
)|Y
(n)
+ var E(Y
n+1
|α
n+1
, α
(n)
, Y
(n)
)|Y
(n)
.
(6.39)
Since the conditional mean and variance are the same in this example, the rst term in (6.39)
coincides with (6.38). As for the second term,
var E(Y
n+1
|α
n+1
, α
(n)
, Y
(n)
)|Y
(n)
= var exp{x
n+1
β + α
n+1
}|Y
(n)
= E exp{2x
n+1
β + 2α
n+1
}|Y
(n)
− E
2
exp{x
n+1
β + α
n+1
}|Y
(n)
and hence
var(Y
n+1
|Y
(n)
) = E(Y
n+1
|Y
(n)
) − E
2
(Y
n+1
|Y
(n)
)
+ exp 2x
n
T
+1
β + (γ(0) − γ
T
n
Vγ
n
) E exp 2γ
n
T
Vα
(n)
|Y
(n)
.
143 State Space Models for Count Time Series
Acknowledgments
The work of the rst author was supported in part by NSF grant DMS-1107031. Travel funds
from a University of New South Wales Faculty of Science Research Fellowship for the rst
author were used in this collaboration.
References
Breslow, N. E. and Clayton, D. G. (1993). Approximate inference in generalized linear mixed models.
Journal of the American Statistical Association, 88(421):9–25.
Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods. Springer, New York.
Brockwell, P. J. and Davis, R. A. (2002). Introduction to Time Series and Forecasting. Springer,
New York.
Chan, K. and Ledolter, J. (1995). Monte Carlo em estimation for time series models involving counts.
Journal of the American Statistical Association, 90(429):242–252.
Davis, R. A., Dunsmuir, W., and Wang, Y. (1999). Modeling time series of count data, Asymptotics,
Nonparametrics, and Time Series, ed. S. Ghosh. Marcel-Dekker, New York.
Davis, R. A., Dunsmuir, W. T., and Wang, Y. (2000). On autocorrelation in a Poisson regression model.
Biometrika, 87(3):491–505.
Davis, R. A. and Rodriguez-Yam, G. (2005). Estimation for state-space models based on a likelihood
approximation. Statistica Sinica, 15(2):381–406.
Davis, R. A. and Rosenblatt, M. (1991). Parameter estimation for some time series models without
contiguity. Statistics and Probability Letters, 11(6):515–521.
Davis, R. A. and Wu, R. (2009). A negative binomial model for time series of counts. Biometrika,
96(3):735–749.
Davis, R. A. and Yau, C. Y. (2011). Comments on pairwise likelihood in time series models. Statistica
Sinica, 21(1):255.
De Jong, P. and Shephard, N. (1995). The simulation smoother for time series models. Biometrika,
82(2):339–350.
Doukhan, P. (1994). Mixing: Properties and Examples. Springer, New York.
Durbin, J. and Koopman, S. J. (1997). Monte carlo maximum likelihood estimation for non-Gaussian
state space models. Biometrika, 84(3):669–684.
Durbin, J. and Koopman, S. J. (2000). Time series analysis of non-Gaussian observations based on
state space models from both classical and Bayesian perspectives. Journal of the Royal Statistical
Society: Series B (Statistical Methodology), 62(1):3–56.
Durbin, J. and Koopman, S. J. (2012). Time Series Analysis by State Space Methods. Oxford University
Press, Oxford, U.K.
Fahrmeir, L. (1992). Posterior mode estimation by extended Kalman ltering for multivariate
dynamic generalized linear models. Journal of the American Statistical Association, 87(418):
501–509.
Fokianos, K. (2015). Statistical analysis of count time series models: A GLM perspective. In R. A.
Davis, S. H. Holan, R. Lund and N. Ravishanker, eds., Handbook of Discrete-Valued Time Series,
pp. 3–28. Chapman & Hall, Boca Raton, FL.
Gamerman, D., Abanto-Valle, C. A., Silva, R. S., and Martins, T.G. (2015). Dynamic Bayesian models
for discrete-valued time series. In R. A. Davis, S. H. Holan, R. Lund and N. Ravishanker, eds.,
Handbook of Discrete-Valued Time Series, pp. 165–186. Chapman & Hall, Boca Raton, FL.
Jung, R. C., Kukuk, M., and Liesenfeld, R. (2006). Time series of count data: modeling, estimation
and diagnostics. Computational Statistics and Data Analysis, 51(4):2350–2364.
144 Handbook of Discrete-Valued Time Series
Jung, R. C. and Liesenfeld, R. (2001). Estimating time series models for count data using efcient
importance sampling. AStA Advances in Statistical Analysis, 85(4):387–408.
Jungbacker, B. and Koopman, S. J. (2007). Monte Carlo estimation for nonlinear non-Gaussian state
space models. Biometrika, 94(4):827–839.
Koopman, S. J., Lucas, A., and Scharth, M. (2015). Numerically accelerated importance sampling
for nonlinear non-Gaussian state space models. Journal of Business and Economic Statistics, 33(1):
114–127.
Kuk, A. Y. (1995). Asymptotically unbiased estimation in generalized linear models with random
effects. Journal of the Royal Statistical Society. Series B (Methodological), 57(2):395–407.
Kuk, A. Y. and Cheng, Y. W. (1999). Pointwise and functional approximations in monte carlo
maximum likelihood estimation. Statistics and Computing, 9(2):91–99.
McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models (Monographs on Statistics and Applied
Probability 37). Chapman & Hall, London, U.K.
McCulloch, C. E. (1997). Maximum likelihood algorithms for generalized linear mixed models.
JournaloftheAmericanstatisticalAssociation, 92(437):162–170.
Nelson, K. P. and Leroux, B. G. (2006). Statistical models for autocorrelated count data. Statistics in
Medicine, 25(8):1413–1430.
Ng, C. T., Joe, H., Karlis, D., and Liu, J. (2011). Composite likelihood for time series models with a
latent autoregressive process. Statistica Sinica, 21(1):279.
Oh, M.-S. and Lim, Y. B. (2001). Bayesian analysis of time series Poisson data. Journal of Applied
Statistics, 28(2):259–271.
Politis, D. N. et al. (2003). The impact of bootstrap methods on time series analysis. Statistical Science,
18(2):219–230.
R Core Team. (2014). R: A Language and Environment for Statistical Computing. R Foundation for
Statistical Computing, Vienna, Austria.
Richard, J.-F. and Zhang, W. (2007). Efcient high-dimensional importance sampling. Journal of
Econometrics, 141(2):1385–1411.
Shephard, N. and Pitt, M. K. (1997). Likelihood analysis of non-Gaussian measurement time series.
Biometrika, 84(3):653–667.
Shumway, R. H. and Stoffer, D. S. (2011). Time Series Analysis and Its Applications: With R Examples.
Springer, New York.
Shun, Z. and McCullagh, P. (1995). Laplace approximation of high dimensional integrals. Journal of
the Royal Statistical Society. Series B (Methodological), 57(4):749–760.
Sinescu, V., Kuo, F. Y., and Sloan, I. H. (2014). On the choice of weights in a function space for quasi-
monte carlo methods for a class of generalised response models in statistics. Springer Proceedings
in Mathematics and Statistics, nizer) Gerhard Larcher, Johannes Kepler University Linz, Austria
Pierre LEcuyer, Université de Montréal, Canada Christiane Lemieux, University of Waterloo,
Canada Peter Mathé, Weierstrass Institute, Berlin, Germany, p. 597.
Skaug, H. J. (2002). Automatic differentiation to facilitate maximum likelihood estimation in nonlin-
ear random effects models. Journal of Computational and Graphical Statistics, 11(2):458–470.
Skaug, H. J. and Fournier, D. A. (2006). Automatic approximation of the marginal likelihood in
non-Gaussian hierarchical models. Computational Statistics and Data Analysis, 51(2):699–709.
Thavaneswaran, A. and Ravishanker, N. (2015). Estimating equation approaches for integer-valued
time series models. In R. A. Davis, S. H. Holan, R. Lund and N. Ravishanker, eds., Handbook of
Discrete-Valued Time Series, pp. 145–164. Chapman & Hall, Boca Raton, FL.
Tjøstheim, D. (2015). Count time series with observation-driven autoregressive parameter dynamics.
In R. A. Davis, S. H. Holan, R. Lund and N. Ravishanker, eds., Handbook of Discrete-Valued Time
Series, pp. 77–100. Chapman & Hall, Boca Raton, FL.
Wu, R. (2012). On variance estimation in a negative binomial time series regression model. Journal of
Multivariate Analysis, 112:145–155.
Wu, R. and Cui, Y. (2013). A parameter-driven logit regression model for binary time series. Journal
of Time Series Analysis, 35(5):462–477.
Zeger, S. L. (1988). A regression model for time series of counts. Biometrika, 75(4):621–629.
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