160 Handbook of Discrete-Valued Time Series
Consider the nonlinear time series model in (7.13). Based on the optimal EF in the
class G of all unbiased EFs g =
n
1
b
t−1
g
y
t
− μ
θ
t
, F
y
, the form of the extended
t=
t−1
GAS model for the location parameter θ
t
is given by
φK
t−1
b
∗
t−1
θ
t
= φθ
t−1
+
1 +[∂μ(
θ
ˆ
t−1
, F
t−1
)/∂θ]K
t−1
b
∗
g(y
t
− μ(θ
t−1
, F
t−1
)),where
t−1
K
t−1
K
t
= ,
1 +[∂μ(θ
t−1
, F
t−1
)/∂θ]φ
2
K
t−1
b
∗
t−1
where b
∗
t−1
is a function of g, θ, and the observations:
b
∗
t−1
=
[∂μ(θ
t−1
, F
t−1
)/∂θ][∂g(θ
t−1
, F
t−1
)/∂μ]
.
Var(g|F
t−1
)
7.5 Hypothesis Testing and Model Choice
Hypothesis testing situations in stochastic modeling are often tests of linear hypotheses
about the unknown parameter θ ∈R
p
. Suppose θ =(θ
1
, θ
2
), and suppose that the optimal
EF g
∗
(θ) and the corresponding unconditional information F
n
(θ) = E[g
∗
(θ)g
∗
(θ)
] have
n n n
conformable partitions, that is,
∗
g
n
∗
1
(θ)
F
n11
(θ) F
n12
(θ)
g
n
(θ) =
, F
n
(θ) =
.
∗
g
n2
(θ)
F
n21
(θ) F
n22
(θ)
Atestof H
0
: θ
2
= θ
20
versus H
1
: θ
2
= θ
20
corresponds to a comparison of a full model
versus a nested model when we test θ
20
= 0. For example, in (7.5), testing θ
20
= 0 could
correspond to testing β = 0,sothat H
0
corresponds to a smaller model with only δ and α
as parameters.
Let θ
n
∗
=(θ
n
∗
1
, θ
∗
n2
) denote the optimal estimate of θ unrestricted by H
0
,and let
θ
n
=
(
θ
n1
,
θ
n2
) denote the optimal estimate under the null hypothesis H
0
. As in Thavaneswaran
(1991), we propose two test statistics, viz., the Wald-type statistic and the score statistic, as
W
n
= (θ
∗
n2
− θ
20
)
A
n22
(θ
n
2
)(θ
∗
n2
− θ
20
), (7.24)
Q
n
= (g
∗
2
(
θ
n
))
A
−1
θ
n
)g
∗
2
(
θ
˜
n
), (7.25)
n22
(
where
A
n22
(θ) = F
n22
(θ) − F
n21
(θ)F
−
n11
1
(θ)F
n12
(θ)
is the inverse of the second diagonal block in the inverse of the partitioned matrix F
n
(θ).
The Wald and score statistics for testing a general linear hypothesis H
0
: Cθ = c
0
versus
H
1
: Cθ = c
0
, where the r × p matrix C has full row rank, are
W
n
= (Cθ
∗
− c
0
[CF
−1
(θ
∗
)C
]
−1
(Cθ
∗
− c
0
) and (7.26)
n n n n
∗
F
−1
∗
Q
n
= (g
n
(
θ
n
))
n
(
θ
n
)g
n
(
θ
n
). (7.27)
161 Estimating Equation Approaches for Integer-Valued Time Series Models
Thavaneswaran (1991) showed that under certain regularity conditions, the test statistics
in (7.26) and (7.27) are asymptotically equivalent, that is, the difference between them con-
verges to zero in probability under H
0
and they have the same limiting null distributions
(which is a χ
2
r
distribution).
7.6 Discussion and Summary
Interest in developing models for integer valued time series, especially for count time
series, is growing. Among these are models discussed in Ferland et al. (2006) and Zhu (2011,
2012a,b), who described classes of INGARCH models with different conditional distribu-
tional specications for the process given its history, and primarily described likelihood
based approaches for estimating model parameters, under parametric assumptions such as
Poisson, negative binomial, or ZIP for the conditional distributions. Although these models
are referred to as INGARCH models in the literature, they model the conditional mean of
the time series and not its conditional variance. These models are similar to the ACP models
discussed in Heinen (2003) and Ghahramani and Thavaneswaran (2009b). Creal et al. (2013)
recently discussed GAS models, while Thavaneswaran and Ravishanker (2015) described
models for circular time series.
This chapter considers modeling the conditional mean and conditional variance of an
integer-valued time series {y
t
}, where conditional moments are functions of θ. We have
described a combined EF approach for estimating θ and have also provided forms for joint
recursive estimates for xed parameters using the most informative combined martingale
difference and provided its corresponding information. In Section 7.4.1, we have shown
how recursive estimation extends to the case with a time-varying parameter. This approach
would be valuable in a study of doubly stochastic models for integer-valued time series,
which we briey discuss below.
Similar to the well-known stochastic volatility (SV) or stochastic conditional duration
(SCD) models described in the literature, a general integer-valued doubly stochastic model
for y
t
with conditional mean E(y
t
|F
t−1
) = μ
t
= exp(λ
t
) is dened via
∞
λ
t
= δ + ψ
j
ε
t−j
, (7.28)
j=0
∞
where δ is a real-valued parameter,
j=0
ψ
2
j
< ∞, ε
t
|F
t−1
are independent N(0, σ
ε
2
) vari-
ables, and ε
s
is independent of y
t
|F
t−1
for all s, t. The conditional moments of y
t
may match
the moments of a known probability distribution for a count random variable, for example,
those corresponding to a Poisson-INDS model, a GP-INDS model, a NB-INDS model, or a
∞
ZIP-INDS model. In lieu of (7.28), if λ
t
is modeled by λ
t
− δ = (1 − B)
−d
ε
t
=
j=0
ψ
j
ε
t−j
where d ∈ (0, 0.5) and ψ
k
= (k + d)/[(d)(k + 1)], where (·) is the gamma function, the
model can handle long-memory behavior. We may also dene an integer-valued quadratic
doubly stochastic (INQDS) model by assuming the rst four conditional moments of y
t
given F
t−1
to be μ
t
= exp(aλ
t
+ bλ
t
2
), σ
2
= μ
t
, γ
t
= μ
−1/2
and κ
t
= μ
−1
, which match the
t
t
t
rst four moments of a Poisson(exp(aλ
t
+bλ
2
t
)) process. Naik-Nimbalkar and Rajarshi (1995)
and Thompson and Thavaneswaran (1999) studied ltering/estimation for state space
162 Handbook of Discrete-Valued Time Series
models and counting processes in the context of EFs. Thavaneswaran and Abraham
(1988) and Thavaneswaran et al. (2015) described combining nonorthogonal EFs following
preltered estimation.
Acknowledgments
The rst author acknowledges support from an NSERC grant.
References
Al-Osh, M. and Alzaid, A. (1987). First-order integer-valued autoregressive (INAR(1)) model. Journal
of Time Series Analysis, 8:261–275.
Bera, A. K., Bilias, Y., and Simlai, P. (2006). Estimating functions and equations: An essay on historical
developments with applications to econometrics. In: T. C. Mills and K. Patterson, (eds.), Palgrave
Handbook of Econometrics, Palgrave MacMillan: New York, Vol. I, pp. 427–476.
Bollerslev, T. (1986). Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics,
31:307–327.
Chan, K. S. and Ledolter, J. (1995). Monte Carlo EM estimation for time series involving counts. Journal
of the American Statistical Association, 90:242–252.
Cox, D. R. (1981). Statistical analysis of time series: Some recent developments. Scandinavian Journal
of Statistics, 8:93–115.
Creal, D. D., Koopman, S. J., and Lucas, A. (2013). Generalized autoregressive score models with
applications. Journal of Applied Econometrics, 28:777–795.
Davis, R. A., Dunsmuir, W. T. M., and Streett, S. B. (2003). Observation-driven models for Poisson
counts. Biometrika, 90:777–790.
Dean, C. B. (1991). Estimating equations for mixed Poisson models. In: V. P. Godambe, (ed.) Estimating
Functions, Oxford University Press: Oxford, U.K., pp. 35–46.
Durbin, J. (1960). Estimation of parameters in time-series regression models. Journal of the Royal
Statistical Society Series B, 22:139–153.
Ferland, R., Latour, A., and Oraichi, D. (2006). Integer-valued GARCH model. Journal of Time Series
Analysis, 27:923–942.
Fisher, R. A. (1924). The conditions under which χ
2
measures the discrepancy between observation
and hypothesis. JournaloftheRoyalStatisticalSociety, 87:442–450.
Fokianos, K., Rahbek, A., and Tjostheim, D. (2009). Poisson autoregression. Journal of the American
Statistical Association, 104:1430–1439.
Ghahramani, M. and Thavaneswaran, A. (2009a). Combining estimating functions for volatility.
Journal of Statistical Planning and Inference, 139:1449–1461.
Ghahramani, M. and Thavaneswaran, A. (2009b). On some properties of Autoregressive Conditional
Poisson (ACP) models. Economic Letters, 105:273–275.
Ghahramani, M. and Thavaneswaran, A. (2012). Nonlinear recursive estimation of the volatility via
estimating functions. Journal of Statistical Planning and Inference, 142:171–180.
Godambe, V. P. (1960). An optimum property of regular maximum likelihood estimation. Annals of
Mathematical Statistics, 31:1208–1212.
Godambe, V. P. (1985). The foundations of nite sample estimation in stochastic process. Biometrika,
72:319–328.
Gourieroux, C. (1997). ARCH Models and Financial Applications. Springer-Verlag: New York.
163 Estimating Equation Approaches for Integer-Valued Time Series Models
Heinen, A. (2003). Modelling time series count data: An Autoregressive Conditional Poisson model.
Discussion Paper, vol. 2003/62. Center for Operations Research and Econometrics (CORE),
Catholic University of Louvain, Louvain, Belgium.
Heyde, C. C. (1997). Quasi-Likelihood and its Application: A General Approach to Optimal Parameter
Estimation. Springer-Verlag: New York.
Jung, R. C. and Tremayne, A. R. (2006). Binomial thinning models for integer time series. Statistical
Modelling, 6:21–96.
Kedem, B. and Fokianos, K. (2002). Regression Models for Time Series Analysis. Wiley: Hoboken, NJ.
Liang, Y., Thavaneswaran, A., and Abraham, B. (2011). Joint estimation using quadratic estimating
functions. Journal of Probability and Statistics, article ID 372512:14 pages.
Lindsay, B. C. (1985). Using empirical partially Bayes inference for increased efciency. The Annals of
Statistics, 13:914–931.
MacDonald, I. L. and Zucchini, W. (2015). Hidden Markov 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. 267–286. Chapman & Hall, Boca Raton, FL.
McKenzie, E. (2003). Some simple models for discrete variate time series. In: C. R. Rao and
D. N. Shanbhag, (eds.), Handbook of Statistics-Stochastic models: Modeling and Estimation, Elsevier
Science: Amsterdam, the Netherlands, Vol. 21, pp. 573–606.
Merkouris, T. (2007). Transform martingale estimating functions. The Annals of Statistics, 35:
1975–2000.
Naik-Nimbalkar, U. V. and Rajarshi, M. B. (1995). Filtering and smoothing via estimating functions.
Journal of the American Statistical Association, 90:301–306.
Neal, P. and Subba Rao, T. (2007). MCMC for integer-valued ARMA models. Journal of Time Series
Analysis, 28:92–110.
Thavaneswaran, A. (1991). Tests based on an optimal estimate. In: V. P. Godambe (ed.), Estimating
Functions, Clarendon Press: Oxford, U.K., pp. 189–198.
Thavaneswaran, A. and Abraham, B. (1988). Estimation of nonlinear time series models using
estimating functions. Journal of Time Series Analysis, 9:99–108.
Thavaneswaran, A. and Heyde, C. C. (1999). Prediction via estimating functions. Journal of Statistical
Planning and Inference, 77:89–101.
Thavaneswaran, A., Ravishanker, N, and Liang, Y. (2015). Generalized duration models and inference
using estimating functions. Annals of the Institute of Statistical Mathematics, 67:129–156.
Thavaneswaran, A. and Ravishanker, N. (2015). Estimating functions for circular models. Technical
Report, Department of Statistics, University of Connecticut: Storrs, CT.
Thompson, M. E. and Thavaneswaran, A. (1999). Filtering via estimating functions. Applied Mathe-
matics Letters, 12:61–67.
West, M. and Harrison, P. J. (1997). Bayesian Forecasting and Dynamic Models. Springer-Verlag:
New York.
Yang, M. (2012). Statistical models for count time series with excess zeros. PhD thesis, University of
Iowa, Iowa City, IA.
Zeger, S. L. and Qaqish, B. (1988). Markov regression models for time series. Biometrics, 44:1019–1031.
Zhu, F. (2011). A negative binomial integer-valued GARCH model. Journal of Time Series Analysis,
32:54–67.
Zhu, F. (2012a). Modeling overdispersed or underdispersed count data with generalized Poisson
integer-valued GARCH models. Journal of Mathematical Analysis and Applications, 389:58–71.
Zhu, F. (2012b). Zero-inated Poisson and negative binomial integer-valued GARCH models. Journal
of Statistical Planning and Inference, 142:826–839.
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