Handbook of Discrete-Valued Time Series

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250 Handbook of Discrete-Valued Time Series

where r

t

= γα

t−1

and p

t

= γβ

t−1

/γβ

t−1

+ 1. The one-step-ahead forecast is obtained as

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α

t−1

E

Y

t

|D

t−1

, γ = .

β

t−1

An interesting property of the model is the long-run behavior of its one-step-ahead fore-

casts. As t gets large, using β

t

= γβ

t−1

+ 1, we can show that β

t

approaches 1/(1 − γ) and

we obtain

E

Y

t

|D

t−1

, γ =

(1 − γ)Y

t−1

+ (1 − γ)γY

t−2

+ ...+ (1 − γ)γ

t

α

0

,

which is an exponentially weighted average of the observed counts.

Although an analytic expression is not available for the k-step-ahead predictive density,

the k-step-ahead predictive means can be easily obtained. ...