13 Statistical Analysis of Count Time Series Models
sequence with Var(e
t
) = 1. It is straightforward to see that under the Poisson assumption,
(1.19) becomes
Y
t
− λ
t
e
t
=
√
, t ≥ 1,
λ
t
while for the case (1.4), we obtain
Y
t
− λ
t
e
t
=
, t ≥ 1.
λ
t
+ λ
2
t
/ν
To compute the Pearson residuals in either case, substitute λ
t
by λ
ˆ
t
≡ λ
t
(θ
ˆ
) and σ
Z
2
by
σˆ
Z
2
. Construction of their autocorrelation function and cumulative periodogram plots (see
Brockwell and Davis 1991, Sec. 10.2) give some clue about the whiteness of the sequence
{e
t
, t ≥ 1}.
Figure 1.1 shows the plots of the autocorrelation function and the cumulative peri-
odogram of the Pearson residuals obtained after tting (1.5) to the transactions data. The
upper panel corresponds to the case of Poisson distribution and the lower panel is con-
structed using the negative binomial assumption. We observe that both models t the data
quite adequately. We have also computed the Pearson residuals for models (1.8) and (1.11).
The corresponding plots are not shown because the results are quite analogous to the case
of the simple linear model.
1.4.2 Goodness-of-Fit Test
A goodness-of-t test for model (1.5), and more generally of (1.9), was recently proposed
by Fokianos and Neumann (2013), by considering two forms of hypotheses. The rst of
these refers to the simple hypothesis
H
0
(s)
: f = f
0
against H
1
(s)
: f = f
0
,
for some completely specied function f
0
which satises (1.10). However, in applications,
the most interesting testing problem is given by the following composite hypotheses
H
0
: f ∈{f
θ
: θ ∈ } against H
1
: f ∈{f
θ
: θ ∈ }, (1.20)
where ⊆ R
m
and the function f
θ
is known up to a parameter θ and again satises (1.10).
The methodology for testing (1.20) is quite general and can be applied to all models
considered so far. Recall that θ
ˆ
denotes the QMLE and λ
ˆ
t
= λ
t
(θ
ˆ
).If e
ˆ
t
are the Pearson
residuals (1.19), the statistic for testing (1.20) is given by
1
n
T
ˆ
n
= sup |G
ˆ
n
(x)|, G
ˆ
n
(x) =
√
e
ˆ
t
w(x − I
ˆ
t−1
), (1.21)
x∈
n
t=1
where x ∈ :=[0, ∞)
2
,
ˆ
I
t
= (λ
ˆ
t
, Y
t
)
,and w(·) is some suitably dened weight function. In
the applications, we can consider the weight function to be of the form w(x) = w(x
1
, x
2
) =
K(x
1
)K(x
2
) where K(·) is a univariate kernel and x = (x
1
, x
2
) ∈ . We can employ the
14 Handbook of Discrete-Valued Time Series
ACF
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
0 5 10
15 20 25
0.0
0.1 0.2 0.3 0.4 0.5
(a)
Lag
(b)
Frequency
ACF
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
0 5
10 15 20 25
0.0
0.1
0.2
0.3
0.4
0.5
(c) Lag (d)
Frequency
FIGURE 1.1
Diagnostic plots for the Pearson residuals (1.19) after tting model (1.5) to the transactions data. (a) Autocorre-
lation function using the Poisson assumption. (b) Cumulative periodogram plot using the Poisson assumption.
(c) Autocorrelation function using the negative binomial assumption. (d) Cumulative periodogram plot using the
negative binomial assumption.
uniform, Gaussian and the Epanechnikov kernels. For instance, when the uniform kernel
is employed, compute the test statistic (1.21) by using the weights
1
w(x − I
t−1
) = K(x
1
− λ
t−1
)K(x
2
− Y
t−1
) =
1(|x
1
− λ
t−1
|≤1)1(|x
2
− Y
t−1
|≤1),
4
where 1(A) is the indicator function of a set A. Then, the test statistic (1.21) becomes
ˆ
T
n
= sup
x∈
|
ˆ
G
n
(x)|,
where
ˆ
G
n
(x
1
, x
2
) =
1
4
√
n
n
t=1
ˆ
e
t
1(|x
1
−
ˆ
λ
t−1
|≤1)1(|x
2
− Y
t−1
|≤1).
15 Statistical Analysis of Count Time Series Models
Obvious formulas hold for other kernel functions. It turns out that (1.21) yields a consis-
tent procedure when testing against Pitman’s local alternatives. It converges weakly with
the usual parametric rate under some regularity conditions on the kernel function; see
Fokianos and Neumann (2013) for more details.
An alternative test statistic for testing goodness of t for count time series can be based
on supremum-type tests of the following form (Koul and Stute 1999):
n
H
ˆ
n
= sup |H
n
(x)|, H
n
(x) = n
−1/2
e
ˆ
t
1(
I
t−1
≤ x), (1.22)
x∈
t=1
using the same notations as before. Although the asymptotic behavior of supremum-type
test statistics based on (1.22) has not been studied in the literature, it is possible to develop
a theory following the arguments of Koul and Stute (1999) and utilizing the recent results
on weak dependence properties obtained by Doukhan et al. (2012), at least for some classes
of models.
Regardless of the chosen statistic and the distributional assumption, we can calcu-
late critical values by using parametric bootstrap; see Fokianos and Neumann (2013) for
details under the Poisson assumption. More specically, to compute the p-value of the test
statistic, (1.21) or (1.22) is recalculated for B parametric bootstrap replications of the data
set. Then, if T
ˆ
n
denotes the observed value of the test statistic and
T
i
;n
denotes the value
of the test statistic in the ith bootstrap run, the corresponding p-value used to determine
acceptance/rejection is given by
p-value =
# i :
T
i
;n
≥
T
n
.
B + 1
A similar result holds for H
n
.
Test statistics (1.21) (with the uniform and Epanechnikov kernels) and (1.22) were com-
puted for the transactions data for testing the goodness of t of the linear model (1.5). Under
the Poisson assumption, the observed values of these test statistics were calculated to be
0.212, 0.234, and 1.390, respectively. Under the negative binomial assumption, the observed
values were equal to 0.146, 0.164, and 0.818, respectively. Table 1.2 shows the bootstrap p-
value of the test statistics which have been obtained by parametric bootstrap as explained
earlier. We note that the test statistics formed by (1.21) yield identical conclusions; that is,
the linear model can be used for tting the transactions data regardless of the assumed dis-
tribution. However, the test statistic (1.22) raises some doubt about the linearity, under the
Poisson assumption.
TABLE 1.2
p-values for the transactions data when testing for the Linear Model (1.5)
Test statistic
Distribution (1.22) (1.21) with Uniform Kernel (1.21) with Epanechnikov Kernel
Poisson 0.024 0.350 0.279
Negative binomial 0.659 0.611 0.585
Note: Results are based on B = 999 bootstrap replications.
16 Handbook of Discrete-Valued Time Series
1.5 Prediction
Following Gneiting et al. (2007), we take the point of view that predictions should be proba-
bilistic in nature. In addition, they should strive to maximize the sharpness of the predictive
distribution subject to calibration. Calibration refers to the statistical consistency between
the predictive distribution and the observations. The notion of sharpness refers to the con-
centration of the predictive distribution and is a property of the forecasts only. It follows
that if the predictive distribution is more concentrated, then the forecasts are sharper. In
this section, we provide diagnostic tools to evaluate the predictive performance. Note that
calculation of all these measures requires an assumption on the conditional distribution of
the process; hence general processes of the form (1.2) cannot be tted without simulating
from the mixing variables Z
t
. Predictive performance based on the following diagnostic
tools has been examined recently by Jung and Tremayne (2011) and Christou and Fokianos
(2015).
1.5.1 Assessment of Probabilistic Calibration
To ascertain whether or not the negative binomial distribution is a better choice than the
Poisson distribution, we use the diagnostic tool of the Probability Integral Transformation
(PIT) histogram, as explained below. This tool is used for checking the statistical consis-
tency between the predictive distribution and the distribution of the observations. If the
observation is drawn from the predictive distribution, then the PIT has a standard uniform
distribution. In the case of count data, the predictive distribution is discrete and therefore
the PIT is no longer uniform. To remedy this, several authors have suggested a randomized
PIT. However, Czado et al. (2009) recently proposed a nonrandomized uniform version of
the PIT. We explain their approach in the context of count time series models. Note that the
approach is quite general and can accommodate various data generating processes.
In our context, we t any model discussed earlier to the data by using the quasi-
likelihood function (1.14). After obtaining consistent estimators for the regression param-
eters, we estimate the mean process λ
t
by
λ
ˆ
t
= λ
t
(
θ
ˆ
) and the parameter ν by νˆ. Then, the
PIT is based on the conditional cumulative distribution
⎧
⎪
0 u ≤ P
y−1
,
⎨
F(u|Y
t
= y) =
(u − P
y−1
)/(P
y
− P
y−1
) P
y−1
≤ u ≤ P
y
,
⎪
⎩
1 u ≥ P
y
,
where P
y
is equal to the conditional c.d.f. either of the Poisson distribution (1.3) evaluated
at
λ
ˆ
t
, or of the negative binomial p.m.f. (1.4) evaluated at
λ
ˆ
t
and νˆ. Subsequently, we form
the mean PIT by
1
n
F
¯
(u) =
F
(t)
(u|y
t
),0 ≤ u ≤ 1.
n
t=1
The mean PIT is compared to the c.d.f. of the standard uniform distribution. The compar-
ison is performed by plotting a nonrandomized PIT histogram, which can be used as a
diagnostic tool. After selecting the number of bins, say J, we compute
17 Statistical Analysis of Count Time Series Models
f
j
= F
¯
j
− F
¯
j − 1
J J
for equally spaced bins j = 1, ..., J. Then we plot the histogram with height f
j
for bin j and
check for uniformity. Deviations from uniformity hint at reasons for forecasting failures and
model deciencies. U-shaped histograms point at underdispersed predictive distributions,
while hump or inverse–U shaped histograms indicate overdispersion.
Figure 1.2 shows nonrandomized PIT histograms with 10 equally spaced bins for two
different situations for the transactions data. The left plots show the PIT histograms when
Linear Poisson prediction Linear negative binomial prediction
0.0
0.5
1.0
1.5
2.0
2.5
Relative frequency
0.0
0.5
1.0
1.5
2.0
2.5
Relative frequency
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Probability integral transform
Probability integral transform
Nonlinear Poisson prediction
for gamma known (γ = 0.5)
Nonlinear negative binomial prediction
for gamma known (γ = 0.5)
0.0
0.5
1.0
1.5
2.0
2.5
Relative frequency
0.0
0.5
1.0
1.5
2.0
2.5
Relative frequency
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
(a) Probability integral transform
(b)
Probability integral transform
FIGURE 1.2
PIT histograms applied to the number of the transactions per minute for the stock Ericsson B during July 2, 2002.
From top to bottom: PIT histograms for model (1.5) and model (1.12) for γ = 0.5. (a) The conditional distribution
is Poisson. (b) The conditional distribution is negative binomial.
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