Detection of Change Points in Discrete-Valued Time Series 229
Assumptions under H
1
:
H
1
(i) The change point is of the form k
0
=λn,0 < λ < 1.
H
1
(ii) For all γ ∈ , is compact and convex, and f
γ
(x) is uniformly Lipschitz in x
with Lipschitz constant L
γ
< 1.
H
1
(iii) The time series before the change is stationary and ergodic such that
E sup
γ∈
Y
t−j
∇f
γ
(Y
t−1
, ..., Y
t−p
) < ∞, j = 0, ..., p.
H
1
(iv) The time series after the change fullls Y
t
=
Y
˜
t
+ R
1
(t), t > λn, where {
Y
˜
t
}
is stationary and ergodic such that E sup
γ∈
Y
˜
t−j
∇f
γ
(
Y
˜
t−1
, ...,
Y
˜
t−p
) < ∞,
j = 0, ..., p, with the remainder term fullling
1
n
R
2
1
(t) = o
P
(1).
n
j=λn+1
H
1
(v) λE∇f
γ
((Y
0
, ..., Y
1−p
))(Y
1
− f
γ
((Y
0
, ..., Y
1−p
))) + (1 − λ)E∇f
γ
(
Y
˜
0
, ...,
Y
˜
1−p
)
(
Y
˜
1
− f
γ
(
Y
˜
0
, ...,
Y
˜
1−p
)) has a unique zero γ
1
∈ in the strict sense of B.2.
The formulation in H
1
(iv) allows for certain deviations from stationarity of the time series
after the change which can, for example, be caused by starting values from the station-
ary distribution before the change, while H
1
(v) guarantees that
γ
n
converges to γ
1
under
alternatives.
The following theorem extends the results of Franke et al. [13].
Theorem 10.6 Let assumptions H
1
(i)–H
1
(iv) be fullled.
(a) For S
PA R
(k, γ) as in (10.10), B.1 and B.2 are fullled.
(b) For
S
˜
PA R
(k, γ) =
j
k
=1
(Y
t
− f
γ
(Y
t−1
, ..., Y
t−p
)) and if k
0
=λn, B.5 is fullled with
F
λ
(t) = E(Y
1
− γ
T
1
Y
0
).
(c) For S
PA R
(k, γ) as in (10.10) with f
γ
(x) = γ
T
x and if k
0
=λn, then B.5 is fullled with
F
λ
(t) = EY
0
(Y
1
− γ
1
T
Y
0
).
From this, we can give assumptions under which the corresponding tests have asymptotic power
one and the point where the maximum is attained is a consistent estimator for the change point in
rescaled time by Theorem 10.2.
B.4 is always fullled for the full score statistic if the time series before and after the
change are correctly specied by the given Poisson autoregressive model. Otherwise,
restrictions apply.
Doukhan and Kengne [7] propose to use several Wald-type statistics based on maxi-
mum likelihood estimators in Poisson autoregressive models. While their statistics are also
designed for the at most one change situation, they explicitly prove consistency under the
multiple change point alternative.
230 Handbook of Discrete-Valued Time Series
10.5 Simulation and Data Analysis
In the previous sections, we have derived the asymptotic limit distribution for various
statistics as well as shown that the corresponding tests have asymptotic power one under
relatively general conditions. In particular, we have proven the validity of these conditions
for two important classes of integer-valued time series: binary autoregressive and Poisson
counts. In this section, we give a short simulation study in addition to some data analy-
sis to illustrate the small sample properties of these tests complementing simulations of
Hudecová [17] and Fokianos et al. [11]. The critical values are obtained from Monte Carlo
experiments of the limit distribution based on 1000 repetitions.
10.5.1 Binary Autoregressive Time Series
In this section, we consider a rst-order binary autoregressive time series (BAR(1)) as
dened in Section 10.3 with Z
t−1
= (1, Y
t−1
). We consider the statistic
1
S
BAR
T
−1
S
BAR
T
n
= max
k, β
n
k, β
n
,
1�k�n
n
1
n
T
Z
t−1
Z
t−1
π
t
β
n
1 − π
t
(β
n
where =
)
n
t=1
and S
BAR
is as in (10.6) and β
n
as in (10.7). Since consistently estimates =
E(Z
t−1
Z
T
t−1
π
t
(β
0
)(1 − π
t
(β
0
))
T
under the null hypothesis, by Theorem 10.3 and Theo-
rem 10.1, the asymptotic null distribution of this statistic is given by
sup
B
1
2
(t) + B
2
2
(t) (10.12)
0�t�1
for two independent Brownian bridges {B
1
(·)} and {B
2
(·)} with a simulated 95% quantile
of 2.53. Table 10.1 reports the empirical size and power (based on 10, 000 repetitions) for
various alternatives, where a change always occurred at n/2. Figure 10.1 shows one sample
path for the null hypothesis and each of the alternatives considered there. The size is always
conservative and gets closer to the nominal level with increasing sample size as predicted
by the asymptotic results. The power is good and increases with the sample size, where
some alternatives have better power than others.
TABLE 10.1
Empirical Size and Power of Binary Autoregressive Model with β
1
= (2, −2) (Parameter before
the Change)
H
0
H
1
: β
2
= (1, −2) H
1
: β
2
= (1, −1) H
1
: β
2
= (2, −1)
n 200
0.032
500
0.040
1000
0.044
200
0.650
500
0.985
1000
1.00
200
0.176
500
0.520
1000
0.871
200
0.573
500
0.961
1000
0.999
231 Detection of Change Points in Discrete-Valued Time Series
1 1
0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6
0.5 0.5
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 20 40
0
60 80 100 120 140 160 180 200 0
0
20 40 60 80 100 120 140 160 180 200
Null hypothesis
Alternative: β
2
= (1, –2)
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
20 40 60 80 100 120 140 160 180 200
0
0
20 40 60 80 100 120 140 160 180 200
Alternative: β
2
= (1, –1) Alternative: β
2
= (2, –1)
FIGURE 10.1
Sample paths for the BAR(1) model, β
1
= (2, −2), k
0
= 100, n = 200.
10.5.1.1 Data Analysis: U.S. Recession Data
We now apply the test statistic mentioned earlier to the quarterly recession data in
Figure 10.2 from the United States for the period 1855–2012
∗
. The datum is 1 if there has
been a recession in at least one month in the quarter and 0 otherwise. The data have been
previously analyzed by different authors; in particular, they have recently been analyzed
in a change point context by Hudecová [17].
We nd a change in the rst quarter of 1933, which corresponds to the end of the great
depression that started in 1929 in the United States and leads to a huge unemployment
rate in 1932. If we split the time series at that point and repeat the change point proce-
dure, no further signicant change points are found. This is consistent with the ndings
in Hudecová [17], who applied a different statistic based on a binary autoregressive time
series of order 3.
∗
This data set can be downloaded from the National Bureau of Economic Research at http://research.
stlouisfed.org/fred2/series/USREC.
232 Handbook of Discrete-Valued Time Series
0
1
1860 1880 1900 1920 1940 1960 1980 2000
FIGURE 10.2
Quarterly U.S. recession data (1855–2012).
10.5.2 Poisson Autoregressive Models
In this section, we consider a Poisson autoregressive model as in (10.9) with λ
t
= γ
1
+
γ
2
Y
t−1
. For this model, we use the following test statistic based on least squares scores:
T
n
= max
1
S
PAR
(k,
γ
n
)
T
−1
S
PAR
(k,
γ
n
),
1�k�n
n
k
where S
PAR
(k, γ) = Y
t−1
(Y
t
− λ
t
), Y
t−1
= (1, Y
t−1
)
T
,
t=1
and
γ
n
as in (10.10) and
−1
is the empirical covariance matrix of {Y
t−1
(Y
t
− λ
t
)}.By
Theorems 10.5b and 10.1, this statistic has the same null asymptotics as in (10.12). Table 10.2
reports the empirical size and power (based on 10, 000 repetitions) for various alternatives,
where a change always occurred at n/2. Figure 10.3 shows one corresponding sample path
for each scenario. The test size is always conservative and gets closer to the nominal level
with increasing sample size as predicted by the asymptotic results. The power is good
TABLE 10.2
Empirical Size and Power of Poisson Autoregressive Model with γ
1
= (1, 0.75) (Parameter before
the Change)
H
0
H
1
: γ
2
= (2, 0.75) H
1
: γ
2
= (2, 0.5) H
1
: γ
2
= (1, 0.5)
n 200
0.028
500
0.0361
1000
0.036
200
0.531
500
0.967
1000
0.999
200
0.252
500
0.683
1000
0.968
200
0.271
500
0.895
1000
0.999
233 Detection of Change Points in Discrete-Valued Time Series
18
16
14
12
10
8
6
4
2
0
16
14
12
10
8
6
4
2
0
18
16
14
12
10
8
6
4
2
0
0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500
Null hypothesis Alternative: γ
2
= (2, 0.75)
0 50 100 150 200 250 300 350 400 450 500
0 50 100 150 200 250 300 350 400 450 500
Alternative: γ
2
= (2, 0.5) Alternative: γ
2
= (1, 0.5)
0
2
4
6
8
10
12
14
16
FIGURE 10.3
Sample paths for the Poisson autoregressive model, γ
1
= (1, 0.75), k
0
= 250, n = 500.
and increases with the sample size. Some pilot simulations suggest that using statistics
associated with partial likelihood scores can further increase the power. While a detailed
theoretic analysis can in principle be done based on the results in Section 10.2, it is beyond
the scope of this work.
10.5.2.1 Data Analysis: Number of Transactions per Minute for Ericsson B Stock
In this section, we use the methods given earlier to analyze the data set that consists of the
number of transactions per minute for the stock Ericsson B during July 3, 2002. The data
set consists of 460 observations instead of 480 for 8 h of transactions because the rst 5 min
and last 15 min of transactions are ignored. Fokianos et al. [12] have analyzed the transac-
tions count from the same stock on a different day with a focus on forecasting the number
of transactions. The data and estimated change points (using a binary segmentation pro-
cedure as described in Theorem 10.2) are illustrated in Figure 10.4. The red vertical lines
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.