Change Point in Panel Data with Small Fixed Panel Size:
Ratio and Non-Ratio Test Statistics
Barbora Peštová1 , Michal Pešta2˚

arXiv:1608.05670v1 [stat.ME] 19 Aug 2016

1. Institute of Computer Science, The Czech Academy of Sciences, Department of Medical Informatics and
Biostatistics, Pod Vodárenskou věží 271/2, 18207 Prague, Czech Republic
2. Faculty of Mathematics and Physics, Charles University in Prague, Department of Probability and
Mathematical Statistics, Sokolovská 49/83, 18675 Prague, Czech Republic

Abstract: The main goal is to develop and, consequently, compare stochastic methods for detection whether a structural change in panel data occurred at some unknown time or not. Panel data of our interest consist of a moderate or relatively
large number of panels, while the panels contain a small number of observations.
Testing procedures to detect a possible common change in means of the panels are
established. Ratio and non-ratio type test statistics are considered. Their asymptotic
distributions under the no change null hypothesis are derived. Moreover, we prove
the consistency of the tests under the alternative. The main advantage of the ratio
type statistics compared to the non-ratio ones is that the variance of the observations
neither has to be known nor estimated. A simulation study reveals that the proposed
ratio statistic outperforms the non-ratio one by keeping the significance level under
the null, mainly when stronger dependence within the panel is taken into account.
However, the non-ratio statistic rejects the null in the simulations more often than it
should, which yields higher power compared to the ratio statistic.
Keywords: Change point, Panel data, Change in mean, Hypothesis testing, Structural change, Fixed panel size, Short panels, Ratio type statistics, Non-ratio type
statistic, CUSUM type statistics.

1

Introduction

The problem of an unknown common change in means of the panels is studied here, where
the panel data consist of N panels and each panel contains T observations over time. Various
values of the change are possible for each panel at some unknown common time τ “ 1, . . . , N .
The panels are considered to be independent, but this restriction can be weakened. In spite of
that, observations within the panel are usually not independent. It is supposed that a common
unknown dependence structure is present over the panels.
Tests for change point detection in the panel data have been proposed only in case when the
panel size T is sufficiently large, i.e., T increases over all limits from an asymptotic point of view,
cf. Chan et al. (2013) or Horváth and Hušková (2012). However, the change point estimation
has already been studied for finite T not depending on the number of panels N , see Bai (2010)
or Peštová and Pešta (2016). The remaining task is to develop testing procedures to decide
whether a common change point is present or not in the panels, while taking into account that
the length T of each observation regime is fixed and can be relatively small.

2

Panel Change Point Model

Let us consider the panel change point model
Yi,t “ µi ` δi Itt ą τ u ` σεi,t ,
˚

Corresponding author: michal.pesta@mff.cuni.cz

1 ď i ď N, 1 ď t ď T ;

(1)

where σ ą 0 is an unknown variance-scaling parameter and T is fixed, not depending on N .
The possible common change point time is denoted by τ P t1, . . . , T u. A situation where τ “ T
corresponds to no change in means of the panels. The means µi are panel-individual. The amount
of the break in mean, which can also differ for every panel, is denoted by δi . Furthermore, it is
assumed that the sequences of panel disturbances tεi,t ut are independent and within each panel
the errors form a weakly stationary sequence with a common correlation structure. This can be
formalized in the following assumption.
Assumption A1. The vectors rεi,1 , . . . , εi,T sJ existing on a probability space pΩ, F, Pq are iid
for i “ 1, . . . , N with Eεi,t “ 0 and Var εi,t “ 1, having the autocorrelation function
ρt “ Corr pεi,s , εi,s`t q “ Cov pεi,s , εi,s`t q ,

@s P t1, . . . , T ´ tu,

which is independent of the lag s, the cumulative autocorrelation function
rptq “ Var

t
ÿ

εi,s “

s“1

ÿ

pt ´ |s|qρs ,

|s|ăt

and the shifted cumulative correlation function
˜
¸
t
v
t
v
ÿ
ÿ
ÿ
ÿ
Rpt, vq “ Cov
εi,s ,
εi,u “
ρu´s ,
s“1

u“t`1

tăv

s“1 u“t`1

for all i “ 1, . . . , N and t, v “ 1, . . . , T .
The sequence tεi,t uTt“1 can be viewed as a part of a weakly stationary process. Note that the
dependent errors within each panel do not necessarily need to be linear processes. For example, GARCH processes as error sequences are allowed as well. The assumption of independent
panels can indeed be relaxed, but it would make the setup much more complex. Consequently,
probabilistic tools for dependent data need to be used (e.g., suitable versions of the central limit
theorem). Nevertheless, assuming, that the claim amounts for different insurance companies
are independent, is reasonable. Moreover, the assumption of a common homoscedastic variance
parameter σ can be generalized by introducing weights wi,t , which are supposed to be known.
Being particular in actuarial practice, it would mean to normalize the total claim amount by the
premium received, since bigger insurance companies are expected to have higher variability in
total claim amounts paid.
It is required to test the null hypothesis of no change in the means
H0 : τ “ T
against the alternative that at least one panel has a change in mean
H1 : τ ă T

3

and Di P t1, . . . , N u : δi ‰ 0.

Test Statistic and Asymptotic Results

We propose a ratio type statistic to test H0 against H1 , because this type of statistic does not
require estimation of the nuisance parameter for the variance. Generally, this is due to the
fact that the variance parameter simply cancels out from the nominator and denominator of
the statistic. For surveys on ratio type test statistics, we refer to Csörgő and Horváth (1997),

Madurkayová (2011), and Horváth et al. (2009). Our particular panel change point CUSUM test
statistic is
ˇ
ˇ
t
N ÿ
ˇ
ˇÿ
`
˘
1
ˇ
ˇ
Yi,s ´ Ysi,T ˇ ,
max ˇ
CN pT q “ ?
ˇ
N t“1,...,T ´1 ˇi“1 s“1
which is going to be compared with the ratio test statistics
ˇř ř
`
˘ˇ
ˇ
s
si,t ˇˇ
Y
´
Y
maxs“1,...,t ˇ N
i,r
r“1
i“1
ˇř ř
¯ˇ ,
´
RN pT q “ max
ˇ
ˇ N
T
t“2,...,T ´2
maxs“t,...,T ´1 ˇ i“1 r“s`1 Yi,r ´ Yri,t ˇ
where Ysi,t is the average of the first t observations in panel i and Yri,t is the average of the last
T ´ t observations in panel i, i.e.,
t
1ÿ
s
Yi,t “
Yi,s
t s“1

and Yri,t “

T
ÿ
1
Yi,s .
T ´ t s“t`1

elaborated in Peštová and Pešta (2015). It will be demonstrated by simulations that RN pT q
keeps the theoretical significance level, however, CN pT q does not.
Firstly, we derive the behavior of the test statistics under the null hypothesis.
Theorem 1 (Under Null). Under hypothesis H0 and Assumption A1
ˇ
ˇ
ˇ
ˇ
t
D
ˇ
CN pT q ÝÝÝÝÑ σ max ˇXt ´ XT ˇˇ
N Ñ8
t“1,...,T ´1
T
and

ˇ
ˇ
maxs“1,...,t ˇXs ´ st Xt ˇ
ˇ
ˇ,
RN pT q ÝÝÝÝÑ max
ˇ
ˇ
N Ñ8 t“2,...,T ´2
maxs“t,...,T ´1 ˇZs ´ TT ´s
Z
ˇ
t
´t
D

where Zt :“ XT ´ Xt and rX1 , . . . , XT sJ is a multivariate normal random vector with zero mean
and covariance matrix Λ “ tλt,v uT,T
t,v“1 such that
λt,t “ rptq

and

λt,v “ rptq ` Rpt, vq, t ă v.

The limiting distribution does not depend on the variance nuisance parameter σ, but it
depends on the unknown correlation structure of the panel change point model, which has to
be estimated for testing purposes. The way of its estimation is shown in Section 4. Note that
in case of independent observations within the panel, the correlation structure and, hence, the
covariance matrix Λ is simplified such that rptq “ t and Rpt, vq “ 0.
Next, we show how the test statistic behaves under the alternative.
ˇ
ˇ
ˇ
1 ˇřN
?
Assumption A2. limN Ñ8 N ˇ i“1 δi ˇ “ 8.
Theorem 2 (Under Alternative). If τ ď T ´ 3, then under Assumptions A1, A2 and alternative
H1
P
P
CN pT q ÝÝÝÝÑ 8 and RN pT q ÝÝÝÝÑ 8.
(2)
N Ñ8

N Ñ8

Assumption A2
? is satisfied, for instance, if 0 ă δ ď δi @i (a common lower change point
threshold) and δ N Ñ 8, N Ñ 8. Another suitable example of δi s for the condition? in
Assumption A2, can be 0 ă δi “ KN ´1{2`η for some K ą 0 and η ą 0. Or δi “ Ciα´1 N
may be used as well, where α ě 0 and C ą 0. The assumption τ ď T ´ 3 means that there

are at least three observations in the panel after the change point. It is also possible to redefine
the test statistic by interchanging the nominator and the denominator of SN pT q. Afterwards,
Theorem 2 for the modified test statistic would require three observations before the change
point, i.e., τ ě 3.
Theorem 2 says that in presence of a structural change in the panel means, the test statistic
explodes above all bounds. Hence, the procedure is consistent and the asymptotic distribution
from Theorem 1 can be used to construct the test.

4

Estimation of the Covariance Structure

The estimation of the covariance matrix Λ from Theorem 1 requires panels as vectors with
elements having common mean (i.e., without a jump). Therefore, it is necessary to construct
an estimate for a possible change point. A consistent estimate of the change point τ in the panel
data is proposed in Peštová and Pešta (2016) as
τpN :“ arg min

t“2,...,T

N t
1 ÿÿ
pYi,s ´ Ysi,t q2 ,
wptq i“1 s“1

(3)

where twptquTt“2 is a sequence of weights specified in Peštová and Pešta (2016).
Since the panels are considered to be independent and the number of panels may be sufficiently
large, one can estimate the correlation structure of the errors rε1,1 , . . . , ε1,T sJ empirically. We
base the errors’ estimates on residuals
"
Yi,t ´ Ysi,pτN , t ď τpN ,
epi,t :“
(4)
Yi,t ´ Yri,pτN , t ą τpN .
Then, the empirical version of the autocorrelation function is
ρpt :“

N T ´t
1 ÿ ÿ
epi,s epi,s`t .
p2 N T i“1 s“1
σ

Consequently, the kernel estimation of the cumulative autocorrelation function and shifted cumulative correlation function is adopted in lines with Andrews (1991):
´s¯
ÿ
rpptq “
ρps ,
pt ´ |s|qκ
h
|s|ăt

p vq “
Rpt,

t
v
ÿ
ÿ
s“1 u“t`1

ˆ
κ

u´s
h

˙
ρpu´s ,

t ă v;

where h ą 0 stands for the window size and κ belongs to a class of kernels
!
κp¨q :

R

ˇ
Ñ r´1, 1s ˇ κp0q “ 1, κpxq “ κp´xq, @x,

ż `8

κ2 pxqdx ă 8,

´8

)
κp¨q is continuos at 0 and at all but a finite number of other points .
Since the variance parameter σ simply cancels out from the limiting distribution
Theorem 1,
ř of ř
T
p2 :“ N1T N
p2i,s .
it neither has to be estimated nor known. Nevertheless, one can use σ
i“1
s“1 e

5

Simulations

A simulation experiment was performed to study the finite sample properties of the test statistics
for a common change in panel means. In particular, the interest lies in the empirical sizes of the
proposed tests (i.e., based on RN pT q and SN pT q) under the null hypothesis and in the empirical
rejection rate (power) under the alternative. Random samples of panel data (5000 each time) are
generated from the panel change point model (1). The panel size is set to T “ 10 and T “ 25 in
order to demonstrate the performance of the testing approaches in case of small and intermediate
panel length. The number of panels considered is N “ 50 and N “ 200.
The correlation structure within each panel is modeled via random vectors generated from
iid, AR(1), and GARCH(1,1) sequences. The considered AR(1) process has coefficient φ “ 0.3.
In case of GARCH(1,1) process, we use coefficients α0 “ 1, α1 “ 0.1, and β1 “ 0.2, which
according to Lindner (2009, Example 1) gives a strictly stationary process. In all three sequences,
the innovations are obtained as iid random variables from a standard normal Np0, 1q or Student
t5 distribution. Simulation scenarios are produced as all possible combinations of the above
mentioned settings.
When using the asymptotic distribution from Theorem 1, the covariance matrix is estimated
as proposed in Section 4 using the Parzen kernel
$
& 1 ´ 6x2 ` 6|x|3 , 0 ď |x| ď 1{2;
2p1 ´ |x|q3 ,
1{2 ď |x| ď 1;
κP pxq “
%
0,
otherwise.
Bartlett window as well
Several values of the smoothing window width h are tried from the interval r2, 5s and all of
them work fine providing comparable results. To simulate the asymptotic distribution of the test
statistics, 2000 multivariate random vectors are generated using the pre-estimated covariance
matrix. To access the theoretical results under H0 numerically, Table 1 provides the empirical
specificity (one minus size) of the asymptotic tests based on RN pT q and SN pT q, where the
significance level is α “ 5%.
Table 1: Empirical specificity (1´size) of the test under H0 for test statistics RN pT q and
CN pT q using the asymptotic critical values, considering a significance level of 5%, weight function
wptq “ t2 , and smoothing window width h “ 2
T

N

innovations

10

50

Np0, 1q
t5
Np0, 1q
t5

.948
.951
.950
.948

.934
.925
.933
.927

.933
.932
.939
.935

.823
.822
.825
.821

.946
.946
.950
.948

.935
.929
.938
.937

Np0, 1q
t5
Np0, 1q
t5

.946
.948
.949
.953

.940
.945
.939
.941

.932
.932
.930
.931

.780
.790
.801
.813

.947
.946
.951
.952

.945
.943
.939
.946

200
25

50
200

IID

AR(1)

GARCH(1,1)

It may be seen that both approaches are close to the theoretical value of specificity .95. As
expected, the best results are achieved in case of independence within the panel, because there

is no information overlap between two consecutive observations. The precision of not rejecting
the null is increasing as the number of panels is getting higher and the panel is getting longer as
well.
The performance of both testing procedures under H1 in terms of the empirical rejection
rates is shown in Table 2, where the change point is set to τ “ tT {2u and the change sizes δi are
independently uniform on r1, 3s in 33%, 66% or in all panels.
Table 2: Empirical sensitivity (power) of the test under H1 for test statistics RN pT q and
CN pT q using the asymptotic critical values, considering a significance level of 5%, weight function
wptq “ t2 , and smoothing window width h “ 2
H1

T

N

innovations

33%

10

50

Np0, 1q
t5
Np0, 1q
t5

.235
.174
.453
.360

1.000
.999
1.000
1.000

.256
.202
.486
.393

.999
.996
1.000
1.000

.193
.201
.387
.389

1.000
.999
1.000
1.000

Np0, 1q
t5
Np0, 1q
t5

.376
.294
.685
.561

1.000
1.000
1.000
1.000

.394
.301
.699
.565

.992
.993
.995
1.000

.312
.312
.584
.590

1.000
1.000
1.000
1.000

Np0, 1q
t5
Np0, 1q
t5

.450
.360
.774
.642

1.000
1.000
1.000
1.000

.491
.377
.807
.692

1.000
1.000
1.000
1.000

.386
.390
.677
.688

1.000
1.000
1.000
1.000

Np0, 1q
t5
Np0, 1q
t5

.688
.558
.951
.874

1.000
1.000
1.000
1.000

.694
.570
.959
.888

1.000
1.000
1.000
1.000

.581
.594
.905
.906

1.000
1.000
1.000
1.000

Np0, 1q
t5
Np0, 1q
t5

.641
.519
.928
.844

1.000
1.000
1.000
1.000

.667
.547
.945
.869

1.000
1.000
1.000
1.000

.563
.546
.868
.872

1.000
1.000
1.000
1.000

Np0, 1q
t5
Np0, 1q
t5

.873
.760
.997
.977

1.000
1.000
1.000
1.000

.884
.771
.997
.982

1.000
1.000
1.000
1.000

.792
.789
.985
.986

1.000
1.000
1.000
1.000

200
25

50
200

66%

10

50
200

25

50
200

100%

10

50
200

25

50
200

IID

AR(1)

GARCH(1,1)

One can conclude that the power of both tests increases as the panel size and the number
of panels increase, which is straightforward and expected. Moreover, higher power is obtained
when a larger portion of panels is subject to have a change in mean. The test power drops when
switching from independent observations within the panel to dependent ones. Innovations with
heavier tails (i.e., t5 ) yield smaller power than innovations with lighter tails. Generally, the newly
defined test statistic SN pT q outperforms RN pT q in all scenarios with respect to the power.
Finally, an early change point is discussed very briefly. We stay with standard normal innovations, iid observations within the panel, the size of changes δi being independently uniform on

r1, 3s in all panels, and the change point is τ “ 3 in case of T “ 10 and τ “ 5 for T “ 25. The
empirical sensitivities of both tests for small values of τ are shown in Table 3.
Table 3: Empirical sensitivity of the test for small values of τ under H1 for test statistics RN pT q
and CN pT q using the asymptotic critical values, considering a significance level of 5%, weight
function wptq “ t2 , and smoothing window width h “ 2
T

τ

N

10

3

50
200

H1 , iid, Np0, 1q
.551
.867

1.000
1.000

T

τ

N

25

5

50
200

H1 , iid, Np0, 1q
.629
.927

1.000
1.000

When the change point is not in the middle of the panel, the power of the test generally
falls down. The source of such decrease is that the left or right part of the panel possesses
less observations with constant mean, which leads to a decrease of precision in the correlation
estimation. Nevertheless, SN pT q again outperforms RN pT q even for early or late change points
(the late change points are not numerically demonstrated here).

6

Conclusions

In this paper, we consider the change point problem in panel data with fixed panel size. Occurrence of common breaks in panel means is tested. We introduce a ratio type test statistic
and derive its asymptotic properties. Under the null hypothesis of no change, the test statistic
weakly converges to a functional of the multivariate normal random vector with zero mean and
covariance structure depending on the intra-panel covariances. As shown in the paper, these
covariances can be estimated and, consequently, used for testing whether a change in means occurred or not. This is indeed feasible, because the test statistic under the alternative converges
to infinity in probability. Furthermore, the whole stochastic theory behind requires relatively
simple assumptions, which are not too restrictive.
A simulation study illustrates that even for small panel size, both investigated approaches—
the newly derived one based on SN pT q and the older one proposed in Peštová and Pešta (2015)—
work fine. One may judge that both methods keep the significance level under the null, while
various simulation scenarios are considered. Besides that, the power of the test is higher in case
of SN pT q compared to RN pT q. Finally, the proposed method is applied to insurance data, for
which the change point analysis in panel data provides an appealing approach.

6.1

Discussion

Our setup can be modified by considering large panel size, i.e., T Ñ 8. Consequently, the whole
theory leads to convergences to functionals of Gaussian processes with a covariance structure
derived in a very similar fashion as for fixed T . However, our motivation is to develop techniques
for fixed and relatively small panel size.
Dependent panels may be taken into account and the presented work might be generalized
for some kind of asymptotic independence over the panels or prescribed dependence among
the panels. Nevertheless, our incentive is determined by a problem from non-life insurance,
where the association of insurance companies consists of a relatively high number of insurance
companies. Thus, the portfolio of yearly claims is so diversified, that the panels corresponding to

insurance companies’ yearly claims may be viewed as independent and neither natural ordering
nor clustering has to be assumed.

A

Proofs

Proof of Theorem 1. Let us define
N t
1 ÿÿ
pYi,s ´ µi q.
UN ptq :“ ?
σ N i“1 s“1

Using
Lindeberg-Lévy CLT for a sequence of T -dimensional iid random vectors
ř1 the multivariate
řT
tr s“1 εi,s , . . . , s“1 εi,s sJ uiPN , we have under H0
D

rUN p1q, . . . , UN pT qsJ ÝÝÝÝÑ rX1 , . . . , XT sJ ,
N Ñ8

ř
řT
J
since Var r 1s“1 εř
1,s , . . . , s“1 ε1,s s “ Λ. Indeed, the t-th diagonal element of the covariance
t
matrix Λ is Var s“1 ε1,s “ rptq and the upper off-diagonal element on position pt, vq is
˜
¸
˜
¸
t
v
t
t
v
ÿ
ÿ
ÿ
ÿ
ÿ
Cov
ε1,s ,
ε1,u “ Var
ε1,s ` Cov
ε1,s ,
ε1,u
s“1

u“1

s“1

s“1

“ rptq ` Rpt, vq,

u“t`1

t ă v.

Moreover, let us define the reverse analogue to UN ptq, i.e.,
T
N
1 ÿ ÿ
VN ptq :“ ?
pYi,s ´ µi q “ UN pT q ´ UN ptq.
σ N i“1 s“t`1

Hence,
N
s
1 ÿ
UN psq ´ UN ptq “ ?
t
σ N i“1

1
“ ?
σ N

#

r“1

N ÿ
s
ÿ
`

ff+
t
1 ÿ
pYi,v ´ µi q
pYi,r ´ µi q ´
t v“1

«

s
ÿ

Yi,r ´ Ysi,t

˘

i“1 r“1

and, consequently,
N
1 ÿ
T ´s
VN psq ´
VN ptq “ ?
T ´t
σ N i“1

1
“ ?
σ N

N
ÿ

#

«

T
ÿ
r“s`1

T
´
ÿ

ff+
T
ÿ
1
pYi,r ´ µi q ´
pYi,v ´ µi q
T ´ t v“t`1

¯
Yi,r ´ Yri,t .

i“1 r“s`1

Using the Cramér-Wold device, we end up with
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ D
ˇ
ˇ
t
t
ˇ
ˇ
ˇ
max ˇUN ptq ´ UN pT qˇ ÝÝÝÝÑ max ˇXt ´ XT ˇˇ
t“1,...,T ´1
N Ñ8 t“1,...,T ´1
T
T

and
ˇ
ˇ
maxs“1,...,t ˇUN psq ´ st UN ptqˇ
ˇ
ˇ
max
ˇ
ˇ
t“2,...,T ´2
maxs“t,...,T ´1 ˇVN psq ´ TT ´s
V
ptq
ˇ
N
´t
ˇ
ˇ
maxs“1,...,t ˇXs ´ st Xt ˇ
ˇ
ˇ.
ÝÝÝÝÑ max
ˇ
ˇ
N Ñ8 t“2,...,T ´2
pX
´
X
q
maxs“t,...,T ´1 ˇpXT ´ Xs q ´ TT ´s
ˇ
t
T
´t
D

Proof of Theorem 2. Considering CN pT q, we have under alternative H1 that
ˇ
ˇ
ˇ
¸ˇ
˜
T
N τ
N ÿ
τ
ˇ
ˇÿ
ÿ
˘ˇˇ
1 ˇˇ ÿ ÿ `
1
1
1
ˇ
ˇ
? ˇ
pµi ` σεi,v q ´ δi ˇ
Yi,s ´ Ysi,T ˇ “ ? ˇ
µi ` σεi,s ´
ˇ
ˇ σ N ˇi“1 s“1
T v“1
T
σ N ˇi“1 s“1
ˇ
ˇ
ˇ
ˇ
N τ
N
N
ˇ
ˇÿ
τ ÿ ˇˇ
1 ˇˇ ÿ ÿ
τ
ˇ P
ˇ
? ˇ δi ˇ Ý
pεi,s ´ εsi,T q ´
Ñ 8, N Ñ 8,
“? ˇ
δi ˇ “ OP p1q `
ˇ
ˇ
ˇ
ˇ
σT
N i“1 s“1
σT
N
i“1
i“1
ř
where εsi,T “ τ1 Tv“1 εi,v .
In case of RN pT q, let t “ τ ` 1. Then under alternative H1 , it holds that
ˇ
ˇ
ˇ
ˇ
s
τ
N ÿ
N ÿ
ˇÿ
ˇÿ
`
`
˘ˇˇ
˘ˇˇ
1
1
ˇ
ˇ
?
Yi,r ´ Ysi,τ `1 ˇ ě ? ˇ
Yi,r ´ Ysi,τ `1 ˇ
max ˇ
ˇ σ N ˇi“1 r“1
ˇ
σ N s“1,...,τ `1 ˇi“1 r“1
ˇ
˜
¸ˇ
N τ
τ `1
ˇ
1 ÿ
1
1 ˇˇ ÿ ÿ
ˇ
µi ` σεi,r ´
pµi ` σεi,v q ´
δi ˇ
“ ? ˇ
ˇ
τ ` 1 v“1
τ `1
σ N ˇi“1 r“1
ˇ
ˇ
ˇ
ˇ
N
N τ
N
ˇ
ˇÿ
ˇ
ÿ
τ
1 ˇˇ ÿ ÿ
τ
ˇ
ˇ
P
? ˇ δi ˇˇ Ý
pεi,r ´ εsi,τ `1 q ´
δi ˇ “ OP p1q `
“? ˇ
Ñ 8,
σpτ ` 1q i“1 ˇ
N ˇi“1 r“1
σpτ ` 1q N ˇi“1 ˇ

N Ñ 8,

1 řτ `1
where εsi,τ `1 “ τ `1
v“1 εi,v .
Since there is no change after τ ` 1 and τ ď T ´ 3, then by Theorem 1 we obtain
ˇ
ˇ
ˇ
ˇ
N
T
ˇÿ
´
¯ˇ
ÿ
ˇ
ˇ
1
T
´
s
ˇ
ˇ
D
ˇZs ´
?
max
Yi,r ´ Yri,τ `1 ˇ ÝÝÝÝÑ
max
Zτ `1 ˇˇ .
ˇ
ˇ
ˇ N Ñ8 s“τ `1,...,T ´1
T ´τ
σ N s“τ `1,...,T ´1 ˇi“1 r“s`1

Acknowledgment
Institutional support to Barbora Peštová was provided by RVO:67985807. The research of Michal
Pešta was supported by the Czech Science Foundation project “DYME – Dynamic Models in
Economics” No. P402/12/G097.

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