This paper focuses on a procedure to test for structural changes in the first two moments of a time series, when no information about the process driving the breaks is available. To approximate the process, an orthogonal Bernstein polynomial is used and testing for the null is achieved either by using an AICu information criterion, or a restriction test. The procedure covers both the pure discrete structural change and the continuous changes models. Running Monte-Carlo simulations, we show that the test has power against various alternatives.Structural changes, Bernstein polynomial, AICu.

Dominique Guegan

Philippe De Peretti

English

Research Papers in Economics

 
Documents de Travail du 
Centre d’Economie de la Sorbonne 
 
 
 
 
 
 
 
 
 
 
Tests of Structural Changes in Conditional Distributions 
with Unknown Changepoints 
 
Dominique GUEGAN, Philippe de PERETTI 
 
2011.42 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Maison des Sciences Économiques, 106-112 boulevard de L'Hôpital, 75647  Paris Cedex 13 
http://centredeconomiesorbonne.univ-paris1.fr/bandeau-haut/documents-de-travail/ 
ISSN : 1955-611X 
 
h
a
l
s
h
s
-
0
0
6
1
1
9
3
2
,
 
v
e
r
s
i
o
n
 
1
 
-
 
2
7
 
J
u
l
 
2
0
1
1Tests of Structural Changes in Conditional
Distributions with Unknown Changepoints
Dominique GuØgan￿ Philippe de Perettiy
July 18, 2011
Abstract
This paper focuses on a procedure to test for structural changes in the
￿rst two moments of a time series, when no information about the process
driving the breaks is available. To approximate the process, an orthogonal
Bernstein polynomial is used, and testing for the null is achieved either by
using an AICu information criterion, or a restriction test. The procedure
covers both the pure discrete structural change and the continuous changes
models. Running Monte-Carlo simulations, we show that the test has
power against various alternatives.
Keywords: Structural Changes ; Bernstein polynomial ; AICu
1 Introduction
This paper deals with models of the form:
A(L)yt = ct (1)
where: A(L) = 1 ￿ ￿1L ￿ ￿2L2 ￿ ::: ￿ ￿pLp;
ct is either de￿ned as ct = f(t) + "t or ct = f(t)"t, where in both cases f(t) is
￿UniversitØ Paris1 PanthØon-Sorbonne, 106-112 Bd de l￿ H￿pital, 75013 Paris, France.
Dguegan@univ-paris1.fr
yUniversitØ Paris1 PanthØon-Sorbonne, 106-112 Bd de l￿ H￿pital, 75013 Paris, France.
philippe.de-peretti@univ-paris1.fr
1
 
Documents de Travail du Centre d'Economie de la Sorbonne - 2011.42
h
a
l
s
h
s
-
0
0
6
1
1
9
3
2
,
 
v
e
r
s
i
o
n
 
1
 
-
 
2
7
 
J
u
l
 
2
0
1
1an unknown, possibly time-varying signal, thus inducing heterogeneity in one of
the two moments of the conditional distribution, "t is an iid term.
For instance, assume the simple case where f(t) is de￿ned as a step-function for
the mean:
f(t) =
8
<
:
￿1 if t ￿ t0
￿2; otherwise
(2)
with ￿1 6= ￿2 and t0 2 (￿1T;T(1 ￿ ￿2)):
Perron (2005) stresses the importance of testing for structural changes. On the
one hand, structural changes are a source of global non-stationarity (Granger
and Starica [2005], GuØgan [2010]) and then of parameter instability, and on the
other hand, ignoring structural changes may lead to erroneous statistical infer-
ence in tests for stationarity (Perron [1989]), and for long memory (Diebold
and Inoue [2001], Char⁄edine and GuØgan [2011]). Thus, testing for time-
heterogeneity of moments in a time series is a prior to modelling.
Several procedures have been developed to test for multiple changes when
the number of changes is unknown. For instance, Bai and Perron (1998) and
Andrews, Lee and Ploberger (1996) have suggested a sequential procedure, while
Bai and Perron (1998) have also introduced the so-called double-maximum test.
In a recent contribution, Heracleous, Koutris and Spanos (2004) have pointed
out that such procedures may not have power against continuous changes. In-
stead they suggest computing rolling moments on (￿ltered) series, and then
testing for time heterogeneity of these moments using an orthogonal polyno-
mial, this latter capturing movements in moments.
In this paper, we present an alternative procedure inspired of that of Her-
acleous, Koutris and Spanos (2004). The procedure tests for the null of no
structural change in the ￿rst two moments of a conditional distribution of a
time series against a pure discrete break model, or continuous changes in mo-
ments. Compared with the above literature, the suggested test di⁄ers in several
ways: i) The procedure requires no estimation of the breaks, and then of f(t),
ii) The procedure is not sequential, iii) The procedure does not use rolling win-
dows estimators for the moments, thus avoiding the di¢ cult choice of choosing
2
 
Documents de Travail du Centre d'Economie de la Sorbonne - 2011.42
h
a
l
s
h
s
-
0
0
6
1
1
9
3
2
,
 
v
e
r
s
i
o
n
 
1
 
-
 
2
7
 
J
u
l
 
2
0
1
1a window, iv) At last the procedure has not the nuisance parameter problem
under the alternative.
Our aim is to estimate model (1), by approximating the unknown function
f(t) by an orthogonal polynomial, here a Bernstein one. With k the degree of
the polynomial, the test of no-structural breaks therefore amounts to testing
k = 0 (constant signal) against k > 0. To perform such a task, we use two
statistical strategies. The ￿rst one consists in using an information criterion to
select the optimal model. We jointly select the order p and the degree k using
the AICu criterion (McQuarrie and Tsai [1988]). Indeed, this criterion ensures
an optimal trade-o⁄ between smoothing and ￿tting. Since the AICu is an ￿ all
or nothing￿decision rule, we also focus on a two-step strategy consisting in i)
Selecting the optimal model using the AICu and then if k > 0, ii) Using a
restriction test to test k = 0 against k > 0.
This note is organized as follows. Section 2 presents the test, Section 3
implements Monte-Carlo simulations, and Section 4 concludes.
2 A test of no structural change
For fytgT
t=1 where yt is real-valued, we de￿ne the following Data Generating
Process:
yt =
p X
i=1
￿iyt￿i + ct (3)
Suppose we are interested in testing for ￿rst-order time homogeneity: H1
0 :
ct = ￿1 + "t; and conditional on H1
0 true, for second-order time homogeneity:
H2
0 : ct = ￿2"t; Under H1
0 and H2
0, (3) can be re-written as:
yt =
p X
i=1
￿iyt￿i + c + "t (4)
where: "t is an iid noise.
Since f(t) is generally unknown in empirical work, we approximate the unknown
signal by an orthogonal Bernstein polynomial. The unconstrained model is thus
3
 
Documents de Travail du Centre d'Economie de la Sorbonne - 2011.42
h
a
l
s
h
s
-
0
0
6
1
1
9
3
2
,
 
v
e
r
s
i
o
n
 
1
 
-
 
2
7
 
J
u
l
 
2
0
1
1given by (5) for the mean:
yt =
p X
i=1
￿iyt￿i + f(t) + "t (5)
=
p X
i=1
￿iyt￿i +
k X
i=0
￿i
￿
k
i
￿
(
t
T
)i(1 ￿
t
T
)k￿i + "t (6)
and (7) for the variance under H1
0 true.
"2
t =
k X
i=0
￿i
￿
k
i
￿
(
t
T
)i(1 ￿
t
T
)k￿i + ￿t (7)
where: "2
t are the squared residuals of model (4), and ￿t is an iid noise
It is straightforward to see that in models (6) and (7), no-structural change in
the conditional distribution of yt implies k = 0; corresponding to a constant
signal. Thus, testing for the null amounts to testing: Hi
0 : k = 0 against k > 0,
i = 1;2.
Two testing strategies are used:
￿ Select the adequate model by minimizing an information criterion. Note
that since we want to extract a signal, a classical mean square error (MSE)
minimization criterion will be inadequate, resulting in overweighting the
￿t. This leads to use a penalized MSE. For an optimal trade-o⁄ between
￿tting and smoothing, we use the AICu criterion, introduced by McQuar-
rie and Tsai (1988), and given by:
AICu = log("0"(T ￿ p ￿ k)￿1) + 2(p + k + 1)(p ￿ k ￿ 2)￿1 (8)
for the mean, and:
AICu = log(￿0￿(T ￿ k)￿1) + 2(k + 1)(k ￿ 2)￿1 (9)
for the variance under Hi
0 true.
￿ Select the adequate model by minimizing the information criterion, if k >
0, test k = 0 against k > 0 using a restriction test. A typical procedure
4
 
Documents de Travail du Centre d'Economie de la Sorbonne - 2011.42
h
a
l
s
h
s
-
0
0
6
1
1
9
3
2
,
 
v
e
r
s
i
o
n
 
1
 
-
 
2
7
 
J
u
l
 
2
0
1
1is then to use tests in a non-nested environment. In what follows, for
instance for the mean, we estimate (10)
yt =
p X
i=1
￿iyt￿i + ￿0 +
k X
i=1
￿i
￿
k
i
￿
(
t
T
)i(1 ￿
t
T
)k￿i + "t (10)
and test H1
0 : ￿1 = ￿2::: = ￿k using a standard Ftest1.
For the variance, under H1
0 true, we estimate (11):
"2
t = ￿0 +
X
i = 1k￿i
￿
k
i
￿
(
t
T
)i(1 ￿
t
T
)k￿i + ￿t (11)
and test H2
0 : ￿1 = ￿2::: = ￿k:
We next turn to Monte-Carlo simulations.
3 Monte Carlo Simulations
In this section, we perform Monte-Carlo simulations to estimate the size and
power of the test for various sample sizes and under di⁄erent kinds of structural
changes. The ￿ve cases for ct = f(t) + "t; "t ￿ N(0;1) are:
￿ f(t) = 0; (iid case),
￿ f(t) = 0 for t ￿ t0 and f(t) = 1 otherwise and t0 is randomly drawn in
(T=4;3T=4) at each iteration (mean break),
￿ f(t) = (1 + 2t=T) (mean trend),
￿ f(t) = f(t ￿ 1) + ￿t;￿t ￿ N(0;1);f(0) = 0 (stochastic trend),
￿ yt = f(t) + "t; f(t) = f(t ￿ 1) +
"
2
t
￿+"2
t ;f(0) = 0 (stop-break model).
Table 1 returns the results of the simulations when one bases the test-
ing strategy on the simple AICu criterion. The iid case is used to study
the empirical size of the procedure, which is computed as 1 ￿ P(k = 0) or
P(k = 1) [ P(k > 1). The four other cases are used to compute the empirical
1For the test, Hetetoscedastic and Autocorrelation Consistent (HAC) matrices are used.
5
 
Documents de Travail du Centre d'Economie de la Sorbonne - 2011.42
h
a
l
s
h
s
-
0
0
6
1
1
9
3
2
,
 
v
e
r
s
i
o
n
 
1
 
-
 
2
7
 
J
u
l
 
2
0
1
1Table 1: AICu based criterion for ￿ve models, the last four ones exhibiting
ruptures in mean
iid case: H0 true
T = 50 T = 100 T = 150 T = 200 T = 500
P(k = 0) 0.813 0.831 0.860 0.872 0.882
P(k = 1) 0.102 0.090 0.088 0.066 0.075
P(k > 1) 0.085 0.079 0.052 0.062 0.043
Single discrete break in mean: H0 false
T = 50 T = 100 T = 150 T = 200 T = 500
P(k = 0) 0.125 0.018 0.006 0.001 0.000
P(k = 1) 0.400 0.329 0.187 0.109 0.006
P(k > 1) 0.475 0.653 0.807 0.890 0.994
Linear trend in mean: H0 false
T = 50 T = 100 T = 150 T = 200 T = 500
P(k = 0) 0.047 0.000 0.000 0.000 0.000
P(k = 1) 0.729 0.849 0.841 0.865 0.878
P(k > 1) 0.224 0.160 0.159 0.135 0.122
Stochastic trend in mean: H0 false
T = 50 T = 100 T = 150 T = 200 T = 500
P(k = 0) 0.229 0.244 0.252 0.233 0.228
P(k = 1) 0.161 0.182 0.182 0.167 0.158
P(k > 1) 0.610 0.574 0.566 0.600 0.614
Stop-break mode: H0 false
T = 50 T = 100 T = 150 T = 200 T = 500
P(k = 0) 0.156 0.059 0.049 0.064 0.000
P(k = 1) 0.202 0.077 0.055 0.078 0.000
P(k > 1) 0.642 0.864 0.896 0.858 1.000
Note 1: The iid case returns the size of the procedure, given by 1 ￿ P(k = 0).
Ideally it should be close to 0
Note 2: The other four cases return the power of the procedure, given by
P(k = 1) [ P(k > 1). Ideally it should be close to 1
power, i.e. P(k = 1)[P(k > 1). Using the AICu criterion returns a low empiri-
cal size, especially for small sample size (T = 50), not exceeding 0:187. Focusing
on the power, results are twofold. For the stop-break model (Engle and Smith
[1999]), the linear trend in mean, and the single discrete break in mean models,
6
 
Documents de Travail du Centre d'Economie de la Sorbonne - 2011.42
h
a
l
s
h
s
-
0
0
6
1
1
9
3
2
,
 
v
e
r
s
i
o
n
 
1
 
-
 
2
7
 
J
u
l
 
2
0
1
1Table 2: Size and power of restriction tests at 4 nominal sizes for ￿ve models,
the last four ones exhibiting ruptures in mean.
iid case: H0 true
size T = 50 T = 100 T = 150 T = 200 T = 500
0.01 0.045 0.039 0.024 0.020 0.011
0.05 0.105 0.086 0.063 0.051 0.045
0.10 0.139 0.120 0.093 0.088 0.070
0.15 0.166 0.148 0.111 0.105 0.092
Single discrete break in mean: H0 false
size T = 50 T = 100 T = 150 T = 200 T = 500
0.01 0.343 0.716 0.910 0.944 1.000
0.05 0.671 0.925 0.970 0.982 1.000
0.10 0.810 0.967 0.989 0.996 1.000
0.15 0.848 0.978 0.992 0.998 1.000
Linear trend in mean: H0 false
size T = 50 T = 100 T = 150 T = 200 T = 500
0.01 0.489 0.918 0.995 1.000 1.000
0.05 0.788 0.986 1.000 1.000 1.000
0.10 0.897 0.994 1.000 1.000 1.000
0.15 0.938 0.998 1.000 1.000 1.000
Stochastic trend in mean: H0 false
size T = 50 T = 100 T = 150 T = 200 T = 500
0.01 0.512 0.467 0.442 0.442 0.432
0.05 0.689 0.652 0.639 0.638 0.654
0.10 0.739 0.718 0.695 0.717 0.729
0.15 0.763 0.741 0.736 0.751 0.749
Stop-break model: H0 false
size T = 50 T = 100 T = 150 T = 200 T = 500
0.01 0.670 0.871 0.880 0.860 0.999
0.05 0.792 0.913 0.936 0.914 1.000
0.10 0.817 0.933 0.946 0.925 1.000
0.15 0.831 0.936 0.947 0.932 1.000
Note 1: The iid case returns the size of the procedure. Ideally it should be
close to the nominal one
Note 2: The four other cases return the power of the procedure. Ideally it
should be close to 1
7
 
Documents de Travail du Centre d'Economie de la Sorbonne - 2011.42
h
a
l
s
h
s
-
0
0
6
1
1
9
3
2
,
 
v
e
r
s
i
o
n
 
1
 
-
 
2
7
 
J
u
l
 
2
0
1
1Table 3: AICu based criterion for four models, the last three ones exhibiting
ruptures in variance
iid case: H0 true
T = 50 T = 100 T = 150 T = 200 T = 500
P(k = 0) 0.932 0.898 0.900 0.897 0.888
P(k = 1) 0.043 0.073 0.074 0.069 0.074
P(k > 1) 0.023 0.029 0.026 0.034 0.038
Single discrete break in variance: H0 false
T = 50 T = 100 T = 150 T = 200 T = 500
P(k = 0) 0.391 0.085 0.001 0.001 0.000
P(k = 1) 0.476 0.635 0.462 0.109 0.145
P(k > 1) 0.133 0.410 0.537 0.890 0.855
Linear trend in variance: H0 false
T = 50 T = 100 T = 150 T = 200 T = 500
P(k = 0) 0.652 0.186 0.111 0.048 0.000
P(k = 1) 0.288 0.154 0.783 0.838 0.862
P(k > 1) 0.060 0.660 0.106 0.114 0.132
Stochastic trend in variance: H0 false
T = 50 T = 100 T = 150 T = 200 T = 500
P(k = 0) 0.380 0.186 0.160 0.110 0.107
P(k = 1) 0.306 0.154 0.101 0.088 0.032
P(k > 1) 0.314 0.660 0.739 0.802 0.861
Note 1: The iid case returns the size of the procedure, given by 1 ￿ P(k = 0).
Ideally it should be close to 0
Note 2: The three other cases return the power of the procedure, given by
P(k = 1) [ P(k > 1). Ideally it should be close to 1
the power ranges from 0:844 to 0:953 for T = 50. For the stochastic trend in
mean model, the power is lower: ranging form 0:771 to 0:748 according to the
sample size. Hence, results remain within an acceptable range.
Focusing now on restriction tests, as presented in Table 2, at the 5% nominal
size, the empirical sizes range from 0:105 (T = 50) to 0:045 (T = 500). Also, as
mentioned above, for the stop-break, the linear trend in mean, and the single
discrete break in mean models, the type II error is close to the nominal size,
except for T = 50. When the data contain a stochastic trend, the power ranges
8
 
Documents de Travail du Centre d'Economie de la Sorbonne - 2011.42
h
a
l
s
h
s
-
0
0
6
1
1
9
3
2
,
 
v
e
r
s
i
o
n
 
1
 
-
 
2
7
 
J
u
l
 
2
0
1
1Table 4: Size and power of restriction tests at 4 nominal sizes for four models,
the last three ones exhibiting ruptures in mean
iid case: H0 true
size T = 50 T = 100 T = 150 T = 200 T = 500
0.01 0.007 0.007 0.013 0.009 0.012
0.05 0.024 0.026 0.030 0.031 0.038
0.10 0.043 0.052 0.055 0.060 0.061
0.15 0.054 0.070 0.076 0.080 0.082
Single discrete break in variance: H0 false
size T = 50 T = 100 T = 150 T = 200 T = 500
0.01 0.109 0.459 0.768 0.913 1.000
0.05 0.305 0.729 0.947 0.985 1.000
0.10 0.447 0.839 0.980 0.995 1.000
0.15 0.538 0.880 0.988 0.998 1.000
Linear trend in variance: H0 false
size T = 50 T = 100 T = 150 T = 200 T = 500
0.01 0.048 0.194 0.399 0.534 0.994
0.05 0.148 0.415 0.659 0.807 1.000
0.10 0.212 0.536 0.786 0.887 1.000
0.15 0.281 0.608 0.835 0.917 1.000
Stochastic trend in variance: H0 false
size T = 50 T = 100 T = 150 T = 200 T = 500
0.01 0.226 0.577 0.666 0.761 0.849
0.05 0.407 0.707 0.765 0.825 0.864
0.10 0.502 0.776 0.795 0.853 0.873
0.15 0.577 0.792 0.818 0.871 0.882
Note 1: The iid case returns the size of the procedure. Ideally it should be
close to the nominal one
Note 2: The three other cases return the power of the procedure. Ideally it
should be close to 1
form 0:639 to 0:689, at 5% suggesting using a higher threshold in empirical work.
Turning now to structural breaks in variances, four cases are considered:
￿ f(t) = 1 and thus ct = "t;"t ￿ N(0;1) (iid case),
￿ f(t) = 1 for t ￿ t0 and
p
f(t) = 2 otherwise, "t ￿ N(0;1); and t0 is
randomly drawn in (T=4;3T=4) at each iteration (variance break),
9
 
Documents de Travail du Centre d'Economie de la Sorbonne - 2011.42
h
a
l
s
h
s
-
0
0
6
1
1
9
3
2
,
 
v
e
r
s
i
o
n
 
1
 
-
 
2
7
 
J
u
l
 
2
0
1
1￿ f(t) = (1 + 2t=T); "t ￿ N(0;1) (variance trend),
￿ f(t) = exp(ht=2), ht = ht￿1 + ￿t, ￿t ￿ N(0;1); "t ￿ N(0;1) (stochastic
volatility model)
Table 3 presents the size and power of the procedure based on the AICu
decision rule. Clearly, the size is low, but unexpectedly doesn￿ t decrease with the
sample size. Considering the power, it is quite low for T = 50; especially when
the variance moves according to a linear trend and generally for all considered
models. It is nevertheless acceptable for sample sizes ranging from T = 100
to T = 500. Turning now to restriction tests, Table 4, it can be seen that the
empirical size is less than the 5% nominal one. Focusing at last on the type
II error; it appears that the test has power against the three models, only for
sample sizes more or less than T = 100 (T = 150 for linear trend in variance).
In all cases under the alternative, the test has low power for small sample sizes
(T = 50).
4 Conclusion
In this note, we have introduced a procedure to test for the null of no struc-
tural change in the ￿rst two moments of a conditional distribution of a time
series. The procedure uses Bernstein polynomials to extract the (noisy) signal
and has not the nuisance parameter problem under the alternative. Two tests
are proposed, a test based on the simple AICu criterion, and a restriction one.
Monte-Carlo simulations suggest that the test is powerful and can be used in
empirical work. Moreover, the procedure could be used as a general misspeci￿-
cation test.
References
[1] Andrews, W.K., Lee, I. & W. Ploberger (1996), Optimal Change Point
Tests for Normal Linear Regression, Journal of Econometrics 70, p. 9-38.
10
 
Documents de Travail du Centre d'Economie de la Sorbonne - 2011.42
h
a
l
s
h
s
-
0
0
6
1
1
9
3
2
,
 
v
e
r
s
i
o
n
 
1
 
-
 
2
7
 
J
u
l
 
2
0
1
1[2] Bai, J. & P. Perron (1998), Estimating and testing linear models with
multiple strucural changes, Econometrica 66, p.47-78.
[3] Charfeddine, L. & D. GuØgan (2011), Which is the best model for the US
in￿ ation rate : A structural changes model or a long memory process ? to
appear in Journal of Applied Econometrics.
[4] Diebold, F.X. & A. Inoue (2001), Long memory and regime switching,
Journal of Econometrics 105, p. 131-159.
[5] Engle, R.F. & A.D. Smith (1999), Stochastic permanent breaks, The Review
of Economics and Statistics 81, p.553-574.
[6] Granger, C,. & C. Starica (2005), Nonstationarities in Stock Returns, The
Review of Economics and Statistics 87, p. 503-522.
[7] GuØgan, D. (2010), Non-stationary samples and meta-distribution. In:
Basu A, Samanta T, SenGupta A, (eds.) ISI Platinium Jubilee Volume:
Statistical sciences and interdisciplinary research, ICSPRAR World Scien-
ti￿c Review, in press.
[8] Heracleous, M.S., Koutris, A. & A. Spanos (2008), Testing for nonsta-
tionarity using maximum entropy resampling: A misspeci￿cation testing
perspective, Econometric Reviews 27, p. 363-384.
[9] McQuarrie, A.D.R. & C.-L. Tsai (1998), Regression and time series model
selection, World Scienti￿c.
[10] Perron, P. (1989), The Great Crash, the oil price shock and the unit root
hypothesis. Econometrica 57, p. 1361-1401.
[11] Perron, P. (2005), Dealing with Structural Breaks, in Palgrave Handbook
of Econometrics, Vol. 1: Econometric Theory.
11
 
Documents de Travail du Centre d'Economie de la Sorbonne - 2011.42
h
a
l
s
h
s
-
0
0
6
1
1
9
3
2
,
 
v
e
r
s
i
o
n
 
1
 
-
 
2
7
 
J
u
l
 
2
0
1
1

Tests of structural changes in conditional distributions with unknown changepoints

http://halshs.archives-ouvertes.fr/docs/00/61/19/32/PDF/11042.pdf

Tests of structural changes in conditional distributions with unknown changepoints

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics