Testing the adequacy of regime-switching time
series models based on aggregation operators
Jozef Komornı́k1 , Magda Komornı́ková2 , and Danuša Szökeová2
1
2
Faculty of Management, Comenius University, Bratislava, Slovakia,
Jozef.Komornik@fm.uniba.sk
Faculty of Civil Engineering, Slovak University of Technology, Bratislava,
Slovakia, (magda, szoke)@math.sk
Summary. The aim of this paper is to enrich modeling procedures for aggregation operators based regime-switching time series models (that has been introduced
and applied in [Bog05] and [KK05]). Namely a test of linearity (against two-regimes
alternative) will be adopted to the specific form of the transition functions constructed via compositions of aggregation operators with a standard logistic shape
function. An application in model building for hydrological data from Slovak rivers
demonstrate promising perspectives of applicability of testing and fitting procedures
developed in this article.
Key words: Time series models, aggregation operators, regime-switching time series model, logistic shape function.
1 Introduction
A new method of construction of regime-switching models based on utilization of aggregation operators has been recently indicated in [Bog05], [KK05]. The goal of this
paper is to provide a test of linearity (against two-regimes alternative) which will be
adopted to the specific form of the transition functions constructed via compositions
of aggregation operator with a standard logistic shape function and to explore its
usability in practical modeling (using the time series of Slovak river stream flow)
2 Theoretical background
2.1 Aggregation operators
Aggregation operators and their basic property have been treated in [CMM02].
Typical continuous aggregation operators on the real line (the map Rn onto R) are
- Arithmetic mean M (x1 , . . . , xn ) =
1
n
n
P
i=1
xi ;
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J.Komornı́k, M.Komornı́ková, D.Szökeová
- Weighted means W (x1 , . . . , xn ) =
n
P
i=1
does not need to be symmetric);
- OWA operators W ′ (x1 , . . . , xn ) =
n
P
n
P
wi xi , where wi ∈ [0, 1],
wi = 1 (it
i=1
wi x′i , where x′i is a non-decreasing per-
i=1
mutation of x1 , . . . , xn , i.e., x′1 ≤ x′2 ≤ . . . ≤ x′n .
In class of OWA operators we can find also M IN (eventually M AX ) operators, corresponding to extremal cases w1 = 1 and wi = 0 otherwise, (eventually
wn = 1 and wi = 0 otherwise) and all order statistics. Similarly projection to kary coordinate with wk = 1 and wi = 0 otherwise is a special weighted mean.
A convenient way of producing a decreasing sequence (w1 , . . . , wn ) of generating
weight coefficients is based on an increasing convex bijection ϕ of [0, 1], when we
put wi = ϕ n−k+1
− ϕ n−k
, k = 1, . . . , n. More about aggregation operators can
n
n
be found, for example, in [CMM02].
2.2 Model building using aggregation operators
Outputs qt = A(yt−1 , . . . , yt−d ) of suitable aggregation operators will play the role
of threshold variables for such regime-switching models. If we apply a so-called
transition function on such threshold variable, we get a usual form of a regimeswitching model
yt = Φ1 (B)yt (1 − F (qt )) + Φ2 (B)yt F (qt ) + εt
(1)
where
Φ1 (B), Φ2 (B) are autoregressive polynomials of orders p1 , p2 in the shift operator B;
A: Rd → R is a continuous aggregation operator for an appropriately chosen
time delay d > 0;
εt ’s are assumed to be a martingale difference sequence with respect to the
history of the time series up to time t − 1, which is denoted as Ωt−1 =
{yt−1 , yt−2 , . . . , yt−(p−1) , yt−p }, p = max(p1 , p2 ), that is, E[εt |Ωt−1 ] = 0. We
also assume that the conditional variance of εt is constant, E[ε2t |Ωt−1 ] = σ 2 ;
F is a so-called transition function, i.e., a non-decreasing surjective map of the
values of a threshold variable qt to the interval [0, 1]:
F (qt ) = h(γ(qt − c)) + 1/2,
(2)
where
h: [−∞, ∞] → [−1/2, 1/2] is a non-decreasing shape function (an odd bijection);
c = F −1 (1/2) is the threshold constant;
γ is the smoothness parameter, that has a critical value γ = 0 that yields
F (c) = (1/2) implying no difference between 2 regimes and thus linearity
of the model (1).
We will use the standard logistic shape function (shifted to h(0) = 0)
h(x) =
1
1
− .
1 + e−x
2
Testing the adequacy of regime-switching models based on agops
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2.3 Testing for regime-switching nonlinearity
For testing the linearity of (1) (or H0 : γ = 0) we adopt the approach from [FD00]
to the specific form of the transition functions constructed via compositions of aggregation operator with a standard logistic shape function
F ∗ (qt ) = F (qt ) − 1/2 = h(γ(qt − c)).
In the reparametrized model equation the linearity can be tested by means of
a Lagrange Multiplier [LM] statistic with a standard asymptotic χ2 -distribution
under the null hypothesis. We denote p = max(p1 , p2 ), xt = (1, x̃′t )′ with x̃t =
(yt−1 , . . . , yt−p )′ and φi = (φi,0 , φi,1 , . . . , φi,p )′ , i = 1, 2. Then we rewrite the model
(1) as
yt = φ′1 xt [1/2 − F ∗ (qt )] + φ′2 xt [1/2 + F ∗ (qt )] + εt
or
yt = 1/2(φ1 + φ2 )′ xt + (φ2 − φ1 )′ xt F ∗ (qt ) + εt =
= 1/2(φ1 + φ2 )′ xt + (φ2 − φ1 )′ xt h(γ(qt − c)) + εt .
In order to derive a linearity test against (1), we approximate the shape function
F ∗ (qt ) with a third-order Taylor approximation around γ = 0. This results in the
auxiliary regression
yt = α′0 + β0′ xt + β1′ xt qt + β2′ xt qt2 + β3′ xt qt3 + et
(3)
where qt = A(yt−1 , . . . , yt−d ), α0 and βi = (βi,0 , βi,1 , . . . , βi,p )′ , i = 0, 1, 2, 3 are
functions of the parameters φ1 , φ2 , γ, c and et = εt + (φ2 − φ1 )′ xt R3 (qt ) with R3 (qt )
the remainder term from the Taylor approximation. Under the linearity hypothesis,
R3 (qt ) ≡ 0 and et = εt (see e. g. [FD00]). Consequently, this remainder term does
not affect the properties of the errors under the null hypothesis and, hence, the
asymptotic distribution properties. Inspection of the exact relationships shows that
the null hypothesis H′0 : γ = 0 corresponds to H′′0 : β1 = β2 = β3 = 0, which can be
tested by a standard LM-type test. Under the null hypothesis of linearity, the test
statistic, to be denoted as LM3 , has an asymptotic χ2 distribution with 3(p + 1)
degrees of freedom.
The LM3 statistic based on (3) can be computed as follows:
1. Estimate the model under the null hypothesis of linearity by regressing yt on
xt qti . Compute the residuals ε̂t and the sum of squared residuals SSR0 =
n
P
ε̂2t .
t=1
2. Estimate the auxiliary regression of yt on xt and xt qt , i = 1, 2, 3. Compute the
residuals êt and the sum of squared residuals SSR1 =
n
P
ê2t .
t=1
3. The LM3 statistic can be computed as
LM3 =
n(SSR0 − SSR1 )
.
SSR0
(4)
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J.Komornı́k, M.Komornı́ková, D.Szökeová
3 Modeling time series of monthly average Slovak rivers
stream flow
In illustrative examples discussed in this article are the 12 univariate time series
of monthly average Slovak rivers stream flow in the period from January 1973 to
December 2003. The data used for testing regime-switching nonlinearity with aggregation operators are residuals obtained after removing annual periodic components.
Appropriate orders p of linear models AR(p) for considered time series have been
determined by minimizing AIC and BIC information criterion (see, e.g. [FD00]).
The number of variables that enter any aggregation operator (for individual time
series models) is h = k − 1, where k is the first value of delay for which the values of the autocorrelation function are not significantly different from 0. In the role
of aggregation operators we used Arithmetic Mean (M ), Weighted average with
the generating functions x2 (W 2 ) and x3 (W 3 ) and OWA operators M IN and
M AX . Usual LSTAR models with threshold variables yt−d can be considered as
products of a special trivial type of aggregation operators (that map the sequence
(yt−1 , . . . , yt−h ) on its single components). The following Table 1 shows the results
of testing of adequacy of 2-regime models for 12 univariate time series of monthly
average stream flow [m3 /s] from observation stations at Slovak rivers. The meaning
of parameters k, p and d has been explained above. For each river, items in the last
7 columns of its row represent p-values for LM3 tests given by (4). The first 2 model
classes are standard LSTAR models (without an explicit aggregation operator). The
next 5 model classes correspond to LSTAR models with the threshold variable qt as
the output from the aggregation operators M AX , M IN , M , W 2 and W 3 .
Table 1. The results of testing of adequacy of 2-regime models.
River
k
p
p-value for LM3 test
for LSTAR with aggregation operator
for standard
LSTAR with
d=1 d=2
M AX
M IN
M
Vlkyňa
4 2 0.0554 0.0018
4 1 0.0002
x
Štı́tnik
Boca
5 2 0.0060 0.1246
Ipel’
3 1 0.0073
x
Kysuca1
2 2 0.4746 0.9972
Kysuca2
2 1 0.1413
x
Litava
15 2 0.2190 0.0005
Bebrava
4 2 0.0006 0.9173
Dobšiná
15 2 0.0013 0.2494
Krupinica 15 2 0.0555 9.4981
Hron1
5 2 0.0011 0.9913
Hron2
6 1 0.0014
x
0.0001
0.0033
0.0039
0.0239
0.7756
0.1613
0.1303
0.0065
0.0056
0.0004
0.0019
0.0079
0.0119
0.4339
0.2048
0.4751
0.4144
0.6591
0.0261
0.4222
0.0059
0.6796
0.2586
0.6757
0.0005
0.0452
0.0535
0.0494
0.8558
0.3093
0.0007
0.0642
0.0004
0.0015
0.0011
0.4191
W2
W3
0.0004 0.0013
0.0024 0.0007
0.0046 0.0019
0.0133 0.0071
0.5313 0.5168
0.1729 0.1532
0.0004 0.0067
0.0366 0.0330
0.0001 0.00001
0.0051 0.0098
0.0002 0.0001
0.0346 0.0102
From results in Table 1 we see that in case of data from Kysuca1 and Kysuca2 the
p-value greatly exceed standard threshold 0.05, which implies that 1-regime models
Testing the adequacy of regime-switching models based on agops
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are superior to 2-regimes alternatives. In case of data from Štitnik, Bebrava and
Hron2 the minimal p-values for LM3 test are attained for standard LSTAR models.
For data from other observation stations the minimum p-values for LM3 test are
attained for aggregation operators M AX (in case of Vlkyňa and Krupinica), W 3
(in case of Boca, Ipel’, Hron1, Dobšiná) and W 2 (in case of Litava). The operators
M IN and M do not provide minimum p-value for any of the investigated river
station data.
The next Table 2 shows an overview of the models characteristics for 10 observation stations, where the adequacy of regime-switching models has been demonstrated by the previous tests. Based on computed p-value, each river flow stream
was modeled by two classes of models, namely by the standard LSTAR with threshold variable yt−d (with delay d determined by the smaller p-value from columns 4
and 5) and by the LSTAR model with threshold variable qt = A(yt−1 , . . . , yt−h ),
where the type of aggregation operator A was determined by the minimal p-value
from columns 6 – 10. In each class we have estimated the parameters for models
with p1 , p2 ≤ 3. The best model of the class is determined by information criteria AIC and
sBIC modified for regime-switching models (see [LCh03]). The criteria
RM SE =
1
m
m
P
(yi − Fi )2 and M AE =
i=1
1
m
m
P
|yi − Fi | (where m is the number
i=1
of predictions, yi is an observed value and Fi a prediction, i = 1, . . . , m) are measures of out-of-sample fitting (for the last year data left out from the model building
sample and used for a subsequent testing, see [FD00]).
Table 2. The measures of out-of-sample fitting for 12 predictions.
River
Vlkyňa
Štı́tnik
Ipel’
Boca
Hron1
Dobšiná
Litava
Bebrava
Krupinica
Hron2
Standard LSTAR
(p1 , p2 , d) RM SE M AE
(2,2,2)
(1,1,1)
(2,1,1)
(2,2,1)
(1,2,1)
(2,2,1)
(2,1,2)
(2,1,1)
(2,2,1)
(1,2,1)
5.068
0.695
1.089
0.960
18.501
0.523
0.473
0.405
0.685
24.08
3.424
0.637
0.985
0.771
15.598
0.457
0.428
0.332
0.533
21.33
Aggregation operator
Type RM SE M AE
M AX
W3
W3
W3
W3
W3
W2
M AX
M AX
M AX
4.835
0.654
1.093
0.954
18.184
0.438
0.419
0.541
0.590
23.67
3.178
0.496
0.917
0.729
15.704
0.358
0.323
0.452
0.532
21.23
The previous results demonstrate that regime-switching models based on aggregation operators provide clearly better fit in 8 of 10 cases and comparable fit in 2
cases.
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J.Komornı́k, M.Komornı́ková, D.Szökeová
4 Conclusion
The above analysis presents an enriched procedure for regime-switching time series
models based on exploiting of aggregation operators and their adequacy testing. The
outcomes of modeling hydrological data yield very promising results.
Acknowledgement
The research summarized in this paper was partly supported by the Grants APVT20-003204, VEGA 1/2032/05 and VEGA 1/1145/04.
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