Transcript Copula approach to modeling of ARMA and GARCH models residuals Anna Petričková FSTA 2012, Liptovský Ján 31.01.2012 Introduction The state-of-art overview      Application on the hydrological data series     Overview of.

Copula approach to modeling of
ARMA and GARCH models
residuals
Anna Petričková
FSTA 2012, Liptovský Ján
31.01.2012
Introduction
The state-of-art overview





Application on the hydrological data series




Overview of the ARMA and GARCH models
The test of homoscedasticity
Copula and autocopula
Goodness of fit test for copulas
Modeling of dependence structure of the ARMA and
GARCH models residuals using autocopulas.
Constructing of improved quality models for the
original time series.
Conclusion
Linear stochastic models – model ARMA

For example ARMA models
Xt - 1Xt-1 - 2Xt-2 - ... - pXt-p = Zt + 1Zt-1 + 2Zt-2 + ... + qZt-q
where Zt, t = 1, ..., n are i.i.d. process, coefficients 1, ..., p (AR
coefficients) and 1, ..., q (MA coefficients) - unknown parameters.

Special cases:


If p = 0, we get MA process
If q = 0, we get AR process
ARCH and GARCH model
ARCH – AutoRegressive Conditional Heteroscedasticity
 Let xt is time series in the form
xt = E[xt | t-1] + et
where





t-1 is information set containing all relevant information up to time t-1
predictable part E[xt | t-1] is modeled with linear ARMA models
et is unpredictable part with E[et| t-1] = 0, E[et2] = 2.
Model of et in the form
with
e t  vt ht2
ht   0  1e t21   2e t22     me t2m
where vt ( i.i.d. process with E(t) = 0 and D(t) = 1 ) is called
ARCH(m), m is order of the model.

Boundaries for parameters:
 0  0, 1 ,,  m1  0,  m  0
1     m  1
ARCH and GARCH model


GARCH – Generalized ARCH model.
et is time series with E[et2] = 2 in the form
where
e t  vt ht2
(1)
ht   0  1ht 1     p ht  p  1e t21     qe t2q
(2)
and {vt} is white noise process with E(t) = 0 and D(t) = 1.
 Time series et generated by (1) and (2) is called generalized
ARCH of order p, q, and denote GARCH(p, q).

Boundaries for parameters:
 0  0,
1 ,,  p 1  0,  p  0
1 ,,  q 1  0,  q  0

and also
(1     p )  (1     q )  1
McLeod and Li test of Homoscedasticity
(1983)

Test statistic
rk2 (eˆ 2 )
McL(m)  n(n  2)
k 1 n  k
m
where n is a sample size, rk2 is the squared sample autocorrelation
of squared residual series at lag k and m is moderately large.

When applied to the residuals from an ARMA (p,q) model, the McL
test statistic follows  2 (m  p  q) distribution asymptotically.
Copula and autocopula
2-dimensional copula is a function
C: [0, 1]2  [0, 1],
C(0, y) = C(x, 0) = 0, C(1, y) = C(x, 1) = x
for all x, y  [0, 1] and
C(x1, y1) + C(x2, y2) − C(x1, y2) − C(x2, y1)  0
for all x1, x2, y1, y2  [0, 1], such that x1  x2, y1  y2.
Let F is joint distribution function of 2-dimensional random vectors (X, Y)
and FX, FY are marginal distribution functions. Then
F(x, y) = C (FX(x), FY(y)).
Copula C is only one, if X and Y are continuous random variables.
Let Xt is strict stationary time series and k Z+, then autocopula CX,k is
copula of random vector (Xt, Xt-k).
Copula and autocopula
In our work we used families:

Archimedean class – Gumbel, strict Clayton, Frank, Joe BB1

convex combinations of Archimedean copulas

Extreme Value (EV) Copulas class – Gumbel A, Galambos
Goodness of fit test for copulas
Let {(xj, yj), j = 1, …, n } be n modeled 2-dimensional observations, FX, FY
their marginal distribution functions and F their joint distribution function.
The class of copulas C is correctly specified if there exists 0 so that
Fx, y   C 0 FX x ,FY y 
White (H. White: Maximum likelihood estimation of misspecified models.
Econometrica 50, 1982, pp. 1 – 26) showed that under correct
specification of the copula class C holds:
 A0  B0
where

 E 

A  E  2 ln c FX x , FY  y 
B


ln c FX x , FY  y  ln c FX x , FY  y 
and c is the density function.
Goodness of fit test for copulas
The testing procedure, which is proposed in A. Prokhorov: A goodness-of-fit
test for copulas. MPRA Paper No. 9998, 2008 is based on the empirical
distribution functions
n
n
ˆF s  1 1x  s a Fˆ s  1 1y  s
X
i
Y
i
n i1
n i1
and on a consistent estimator ˆ of vector of parameters 0.
To introduce the sample versions of A and B put:

  


Ai     2 ln c  Fˆ X x i ,Fˆ Y y i 
Bi      ln c  Fˆ X x i ,Fˆ Y y i   ln c  Fˆ X x i ,Fˆ Y y i 
n
1
Aˆ  
n
 A ,
1
Bˆ  
n
 B 
i
i 1
n
i
i 1


Goodness of fit test for copulas
Put:
n
1
Dˆ    d i  
n i1
Under the hypothesis of proper specification the statistics
asymptotical distribution N(0, V), where V is estimated by
1
Vˆ 
di  . di  

n 1
Statistics
1
Vˆ 
di  . di  

n 1
is asymptotically as
k2 k 1
2
ˆ
nD
has
Takeuchi criterion TIC
S. Grønneberg, N. L. Hjort : The copula information criterion.
Statistical Research Report , E-print 7, 2008


TIC n  2 L(θ)  2 Tr Bˆ θ  Aˆ θ1

Application
Modeling of dependence of residuals of the ARMA and GARCH
models with autocopulas

performed using the system MATHEMATICA, version 8

applied the significance level 0.05

from each of the considered time series omitted 12 the most recent
values (that were left for purposes of subsequent investigations of
the out-of-the-sample forecasting performance of the resulting
models)

14 hydrological data series – (monthly) Slovak rivers‘ flows
Sequence of procedures

At first, we have ‘fitted’ these real data series with the ARMA
models (seasonally adjusted). We have selected the best model
on the basis of the BIC criterion (case 1).

We have fitted autocopulas to the subsequent pairs of the
above mentioned residuals of time series. Then we have
selected the optimal models that attain the minimum of the TIC
criterion. Finally we have applied the best autocopulas instead
of the white noise into the original model (case 2).

The residuals of the ARMA models should be homoscedastic,
that was checked with McLeod and Li test of homoscedasticity.

When homoscedasticity in residuals has been rejected, we
have fitted them with ARCH/GARCH models (case 3).
The best copulas
Archimedean copulas (AC)
1
convex combinations of AC
13
extreme value copulas
0
Improved models
Instead of et , (which is the strict white noise process with E[et] = 0, D[et]
=2e), we have used the autocopulas that we have chosen as the best
copulas above (for each real time series).
For all 14 (seasonally adjusted) data series fitted with ARMA models
McLeod and Li test rejected homoscedasticity in residuals, so we fitted
them with ARCH/GARCH models.
To compare the quality of the optimal models in all 3 categories we have
computed their standard deviations () as well as prediction error RMSE
(root mean square error).
Comparison - description
sigma of residuals
data
ARMA without copula
ARMA with copula
ARCH/GARCH
Belá - Podbanské
2,40201
2,40913
2,35738
Čierny Váh
1,74083
1,7688
1,86467
Dunaj - Bratislava
0,55422
0,55372
0,86239
Dunajec - Červený Kláštor
1,18901
1,19725
2,37627
Handlovka - Handlová
0,24818
0,25379
0,54604
Hnilec - Jalkovce
0,34886
0,34985
0,71544
Hron - BB
0,13626
0,13072
0,18785
Kysuca - Čadca
0,36173
0,36515
0,82324
Litava - Plastovce
0,10898
0,09281
0,18617
Morava - Moravský Ján
0,59758
0,60372
0,86566
Orava - Drieňová
0,10661
0,09083
0,27903
Poprad - Chmelnica
0,6653
0,66518
1,23505
Topľa - Hanušovce
0,42021
0,42648
0,61885
Torysa - Košické Olšany
0,41571
0,41538
0,79986
Comparison - prediction
RMSE
data
ARMA without copula
ARMA with copula
ARCH/GARCH
Belá - Podbanské
1,63557
1,66597
2,72753
Čierny Váh
1,25157
1,40243
1,36957
Dunaj - Bratislava
0,37416
0,31338
0,94412
Dunajec - Červený Kláštor
1,82441
1,85133
2,48126
Handlovka - Handlová
0,27643
0,25723
0,35238
Hnilec - Jalkovce
0,43378
0,42289
0,88977
Hron - BB
0,07929
0,07902
0,22236
Kysuca - Čadca
0,37337
0,44147
0,97517
Litava - Plastovce
0,08067
0,08057
0,29661
Morava - Moravský Ján
0,26169
0,35511
0,79046
Orava - Drieňová
0,09936
0,09888
0,26276
Poprad - Chmelnica
0,50249
0,49729
1,1649
Topľa - Hanušovce
0,57753
0,57612
0,56784
Torysa - Košické Olšany
0,57104
0,57075
0,97139
Improved models
The best descriptive properties belonged to classical ARMA models
and ARMA models with copulas, only in 1 case to ARCH/GARCH
model.
The best predictive properties had ARMA models with copulas (9) and
5 classical ARMA models. ARCH/GARCH models had the worst
RMSE of residuals for all 14 time series.
Torysa
KošickéOlšany;
3
flow m s
6
5
4
3
2
1
1936
1946
1956
1966
Dunaj
1976
1986
1996
2006
time Month
Bratislava
3
flow m s
6
4
2
1888
1898
1908
1918
1928
1938
1948
1958
1968
1978
1988
1998
2008
time Month
Torysa
KošickéOlšany;
flow m3 s
4
3
2
1
0
2
4
6
Dunaj
8
10
12
time Month
Bratislava
flow m3 s
4
3
2
1
0
2
4
6
8
10
12
time Month
Conclusions
We have found out that ARCH/GARCH models are not very
suitable for fitting of rivers’ flows data series. Much better attempt
was fitting them with classical linear ARMA models and also ARMA
models with copulas, where copulas are able to capture wider
range of nonlinearity.
In future we also want to describe real time series with nonArchimedean copulas like Gauss, Student copulas, Archimax
copulas etc. We also want to use regime-switching model with
regimes determined by observable or unobservable variables and
compare it with the others.
Thank you for your attention.