Copula approach to modeling of ARMA and GARCH models residuals
description
Transcript of Copula approach to modeling of ARMA and GARCH models residuals
Copula approach to modeling of ARMA and GARCH modelsresiduals
Anna Petričková
FSTA 2012, Liptovský Ján
31.01.2012
Introduction
The state-of-art overview Overview of the ARMA and GARCH models The test of homoscedasticity Copula and autocopula Goodness of fit test for copulas
Application on the hydrological data series 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] + t
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
t is unpredictable part with E[t| t-1] = 0, E[t2] = 2.
Model of t in the form
with
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:
2ttt hv
2222
2110 mtmttth
1
0,0,,,0
1
110
m
mm
ARCH and GARCH model
GARCH – Generalized ARCH model. t is time series with E[t
2] = 2 in the form
(1)
where
(2)
and {vt} is white noise process with E(t) = 0 and D(t) = 1.
Time series t generated by (1) and (2) is called generalized ARCH of order p, q, and denote GARCH(p, q).
Boundaries for parameters:
and also
2ttt hv
2211110 qtqtptptt hhh
0,0,,
0,0,,,0
11
110
pp
1)()( 11 qp
McLeod and Li test of Homoscedasticity (1983)
Test statistic
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 distribution asymptotically.)(2 qpm
m
k
k
kn
rnnmMcL
1
22 )ˆ()2()(
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 autocopulautocopula 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
Copula and autocopula
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
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:
where
00 BA
and c is the density function.
yFxFcyFxFcE
yFxFcE
YXYX
YX
,ln,ln
,ln2
B
A
yF,xFCy,xF YX0
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
and on a consistent estimator of vector of parameters 0.
To introduce the sample versions of A and B put:
n
iiY
n
iiX sy
nsF a sx
nsF
11
11
11
iYiXiYiX
iYiX
yF,xFclnyF,xFcln
yF,xFcln
i
i
B
A 2
n
ii
n
ii
n
n
1
1
1
1
BB
AA
ˆ
,ˆ
Goodness of fit test for copulas
Put:
Under the hypothesis of proper specification the statistics has
asymptotical distribution N(0, V), where V is estimated by
n
iin
ˆ1
1 dD
ii .n
ˆ ddV1
1
Statistics
ii .n
ˆ ddV1
1
is asymptotically as
Dn
2
21 kk
Goodness of fit test for copulas
S. Grønneberg, N. L. Hjort : The copula information criterion. Statistical Research Report , E-print 7, 2008
122
θθn ABTr)θ(LTIC
Takeuchi criterion TIC
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
Application
At first, we have ‘fitted’ these real data series with the ARMAARMA 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 LiMcLeod and Li test test of homoscedasticity of homoscedasticity.
When homoscedasticity in residuals has been rejected, we have fitted them with ARCH/GARCHARCH/GARCH models (case 3).
Sequence of procedures
Archimedean copulas (AC) 1
convex combinations of AC 13
extreme value copulas 0
The best copulas
Instead of , (which is the strict white noise process with E[et] = 0, D[et] =
e), we have used the autocopulas that we have chosen as the best copulas above (for each real time series).
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).
te
Improved models
For all 14 (seasonally adjusted) data series fitted with ARMA models McLeod and Li test rejectedrejected homoscedasticity in residuals, so we fitted them with ARCH/GARCH models.
Comparison - description
data
sigma of residuals
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
data
RMSE
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
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.
Improved models
1936 1946 1956 1966 1976 1986 1996 2006time Month 1
2
3
4
5
6
flow m 3s Tory s a K oš ic k é O lš any;
1888 1898 1908 1918 1928 1938 1948 1958 1968 1978 1988 1998 2008time Month
2
4
6
flow m 3s Dunaj B rat is lava
0 2 4 6 8 10 12time Month
1
2
3
4flow m 3s Tory s a K oš ic k é O lš any;
0 2 4 6 8 10 12time Month
1
2
3
4flow m 3s Dunaj B rat is lava
ConclusionsConclusions
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 non-Archimedean 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.