Copula approach to modeling of ARMA and GARCH models residuals Anna Petričková FSTA 2012, Liptovský Ján

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 X t -  1 X t-1 -  2 X t  p X t-p = Z t +  1 Z t-1 +  2 Z t  q Z t-q where Z t, 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 x t is time series in the form x t = E[x t |  t-1 ] +  t where   t-1 is information set containing all relevant information up to time t-1  predictable part E[x t |  t-1 ] is modeled with linear ARMA models   t is unpredictable part with E[  t |  t-1 ] = 0, E[  t 2 ] =  2. Model of  t in the form with where v t ( 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:

ARCH and GARCH model GARCH – Generalized ARCH model.  t is time series with E[  t 2 ] =  2 in the form (1) where (2) and {v t } 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

McLeod and Li test of Homoscedasticity (1983) Test statistic where n is a sample size, r k 2 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-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(x 1, y 1 ) + C(x 2, y 2 ) − C(x 1, y 2 ) − C(x 2, y 1 )  0 for all x 1, x 2, y 1, y 2  [0, 1], such that x 1  x 2, y 1  y 2. Let F is joint distribution function of 2-dimensional random vectors (X, Y) and F X, F Y are marginal distribution functions. Then F(x, y) = C (F X (x), F Y (y)). Copula C is only one, if X and Y are continuous random variables. utocopula Let X t is strict stationary time series and k  Z +, then autocopula C X,k is copula of random vector (X t, X t-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 {(x j, y j ), j = 1, …, n } be n modeled 2-dimensional observations, F X, F Y 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 and c  is the density function.  yF,xFCy,xF YX 0   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:  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 Statistics is asymptotically as  D ˆ n    kk Goodness of fit test for copulas

S. Grønneberg, N. L. Hjort : The copula information criterion. Statistical Research Report, E-print 7, 2008 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

ARMA 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). McLeod and Li test of homoscedasticity The residuals of the ARMA models should be homoscedastic, that was checked with McLeod and Li test of homoscedasticity. ARCH/GARCH When homoscedasticity in residuals has been rejected, we have fitted them with ARCH/GARCH models (case 3). Sequence of procedures

Archimedean copulas (AC)1 convex combinations of AC13 extreme value copulas0 The best copulas

Instead of, (which is the strict white noise process with E[e t ] = 0, D[e t ] =   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). Improved models rejected 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.

Comparison - description data sigma of residuals ARMA without copulaARMA with copulaARCH/GARCH Belá - Podbanské2,402012,409132,35738 Čierny Váh1,740831,76881,86467 Dunaj - Bratislava0,554220,553720,86239 Dunajec - Červený Kláštor1,189011,197252,37627 Handlovka - Handlová0,248180,253790,54604 Hnilec - Jalkovce0,348860,349850,71544 Hron - BB0,136260,130720,18785 Kysuca - Čadca0,361730,365150,82324 Litava - Plastovce0,108980,092810,18617 Morava - Moravský Ján0,597580,603720,86566 Orava - Drieňová0,106610,090830,27903 Poprad - Chmelnica0,66530,665181,23505 Topľa - Hanušovce0,420210,426480,61885 Torysa - Košické Olšany0,415710,415380,79986

Comparison - prediction data RMSE ARMA without copulaARMA with copulaARCH/GARCH Belá - Podbanské1,635571,665972,72753 Čierny Váh1,251571,402431,36957 Dunaj - Bratislava0,374160,313380,94412 Dunajec - Červený Kláštor1,824411,851332,48126 Handlovka - Handlová0,276430,257230,35238 Hnilec - Jalkovce0,433780,422890,88977 Hron - BB0,079290,079020,22236 Kysuca - Čadca0,373370,441470,97517 Litava - Plastovce0,080670,080570,29661 Morava - Moravský Ján0,261690,355110,79046 Orava - Drieňová0,099360,098880,26276 Poprad - Chmelnica0,502490,497291,1649 Topľa - Hanušovce0,577530,576120,56784 Torysa - Košické Olšany0,571040,570750,97139

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

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 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.