A One Line Derivaion of DCC: Alicaion of a Vecor Random Coefficien Moving Average Process* Chrisian M. Hafner Insiu de saisique, biosaisique e sciences acuarielles Universié caholique de Louvain Michael McAleer Dearmen of Quaniaive Finance Naional Tsing Hua Universiy Taiwan and Economeric Insiue Erasmus School of Economics Erasmus Universiy Roerdam and Tinbergen Insiue The Neherlands and Dearmen of Quaniaive Economics Comluense Universiy of Madrid Revised: July 04 * The auhors are mos graeful o Gian Piero Aielli, Massimiliano Caorin and Yuk Tse for helful commens and suggesions. For financial suor, he second auhor wishes o acknowledge he Ausralian Research Council and he Naional Science Council, Taiwan. An earlier version of he aer was resened a he Inernaional Conference on Froniers of Time Series Economerics and Relaed Fields, Hong Kong, July 03.
Absrac One of he mos widely-used mulivariae condiional volailiy models is he dynamic condiional correlaion (or DCC) secificaion. However, he underlying sochasic rocess o derive DCC has no ye been esablished, which has made roblemaic he derivaion of asymoic roeries of he Quasi-Maximum Likelihood Esimaors. The aer shows ha he DCC model can be obained from a vecor random coefficien moving average rocess, and derives he saionariy and inveribiliy condiions. The derivaion of DCC from a vecor random coefficien moving average rocess raises hree imoran issues: (i) demonsraes ha DCC is, in fac, a dynamic condiional covariance model of he reurns shocks raher han a dynamic condiional correlaion model; (ii) rovides he moivaion, which is resenly missing, for sandardizaion of he condiional covariance model o obain he condiional correlaion model; and (iii) shows ha he aroriae ARCH or GARCH model for DCC is based on he sandardized shocks raher han he reurns shocks. The derivaion of he regulariy condiions should subsequenly lead o a solid saisical foundaion for he esimaes of he DCC arameers. Keywords: Dynamic condiional correlaion, dynamic condiional covariance, vecor random coefficien moving average, saionariy, inveribiliy, asymoic roeries. JEL classificaions: C, C5, C58, G3.
. Inroducion Among mulivariae condiional volailiy models, he dynamic condiional correlaion (or DCC) secificaion of Engle (00) is one of he mos widely used in racice. The basic DCC modelling aroach has been as follows: (i) esimae he univariae condiional variances using he GARCH(,) model of Bollerslev (986), which are based on he reurns shocks; and (ii) esimae wha is urored o be he condiional correlaion marix of he sandardized residuals. The firs se is enirely arbirary as he condiional variances could us as easily be based on he sandardized residuals hemselves, as will be shown in Secion 4 below. A similar commen alies o he varying condiional correlaion model of Tse and Tsui (00), where he firs sage is based on a sandard GARCH(,) model using reurns shocks. The second sage is slighly differen from he DCC formulaion as he condiional correlaions are defined aroriaely. However, no regulariy condiions are resened, and hence no saisical roeries are given. The DCC model has been analyzed criically in a number of aers as is underlying sochasic rocess has no ye been esablished, which has made roblemaic he derivaion of he asymoic roeries of he Quasi-Maximum Likelihood Esimaors (QMLE). To dae, he saisical roeries of he QMLE of he DCC arameers have been derived under highly resricive and unverifiable regulariy condiions, which in essence amouns o roof by assumion. This aer shows ha he DCC secificaion can be obained from a vecor random coefficien moving average rocess, and derives he condiions for saionariy and inveribiliy. The derivaion of regulariy condiions should subsequenly lead o a solid saisical foundaion for he esimaes of he DCC arameers. The derivaion of DCC from a vecor random coefficien moving average rocess raises hree imoran issues: (i) demonsraes ha DCC is, in fac, a dynamic condiional covariance model of he reurns shocks raher han a dynamic condiional correlaion model; (ii) rovides he moivaion, which is resenly missing, for sandardizaion of he condiional covariance model o 3
obain he condiional correlaion model; and (iii) shows ha he aroriae ARCH or GARCH model for DCC is based on he sandardized shocks raher han he reurns shocks. The remainder of he aer organized is as follows. In Secion, he sandard ARCH model is derived from a random coefficien auoregressive rocess o rovide a background for he remainder of he aer. In Secion 3, he DCC model is discussed. Secion 4 resens a vecor random coefficien moving average rocess, from which DCC is derived in Secion 5. The condiions for saionariy and inveribiliy are given in Secion 6. Some concluding commens are given in Secion 7.. Random Coefficien Auoregressive Process Consider he following a random coefficien auoregressive rocess of order one: i () where ~ iid ( 0, ), ~ iid ( 0, ). The ARCH() model of Engle (98) can be derived as (see Tsay (987)): h E. () ( I ) where h is condiional volailiy, and I is he informaion se a ime -. The use of an infinie lag lengh for he random coefficien auoregressive rocess leads o he GARCH model of Bollerslev (986). 4
The scalar BEKK and diagonal BEKK models of Baba e al. (985) and Engle and Kroner (995) can be derived from a vecor random coefficien auoregressive rocess (see McAleer e al. (008)). As he saisical roeries of vecor random coefficien auoregressive rocesses are well known, he saisical roeries of he arameer esimaes of he ARCH, GARCH, scalar BEKK and diagonal BEKK models are sraighforward o esablish. 3. DCC Secificaion Le he condiional mean of financial reurns be given as: y E( y I ) (3) where y y y )', (,..., m y i = log Pi reresens he log-difference in sock rices ( P i ), i =,,m, I is he informaion se a ime -, and is condiionally heeroskedasic. Wihou disinguishing beween dynamic condiional covariances and dynamic condiional correlaions, Engle (00) resened he DCC secificaion as: Q ' ( ) Q Q (4) where Q is assumed o be osiive definie wih uni elemens along he main diagonal, he scalar arameers are assumed o saisfy he sabiliy condiion, <, he sandardized shocks, (,..., m)' are given as i i / hi, wih D, and D is a diagonal marix wih yical elemen h i, i =,,m. As he marix in equaion (4) does no saisfy he definiion of a correlaion marix, Engle (00) uses he following sandardizaion: / / R ( diag( Q )) Q ( diag( Q )) (5) 5
There is no clear exlanaion given in Engle (00) for he sandardizaion in equaion (5) or, more recenly, in Aielli (03). The sandardizaion in equaion (5) migh make sense if he marix Q were he condiional covariance marix of or, hough his is no made clear. Desie he ile of he aer, Aielli (03) also does no rovide any saionariy condiions for he DCC model, and does no menion inveribiliy. Indeed, in he lieraure on DCC, i is no clear wheher equaion (4) refers o a condiional covariance or a condiional correlaion marix. Some caveas regarding DCC are given in Caorin and McAleer (03). 4. Vecor Random Coefficien Moving Average Process Marek (005) roosed a linear moving average model wih random coefficiens (RCMA), and esablished he condiions for saionariy and inveribiliy. In his secion, we derive he saionariy and inveribiliy condiions of a vecor random coefficien moving average rocess. Consider a univariae random coefficien moving average rocess given by: (6) where ~ iid ) ( 0,. The condiional and uncondiional execaions of given by: are zero. The condiional variance of is h E (7) ( I ) 6
which differs from he ARCH() model in equaion () in ha he reurns shock is relaced by he sandardized shock. The use of an infinie lag lengh for he random coefficien moving average rocess in equaion (6) would leads o generalized ARCH model ha differs from he GARCH model of Bollerslev (986). The univariae ARCH() model in equaion (7) is conained in he family of GARCH models roosed by Henschel (995), and he augmened GARCH model class of Duan (997). I can be shown seen from he resuls in Marek (005) ha a sufficien condiion for saionariy is ha he vecor sequence, )' is saionary. Moreover, by Lemma. of Marek ( (005), a sufficien condiion for inveribiliy is ha: log 0 E. (8) The saionariy of (, )' and he inveribiliy condiion in equaion (8) are new resuls for he univariae ARCH() model given in equaion (7), as well as is direc exension o GARCH models. Exending he analysis given above o he mulivariae case and o a vecor random coefficien moving average (RCMA) model of order, we can derive a secial case of DCC(,q), namely DCC(,0), as follows: (9) where and are boh m vecors and, =,, are random iid m m marices. As ~ iid ( 0, ), he uncondiional variance of i is given as: 7
E ( ) ( ). h For he mulivariae case in equaion (9), i is assumed ha he vecor ~ iid ( 0, ). As he diagonal elemens of are equal o uniy, his is also he correlaion marix of. I follows ha: E( H ). This aroach can easily be exended o include auoregressive erms. For examle, in a model analogous o GARCH(,q), namely: q ' i i i i H H q where [0, ) and <, i follows ha: E( H ) i q i. The derivaion given above shows ha, as comared wih he sandard DCC formulaion, our formulaion ermis sraighforward comuaion of he uncondiional variances and covariances. I should also be noed ha in Aielli s (03) variaion of he sandard DCC model, i is ossible o calculae he uncondiional execaion of he o he uncondiional covariance marix of Q marix, as in equaion (4), bu his is no equal, which is analyically inracable. This is an addiional advanage of using he vecor random coefficien moving average rocess given in equaion (9). 8
5. One Line Derivaion of DCC If in equaion (9) is given as: I m, wih iid (0, ), =,,, ~ where is a scalar random variable, hen he condiional covariance marix can be shown o be: H ' E( I ). (0) ' The DCC model in equaion (4) is obained by leing and sandardizing H o obain a condiional correlaion marix. For he case = in equaion (0), he aroriae univariae condiional volailiy model is given in equaion (7), which uses he sandardized shocks, raher han in equaion (), which uses he reurns shocks. The derivaion of DCC in equaion (0) from a vecor random coefficien moving average rocess is imoran as i: (i) demonsraes ha DCC is, in fac, a dynamic condiional covariance model of he reurns shocks raher han a dynamic condiional correlaion model; (ii) rovides he moivaion, which is resenly missing, for sandardizaion of he condiional covariance model o obain he condiional correlaion model; and (iii) shows ha he aroriae ARCH or GARCH model for DCC is be based on he sandardized shocks raher han he reurns shocks. 6. Derivaion of Saionariy and Inveribiliy This secion derives he saionariy and inveribiliy condiions for he DCC model. Assumion. E m log () k 9
where is he Frobenius norm, and is given by:. 0 0........... 0. 0 Theorem. A sufficien condiion for saionariy is ha he vecor sequence: ' (,,..., ) is saionary. Furhermore, under Assumion, he vecor random coefficien moving average rocess,, is inverible. Proof: The roof of saionariy is similar o ha given above for he univariae random coefficien moving average rocess. For inveribiliy, noe ha: which can be wrien as: ~ ~ ~ where ~ (, and ',..., ) ~ (,. ',..., ) Hence, 0
~ n 0 k k n k k n. Now le: ( n) n k 0 k ~ Consider log n m n log n m n k k n n log k log n m n m k n n log k log n k m n m n E log a. s. k m 0 as n E log m, by assumion. This imlies ha. 0 and, hence, k a s. is asymoically measurable wih resec o {,,... }, and is inverible. Noe ha a sufficien condiion for equaion () is ha: E m ()
as E log m k log E m k log E m ( ) m log E m ( ) / log E m ( ) / 0. The condiion given in equaion () may be easier o check ha ha in equaion (). For he secial case I, wih ~ iid (0, ), =,,, discussed in Secion 5 m above, he condiion in equaion () simlifies o he wel-known condiion on he long-run ersisence o reurns shocks, namely: E. 7. Conclusion The aer is concerned wih one of he mos widely-used mulivariae condiional volailiy models, namely he dynamic condiional correlaion (or DCC) secificaion. As he underlying sochasic rocess o derive DCC has no ye been esablished, he aer showed ha he DCC secificaion could be obained from a vecor random coefficien moving average rocess, and
derived he saionariy and inveribiliy condiions. The derivaion of he regulariy condiions should evenually lead o a solid foundaion for he saisical analysis of he esimaes of he DCC arameers. The derivaion of DCC from he vecor random coefficien moving average rocess demonsraed ha DCC is, in fac, a dynamic condiional covariance model of he reurns shocks raher han a dynamic condiional correlaion model. Moreover, he derivaion rovided he moivaion, which is resenly missing, for sandardizaion of he condiional covariance model o obain he condiional correlaion model. Finally, he derivaion also showed ha he aroriae ARCH or GARCH model for DCC is based on he sandardized shocks raher han he reurns shocks. The derivaion of regulariy condiions should subsequenly lead o a solid saisical foundaion for he QMLE of he DCC arameers. 3
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