Chaper 5 Heerocedasic Models Inroducion o ime series (2008) 1
Chaper 5. Conens. 5.1. The ARCH model. 5.2. The GARCH model. 5.3. The exponenial GARCH model. 5.4. The CHARMA model. 5.5. Random coefficien auoregressive (RCA) model. 5.6. Sochasic volailiy. Inroducion o ime series (2008) 2
In opions rading and in he foreign exchange rae marke, volailiy plays an imporan role. Here volailiy means condiional variance of he underlaying asse reurn. In his chaper we discuss economeric and saisical models available in he lieraure o model he evoluion of volailiy over ime. Inroducion o ime series (2008) 3
Alhoug volailiy is no direcly measurable, i has some properies ha are common in asse reurns: 1. There are volailiy clusers, ha is volailiy may be high for cerain ime periods. 2. Volailiy evolves over ime in a coninuous manner here are no volailiy jumps. 3. Volailiy does no diverge o infiniy. 4. Volailiy seems o reac differenly o a big posiive reurn and a big negaive reurn. Inroducion o ime series (2008) 4
Assumpions and noaion. Le be he reurn series of an asse, specifically, ln( p ) ln( p 1) If he asse is a sock wih dividend paymen ln( p ) ln( d p 1 d 1) Inroducion o ime series (2008) 5
I is informaive o consider he condiional mean and variance E( / Z ) h Var( / Z ) E[( ) 2 / Z ] For simpliciy, we will assume 0 All volailiy models we will menion are models for h Inroducion o ime series (2008) 6
Chaper 5. Heerocedasic models. 5.1. The ARCH model. Inroducion o ime series (2008) 7
The ARCH model. Engle (1982). The basic idea of ARCH models is ha 1) he asse reurn is serially uncorrelaed bu dependen and 2) he dependence can be described wih a cuadraic funcion. An ARCH(r) model is h 2 2, h 0 1 1... r r where iid ( 0,1) 0 0 i 0 Inroducion o ime series (2008) 8
The ARCH model. Inerpreaion. Large pas square reurns imply a large condiional variance for he reurn. Consequenly, he reurn ends o assume a large value (in modulus). Under he ARCH framework, large reurns end o be followed by anoher large reurn. This is similar o he volailiy clusering propery. Inroducion o ime series (2008) 9
Inroducion o ime series (2008) 10 The ARCH model. Properies. Consider he ARCH(1) model (1) he condiional mean remains ero (2) uncondiional variance 2 1 1 0, h h 0 )] / ( [ ) ( Z E E E ) ( ) ( )] / ( [ ) ( ) ( 2 1 1 0 2 1 1 0 2 2 E E Z E E E Var
Because The ARCH model. is a saionary process, E( 2 ) E( 2 1 ) Var( ) and, herefore, Var( ) 0 1 ( ) ( ) 0 /(1 Var Var 1) because variance mus be posiive, we need 0 1 1 Inroducion o ime series (2008) 11
The ARCH model. Under he ARCH assumpions, he ail disribuion of is heavier han ha of a normal disribuion. Therefore, he probabiliy of ouliers is higher. This is in agreemen men wih he empirical finding ha ouliers appear more ofen in asse reurns han ha implied by an iid sequence of normal random variaes. Inroducion o ime series (2008) 12
Weakness of ARCH model. The ARCH model. (1) reas posiive and negaive reurns in he same way (by pas square reurns) (2) is very resricive (in parameers) (3) does no provide any new insigh for undersanding financial ime series (jus mechanical way o describe behaviour) (4) ofen over-predics he volailiy, because i respond slowly o large shocks. Inroducion o ime series (2008) 13
The ARCH model. Building ARCH models: (1) an ARIMA model is buil for he observed ime series o remove any serial correlaion in he daa. (2) examine he squared residuals o check for condiional heerocedasiciy. (3) use he pacf of squared residuals o deermine he ARCH order and o perform he MLE of he specified model. Inroducion o ime series (2008) 14
Chaper 5. Heerocedasic models. 5.2. The GARCH model. Inroducion o ime series (2008) 15
Inroducion o ime series (2008) 16 The GARCH model. Bollerslev (1986). ARCH models ofen require o many parameers o describe he evoluion of volailiy. Al alernaive is he GARCH(r,s) model where r i s j j j i i h h h 1 1 2 0, ), max( 1 0 1 ) ( 0, 0, 0, s r i j i j i
Inroducion o ime series (2008) 17 The GARCH model. he GARCH model implies ha he uncondiional variance of is finie whereas is condiional variance evolves over ime. Le We can rewrie he GARCH model as h h h 2 2 ), max( 1 1 2 0 2 ) ( s r i s j j j i i i
The GARCH model. The equaion before is an ARMA form of he squared series 2. I is hen clear Var 2 0 ( ) E( ) max( r, s) 1 And herefore, he uncondiional variance is finie whereas he condiional one evolves over ime. i1 ( i ) i Inroducion o ime series (2008) 18
The GARCH model. Some properies. (1) recreaes he clusering behavior. (2) heavier ail han ha of a normal dis. (3) simple parameric form ha can be used o describe he evoluion of volailiy bu no undersanding. (4) do no reflec asimeric behavior. (5) ails are sill oo shor. Inroducion o ime series (2008) 19
The GARCH model. Example: Monhly reurns of S&P500 for 792 observaions, saring from 1926. An AR (3) (or MA(3)) model is suggesed by he correlogram. AR(3) esimaion:.0066.088 2 1.0232.1233 a ˆ a.0033 bu none of he parameers is significan. Inroducion o ime series (2008) 20
The GARCH model.. Inroducion o ime series (2008) 21
The GARCH model.. Inroducion o ime series (2008) 22
The GARCH model. Join esimaion of AR(3)-ARCH(1,1). The AR(3) parameers are sill non significan, and herefore, dropped, h.0085 a.0001.8470h 1. 1221a 2 1 he uncondiional variance is.0001 E( 2 ) 1.847.1221.0036 Inroducion o ime series (2008) 23
Chaper 5. Heerocedasic models. 5.3. The exponenial GARCH model. Inroducion o ime series (2008) 24
The exponenial GARCH model. To overcome some weakness of he GARCH models in handling financial daa, Nelson (1991) proposed he EGARCH. To allow for asymmeric effecs, he considers he weighed innovaion: g( ) [ E( )] where, are real consans, and, E( ) are ero mean iid sequences wih coninuous disribuions. Inroducion o ime series (2008) 25
The exponenial GARCH model. An EGARCH (r,s) model can be wrien as, h 1 1B... sb ln( ) 0 g( 1) r 1 B... B 1 r s Differences wih GARCH: Uses logged condiional variance o relax he posiveness consrain of model coefficiens. The model can respond asymmerically o posiive and negaive values of he reurns. Inroducion o ime series (2008) 26
Chaper 5. Heerocedasic models. 5.4. The CHARMA model. Inroducion o ime series (2008) 27
The CHARMA model. Condiional Heerocedasic ARIMA model. Uses random coefficiens o produce condiional heerocedasiciy. Have similar second order condiional properies han GARCH. Where ( B) ( B) a ( B) a Niid(0, 2 ) Inroducion o ime series (2008) 28
The CHARMA model. and ( B ) 11, B... r, B r is a purely random coefficien polynomial in B. The random coefficien vecor is a sequence of iid random vecors wih mean ero and nonnegaive definie covariance marix.the condiional variance of a h 2 ( a 1,..., a r ) ( a 1,..., a r )' Inroducion o ime series (2008) 29
The CHARMA model. The condiional variance is equivalen o ha of an ARCH(r) model if he marix is diagonal. The CHARMA model uses cross-producs of he lagged values of For example, in modelling asse reurns, he cross produc erms denoe ineracions beween previous reurns. Inroducion o ime series (2008) 30
Chaper 5. Heerocedasic models. 5.5. Random Coefficien Auoregressive (RCA) model. Inroducion o ime series (2008) 31
The RCA model. Is inroduced o accoun for variabiliy among differen subjecs under sudy, similar o panel daa in economerics.is a condional heerocedasic model, bu i is used o obain a beer descripion of he condiional mean of he process by allowing for he parameers o evolve over ime. A RCA(p) model is p ( i1 i i ) i a Inroducion o ime series (2008) 32
Inroducion o ime series (2008) 33 The RCA model. where is a sequence of independen random vecors wih mean ero and covariance The condiional mean and variance are )',..., ( ),..., ( ), ( 1 1 2 1 1 p p a p i i i h Z E
Chaper 5. Heerocedasic models. 5.6. Sochasic Volailiy model. Inroducion o ime series (2008) 34
The sochasic volailiy model. An alernaive approach o describe he evoluionnof volailiy is o inroduce an innovaion o he condiional variance equaion of. A simple SV model is r h (1 1 B... rb )ln( h ) 0 Inroducing he innovaion makes he SV model more flexible in describing he evoluion of, bu also increases he h Inroducion o ime series (2008) 35
The sochasic volailiy model. difficuly in parameer esimaion. A quasilikelihood mehod wih Kalman filer is needed. Inroducion o ime series (2008) 36
Example Daily price observaions of he Deuschemark/Briish Pound foreign exchange rae. Inroducion o ime series (2008) 37
Example Conver he prices o a reurn series. Inroducion o ime series (2008) 38
Example. Check for correlaion in he reurn series Inroducion o ime series (2008) 39
Example. Inroducion o ime series (2008) 40
Example. Check for correlaion in he squared reurns Inroducion o ime series (2008) 41
Example This figure shows ha, alhough he reurns hemselves are largely uncorrelaed, he variance process exhibis some correlaion. Noe ha he ACF shown in his figure appears o die ou slowly, indicaing he possibiliy of a variance process close o being nonsaionary. Inroducion o ime series (2008) 42
Example Ljung-Box-Pierce Q-Tes. no significan correlaion is presen in he raw reurns when esed for up o 10, 15, and 20 lags of he ACF a he 0.05 level of significance. [H,pValue,Sa,CriicalValue] = 0 0.7278 6.9747 18.3070 0 0.2109 19.0628 24.9958 0 0.1131 27.8445 31.4104 However, here is significan serial correlaion in he squared reurns when you es hem wih he same inpus. [H,pValue,Sa,CriicalValue] = 1.0000 0 392.9790 18.3070 1.0000 0 452.8923 24.9958 1.0000 0 507.5858 31.4104 Inroducion o ime series (2008) 43
Example. Engle's ARCH Tes. [H,pValue,Sa,CriicalValue] = 1.0000 0 192.3783 18.3070 1.0000 0 201.4652 24.9958 1.0000 0 203.3018 31.4104 2 h* R *Given sample residuals obained from a curve fi, ess for he presence of h-h order ARCH effecs by regressing he squared residuals on a consan and he lagged values of he previous h squared residuals. Under he null hypohesis, he asympoic es saisic (h*r^2), is asympoically chi-square disribued wih h degrees of freedom. Inroducion o ime series (2008) 44
. Examine he Esimaed GARCH Model. Example Mean: ARMAX(0,0,0); Variance: GARCH(1,1) Condiional Probabiliy Disribuion: Gaussian Number of Parameers Esimaed: 4 Sandard T Parameer Value Error Saisic ----------- ----------- ------------ ----------- C -6.1919e-005 8.4331e-005-0.7342 K 1.0761e-006 1.323e-007 8.1341 GARCH(1) 0.80598 0.016561 48.6685 ARCH(1) 0.15313 0.013974 10.9586 h 1.0761e - 006 0.80598 h -1 0.15313 2-1 Inroducion o ime series (2008) 45
Example. Compare he Residuals, Condiional Sandard Deviaions, and Reurns Inroducion o ime series (2008) 46
Example. Plo and Compare he Correlaion of he Sandardied Innovaions Inroducion o ime series (2008) 47
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Example. Quanify and Compare Correlaion of he Sandardied Innovaions [H,pValue,Sa,CriicalValue] = 0 0.5262 9.0626 18.3070 0 0.3769 16.0777 24.9958 0 0.6198 17.5072 1.4104 [H,pValue,Sa,CriicalValue] = 0 0.5625 8.6823 18.3070 0 0.4408 15.1478 24.9958 0 0.6943 16.3557 31.4104 Inroducion o ime series (2008) 49
Example.MODELOS NO LINEALES EN LA MEDIA Inroducion o ime series (2008) 50
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.MODELOS BILINEALES Example Inroducion o ime series (2008) 53
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Example.MODELOS TAR Inroducion o ime series (2008) 56
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