Modeling and Forecasing Volailiy Auoregressive Condiional Heeroskedasiciy Models Anhony Tay Slide 1
smpl @all line(m) sii dl_sii S TII D L _ S TII 4,000. 3,000.1.0,000 -.1 1,000 -. 0 86 88 90 9 94 96 98 00 0 04 06 08 10 -.3 86 88 90 9 94 96 98 00 0 04 06 08 10 Anhony Tay Slide
Auoregressive Condiional Heeroskedasiciy (ARCH) Model Pure ARCH(1) [Mean equaion] y 1/ iid, ~(0,1) h u u N [Variance equaion] h 1 1 Anhony Tay Slide 3
y 1/ iid, ~(0,1) h u u N h 1 1 E[ y y, y,...] E[ y, y,...] 1 1 E h u y y 1/ [ 1,,...] h E[ u y, y,...] 0 1/ 1 This resul implies - uncondiional mean is zero - serially uncorrelaed process E[ E[ y y y,...]] E[ y E[ y y,...]] E[ y.0] 0. 1 1 1 1 1 Pure ARCH(1) model herefore does no (and canno) describe any sor of cyclicaliy or predicabiliy in he mean Anhony Tay Slide 4
y iid h u, u ~(0,1) N, 1/ h 1 1 var[ y y, y,...] var[ y y, y,...] 1 1 i.e. h is he condiional variance of 1/ var[ h u y 1, y,...] h var[ u y, y,...] h 1 Condiional variance depends on he pas values of y (We usually assume and 1 o be posiive o ensure posiive variances) y Anhony Tay Slide 5
Incidenally, he uncondiional variance is: Var y [ ] E[ y ] E[ h ](since E[ y ] E[ h u ] E[ E[ h u y ]] E[ h E[ u y ]] E[ h]) 1 1 E[ y ] 1 1 E[ y ] 1 1 E[ h ] 1 1 E[ y ] 1 1 E[ y ] 1 1 E[ h ] 1 1...(1 1...) 1 1 1 herefore consan and finie if 1 1 Anhony Tay Slide 6
[ Mean equaion ] y 1/ iid, ~(0,1) h u u N h 1 1 We can have a non-rivial mean equaion: AR(1) - ARCH(1) model: y y, 0 1 1 1/ iid, ~(0,1) h u u N h 1 1 Anhony Tay Slide 7
y y 0 1 1 1/ iid, ~(0,1) h u u N h 1 1 Can show: E[ y y 1,...] 0 1y 1 ( E[ y 1,...] 0) var[ y y,...] var[ y,...] h 1 1 1 1 Noice ha y 0 1y 1, so in your compuaions Same wih condiional variance E[. y, y,...] E[.,,...] 1 1 Anhony Tay Slide 8
How he ARCH model capures volailiy-clusering y 1/ iid, ~(0,1) h u u N h 1 1 Focus on volailiy Suppose If y 1 y small (so happens) hen h 1 is small (which means ha var[ y 1 y,...] so y 1 large is small) ends o be small; if so, hen cycle coninues hen h 1 is large (which means ha var[ y 1 y,...] so y 1 is large) ends o be large; if so, hen cycle coninues Anhony Tay Slide 9
Anoher example y y 0 1 1 1/ iid, ~(0,1) h u u N h 1 1 Suppose small (so happens) If 1 hen h 1 is small (which means ha var[ 1 y,...] so 1 large is small) ends o be small; if so, hen cycle coninues hen h 1 is large (which means ha var[ 1 y,...] so 1 is large) ends o be large; if so, hen cycle coninues Anhony Tay Slide 10
Volailiy clusering can also accoun, a leas parially, for excess kurosis or faails dl_sii.his 3,600 3,00,800,400,000 1,600 1,00 800 S eries : DL_S TII S am ple 1/04/1985 10/6/011 O bs ervaions 6993 M ean 0.0001 M edian 0.000000 M axim um 0.154809 M inim um -0.9186 S d. Dev. 0.013707 S kewness -1.54600 K urosis 44.60801 K E[( Y E[ Y ]) ] () 4 400 0-0.3-0. -0.1 0.0 0.1 Jarque-B era 507.0 P robabiliy 0.000000 This is parially due o he presence of volailiy clusering (large observaions cluser, so you ge oo many of hem) ARCH processes imply excess kurosis even when condiionally normal Anhony Tay Slide 11
Anoher feaure of series wih volailiy clusering: square of residuals ofen show ARMA-ype properies Ignoring he MA(1), which is negligible in he dl_sii series, regress dl_sii on a consan, and compue he correllogram of squared residuals: equaion eq1.ls dl_sii c eq1.correlsq Anhony Tay Slide 1
ARCH(1)-ype errors possesses his propery h u, 1/ h 1 1 Rewrie variance equaion as () h 1 1 which akes he form of an AR(1) in squares of (You can view h as a zero-mean error erm) Anhony Tay Slide 13
Applicaion: Forecasing STII wihou ARCH You can show: log(stii) clearly I(1) work wih DL_STII The correllogram shows MA(1)? Anhony Tay Slide 14
smpl @firs 1/31/007 equaion eq1.ls dl_sii c ma(1) eq1.correl Dependen Variable: DL_STII residual correllogram Sample (adjused): 1/07/1985 1/31/007 Included observaions: 5996 afer adjusmens Convergence achieved afer 6 ieraions MA Backcas: 1/04/1985 Variable Coefficien Sd. Error -Saisic Prob. C 0.00086 0.000 1.4313 0.15 MA(1) 0.165015 0.0174 1.9516 0 R-squared 0.0449 Anhony Tay Slide 15
In-sample fi as well as he one-sep ahead forecass..08.1..1.0 -.1 -..0 -.1 -. -.3.04.00 -.04 -.08 -.3 86 88 90 9 94 96 98 00 0 04 06 -.1 004 005 006 007 008 009 010 011 Residual Acual Fied Y Y_F Y_UP Y_DOWN Anhony Tay Slide 16
The sandardized residuals also show evidence of volailiy clusering: In addiion o he correllogram of squared residuals, we can es for ARCH errors using Engle s LM es: Esimae his mean equaion, calculae he residuals ˆ, regress and calculae he TR ˆ on ˆ 1,,, ˆ ˆ 3 Anhony Tay Slide 17
This saisic follows a chi-square disribuion wih m degrees of freedom under he null of no condiional heeroskedasiciy in he errors. eq1.arches Heeroskedasiciy Tes: ARCH F-saisic 574.499 Prob. F(1,5993) 0.000 Obs*R-squared 54.419 Prob. Chi-Square(1) 0.000 Tes Equaion: Dependen Variable: RESID^ Variable CoefficienSd. Error -Saisic Prob. C 0.000 1.49E-05 8.305 0.000 RESID^(-1) 0.96 0.0134 3.969 0.000 The es clearly rejecs he null of no condiional heeroskedasiciy Anhony Tay Slide 18
Forecasing Volailiy Forecasing an ARCH processes is no as sraighforward as AR processes y 1/ iid, ~(0,1) h u u N h 1 1 The one-sep ahead forecas y 1 is simply y 1 0 To compue he one-sep ahead forecas of variance of y 1, use var[ y 1 y,...] var[ y y,...] h y 1 1 1 1 : Anhony Tay Slide 19
To ge he wo-sep ahead forecas for he variance, noe ha var[ y y,...] var[ y,...] E[ y,...] E[ h u y,...] E[ u y,...] E[ h y,...] 1 Since we have h h u, E[ h 1 y,...] h 1, 1 1 1 1 1 E[ u,...] 1 1 y, var[ y y,...]()(1) h y y 1 1 1 1 1 1 Anhony Tay Slide 0
In general, var[ y y,...] h...(1...) y, k 1 k k 1 k 1 1 1 1 As k, var[ y k y,...] as long as 1 1 1 1 Tha is, he k -sep ahead forecas of he variance of variance as he forecas horizon increases oward infiniy y converges o he uncondiional Anhony Tay Slide 1
Back o DL_STII: Fiing MA(1)-ARCH(1) and Forecas equaion.eq.arch(1,0) dl_sii c ma(1) Dependen Variable: DL_STII Mehod: ML - ARCH (Marquard) - Normal disribuion Sample (adjused): 1/07/1985 1/31/007 Included observaions: 5996 afer adjusmens Variable Coefficien Sd. Error z-saisic Prob. C 0.000519 0.000156 3.3519 0.0009 MA(1) 0.17774 0.006786 6.1911 0.0000 Variance Equaion C 0.000107 7.65E-07 140.959 0.0000 RESID(-1)^ 0.347641 0.011511 30.0077 0.0000 R-squared 0.0413 Mean dependen var 0.00086 Adjused R-squared 0.0396 S.D. dependen var 0.013433 S.E. of regression 0.01371 Akaike info crierion -6.0319 Sum squared resid 1.055603 Schwarz crierion -6.0683 Log likelihood 18085.81 Hannan-Quinn crier. -6.0974 Durbin-Wason sa.03671 Invered MA Roos -0.18 Anhony Tay Slide
There are no imporan changes o he esimae of he mean equaion The coefficien of ˆ 1 The fied sandard deviaion of in he variance equaion is clearly significan y is shown on he lef The fied and acuals are shown on he righ wih he +/- sandard deviaions. eq.makegarch h_ha line h_ha^0.5 genr dl_sii_ha = dl_sii-resid genr upp = dl_sii_ha + *h_ha^0.5 genr low = dl_sii_ha - *h_ha^0.5 line dl_sii dl_sii_ha upp low Anhony Tay Slide 3
H_HAT^0.5.16.3.14..1.1.10.0.08.06 -.1.04 -..0 -.3.00 86 88 90 9 94 96 98 00 0 04 06 -.4 86 88 90 9 94 96 98 00 0 04 06 The one-sep ahead forecass are shown below..1.08.04.00 -.04 -.08 -.1 004 005 006 007 008 009 010 011 Y Y_F Y_UP Y_DOWN Anhony Tay Slide 4
There is some improvemen in ha we are capuring some of he changes in volailiy, bu clearly he model sill does no properly describe and forecas volailiy here does no seem o be enough flexibiliy, and for long periods he inervals are sill far oo wide. Anhony Tay Slide 5
How do we evaluae he fi of ARCH model? The R says nohing abou he volailiy fi i only ells us how well he mean equaion fis he daa. The R here is acually smaller han in he pure MA(1) esimaion oupu Wih consan plus pure ARCH models we will usually ge negaive R s (why?) Anhony Tay Slide 6
To evaluae how well he ARCH model fis he volailiy, we make use of 1/ Our ARCH model produces esimaes of iid, ~(0,1) h u u N 1/ h (h_ha^0.5) We also have he residuals ˆ from he MA(1) model ha was fi o he daa Compue If h ˆ are good esimaes of he rue sandardized residuals should be iid uˆ ˆ ˆ 1/ h ( sandardized residuals ) h and mean equaion is correcly specified, hen he The correllogram of he sandardized residuals and heir squares should no show any dynamics: Anhony Tay Slide 7
eq.correl correl. of sandardized res. eq.correlsq correllogram of squared residuals I appears he mean equaion has done a good job (Q-sas on he lef) whereas he volailiy equaion could use some improvemens (again Q-sas, on he righ-hand figure) alhough he correlaions appear small Anhony Tay Slide 8
The hisogram of he sand. res. also ells us somehing useful abou he model. eq.his,500,000 S eries : S andardiz ed Res iduals S am ple 1/07/1985 1/31/007 O bs ervaions 5996 1,500 1,000 500 0-10 -8-6 -4-0 4 6 8 10 1 14 M ean -0.017997 M edian -0.014 M ax im um 14.30354 M inim um -10.4450 S d. Dev. 0.999893 S k ewnes s -0.104807 K uros is 18.3639 Jarque-B era 58009.18 P robabiliy 0.000000 Kurosis is much smaller han for he raw daa Some excess kurosis was removed by sandardizaion by 1/ h Bu sandardized residuals sill no normally disribued Anhony Tay Slide 9
ARCH(1) specificaion ofen no rich enough GARCH Models y 1/ iid, ~(0,1) h u u N h h 1 1 1 1 Consider he variance equaion for he GARCH(1,1) process Recursively subsiuing backwards gives h h h 1 1 1 1 () h 1 1 1 1 1 (1) h... 1 1 1 1 1 1 (1...)... 1 1 1 1 1 1 1 3 Anhony Tay Slide 30
equaion eq3.arch(1,1) dl_sii c ma(1) Dependen Variable: DL_STII Mehod: ML - ARCH (Marquard) - Normal disribuion Sample (adjused): 1/07/1985 1/31/007 Included observaions: 5996 afer adjusmens Convergence achieved afer 36 ieraions eq3.correlsq Variable Coefficien Sd. Error z-saisic Prob. C 0.001 0.000 4.71 0.000 MA(1) 0.145 0.014 10.079 0.000 Variance Equaion C 5.35E-06.51E-07f 1.34 0.000 RESID(-1)^ 0.139 0.004 34.735 0.000 GARCH(-1) 0.838 0.004 06.811 0.000 1,600 eq3.his R-squared 0.04 Mean dependen var 0.000 Adj R-squared 0.03 S.D. dependen var 0.013 S.E. of regression 0.013 Akaike info crierion -6.181 Sum sq resid 1.056 Schwarz crierion -6.175 Log likelihood 18534.360 Hannan-Quinn crier. -6.179 DW sa 1.973 Invered MA Roos -0.150 1,400 1,00 1,000 800 600 400 00 0-14 -1-10 -8-6 -4-0 4 6 Series : Sandardized Residuals Sample 1/07/1985 1/31/007 O bs ervaions 5996 M ean -0.0353 M edian -0.037406 M ax im um 5.959593 M inim um -13.53757 S d. Dev. 0.999539 S k ewnes s -0.987643 K uros is 16.31053 Jarque-B era 4537.79 P robabiliy 0.000000 Anhony Tay Slide 31
The one-sep ahead forecass over he forecas sample shows a subsanial improvemen:.1.08.04.00 -.04 -.08 -.1 004 005 006 007 008 009 010 011 Y Y_F Y_UP Y_DOWN Anhony Tay Slide 3
GARCH-in-Mean y h 0 1 1/ iid, ~(0,1) h u u N h... h... h 1 1 p p 1 1 q q This allows he condiional mean of y o be correlaed wih he condiional variance In he following, sandard deviaion h 1/ equaion: is included as a regressor in he mean Anhony Tay Slide 33
equaion eq3.arch(1,1, archm=sd) dl_sii c ma(1) Dependen Variable: DL_STII Mehod: ML - ARCH (Marquard) - Normal disribuion Sample: 1/01/004 10/6/011 Included observaions: 040 Convergence achieved afer 1 ieraions Variable Coefficien Sd. Error z-saisic Prob. @SQRT(GARCH) 0.000 0.060 0.008 0.994 C 0.001 0.001 1.37 0.185 MA(1) 0.008 0.05 0.309 0.758 Variance Equaion C 0.000 0.000 3.851 0.000 RESID(-1)^ 0.106 0.010 10.176 0.000 GARCH(-1) 0.889 0.010 86.50 0.000 R-squared -0.001 Mean dependen var 0.000 Adjused R-squared -0.00 S.D. dependen var 0.013 S.E. of regression 0.013 Akaike info crierion -6.380 Sum squared resid 0.31 Schwarz crierion -6.363 Log likelihood 6513.38 Hannan-Quinn crier. -6.374 Durbin-Wason sa 1.968 Invered MA Roos -0.010 Anhony Tay Slide 34