SOME RECENT DEVELOPMENTS IN MODELLING UNIVARIATE FINANCIAL TIME SERIES TERENCE C. MILLS PROFESSOR OF APPLIED STATISTICS AND ECONOMETRICS

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1 SOE RECENT DEELOPENTS IN ODELLING UNIARIATE FINANCIAL TIE SERIES TERENCE C. ILLS PROFESSOR OF APPLIED STATISTICS AND ECONOETRICS LOUGHBOROUGH UNIERSITY, U.K. 5 h Noon o Noon eeing on Saisics and ahemaical Applicaions in Finance Deparmen of ahemaics and Saisics, Universiy of aasa, Finland December 8-9, 005

2 . SIGNAL EXTRACTION, FILTERING AND ODELLING TRENDS S + N ACGF of γ γ is defined as ( ) B γ, B ( µ )( ), E µ µ E( ) Similarly, define he ACGF of γ S B. The SE esimaor of S, based on he infinie sample K,,, K, is hen, + S o be ( ) S ϖ ϖ ( B), ( B) γ ( B) ( B) S ϖ () γ ϖ ( B) is ofen referred o as he Wiener-Kolmogorov (W-K) filer and he resul () has been exended o incorporae non-saionary and S, and indeed N. ( ) If has he Wold decomposiion ψ B e, where e is serially uncorrelaed (whie noise) wih zero mean and variance σ e, hen is ACGF is γ ( B) σ ψ ( B) ψ ( B ) σ ψ ( B) e e

3 Signal Exracion from ARIA models µ + ψ ( B) e µ + ψ e 0 Beveridge-Nelson (BN) decomposiion µ + ψ () e ~ ψ ( B) e ~ ψ ( B) ψ ( B) ψ ( ) + BN S µ + ψ e µ + ψ 0 () e N BN ψ K ~ ψ e ψ e ψ e 3 ( B) e Since e is whie noise, he BN signal is herefore a random walk wih rae of drif equal o µ and an innovaion equal o ψ () e, which is hus proporional o ha of he original series. The BN noise componen is clearly saionary, bu since i is driven by he same innovaion as he rend componen, correlaed. S BN and BN N mus be perfecly ARIA(0,,) µ θ () + e e The BN decomposiion of his process is, since ψ () θ and ~ ψ θ S BN ( θ ) e µ + N θ BN e Wriing () as shows ha θ µ + e θb S θ θb BN ( θ ) 3

4 An esimae of he signal is hus or Sˆ Sˆ ω BN ( B) ( ) ( ) ( )( ) ( ) θb θ θb θ B K θ θ θ θsˆ + ( θ ) 0 {,,,K} {, +, +,K} { + m, +,K} + + m m FTSE All Share Index: Jan 95 Augus 005; Logarihms e e ˆ σ 0.05 e ( 0.004) ( 0.039) Sˆ 0.5Sˆ

5 A BN smooher Two-sided esimaor (Proiei and Harvey, 000) BN ( B) ω ( B) [ ψ () ] BNS ω φ θ ( B) ( B) i.e. Sˆ ω BNS ( B) This filer will be opimal for S β + v N λ( B) u v and u are muually uncorrelaed whie noise if () ψ (admissibiliy condiion) For he ARIA(0,,) Sˆ θ ( ) ( )( ) θ θb θb + θ θ and for his o be admissible, θ > 0. Noe ha Sˆ ˆ θ θ S + θ + θ + θ + θ + θ ˆ + ˆ S S For he FTSE All Share index, he decomposiion is inadmissable since ˆ θ 0. 5, bu, neverheless, S ˆ S S ˆ 0.8 ˆ For finie samples, {, K,,, K} i.e Sˆ θ, + m, + m θb ( θ ) θ + m + m + m 0 m θ 0 + 5

6 Sˆ m θ Sˆ + m + ( θ ) ~ S + m m + ( m) where ~ S ( m) ( θ ) θ ( m) ( ) m θ + 0 ~ ~ m 0 Trend Filers Hodrick-Presco Filer (Hodrick and Presco, 997) inimises he variaion in T N + λ T N subec o a smoohness condiion on S : ( ( S+ S ) ( S S )) This leads o ( B) HP S h + λ ( B) ( B ) This will be an opimal filer (in he WK sense) for he model S v N u λ σ u σ v Buerworh Filer (Pollock, 000, 00) S B b ( B) n n ( + B) ( + B ) n n n ( + B) ( + B ) + λ( B) ( B ) n This will be an opimal filer for he model 6

7 d n nd S ( + B) v N ( B) u A general model n nd ( ) v N ( B) u d + S ΘB d, n Θ 0 HP n Θ 0 Gomez (00) Buerworh sine filer n d Θ Gomez Buerworh angen filer Θ Pollock Buerworh HP S has a frequency response funcion G S ( ω) + 4λ ( cosω) B S has a frequency response funcion ( ω) + λ GT ( an( ω ) ) n ω is he frequency for which G( ω ) 0. 5 c sharper. For a given ω c, c ; as d and n increase, so he filer becomes λ HP ( cos ) 4 ω c λ B an ( ω ) n c Seing ω c 0. 0 implies λ 6,50, HP 000 (and a period of approximaely 5 years), while seing ω 0. 4 implies λ 500 (and a period of approximaely 4 years). c HP 7

8 HP rend wih λ HP HP rend wih λ HP 500 8

9 Nonparameric Trends Local Polynomial Trend (Lo, amaysky and Wang, 000) () N f + A popular choice for f () is a local polynomial, so ha a each poin in ime τ he parameers of he local ph order polynomial are chosen o solve he weighed leas squares problem Q p T p ( δ δ ( τ ) K δ ( τ ) ) Κ ( τ b) 0 p ; where ( τ ;b) Κ is a symmeric posiive funcion ha decreases as τ increases. Here b > 0 is a smoohing parameer, usually referred o as he bandwidh. The larger is b, he greaer he degree of averaging and he smooher will be he esimaed ˆ τ. f b, p funcion ( ) Wih regard o he choice of polynomial order, heoreical research has shown ha even values of p lead o greaer asympoic bias in f ˆ b, p ( τ ) near he ends of he sample han do odd values. Since accuracy a he end of he sample is imporan here, his suggess ha seings of p or 3 should be preferred over p 0 or. The bandwidh can be chosen hrough cross-validaion. This uses he idea of leaveone-ou predicion and minimizes he crierion where C ( ) τ b, p T ( ( b) fˆ ) () T ˆ ( τ ) () is he esimae of () f b, p, K τ τ + K,,,,, T ˆ based on he sample f b, p, i.e., wih observaion τ removed. Local polynomial fiing by kernel regression chooses ( τ ;b) Κ o be a densiy funcion (more precisely a funcion ha inegraes o uniy). Gaussian kernel, 9

10 Κ ( ) ; b b π ( τ b) exp ( τ ) Local cubic polynomial fis o FTSE All Share Index: b and b 00 0

11 Long emory and Srucural Shifs FTSE All Share monhly reurns ( ) Auocorrelaions Lag SACF of

12 Absolue FTSE All Share reurns ( ) Auocorrelaions Lags SACF of

13 Long memory/fracional differencing: d d θ ( B) a a e φ( B) Geweke and Porer-Hudak (983, GPH) nonparameric esimaor ln f ˆ α + dx + u fˆ : periodogram X ln ( cosω ) ω π,, K, J < T T 4 5 Hurvich e al (998) sugges seing J K T. d ˆ 0.05( 0.08) J 8 Absolue Reurns d d ( ) a ( 0.08) d ˆ 0.30 J 8 3

14 Level Shifs and Long emory Long memory may be an illusion generaed by occasional level shifs: e.g., Diebold and Inoue (00); Granger and Hyung (004), Hsu (005), Hyung and Franses (005), Smih (005). Smih (005) assumes a slowly changing mean process of he form ( p) S + p v S p small and shows ha he GPH esimaor is biased upwards. (Bu wha abou a slowly changing rend process?) He proposes a modified GPH esimaor from he exended regression ln f ˆ α + dx + βy + u Y ln + ω J K T T which has a much smaller bias. ~ d 0.00( 0.) J 69 Absolue Reurns d d ( ) a ~ d 0.4 J 69 ( 0.) 4

15 Ohanissian, Russell and Tsay (004) propose a es o disinguish beween rue long memory and spurious long memory based on he invariance of d for emporal aggregaes under he null of rue long memory. Thus; define ( m) m s m + s T m ( s ) and calculae esimaes d ˆ ( m ), m, ( ) ˆ m, m, K m. Using he resul ha (asympoically) d,, ( m dˆ ) for alernaive aggregaion levels ( m ) ( m ) i ( mi ) Cov( d ˆ, dˆ ) ar( dˆ ) < i < W ( Td) ( TΛ T ) ( Td) ~ χ ( ) T K K K K Λ 3 3 K K K K 3 i ar ( m ) ( dˆ i ) d ( ) ( ) ( ) m m dˆ, K, dˆ Wih m, m, 4 m 3, m 4 8 () d ˆ 0.9( 0.08) ˆ d ( ) 0.48( 0.9) ( ) d ˆ ( 0.5) ˆ 4 d ( ) 0.4( 0.) W 6.9 ~ χ () 3 5

16 BN decomposiion for fracional differencing case d For X ( B) ~ ARA( p, q), d ( 0.5,.5) Γ( + ) Γ( ) δ, Ariño and armol, 004, show ha S + Γ ( d ) X ( ) ˆ, and he gamma funcion Thus d T X + Γ δ ( d ) Xˆ ( ) and, using (4) and (6), his can be wrien as δ d S X + Γ ( d ) ( X + ψ () e ) 6

17 Tesing changes in Persisence Raio based ess K τt ( ( v ) τ τ + iτ τt ) τ T ( vi τ ) ( ) T T τt T + i T i ) x β, K, τ ) ' vτ y ( x β τ +, K, T ( ' vτ y x ( ), { } Unknown change poin: calculae sequence of saisics K τ T, τ l τ τ u H : ( 0) agains H : I( 0) ( ) 0 I 0 I ean Score S T τ ut * K τ lt ean Exponenial τ ut E ln T* exp( 0.5K ) τ lt ax X max K, τ T l τ T u H : () agains H : I() ( 0) I SR, ER, XR : consruced wih 0 I K H 0 or H agains H 0 or H0 7

18 S max, ( S SR) odified Tess (Harvey e al, 005) E X A problem wih hese ess is ha, if we es H 0 : I( 0) agains 0 : I( 0) I( ) he series is () spuriously reec : ( 0) H and I hroughou, hen hese saisics display a very srong endency o H in favour of a change in persisence, even asympoically. 0 I The following modificaion correcs his weakness: S m min exp( bj )S min J min is an adusmen facor based on a variable addiion echnique and b is a scaling facor ha enables he unmodified criical values o be used. Applying hese ess o he FTSE All Share index, wih τ 0. and τ 0. 8: S 8.90 S m min. 33 5%:.87 E.89 E m min. 8 5%:.96 X X m min %: 8.58 SR.6 SR m min %:.88 ER.37 ER m min %:.96 XR.65 XR m min %: 8.5 S 8.90 S m min. 4 5%: 3.4 E.89 E m min %:.47 X X min. 6 5%: 0.6 m The unmodified ess consisenly reec H 0 : I( 0) in favour of 0 : I( 0) I( ) while here is evidence of a reecion of H : () in favour of H : I() ( 0). The modified ess never reec 0 : I( 0) bes described as () I l µ H, 0 I H or H : I(), suggesing ha he series is I hroughou (given he resuls from sandard uni roo ess). 8

19 Yield on UK 0 year gils: Jan 95 Sep 005 For his series here looks o be an abrup change in persisence during he auumn of 998. Calculaing he es saisics over he window confirms his, wih es saisics of H 0 : I( 0) agains 0 : I( 0) I( ) saisics of H : () agains : I() ( 0) I 0 I H being close o zero and es H being very large. 9

20 Deecing Nonlineariy There are many well known ess of nonlineariy. A simple approach suggesed by Peña and Rodriguez (005) is o fi an AR(p) o he squared residuals from a linear fi and o use he BIC o deermine he value of p. They find ha his es has good power properies compared o a range of exan ess, alhough i has no power agains hreshold behaviour. Noe ha his is effecively an L es for ARCH, bu has good power agains a range of nonlinear alernaives For he UK long ineres rae ( 0.039) ( 0.039) eˆ eˆ α + α eˆ 0 8 i i i + error i.e., p 8. Asymmeric PARCH(,) wih GED errors: Ding, Granger and Engle (993). + e 0.00 ( 0.043) ˆ e ~ GED ( ) ˆ σ ˆ λ ˆ λ ˆ λ ˆ ˆ 0.6 ˆ ˆ 0.79 e e e + σ ( 0.070) ( 0.08) ( 0.068) ( 0.00) ˆ λ ˆ λ.55 ( 0.49) 0

21 Changes in gil yields Squared residuals from AR() fi

22 Condiional Sandard Deviaion

23 References Ariño,.A. and armol, F. (004). A permanen-ransiory decomposiion for ARFIA processes. Journal of Saisical Planning and Inference 4, Diebold, F.X. and Inoue, A. (00). Long memory and regime swiching, Journal of Economerics 05, Ding,., Granger, C.W.J. and Engle, R.F. (993). A long memory propery of sock reurns and a new model. Journal of Empirical Finance, Engle, R.F. and Kozicki, S. (993). Tesing for common feaures. Journal of Business and Economic Saisics, Geweke, J. and Porer-Hudak, S. (983). The esimaion and applicaion of long memory ime series models. Journal of Time Series Analysis 4, -38. Gómez,. (00). The use of Buerworh filers for rend and cycle esimaion in economic ime series. Journal of Business and Economic Saisics 9, Granger, C.W.J. and Hyung, N. (004). Occasional srucural breaks and long memory wih an applicaion o he S&P 500 absolue sock reurns. Journal of Empirical Finance, Hansen, B.E. and Seo, B. (00). Tesing for wo-regime hreshold coinegraion in vecor error-correcion models. Journal of Economerics 0, Hansen, P.R. (003). Srucural changes in he coinegraed vecor auoregressive model. Journal of Economerics 4, Harvey, D.I., Leybourne, S.J. and Taylor, A..R. (005). odified ess for a change in persisence. Journal of Economerics. Hodrick, R.J. and Presco, E.C. (997). Poswar US business cycles: an empirical invesigaion. Journal of oney, Credi and Banking 9, -6. Hsu, C.-C. (005). Long memory or srucural changes: An empirical examinaion on inflaion raes. Economics Leers 88, Hurvich, C, Deo, R. and Brodsky, J. (998). The mean squared error of Geweke and Porer-Hudak s esimaor of he memory parameer of a long memory ime series. Journal of Time Series Analysis 9, Hyung, N. and Franses, P.H. (005). Forecasing ime series wih long memory and level shifs. Journal of Forecasing 4, -6. Lo, A.W., amaysky, H. and Wang, J. (000). Foundaions of echnical analysis: compuaional algorihms, saisical inference and empirical implemenaion. Journal of Finance 55, Ng, S. and ogelsang, T.J. (00). Analysis of vecor auoregressions in he presence of shifs in mean. Economeric Reviews,

24 Ohanissian, A., Russell, J.R. and Tsay, R.S. (004). True or spurious long memory? A new es. Universiy of Chicago, mimeo. Peña, D. and Rodriguez, J. (005). Deecing nonlineariy in ime series by model selecion crieria. Inernaional Journal of Forecasing, Pollock, D.S.G. (000). Trend esimaion and de-rending via raional square wave filers. Journal of Economerics, 99, Pollock, D.S.G. (00). Filers for shor nonsaionary sequences. Journal of Forecasing 0, Proiei, T. and Harvey, A.C. (000). A Beveridge-Nelson smooher. Economics Leers, 67, Smih, A. (005). Level shifs and he illusion of long memory in economic ime series. Journal of Business and Economic Saisics 3,