FINANCIAL ECONOMETRICS
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1 FINANCIAL ECONOMETRICS SPRING 07 WEEK IV NONLINEAR MODELS Prof. Dr. Burç ÜLENGİN
2 Nonlner NONLINEARITY EXISTS IN FINANCIAL TIME SERIES ESPECIALLY IN VOLATILITY AND HIGH FREQUENCY DATA LINEAR MODEL IS DEFINED AS 0 0 s 0 n d(0, re rel numbers
3 Nonlner model F nformon vlble me Condonl men E[ F ] g(f F collecon of......} Condonl vrnce Vr[ - F - ] - h(f - where g(. nd h(. well defned funcons nd h(. 0 In generl g( F h( F sndrdzed shocks
4 Blner Men Models Proposed b Grnger nd Andersen(978 0,,, m s q p c p q j m s j j j j j ( ] [ ] [ s s F Vr F E Specl form Condonl Heeroscedsc model ARIMA(p,q
5 Blner models: Emple R R R R ( R monhl smple reurn of CRSP equll weghed nde from Jn.96 o Dec.997 R sgnfcn PACF lgs nd 3, nd R hs sgnfcn PACF ( R R R (0.003 (0.06 (0.08 (0.083 (0.084 ( R R R Usng sme d AR(3-ARCH(3 esmon
6 Threshold Regresson Model:TR Sndrd Lner Regresson model wh T observons nd m poenl hresholds (crees m+ regmes. Y = X + Z d j + =,,..,T j=0,, m X: vrbles h prmeers do no vr crosss regmes Z: vrbles h prmeers vr cross regmes. Observble hreshold vrble q nd srcl ncresng hreshold vlues (g <g < g m such h we re n regme j f nd onl f g q <g j+ where g 0 =- nd g m+ =+. For emple wo regme - sngle hreshold model Y = X + Z d + - <q <g Y = X + Z d + g q <+
7 Threshold Regresson Model:TR Indcor funcon I(.= f he epresson s rue nd 0 oherwse nd defnng I j (q,g= f ( g q < g j+ Y = X + I j (q,g Z d j +e Mmze S(d,,g= (Y - X - I j (q,g Z d j usng nonlner les squres. Iden of hreshold vrble q nd he regressors X nd Z deermne he pe of TR specfcon. If q s he dh lgged vlue of Y TR model s «self-ecng» model wh del d. If he regressors X nd Z conn onl consn nd lgs of he Y, s known s «Auoregressve» model.
8 Threshold AuoRegressve Model:TAR A pecewse lner model n he hrehold. Regme AR( model (0, ~ 0 0 N d f f Del s me perod, - s he hreshold vrble nd he hrehold s 0 (0, ~ N d f f X
9 Properes of TAR Asmer n rsng nd declnng perns, more observons re posve hn negve. The men of s no zero even hough here s no consn erm n he model. E[] s weghed verge of wo regmes. Weghs re he probbles of beng he regme n sonr dsrbuon
10 TAR Esmon =IF(A<0.5; (-0.75*A+RAND(; (0.5*A+RAND( Y Y Dependen Vrble: Y Mehod: Threshold Regresson Smple (djused: 50 Included observons: 49 fer djusmens Threshold pe: B-Perron ess of L+ vs. L sequenll deermned Threshold vrble: Y(- Threshold selecon: Trmmng 0.5,, Sg. level 0.05 Threshold vlue used: Vrble Coeffcen... Sd. Error -Ssc Prob. Y(- < obs Y( <= Y( obs Y( Non-Threshold Vrbles C R-squred Men dependen vr Adjused R-squred S.D. dependen vr S.E. of regresson Akke nfo creron Sum squred resd Schwrz creron Log lkelhood Hnnn-Qunn crer F-ssc Durbn-Wson s Prob(F-ssc
11 TAR : EXAMPLE Dl log reurn of IBM sock from Jn. 96 o Dec. 999 for 944 obs. AR(-GARCH(, r r Condonl men Smple men Condonl men s hgher hn smple men. I ndces he model msspefcon. AR(-TAR-GARCH(, model r r Condonl men 0.0 Smple men ( I(. s I 0 I( f 0 bnr vrble 0 oherwse
12 TAR : EXAMPLE 0 ( ( ( I r r Furher smplcon IGARCH GARCH(, 0 f f
13 .04 TAR EXAMPLE Dependen Vrble: D(LUS Mehod: Threshold Regresson Smple (djused: /07/05 //07 Included observons: 57 fer djusmens Threshold pe: B-Perron ess of L+ vs. L sequenll deermned Threshold vrble: USA$ Threshold selecon: Trmmng 0.5,, Sg. level 0.05 Threshold vlue used: Vrble Coeffcen... Sd. Error -Ssc Prob USA$ < obs I II III IV I II III IV I C D(LUS( D(LUS( D(LUS USA$ <= USA$ obs C D(LUS( D(LUS( R-squred Men dependen vr Adjused R-squred S.D. dependen vr S.E. of regresson Akke nfo creron Sum squred resd Schwrz creron Log lkelhood Hnnn-Qunn crer F-ssc Durbn-Wson s Prob(F-ssc
14 TAR : EXAMPLE Dl log reurn of IMKB sock nde from Dec. 003 o Nov. 004 for 5 obs
15 TAR : EXAMPLE Vrble Coeff Sd Error T-S Sgnf MU A A A B B B r ( I( f f
16 TAR : EXAMPLE Correlons of Seres RESID Auocorrelons : : : Ljung-Bo Q-Sscs Q(0-0= 0.30 SgnfcnLevel 0.49 Q(0-0=.767 SgnfcnLevel Correlons of Seres RESIDSQ Auocorrelons : : : Ljung-Bo Q-Sscs Q(0-0= 6.05 SgnfcnLevel Q(0-0= SgnfcnLevel 0.09
17 Smooh Trnson AR model:star p p d c s F c ( ( 0 0 STAR s sochsc mure of wo lner AR models d del prmeer loce model rnson s scle of model rnson F m be logsc, Eponenl cumulve ds. funcon p p c c c ( ( Mn equon s weghed lner combnon of wo equons. Weghs deermned b ( s F d I s dffcul o esme nd s. The hve lrge sndr errors nd low -vlues. Therefore, generll hese vlues re fed before he esmon.
18 STAR Emple r r Monhl smple sock reurn for 3M sock from Feb 946 o Dec 997. ARCH( model 0.04 (0.00 ( (0.00 (0.00 ( STAR model 0.08 ( ( ( (0.056 F e e 000 logsc ( (0.00 funcon (0.0 When chnges, F chnges herefore s mure of he frs pr nd second pr. Wegh s no consn.
19 STAR Emple ( For lrge negve - F e e 000 ( For lrge posve - F e
20 STAR Emple : Indusrl Producon _USA IP
21 STAR Emple : Indusrl Producon _USA dlog(p=log(p-log(p( DLOG(IP
22 STAR Emple : Indusrl Producon _USA Yerl growh re D4Y=log(p-log(p( D4Y
23 Smooh Trnson AR STAR model F( e g ( 3 3 g F( 4 ( e 4 5 g ( 5 6 g The prmeer g deermnes he speed of he prmeer swches due o vrons n - round g. For lrge g, he swchng funcon becomes ver seep nd he model converges o he TAR model. If he g s relvel smll, he he rnsons re more smooh nd f g =0 he prmeers do no chnge ll. 6
24 Logsc STAR( Esmon Emple : Ind. Prod. _USA D g ( g 0.0.7D4 0.7D D4 e 5.5(D4 Dependen Vrble: D4Y Mehod: Les Squres Smple: 96: 994:4 Included observons: 36 Convergence cheved fer 8 erons D4Y=C(+C(*D4Y(-+C(3*D4Y(-+(C(4+C(5*D4Y(-+C(6 *D4Y(-/(+@EXP(-C(7*(D4Y(--C(8 Coeffcen Sd. Error -Ssc Prob. C( C( C( C( C( C( C( C( R-squred Men dependen vr 0.03 Adjused R-squr S.D. dependen vr S.E. of regresso Akke nfo creron Sum squred res Schwrz creron Log lkelhood Durbn-Wson s.73 e 0.4D Swchng vrble D4 - The vlue of g s ver lrge nd leds o ver fs rnsons. Therefore model converges o he TAR model. Bu boh g nd g esmes re no sgnfcn.
25 Threshold AR -TAR model esme Emple : Ind. Prod. _USA D ( ( Here D s dumm vrble wh D =0 for < nd D = for >. So he regme swches brupl he me nsn =. I m lso be h he regme s deed b ps vlues of he process. For nsnce, f recesson corresponds o negve vlue of -, hen he regme m be defned b he sgn of -. D ( 0 f f 0 0 D ( (
26 Threshold AR -TAR model esme Emple : Ind. Prod. _USA Dependen Vrble: D4Y Mehod: Les Squres Smple: 96: 994:4 Included observons: 36 D4Y=C(+C(*D4Y(-+C(3*D4Y(-+DUMPLUS*(C(4+C(5*D4Y( -+C(6*D4Y(- Coeffcen Sd. Error -Ssc Prob. C( C( C( C( C( C( R-squred Men dependen vr 0.03 Adjused R-squred S.D. dependen vr S.E. of regresson Akke nfo creron Sum squred resd Schwrz creron Log lkelhood Durbn-Wson s.580 DUMPLUS 0 f D4 f D4 D4 D4 0 0 f f D D4 D4 D D D4 Generl equon: No regme swchng Dependen Vrble: D4Y Coeffcen Sd. Error -Ssc Prob. C( C( C( R-squred 0.8 Men dependen vr 0.03 Adjused R-squred 0.89 S.D. dependen vr S.E. of regresson 0.0 Akke nfo creron Sum squred resd Schwrz creron Log lkelhood Durbn-Wson s.050
27 Swchng Regresson Model Y follows process h depend on he vlue of n unobserved dscree se vrble s. There re M possble regmes nd we re sd o be n se or regme m n he perod when s = m for m=,,,m The swchng model ssumes h here s dfferen regresson model ssoced wh ech regme. Y = (m+ (m (m= X m + Z g m coeffcens for X re ndeed b regme whle he g coeffcens of Z re regme nvrn. The regresson errors re normll dsrbued..d.- wh vrnce h m depend on he regme.
28 Swchng Regresson Model Lkelhood funcon
29 Smple Swchng Regresson Model The regme probbles re ssumed consn vlues The m be llowed vrng probbles b ssumng h p m s funcon of eogenous vrbles G - nd coeffcens d prmerzed usng mulnomnl log specfcon. P s = m ξ, δ = p m G, δ = ep(g δ m M ep(g δ j j=
30 Mrkov Swchng Regresson Model The Mrkov swchng regresson model eends he smple eogenous probbl frmework b specfng frs order mrkov process for he regme probbles. P s j = j s j = = P j ( Trnson mr for M regme p ( p M ( p M ( p MM (
31 Mrkov Swchng Model s ssumes vlues n {,} nd s frs-order Mrkov chn wh rns. prob. P(s = s - = = w P(s = s - = = w Where 0<w < s he probbl of swchng ou se from me - o me. A lrge w mens h s es o swch ou Se. X cn no s n Se for long. The nverse, /w s he epeced duron number of me perods- o s n Se To esme MSA s dffcul, becuse he ses re no observble. p p c c s f s f Two-se MS model
32 Mrkov Swchng Model In STAR model, he rnson s deermned b prculr lgged vrble. In MSA, he sochsc nure of he ses mples h one s never cern bou whch se belongs o. Sr uses deermnsc scheme o govern model shfs, wheres MSA model uses sochsc scheme. Ths dfference s mporn n forecsng. The forecsed vlue of T+h n MSA model comes from lner combnon of sub-models. Bu hose n STAR model, forecs come from one of he sub-model.
33 Esmon of Mrkov Swchng Model Mrkov Chn Mone Crlo smulons. Predc he probbl of ses b Prob or Log model. Some eplnor vrbles vlble me - n he Prob or Log model ws used.
34 Emple: Mrkov Swchng Model Sesonll djused growh re of US qurerl rel GNP STATE Pr. c 3 4 w Es Se Men growh re = 0.909/( =0.965 Se Men growh re = -0.40/( =-.88 Se Se S.E STATE Es S.E /w =/0.8=8.5 qurers Growh, boom perod /w =/0.86=3.5 qurers Conrcon, recesson perod
35 Dependen Vrble: DLUS Mehod: Mrkov Swchng Regresson (BFGS / Mrqurd seps Smple (djused: /06/05 //07 Included observons: 58 fer djusmens Number of ses: Convergence cheved fer 3 erons Tme-vrng Mrkov Trnson Probbles Pr(S(= S(-= Pr(S(= S(-= Vrble Coeffcen Sd. Error z-ssc Prob Regme C DLUS( I II III IV I II III IV I I II III IV I II III IV I LOG(SIGMA Pr(S(= S(-= Pr(S(= S(-= Regme.0.0 C DLUS( LOG(SIGMA Trnson Mr Prmeers P-C P-LUS( P-C P-LUS( Men dependen vr S.D. dependen vr S.E. of regresson Sum squred resd Durbn-Wson s.9860 Log lkelhood Akke nfo creron Schwrz creron Hnnn-Qunn crer Equon: UNTITLED Trnson summr: Tme-vrng Mrkov rnson probbles nd epeced durons 0. Smple (djused: /06/05 //07 Included observons: 58 fer 0.0 djusmens I II III IV I II III IV I Tme-vrng rnson probbles: P(, k = P(s( = k s(- = (row = / column = j Men Sd. Dev I II III IV I II III IV I Tme-vrng epeced durons: Men Sd. Dev
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