Chapter 8 Dynamic Models

Transcription

1 Chaper 8 Dnamc odels 8. Inroducon 8. Seral correlaon models 8.3 Cross-seconal correlaons and me-seres crosssecon models 8.4 me-varng coeffcens 8.5 Kalman fler approach

2 8. Inroducon When s mporan o consder dnamc ha s emporal aspecs of a problem? For forecasng problems he dnamc aspec s crcal. For oher problems ha are focused on undersandng relaons among varables he dnamc aspecs are less crcal. Sll undersandng he mean and correlaon srucure s mporan for achevng effcen parameer esmaors. How does he sample sze nfluence our choce of sascal mehods? For man panel daa problems he number of cross-secons (n) s large compared o he number of observaons per subjec ( ). hs suggess he use of regresson analss echnques. For oher problems s large relave o n. hs suggess borrowng from oher sascal mehodologes such as mulvarae me seres.

3 Inroducon connued How does he sample sze nfluence he properes of our esmaors? For panel daa ses where n s large compared o hs suggess he use of asmpoc approxmaons where s bounded and n ends o nfn. In conras for daa ses where s large relave o n we ma acheve more relable approxmaons b consderng nsances where n and approach nfn ogeher or where n s bounded and ends o nfn.

4 Alernave approaches here are several approaches for ncorporang dnamc aspecs no a panel daa model. Perhaps he eases wa s o le one of he explanaor varables be a prox for me. For example we mgh use x j for a lnear rend n me model. Anoher sraeg s o analze he dfferences eher hrough lnear or proporonal changes of a response. hs echnque s eas o use and s naural s some areas of applcaon. o llusrae when examnng sock prces because of fnancal economcs heor we alwas look a proporonal changes n prces whch are smpl reurns. In general one mus be war of hs approach because ou lose n (nal) observaons when dfferencng.

5 Addonal sraeges Seral Correlaons Secon 8. expands on he dscusson of he modelng dnamcs hrough he seral correlaons nroducon n Secon.5.. Because of he assumpon of bounded one need no assume saonar of errors. me-varng parameers Secon 8.4 dscusses problems where model parameers are allowed o var wh me. he classc example of hs s he wo-wa error componens model nroduced n Secon 3.3..

6 Addonal sraeges he classc economerc mehod handlng of dnamc aspecs of a model s o nclude a lagged endogenous varable on he rgh hand sde of he model. Chaper 6 descrbed approach hnkng of hs approach as a pe of arkov model. Fnall Secon 8.5 shows how o adap he Kalman fler echnque o panel daa analss. hs a flexble echnque ha allows analss o ncorporae me-varng parameers and broad paerns of seral correlaon srucures no he model. Furher we wll show how o use hs echnque o smulaneousl model emporal and spaal paerns. Cross-seconal correlaons Secon 8.3 When s large relave o n we have more opporunes o model cross-seconal correlaons.

7 8. Seral correlaon models As becomes larger we have more opporunes o specf R Var he emporal varance-covarance marx. Secon.5. nroduced four specfcaons of R: () no correlaon () compound smmer () auoregressve of order one and (v) unsrucured. ovng average models sugges he oeplz specfcaon of R: R rs σ r-s. hs defnes elemens of a oeplz marx. R rs σ r-s for r-s < band and R rs for r-s band. hs s he banded oeplz marx. Facor analss suggess he form R ΛΛ Ψ where Λ s a marx of unknown facor loadngs and Ψ s an unknown dagonal marx. Useful for specfng a posve defne marx.

8 Nonsaonar covarance srucures Wh bounded we need no f a saonar model o R. A saonar AR() srucure - η elds A (nonsaonar) random walk model - η Wh we have Var σ η nonsaonar ) ( 3 3 O AR R RW O Var η σ η σ R

9 Nonsaonar covarance srucures However hs s eas o nver (Exercse 4.6) and hus mplemen. One can easl exend hs o nonsaonar AR() models ha do no requre < use hs o es for a un-roo Has desrable roo-n rae of asmpocs here s a small leraure on un-roo ess ha es for saonar as becomes large hs s much rcker O R RW

10 Connuous me correlaon models When daa are no equall spaced n me consder subjecs drawn from a populaon e wh responses as realzaons of a connuous-me sochasc process. for each subjec he response s { () for R}. Observaons of he h subjec a aken a me j so ha j ( j ) denoes he jh response of he h subjec Parcularl for unequall spaced daa a paramerc formulaon for he correlaon srucure s useful. Use R rs Cov ( r s ) σ ( r s ) where s he correlaon funcon of { ()}. Consder he exponenal correlaon model (u) exp ( φ u ) for φ > Or he Gaussan correlaon model (u) exp ( φ u ) for φ >.

11 Spaall correlaed models Daa ma also be clusered spaall. If here s no me elemen hs s sraghforward. e d j o be some measure of spaal or geographcal locaon of he jh observaon of he h subjec. hen d j d k s he dsance beween he jh and kh observaons of he h subjec. Use he correlaon funcons. Could also gnore he spaal correlaon for regresson esmaes bu use robus sandard errors o accoun for spaal correlaons.

12 Spaall correlaed models o accoun for boh spaal and emporal correlaon here s a wo-wa model α x β Sackng over we have where n s a n vecor of ones. We re-wre hs as α n X β. Defne H Var o be he spaal varance marx H j Cov ( j ) σ ( d d j ). Assumng ha { } s..d. wh varance σ we have Var Var α σ n n Var σ α I n σ J n H σ α I n V H. Because Cov ( r s ) σ α I n for r s we have V Var σ α I n J V H I. Use GS from here. n α α α n n x x x n β n.

13 V j 8.3 Cross-seconal correlaons and me-seres cross-secon models When s large relave o n he daa are somemes referred o as me-seres cross-secon (SCS) daa. Consder a SCS model of he form X β we allow for correlaon across dfferen subjecs hrough he noaon Cov( j ) V j. Four basc specfcaons of cross-seconal covarances are: he radonal model se-up n whch ols s effcen. Heerogene across subjecs. Cross-seconal correlaons across subjecs. However observaons from dfferen me pons are uncorrelaed. σ I j j V j σ I j j Cov ( ) js σ j s s

14 me-seres cross-secon models he fourh specfcaon s (Parks 967): Cov( js ) σ j for s and - η. hs specfcaon perms conemporaneous crosscorrelaons as well as nra-subjec seral correlaon hrough an AR() model. he model has an eas o nerpre cross-lag correlaon funcon of he form for s < ( ) s σ Cov js he drawback parcularl wh specfcaons 3 and 4 s he number of parameers ha need o be esmaed n he specfcaon of V j. j j

15 Panel-correced sandard errors Usng OS esmaors of regresson coeffcens. o accoun for he cross-seconal correlaons use robus sandard errors. However now we reverse he roles of and. In hs conex he robus sandard errors are known as panel-correced sandard errors. Procedure for compung panel-correced sandard errors. Calculae OS esmaors of β b OS and he correspondng resduals e x b OS. Defne he esmaor of he (j)h cross-seconal covarance o be σˆ e e Esmae he varance of b OS usng n j n n n X ˆ X σ jxx j XX j j

16 8.4 me-varng coeffcens he model s z α α z x β A marx form s Z α α Z X β. Use R Var DVar α and V α Z DZ R Example : Basc wo-wa model α x β Example : me varng coeffcens model x β e z x and β - β.

17 Forecasng We wsh o predc or forecas he BUP forecas urns ou o be β x z α z α ( ) BUP BUP GS Σ z α z b x α ) Cov( ˆ σ ( ) BUP ) Cov( e R σ

18 Forecasng - Specal Cases No me-specfc Componens ˆ x b GS z α a BUP Cov( σ ( ) ) R e BUP Basc wo-wa Error Componens Balag (988) and Konng (988) (balanced) ˆ ( ζ ) σ n( ζ ) n x b GS ζ ( x b GS ) GS σ σ ( x b ) Random Walk model ˆ x b GS s BUP z α α BUP Cov( σ ) ( ) BUP R e

19

20 oer Sales odel Selecon In-sample resuls show ha One-wa error componens domnaes pooled crossseconal models An AR() error specfcaon sgnfcanl mproves he f. he bes model s probabl he wo-wa error componen model wh an AR() error specfcaon

21 8.5 Kalman fler approach he Kalman fler s a echnque used n mulvarae me seres for esmang parameers from complex recursvel specfed ssems. Specfcall consder he observaon equaon W δ and he ranson equaon δ δ - η. he approach s o consder condonal normal of gven - and use lkelhood esmaon. he basc approach s descrbed n Appendx D. We exend hs b consderng fxed and random effecs as well as allowng for spaal correlaons.

22 Kalman fler and longudnal daa Begn wh he observaon equaon. z α α z x β he me-specfc quanes are updaed recursvel hrough he ranson equaon Φ - η. Here {η } are..d mean zero random vecors. As anoher wa of ncorporang dnamcs we also assume an AR(p) srucure for he dsurbances auoregressve of order p ( AR(p) ) model φ - φ - φ p -p ζ. Here {ζ } are..d mean zero random vecors.

23 ranson equaons We now summarze he dnamc behavor of no a sngle recursve equaon. Defne he p vecor ξ ( - -p ) so ha we ma wre Sackng hs over n elds Here ξ s an np vecor I n s an n n den marx and s a Kronecker (drec) produc (see Appendx A.6). p p η ξ Φ ξ ξ O ζ φ φ φ φ ( ) n n n n η ξ Φ I η η ξ Φ ξ Φ ξ ξ ξ

24 Spaal correlaon he spaal correlaon marx s defned as H n Var(ζ ζ n )/ σ for all. We assume no cross-emporal spaal correlaon so ha Cov(ζ s ζ j ) for s. hus Recall ha φ - φ p -p ζ and Var p n H η σ p p η ξ Φ ξ ξ O ζ φ φ φ φ

25 wo sources of dnamc behavor We now collec he wo sources of dnamc behavor and no a sngle ranson equaon. Assumng ndependence we have o nalze he recurson we assume ha δ s a vecor of parameers o be esmaed. ( ) ( ) n n η δ η ξ Φ I Φ η η ξ Φ I Φ ξ δ * Var Var Var p n Q H Q η η η Q σ σ

26 easuremen equaons For he h me perod we have ha we express as Wh ha s fxed and random effecs wh a dsurbance erm ha s updaed recursvel. n n n n n n O z z z α α α z z z β x x x α α α ξ W Z α Z X β α δ W α Z β X α n x x x X ( ) q n I z z z Z α α α α O α n α α α n z z z Z [ ] ξ W Z ξ W Z δ W

27 Capal asse prcng model We use he equaon β β x m where s he secur reurn n excess of he rsk-free rae x m s he marke reurn n excess of he rsk-free rae. We consder n 9 frms from he nsurance carrers ha were lsed on he CRSP fles as a December he nsurance carrers consss of hose frms wh sandard ndusral classfcaon SIC codes rangng from 63 hrough 633 nclusve. For each frm we used sx monhs of daa rangng from Januar 995 hrough December 999.

28 able 8.. Summar Sascs for arke Index and Rsk Free Secur Based on sx monhl observaons Januar 995 o December 999. Varable ean edan nmum axmum Sandard devaon VWRED (Value weghed ndex) RISKFREE (Rsk free) VWFREE (Value weghed n excess of rsk free) able 8.3. Summar Sascs for Indvdual Secur Reurns Based on 54 monhl observaons Januar 995 o December 999 aken from 9 frms. Varable ean edan nmum axmum Sandard devaon RE (Indvdual secur reurn) REFREE (Indvdual secur reurn n excess of rsk free)

29 able 8.4. Fxed effecs models Summar measure Homogeneous model Varable nerceps model Varable slopes model Varable nerceps and slopes model Varable slopes model wh AR() erm Resdual sd devaon (s) ln kelhood AIC AR() corr ( ) sasc for -5.98

30 me-varng coeffcens models We nvesgae models of he form: β β x m where - η and β - β β (β - - β ) η. We assume ha { } and {β } are saonar AR() processes. he slope coeffcen β s allowed o var b boh frm and me. We assume ha each frm has s own saonar mean β and varance Var β.

31 Expressng CAP n erms of he Kalman Fler Frs defne j n o be an n vecor wh a one n he h row and zeroes elsewhere. Furher defne x j n x m β β β β hus wh hs noaon we have β β x m z x β. no random effecs. n z jn x m β β n β β n

32 Kalman fler expressons For he updang marx for me-varng coeffcens we use Φ I n β. AR() error srucure we have ha p and Φ. hus we have and n n n β β β β ξ δ n n n I I I β β n n I I η η Q ) ( ) ( Var Var β β σ σ

33 able 8.5 me-varng CAP models Parameer σ odel f wh parameer Esmae 9.57 Sandard Error.4 odel f whou parameer β σ β Esmae Sandard Error he model wh boh me seres parameers provded he bes f. he model whou he elded a sascall sgnfcan esmae of he β parameers he prmar quan of neres.

34 BUPs of β Pleasan calculaons show ha he BUP of β s b β m BUP b GS σ β β xm Var GS GS where x ( ) ( b b x ) m x m ( ) x m xm Var σ X R ( ) X σ R β m AR β m AR ( ) X m dag ( x ) x m m

35 BUP predcors me seres plo of BUP predcors of he slope assocaed wh he marke reurns and reurns for he ncoln Naonal Corporaon. he upper panel shows ha BUP predcor of he slopes. he lower panels shows he monhl reurns. BUP renc

36 Appendx D. Sae Space odel and he Kalman Fler Basc Sae Space odel Recall he observaon equaon W δ and he ranson equaon δ δ - η. Defne Var - H and Var - η Q. d E δ P Var δ and P Var δ. Assume ha { } and {η } are muuall ndependen. Sackng we have Wδ δ δ δ W W W W δ W δ W δ O

37 Kalman Fler Algorhm akng a condonal expecaon and varance of he ranson equaon elds he predcon equaons d /- E - δ d - and P /- Var - δ P - Q. akng a condonal expecaon and varance of measuremen equaon elds E - W d /- and F Var - W P /- W H. he updang equaons are d d /- P /- W F - ( - W d /- ) and P P /- - P /- W F - W P /-. he updang equaons are movaed b jon normal of δ and.

38 kelhood Equaons he updang equaons allows one o recursvel compue E - and F Var - he lkelhood of { } ma be expressed as ln f(... ) ln f( ) f(... ) N ln π ln de( F ) ( ) E F ( E ) hs s much smpler o evaluae (and maxmze) han he full lkelhood expresson.

39 From he Kalman fler algorhm we see ha E - s a lnear combnaon of { - }. hus we ma wre where s a N N lower rangular marx wh one s on he dagonal. Elemens of he marx do no depend on he random varables. Componens of are mean zero and are muuall uncorrelaed. ha s condonal on { - } he h componen of v has varance F. E E E

40 Exensons Appendx D provdes exensons o he mxed lnear model he lnear of he ransform urns ou o be mporan Secon 8.5 shows how o exend hs o he longudnal daa case. We can esmae nal values as parameers Can ncorporae man dfferen dnamc paerns for boh and Can also ncorporae spaal relaons