Estimation for State-Space Space Models: an Approximate Likelihood Approach

Transcription

1 Esimaio for Sae-Sace Sace Models: a Aroximae Likelihood Aroach Richard A. Davis ad Gabriel Rodriuez-Yam Colorado Sae Uiversiy h:// Joi work wih: William Dusmuir Uiversiy of New Souh Wales Yi Wa De of Public Healh W. Viriia 1

2 Examle: Daily Ashma Preseaios 199: Ja Feb Mar Ar May Ju Jul Au Se Oc Nov Dec Year Ja Feb Mar Ar May Ju Jul Au Se Oc Nov Dec Year Ja Feb Mar Ar May Ju Jul Au Se Oc Nov Dec Year Ja Feb Mar Ar May Ju Jul Au Se Oc Nov Dec Year

3 Examle: Poud-Dollar Exchae Raes Oc Ju ; Kooma websie day la 3 lo reurs exchae raes ACF la la ACF of squares ACF of abs values

4 Moivai Examles Time series of cous Sochasic volailiy Geeralized sae-sace models Observaio drive Parameer drive Model seu ad esimaio Exoeial family 2 examles Esimaio Imorace samli Aroximaio o he likelihood Simulaio ad Alicaio Time series of cous Sochasic volailiy How ood is he oserior aroximaio? Poserior mode vs oserior mea 4

5 Observaios: y 1... y Saes: 1... Observaio equaio: : -1 y -1 Geeralized Sae-Sace Models Sae equaio: -observaio drive +1 y : +1-1 y -arameer drive +1 : +1-1 y 5

6 Exoeial Family Seu for Parameer-Drive Model Time series daa: Y 1... Y Reressio exlaaory variable: x Observaio equaio: ex{ + β Τ x y b + β Τ x + c }. Sae equaio: { } follows a auoreressive rocess saisfyi he recursios γ+φ 1-1 +φ φ - + ε where {ε } ~ IID Nσ 2. Noe: corresods o sadard eeralized liear model. Oriial rimary objecive: Iferece abou β. 8

7 Examles of arameer drive models Poisso model for ime series of cous Observaio equaio: T β T x x -e + β + y e e y y y! 1... Sae equaio: Sae variables follow a Gaussia AR1 rocess AR1 oise: φ -1 + ε {ε }~IID Nσ 2 The resuli rasiio desiy of he sae variables is is ; φ σ 2 Remark: The case σ 2 corresods o a lo-liear model wih Poisso oise. 9

8 Examles of arameer drive models-co A sochasic volailiy model for fiacial daa Taylor `86: Model: Y σ Z {Z }~IID N1 φ -1 + ε {ε }~IID Nσ 2 where 2 lo σ. The resuli observaio ad sae rasiio desiies are ; ex ; φ σ 2 Proeries: Mariale differece sequece. Saioary. Sroly mixi a a eomeric rae. 1

9 Esimaio Mehods for Parameer Drive Models Esimai equaios Zeer `88: Le βˆ be he soluio o he equaio µ β Γ y µ where µ exx β ad Γ vary. Moe Carlo EM Cha ad Ledoler `95 GLM iores he resece of he lae rocess i.e.. Imorace samli Durbi & Kooma `1 Kuk `99 Kuk & Che `97: Aroximae likelihood Davis Dusmuir & Wa 98 11

10 15 Esimaio Mehods Imorace Samli Durbi ad Kooma Model: Y x Poisexx T β + φ -1 + ε {ε }~IID N σ 2 Relaive Likelihood: Le β φ σ 2 ad suose ; is a aroximai joi desiy for Y Y 1... Y ' ad 1... '. d L d L L d ; y ; ; ; y ; d ; ; d ; ; d L L d ; y ; ; ; y ;

11 Imorace Samli co L L ; y ; d E ; y ; where Noes: ~ 1 N N j j ; j 1 j { ; j 1... N} ~ iid y;. Aroximaio is oly ood i a eihborhood of. Geyer suess maximizi raio wr ad ierae relaci wih ˆ. This is a oe-samle aroximaio o he relaive likelihood. Tha is for oe realizaio of he s we have i ricile a aroximaio o he whole likelihood fucio. j 16

12 Imorace Samli examle Simulaio examle: Y Poisex ε {ε }~IID N.3 2 N 1 lo likelihood hi 18

13 Imorace Samli examle Simulaio examle: Y Poisex ε {ε }~IID N.3 2 N 1 likelihood likelihood hi_ hi_-.367 likelihood likelihood hi_ hi_

14 Imorace Samli examle Simulaio examle: β.7 φ.5 σ N 1 5 realizaios loed likelihood likelihood likelihood hi_-.5 hi_-.25 hi_ likelihood likelihood likelihood hi_ hi_ hi_.75 21

15 Choice of imorace desiy : Imorace Samli co Durbi ad Kooma sues a liear sae-sace aroximai model wih Y µ + x T β+ +Z Z ~NH µ y ˆ x' y e ˆ + x' β + 1 ˆ + x' β H e where he ˆ E y are calculaed recursively uder he aroximai model uil coverece. Wih his choice of aroximai model i urs ou ha where y~ y y ; e ~ N Γ Xβ + ˆ + e 1 y~ Xβ + ˆ Γ ˆ 1 Γ dia e Xβ + ˆ + E ' 1. 23

16 Comoes required i he calculaio. 1 y~ ' ~ Γ y de Γ simulae from comue simulae from N N Γ 1 Γ y~ Imorace Samli co y~ Γ Γ Remark: These quaiies ca be comued quickly usi a versio of he iovaios alorihm or he Kalma smoohi recursios. 24

17 Esimaio Mehods Aroximaio o he likelihood Geeral seu: where de G 1/ 2 ex{ µ T G µ / 2} 1 T G E µ µ Likelihood: L d Cosider a Gaussia aroximaio a y φ ; µ Σ o he oserior y y Sei equal he resecive oserior modes a ad of a y ad y we have µ where is he soluio of he equaio lo G µ 25

18 26 Noes: 1. This aroximai oserior is ideical o he imorace samli desiy used by Durbi ad Kooma. Esimaio Mehods Aroximaio o he likelihood co Machi Fisher iformaio marices: Aroximai oserior: 2. I radiioal Bayesia sei oserior is aroximaely a for lare see Berardo ad Smih Obai same resul if oe alies a Taylor series exasio o he joi likelihood ad iore erms of order > lo + Σ T G 1 2 lo + Σ T G 1 2 lo y + φ T a G 1 2 lo y + φ T a G

19 27 Esimaio Mehods Aroximaio o he likelihood co Aroximae likelihood: Noe ha which by solvi for L i he exressio a y y ;y y L we obai ;y y L 2 1/ 2 2 1/ lo de 2} / ex{ y / ;y + µ µ T T a a G G G L 2 1/ 2 2 1/ lo de 2} / ex{ y / ;y + µ µ T T a a G G G L

20 Esimaio Mehods Aroximaio o he likelihood co Case of exoeial family: L a 1/ 2 G T T T ;y ex{y 1 { b c} µ G µ / 2} K + G 1/ 2 where 2 K dia{ ad is he soluio o he equaio 2 b } y b G µ. Usi a Taylor exasio he laer equaio ca be solved ieraively. 28

21 Esimaio Mehods Aroximaio o he likelihood Imlemeaio: 1. Le be he covered value of j where j+ 1 b && j + G ad j j j j y~ y b& + & b + Gµ. -1 y~ j 2. Maximize La ;y wih resec o. 29

22 Simulaio Resuls Model: Y Poisex ε {ε }~IID N.3 2 Esimaio mehods: Imorace samli N1 udaed a maximum of 1 imes bea hi sima2 mea sd Aroximaio o likelihood bea hi sima2 mea sd

23 Model: Y Poisex ε {ε }~IID N.3 2 Arox likelihood desiy desiy desiy bea hi Imorace Samli sima^2 desiy desiy desiy bea hi sima^2 31

24 Alicaio o Model Fii for he Polio Daa Model for { }: φ -1 +ε {ε }~IID N σ 2. Imorace samli udaed 5 imes for each N1 5 1 Simulaio based o 1 relicaios ad he fied AL model. Imor Samli Simulaio ˆβ IS Mea SD Ierce Tred cos2π/ si2π/ cos2π/ si2π/ φ σ Arox Like GLM Simulaio ˆβ AL Mea SD ˆβ GLM SD

25 Sochasic volailiy model: Y σ Z {Z }~IID N1 Simulaio Resuls γ + φ -1 + ε {ε }~IID Nσ 2 where 2 lo σ ; 1 NR5 CV1 CV1 True AL γ φ σ True AL γ φ σ RMSE IS RMSE RMSE IS RMSE

26 Alicaio o Sydey Ashma Cou Daa Daa: Y 1... Y 1461 daily ashma reseaios i a Cambellow hosial. Prelimiary aalysis ideified. o uward or dowward red aual cycle modeled by cos2π/365 si2π/365 seasoal effec modeled by Tij Tij Pij B where B2.55 is he bea fucio ad T ij is he sar of he j h school erm i year i. day of he week effec modeled by searae idicaor variables for Suday ad Moday icrease i admiace o hese days comared o Tues-Sa. Of he meeoroloical variables max/mi em humidiy ad olluio variables ozoe NO NO 2 oly humidiy a las of 12-2 days ad NO 2 max aear o have a associaio. 5 35

27 Resuls for Ashma Daa IS & AL Term IS Ierce.59 Suday effec.138 Moday effec.229 cos2π/ si2π/365.2 Term Term Term Term Term Term Term Term Humidiy H /2.9 NO 2 max AR1 φ.385 σ 2.53 AL Mea SD

28 Is he oserior disribuio close o ormal? Comare oserior mea wih oserior mode: Ca comue he oserior mea usi SIR samli imorace-resamli Poserior mode: The mode of y is foud a he las ieraio. Poserior mea: The mea of y ca be foud usi SIR. Le N be ideede draws from he mulivariae disr a y. For N lare a aroximae iid samle from y ca be obaied by drawi a radom samle from N wih robabiliies i N w i 1 i w i y y L ;y i wi i i a a y i i 1 K N. 38

29 Polio daa: blue mea red mode Poserior mea vs oserior mode? smoohed sae vecor

30 Poserior mea vs oserior mode? Poud/US exchae rae daa: blue mea red mode oserior mea ime 4

31 Is he oserior disribuio close o ormal? Suose M are ideede draws from he mulivariae disr y eeraed usi SIR. The d 2 j j T K + G j iid ~ χ 2 5 M1 5 M1 5 M1 chi-square quaile chi-square quaile chi-square quaile Observed quaile Observed quaile Observed quaile 1 M15 1 M15 1 M15 chi-square quaile chi-square quaile chi-square quaile Correlaios are all siifica Observed quaile Observed quaile Observed quaile 2 M25 2 M25 2 M25 chi-square quaile chi-square quaile chi-square quaile Observed quaile Observed quaile Observed quaile 41

32 Summary Remarks 1. Imorace samli offers a ice clea mehod for esimaio i arameer drive models. 2. Relaive likelihood aroach is a oe-samle based rocedure bu may have coverece roblems. 3. Aroximaio o he likelihood is a o-simulaio based rocedure which may have rea oeial esecially wih lare samle sizes ad/or lare umber of exlaaory variables. 5. Aroximaio likelihood aroach is ameable o boosrai rocedures for bias correcio. 6. Poserior mode maches oserior mea reasoably well. 7. Exesio o more eeral lae rocess models e.. lo memory is i roress. 42