A Note on Prediction with Misspecified Models

Transcription

1 ITB J. Sci., Vol. 44 A, No. 3,, A Noe o Predicio wih Misspecified Models Khresha Syuhada Saisics Research Divisio, Faculy of Mahemaics ad Naural Scieces, Isiu Tekologi Badug, Jala Gaesa Badug, Jawa Bara 43 Idoesia khresha@mah.ib.ac.id Absrac. Suppose ha a ime series model is fied. I is likely ha he fied model is o he rue model. I oher words, he model has bee misspecified. I his paper, we cosider he predicio ierval problem i he case of a misspecified firs-order auoregressive or AR() model. We have calculaed he coverage probabiliy of a upper oe-sep-ahead predicio ierval for boh properly specified ad misspecified models hrough Moe Carlo simulaio. I was foud ha dealig wih predicio ierval for misspecified model is complicaed: he disribuio of a fuure observaio codiioal o he las observaio ad he parameer esimaor is o ideical o he disribuio of his fuure observaio codiioal o he las observaio aloe. Keywords: auoregressive; coverage probabiliy; model misspecificaio; ime series predicio. Iroducio A very iformaive way of specifyig he accuracy of a ime series predicio is o use a predicio ierval. To do his, some auhors, e.g. [-5], have compued he predicio ierval or limi for he auoregressive (AR) process ad he auoregressive codiioal heeroscedasic (ARCH) process. For hese models, hey have assumed ha he fied model is he rue model. I oher words, he auhors have employed properly specified models. I pracice, however, ime series models are likely o be misspecified. For isace, a AR() model is fied bu he rue model is a AR() or MA() model. This paper deals wih he predicio ierval i he case of a misspecified AR() model. I he coex of he predicio problem, here have bee a few papers, such as [6] ad [7], ha provide derivaios of expressios for measquared-error of predicio i he case of a misspecified Gaussia AR(p) model. A ivesigaio of he coverage probabiliy, codiioal o he las observaios, of a predicio ierval based o a misspecified Gaussia AR(p) model was preseed firs by [8]. I he case of a misspecified Gaussia AR(p) model, he derivaio of he asympoic expasio of he coverage probabiliy, codiioal o he las Received November 7 h,, Revised April h,, Acceped for publicaio May 8 h,. Copyrigh Published by LPPM ITB, ISSN: , DOI:.564/ibj.sci

2 8 Khresha Syuhada observaios, of a predicio ierval requires a grea deal of care. To make clear he issues ivolved i his derivaio, we cosider he simple sceario described i Secio. Namely, a misspecfied zero-mea Gaussia AR() process fied o a saioary zero-mea Gaussia process. We also cosider a oe-sep-ahead predicio ierval wih arge coverage probabiliy. Similarly o [8], we aim o fid he asympoic expasio for he coverage probabiliy of he predicio ierval uder cosideraio, codiioal o he las observaio. I Secio 3, we compue he codiioal coverage probabiliies for boh properly ad misspecified firs order auoregressive models. This is aimed o examie he effec of misspecificaio o he coverage probabiliy of he predicio ierval. Furhermore, he echical argume for fidig he codiioal coverage probabiliy of a predicio ierval as proposed by de Lua (which differs from he esimaive predicio ierval) is reviewed i deail i Secio 4. A umerical example o illusrae a gap i his argume is give i Secio 5. Descripio of he Time Series ad Fied Models Cosider a discree ime saioary Gaussia zero-mea process { }. Le k E( k ) deoe he auocovariace fucio of { } evaluaed a lag k. Suppose ha < j j. We do o require ha { } is a AR() process. The available daa are,,,. We will be careful o disiguish bewee radom variables, wrie i upper case, ad heir observed values, wrie i lower case. We cosider oe-sep-ahead predicio usig a saioary Gaussia zero-mea AR() model. This model may be misspecified. Le k k/. Defie o be he value of a ha miimizes E( a ). Thus /. Defie var( y ). Noe ha ad as E( ). Le k deoe a cosise esimaor of k such j k j jk. Le / ad oe ha is a cosise

3 A Noe o Predicio wih Misspecified Modelo 9 esimaor of. We use he oaio [ a b] for he ierval [ a b, a b] ( b > ). 3 The Coverage Properies of Misspecified AR() Model I his secio, we examie he effec of misspecificaio o he coverage probabiliy of he predicio ierval. The rue model { } is a zero-mea Gaussia AR() process, saisfyig () for all ieger, where are idepede ad ideically N (, ) disribued. The roos of m m are ouside he ui circle o esure saioariy of he process. We cosider a upper oe-sep-ahead predicio ierval wih codiioal mea square predicio error equal o V, i.e. I V V u ( ;, ) ( ), where he esimaors ad V are obaied by maximizig he loglikelihood fucio codiioal o y, ha are () ad ( ). V The coverage probabiliy of I u ( ;, V ), codiioal o y is calculaed as follows. Firs, we observe ha

4 Khresha Syuhada P( () V y, y ) P( () V y, y ) P( ( ) y y () V y, y ) y y V E ( ) ( ) y, y ad we esimae his codiioal expecaio by simulaio. We use he backward represeaio: (4) for all ieger, where disribued. I is oed ha ad (3) are idepede ad ideically N (, ),, are idepede. We begi he (, ) ( y, y simulaio ru by seig ) ad he use (4) o ru he process backwards i ime. Table Esimaed coverage probabiliies, codiioal o (, ) (,), of he upper.95 oe-sep-ahead forecasig iervals for rue ad misspecified models. Sadard errors are i brackes. 5 a, a, Model a, a.5, True (.79) (.55) Misspecified.9999 (.) (.) a., a.4, True (.9) (.56) Misspecified.93 (.).936 (.9) a.3, a.3, True (.88) (.57) Misspecified.9656 (.).978 (.7) a.75, a.5, True (.74) (.49) Misspecified.9988 (.6).9993 (.)

5 A Noe o Predicio wih Misspecified Modelo Table preses he esimae of codiioal coverage probabiliy (3). Sadard errors of hese esimaes are give i brackes. We also provide he esimae of codiioal coverage probabiliy of whe he predicor is also a zero-mea Gaussia AR() model (properly specified model) as i [4]. 4 The Codiioal Disribuio for Misspecified Model de Lua [8] cosiders he daa geeraig mechaism }, as described earlier, ad h -sep-ahead predicio usig a zero-mea Gaussia AR( p ) model. This model may be misspecified. Cosider he case ha h ad p. His predicio ierval for is J [ ( ) z, ( ) z ], where ) w, w { / / ( ( k k k ) k ad / z is he ( /) quaile of he N (,) disribuio. I pracice, ad w are ukow sice hey deped o a ukow auocovariace fucio of he process }. However, de Lua assumes ha boh ad w are kow. { de Lua claims ha P( J y) O( ). His jusificaio for his claim, wih iermediae seps added, is he followig. P( J y) P( [ y ( y ) z ] y ) / ( by he subsiuio heorem for codiioal expecaio s) E( P( [ y ( y) z /] y, ) y by he double expecaio heorem. To evaluae his codiioal expecaio, de Lua proceeds as follows. Defie F () ad f () o be he cdf ad pdf of codiioal o y, respecively. He he assumes ha is equal o P( [ y ( y ) z ] y, ) / ),

6 Khresha Syuhada F( y ( y ) z ) F( y ( y ) z ), / / (5) sice he Taylor expasios are applied o F ad o some oher fucio. The, by Taylor expasio of F aroud y z /, F( y ( y) z /) F( y z /) f ( y z )( y ( ) ( ( y ) ) z ) / / Now, F( y ) z / /. Thus E F y y z y ( ( ( ) /) ) is equal o f ( y z /)( y E( y) ( ( y) ) z /) Similarly, E F y y z y ( ( ( ) /) ) f ( y z /)( y E( y) ( ( y) ) z ) / If i is assumed ha P( X J y) O( ). E y O ( ) ( ) he The assumpio (5) is crucial. I is rue if { } is also a AR() process (i.e. if he model is properly specified), bu i is o ecessarily rue if he model is misspecified. Le F ( ; ˆ ) deoe he cdf of codiioal o y ad ˆ. Now

7 A Noe o Predicio wih Misspecified Modelo 3 P( [ y ( y ) z ] y, ) / F ( y ( y) z /; ) F ( y ( y) z /; ). If { } is also a AR() process he, by he Markov propery, F ( ; ˆ) F( ) ad so (5) is saisfied. If, however, he model is misspecified he F ( ; ˆ ) is o ecessarily equal o F (). 5 Compuaioal Example o Illusrae he Error i he Calculaig Codiioal Disribuio I his secio, we provide a umerical example o suppor he followig argume. For a misspecified AR() model, he disribuio of codiioal o y is o ideical o he disribuio of codiioal o y ad ˆ. I doig so, we seek a esimaor, a observed value ˆ ad a misspecified AR() model such ha he codiio ˆ, whe added o he codiio y, ells us a grea deal more abou he value of ha he codiio y aloe. Cosider he followig esimaor: T T T ad le ( y,, y) (,, T ) ad ( x,, x) (,, T ). Thus, we wrie he esimaor (6) as yx i i i yixj i j (6)

8 4 Khresha Syuhada which is less ha or equal o, accordig o he Cauchy-Schwarz iequaliy. * * Equaliy occurs if x a y, i.e. a, where Thus, a i y x i i * i T yi i T * a i where a *. Meawhile, ad y y. Now, cosider a saioary zero-mea Gaussia AR() process { } saisfyig are idepede ad ideically ) where N (, disribued, is close o bu saisfies <. We fid he auocovariace properies of } as follows: k, k,, k whils, for l,3,. By he properies above, / for he l { above process. Thus he disribuio of codiioal o y is N (, ), i.e. ) (, ),. y N N ( If we choose,.95 ad y fairly large, (7)

9 A Noe o Predicio wih Misspecified Modelo 5 he y ( 6.45, say, ) N(,/(.95 )) N(,.56). Now, if we codiio o <, where., say, ad is o oo large he we expec ha, o a good approximaio, ( y, < ) N( y, ) N(.95y which is quie differe from (7).,). (8) As for illusraio, a simulaio is carried ou as follows. Iiialize he couer: se l. The k h simulaio ru is simulae a observaio of (,, ) codiioal o y. calculae if <, he l l ad sore x l y Afer M simulaio rus, suppose ha of l N. The probabiliy desiy fucio codiioal o y ad < is approximaed by N f ( y;.95 xl ) N l where f ( y;.95xl ) deoes he N (.95x l,) probabiliy desiy fucio evaluaed a y. Noe ha o simulae a observaio of,, ) codiioal o y (

10 6 Khresha Syuhada Figure The pdf of codiioal o y (op) ad he pdf of codiioal o y ad (boom), M.. we shall kow ha he vecor T ),, ( follows a mulivariae ormal disribuio wih mea vecor ad he covariace marix i.e. N, Now if we pariio he mea vecor ad he covariace marix such ha wih sizes for ad are ) ( ad, respecively, ad

11 A Noe o Predicio wih Misspecified Modelo 7 wih sizes ( ) ( ) ( ) ( ) Figure The pdf of codiioal o y (op) ad he pdf of codiioal o y ad (boom), M5.. The he disribuio of (,, ) codiioal o y is mulivariae ormal wih mea vecor ad covariace marix ([9]), i.e. (,, y ) N(, ) where he mea vecor ad he covariace marix

12 8 Khresha Syuhada y ( ) Figures -3 show he pdf of codiioal oly o y (op figure) ad he pdf of codiioal o y ad < (boom figure). We carry ou 3 ad M,,5, ad, simulaio rus. 6 Discussio The derivaio of he asympoic codiioal coverage probabiliy for a misspecified model requires a grea deal more care ha he derivaio of he asympoic codiioal coverage probabiliy for a properly specified model. Resuls similar o hose of de Lua ca be derived by defiig he radom variable X E( X X, X, ) ad oig ha ad ( X,, ) are idepede radom vecors. X Figure 3 The pdf of codiioal o y (op) ad he pdf of codiioal o y ad (boom), M..

13 A Noe o Predicio wih Misspecified Modelo 9 X E( X X, X, ) ad oig ha ad ( X, X, ) are idepede radom vecors. Ackowledgeme I am graeful o Paul Kabaila for his valuable commes ad suggesios o clarify may aspecs o his paper. Refereces [] Bardorff-Nielse, O.E. & Cox, D.R., Predicio ad Asympoics, Beroulli,, pp , 996. [] Kabaila, P., The Relevace Propery for Predicio Iervals, Joural of Time Series Aalysis, (6), pp , 999. [3] Kabaila, P. & Syuhada, K., The Relaive Efficiecy of Predicio Iervals, Commuicaio i Saisics: Theory ad Mehods, 36(5), pp , 7. [4] Kabaila, P. & Syuhada, K., Improved Predicio Limis for AR( p ) ad ARCH( p ) Processes, Joural of Time Series Aalysis, 9, pp. 3-3, 8. [5] Vidoi, P., Improved Predicio Iervals for Sochasic Process Models, Joural of Time Series Aalysis, 5, pp , 4. [6] Davies, N. & Newbold, P., Forecasig wih Misspecified Models, Applied Saisics, 9, pp. 87-9, 98. [7] Kuiomo, N. & amamoo, T., Properies of Predicors i Misspecified Auoregressive Time Series Models, Joural of he America Saisical Associaio, 8, pp , 985. [8] de Lua, X., Predicio Iervals Based o Auoregressio Forecass, The Saisicia, 49, pp ,. [9] Graybill, F.A., Theory ad Applicaio of The Liear Model, Norh Sciuae, Mass: Duxbury Press, 976.