SUBSAMPLING INTERVALS IN AUTOREGRESSIVE MODELS WITH LINEAR TIME TREND

Transcription

1 Ž. Ecoomerica, Vol. 69, No. 5 Sepember, 200, SUBSAMPLING INTERVALS IN AUTOREGRESSIVE MODELS WITH LINEAR TIME TREND BY JOSEPH P. ROMANO AND MICHAEL WOLF A ew mehod is proposed for cosrucig cofidece iervals i auoregressive models wih liear ime red. Ieres focuses o he sum of he auoregressive coefficies because his parameer provides a useful scalar measure of he log-ru persisece properies of a ecoomic ime series. Sice he ype of he limiig disribuio of he correspodig OLS esimaor, as well as he rae of is covergece, deped i a discoiuous fashio upo wheher he rue parameer is less ha oe or equal o oe Ž ha is, red-saioary case or ui roo case., he cosrucio of cofidece iervals is ooriously difficul. The crux of our mehod is o recompue he OLS esimaor o smaller blocks of he observed daa, accordig o he geeral subsamplig idea of Poliis ad Romao Ž 994a., alhough some exesios of he sadard heory are eeded. The mehod is more geeral ha previous approaches i ha i works for arbirary parameer values, bu also because i allows he iovaios o be a marigale differece sequece raher ha i.i.d. Some simulaio sudies examie he fiie sample performace. KEYWORDS: Auoregressive ime series, local-o-uiy asympoics, subsamplig, redsaioariy, ui roos.. INTRODUCTION MUCH WORK IN THE RECENT LITERATURE has bee devoed o he quesio of wheher a macroecoomic ime series is red-saioary or wheher i possesses a ui roo. Whe he ime series is modeled by a ARŽ p. sequece wih liear ime red, he aswer depeds o wheher he sum of he ARŽ p. coefficies, a parameer usually deoed by, is less ha oe or equal o oe. This parameer ca be cosisely esimaed by applyig ordiary leas squares Ž OLS. o he usual Dickey-Fuller form regressio model. Uforuaely, he cosrucio of cofidece iervals is orivial, sice he ype of he asympoic disribuio of he OLS esimaor, as well as is rae of covergece, are differe i he red-saioary case as compared o he ui roo case. Whe, he limiig disribuio is ormal ad he rae of covergece is he square roo of he sample size. O he oher had, whe, he limiig disribuio is osadard ad he rae of covergece equals he sample size. This difficuly explais he emphasis i he ui roo lieraure placed o hypohesis esig. However, cofidece iervals provide much more iformaio ha kowig wheher he ull hypohesis of a ui roo ca be rejeced or o, amely hey serve as a measure of samplig uceraiy ad describe he We hak Graham Ellio, Guido Imbes, ad Halber Whie for helpful discussios. The fial versio of his paper has subsaially beefied from commes of hree aoymous referees ad he co-edior. Research of he secod auhor was parly fuded by he Spaish Direccio Geeral de Eseaza Superior Ž DGES., referece umber PB

2 284 J. P. ROMANO AND M. WOLF rage of models ha are cosise wih he observed daa. This poi was made i Sock Ž 99. ad Adrews ad Che Ž 994., amog ohers. I his paper, we propose a ovel approach for cosrucig cofidece iervals for he parameer, based o he subsamplig mehod of Poliis ad Romao Ž 994a.. The crux of he ew approach is o recompue he OLS esimaor o smaller blocks, or subsamples, of he observed daa sequece. The, he empirical disribuio of hese subsample esimaes, afer a appropriae ormalizaio, is used o approximae he samplig disribuio of he esimaor based o he eire daa. Ulike he coveioal boosrap, he subsamplig mehod ca hadle he discoiuiy of he limiig disribuio of he OLS esimaor Ž as a fucio of., sice he subsamples are all geeraed by he rue model raher ha a approximaig boosrap disribuio. While we will focus o he parameer, he proposed mehod ca equally well be applied o cosruc cofidece iervals for aleraive parameers of ieres, such as he larges roo of he ARŽ p. model, a paricular AR coefficie, or he coefficie o he ime red. I Secio 2, he model ad he parameer of pricipal ieres are preseed. Also, some previous mehods for cofidece ierval cosrucio are briefly described. The basic subsamplig mehodology ad some ecessary exesios are discussed i Secio 3. The geeral approaches are applied o he problem a had i Secio 4, while Secio 5 deals wih issues cocerig he pracical implemeaio. Two simulaio sudies are preseed i Secio 6. Some coclusios are saed i Secio 7. The proofs appear i a Appedix. 2. DEFINITIONS AND BACKGROUND 2.. Defiiio of he Model The model uder cosideraio is a ARŽ p. model wih iercep ad liear ime red. The exposiio of he model ad he oaio closely follow Adrews ad Che Ž 994. wih he excepio ha we do o require he iovaios o be i.i.d. ad ormal. The model ca be wrie i a uobserved-compoes form ad i a regressio form. I he former, i is give by Ž. Y * *Y for,...,, Y Y Y Y, p where, p,...,4 is a sricly saioary, marigale differece iova- io sequece ad Y,,...,4 is he observed series. The variable Y deoes Y Y. The parameer saisfies Ž, ; whe, he model is osaioary. The parameers Ž,...,. p are such ha he AR model for Y is saioary whe Ž,. ad he AR model for Y is saioary whe. The sarig values of Y ha is, ŽY,...,Y. p are ake such ha Y 4 is saioary whe Ž,. ad Y 4 is saioary

3 AUTOREGRESSIVE MODELS 285 whe. The level of he Y series is arbirary whe Ž ha is, whe, he iiial radom variable Y ca be fixed or ca have ay disribuio provided he subseque Y values are such ha Y is saioary.. The regressio form of Model Ž. is give by Ž 2. Y YY pyp for,...,, * Ž. Ž. * ad * Ž. p, where Ž Y,...,Y. ad, p,...,t4 are defied as i Ž.. Model Ž. p 2 is he well-kow augmeed Dickey-Fuller regressio form of he ARŽ p. model. The correspodig sadard ARŽ p. regressio form is give by Ž 3. Y Y pyp. As is easy o see, he parameer i he augmeed Dickey-Fuller form equals he sum of he AR coefficies i he sadard form, ha is, p. Moreover, i follows ha Ž. j j p for j,..., p. I should be poied ou ha he ime red parameer is ecessarily equal o 0 whe i boh models Ž. 2 ad Ž. 3. This desirable feaure esures ha EY Ž. is a liear fucio of for all Ž,. If 0 were allowed whe, EY Ž. would be a quadraic fucio of whe, so his discoiuiy is aurally avoided The Parameer of Ieres ad is Iferece Problems The remaider of his paper will maily focus o cosrucig cofidece iervals for he parameer. The moivaio is ha i provides a useful scalar measure for he log-ru persisece properies of he ime series Y 4. Ideed, i ARŽ p. models, Ž. equals he sum of he impulse respose fucios over all ime horizos, ha is, he cumulaive impulse respose; see Adrews ad Che Ž A aleraive scalar measure ha has bee cosidered i he lieraure is he larges roo of he ArŽ p. model, usually deoed by. For example, Sock Ž 99. derived asympoic cofidece iervals for based o a local-o-uiy model ad DeJog ad Whiema Ž 99a, b. discussed Bayes esimaors of. Bu, as was demosraed i Adrews ad Che Ž 994., he persisece of wo ime series wih he same value of ca be very differe depedig o he values of he oher roos. Therefore, we have decided o focus o he parameer isead. Noe, however, ha he mehodology developed i his paper o cosruc cofidece iervals for ca equally well be employed o cosruc cofidece iervals for. As usual whe iferece for a ukow scalar parameer is desired, here exis wo mai aveues, amely hypohesis ess ad cofidece iervals. I accordace wih may oher auhors, we feel ha a cofidece ierval is much more iformaive ha a es, sice i o oly saes wheher a specific parameer value is rejeced or o by he observed daa, bu because i also

4 286 J. P. ROMANO AND M. WOLF provides he rage of all parameer values cosise wih he daa. I paricular, his allows judgme of he degree of uceraiy abou poi esimaes of he ukow parameer. So why is i ha he mai par of he ui roo lieraure has bee cocered wih hypohesis ess for, wih he ull hypohesis ypically give by? The reaso for his preoccupaio wih he wrog mehod is ha hypohesis ess for are by a order of magiude easier o cosruc ha cofidece iervals. While ca be cosisely esimaed by applyig OLS o he Dickey-Fuller form regressio model Ž. 2, he form of he limiig disribuio of he OLS esimaor, as well as is rae of covergece deped i a discoiuous way o wheher or o equals ; see he proof of Theorem 4. for deails. Exacly his fac makes he cosrucio of cofidece iervals difficul. Clearly, he sadard asympoic approachusig he quailes of he Ž esimaed. limiig disribuiois redered useless, sice oe has o kow wheher is equal o or o i order o kow he quailes of which disribuio oe should use. Moreover, he discoiuiy of he form of he limiig disribuio, as a fucio of, causes he sadard, residual-based boosrap cofidece iervals o fail; see Basawa e al. Ž 99.. O he oher had, his dilemma does o affec hypohesis ess, sice hey oly require specificaio of he limiig disribuio of he es saisic uder he ull hypohesis. Despie he ihere difficulies i cosrucig cofidece iervals for he parameer, some oable progress has bee made recely. Sock Ž 99., focusig o he larges roo of he ARŽ p. model raher ha o he sum of he ARŽ p. parameers, made use of local-o-uiy asympoics. To be more specific, he assumed ha shriks owards oe as he sample size eds o ifiiy i he liear fashio c, for some cosa c0; oe ha he heory also works whe c0. This model allows esig of he ull hypohesis cc0 for ay value c0 ad hereby fidig a cofidece ierval for c as he collecio of c values ha are o rejeced by he es. Usig 0 he relaio c, a cofidece ierval for immediaely esues. The dowside of his approach may be cosidered is breakdow problem. The cofidece iervals work well whe is close o oe, where close depeds o he sample size. Judgig from he simulaio sudies i Sock Ž 99., he acual coverage is Ž early. equal o he omial oe whe, ha is, whe c0 bu deerioraes as moves away from, ha is, as c ador decrease. Adrews ad Che Ž 994. based cofidece iervals o approximaely media-ubiased esimaio i ARŽ p. models. This is a exesio of previous work of Adrews Ž 993., where exacly media-ubiased esimaio i ARŽ. models was developed. The idea is o compue Žor o simulae wih arbirary precisio. he samplig disribuio of he OLS esimaor usig model Ž 2. ˆ bu wih i.i.d. iovaios from a NŽ0, 2. disribuio. I he ARŽ. case, his disribuio ca be show o deped o oly, bu o o,, ad 2 ; see Adrews Ž I he geeral ARŽ p. case, he disribuio also depeds o Ž,...,., so i ca oly be approximaed. Give ha oe ca Ž p approximaely. compuesimulae he samplig disribuio of ˆ for ay value of 0, a cofidece ierval for is obaied as he collecio of all values whose 0

5 AUTOREGRESSIVE MODELS 287 samplig disribuio is cosise wih he observed value of ˆ ; see Adrews ad Che Ž 994. for deails. The obvious criicism of his mehod is ha oe has o specify he disribuio of he iovaios Ž such as ormal. i order o calculaesimulae he samplig disribuio of ˆ. However, he mehod seems fairly robus agais misspecificaio of his disribuio, as appears from some simulaios i Adrews ad Che Ž 994., as log as he iovaios remai i.i.d. I will be clear from he proof of Theorem 4. ha he mehod i geeral does o work whe he iovaios are depede; specifically, see Remark 4.3. As meioed before, eve whe he iovaios are assumed i.i.d., he coveioal, residual-based boosrap cofidece iervals fail whe ; see Basawa e al. Ž 99.. O he oher had, i was show by Hase Ž 999. ha oe ca cosruc idirec boosrap cofidece iervals ha are guaraeed o work for ay Ž,. The rick is o iver boosrap ess, ha is, o obai a cofidece ierval for as he collecio of all 0 values ha are o rejeced by a boosrap es of he ull hypohesis 0. Hase coied his mehod he grid boosrap bu i should be poied ou ha he idea of iverig boosrap ess o cosruc cofidece iervals is ime hoored; for example, see DiCiccio ad Romao Ž A shorcomig of his approach is ha i, oo, is resriced o i.i.d. iovaios. The aim of his paper is o provide a ew way for cosrucig cofidece iervals for ha works for ay Ž, ad allows for saioary, depede raher ha i.i.d. iovaios, hough eve he assumpio of saioariy could be relaxed; see Remark 3.2. The ew approach is based o he subsamplig mehod of Poliis ad Romao Ž 994a.. To make he paper self-coaied, he geeral mehod peraiig o uivariae parameers will be briefly described; broader mehods, peraiig o mulivariae or fucio-valued parameers ca be foud i Poliis, Romao, ad Wolf Ž The, some exesios of he sadard heory ha are ecessary for he case will be preseed. 3. THE SUBSAMPLING METHOD 3.. The Basic Mehod Subsamplig is a geeral ool ha allows oe o cosruc asympoically valid cofidece iervals for ukow parameers uder very weak assumpios. Suppose X, X, is a sequece of vecor-valued radom variables defied o a commo probabiliy space. Deoe he joi probabiliy law goverig he ifiie sequece by P. The goal is o cosruc a cofidece ierval for some real-valued parameer Ž P., o he basis of observig X,..., X 4. We assume he exisece of a sesible esimaor ˆ ˆŽ X,..., X.. For ime series daa, he gis of he subsamplig mehod is o recompue he saisic of ieres o smaller blocks of he observed sequece X,..., X 4. Defie ˆ ˆ Ž X,..., X. b, b b, he esimaor of based o he subsample X,..., X 4 b. I his oaio, b is he block size ad is he sarig idex of he block; oe ha ˆ ˆ. Le J Ž P. be he samplig disribuio of, b

6 288 J. P. ROMANO AND M. WOLF Ž ˆ. b b,, assumig ha his disribuio is idepede of. Here, b is a appropriae ormalizig cosa. Also, defie he correspodig cumulaive disribuio fucio: ½ Ž. 5 J Ž x, P. Prob ˆ x. b P b b, Noe ha wih his oaio J Ž P. is he samplig disribuio of Ž ˆ., ha is, he samplig disribuio of he Ž properly ormalized. esimaor based o he eire sample. A major assumpio ha is eeded o cosruc asympoically valid cofidece iervals for is he followig. ASSUMPTION 3.: There exiss a odegeerae limiig law JŽ P. such ha J Ž P. coerges weakly o JŽ P.. This assumpio saes ha he esimaor, properly ormalized, has a limiig disribuio. I is hard o coceive of ay asympoic heory free of such a requireme. Also, i follows ha he proper ormalizig cosa is he oe esurig a limiig disribuio. I regular cases, he limiig disribuio is ormal ad 2. The subsamplig approximaio o J Ž x, P. is defied by b Ž 4. L Ž x. ˆ ˆ, b bž b, / x. b Ý ½ 5 The moivaio behid he mehod is he followig. For ay, X,..., X 4 b is a rue subsample of size b. Hece, he exac disribuio of Ž ˆ. b b, is J Ž P. b. If boh b ad are large, he he empirical disribuio of he b values of Ž ˆ. should serve as a good approximaio o J Ž P. b b,. Replacig by ˆ is permissible because Ž ˆ. b is of order b i probabiliy ad we will assume ha b0. For J Ž x, P. o be approximaed cosisely by L Ž x., b, boh should have he same limi, amely JŽ x, P.. To esure ha L Ž x. coverges o JŽ x, P., b i probabiliy, i is ecessary ha he iformaio i he b subsample saisics Ž ˆ ˆ. b b, ed o ifiiy wih he sample size. I previous heory ŽPoliis ad Romao Ž 994a.; Poliis, Romao, ad Wolf Ž 997.., his followed from a weak depedece codiio o he uderlyig sequece Y 4, amely a -mixig codiio ŽRosebla Ž s DEFINITION 3.: Give a saioary radom sequece X, le F be he -algebra geeraed by he segme X, X,..., X 4 s ad defie he corre- spodig -mixig sequece by Ž h. sup PŽ AB. PŽ A. PŽ B., X A, B where A ad B vary over he -fields F ad Fh, respecively. The sequece X 4 is called -mixig or srog mixig if Ž h. 0 as h. X

7 AUTOREGRESSIVE MODELS 289 For our applicaios, i will be coveie o have a more geeral heory ha imposes a mixig codiio o he subsample saisics oly raher ha o he uderlyig sequece. To his ed, le Z Ž ˆ. ad deoe by Ž., b, b b, a, b he mixig coefficies correspodig o he sequece Z, b,,,..., b 4. The followig heorem shows how subsamplig ca be used o cosruc asympoically valid cofidece iervals for. THEOREM 3.: Assume Assumpio 3. ad ha b0, b0 ad b as. Also assume ha Ý Ž h. h, b 0 as. Ž. i If x is a coiuiy poi of JŽ, P., he L Ž x. JŽ x, P., b i probabiliy. Ž ii. If JŽ, P. is coiuous, he sup L Ž x. JŽ x, P. x, b 0 i probabiliy. Ž iii. For Ž 0,., le c Ž. if x : L Ž x. 4, b, b. I oher words, c Ž. seres as a Ž. quaile of he subsamplig disribuio L Ž., b, b. Correspodigly, defie cž, P. if x : JŽ x, P. 4. If JŽ, P. is coiuous a cž, P., he ½ Ž. 5 Prob ˆ c Ž. as. P, b Thus, he asympoic coerage probabiliy uder P of he ieral ˆ I c Ž., b,. is he omial leel. REMARK 3.: The sufficie codiios o he block size b are very weak. I mos applicaios,, for some cosa 0 ad he codiios reduce o b ad b0 as. As show i Poliis ad Romao Ž 994a., he laer wo codiios are i geeral o oly sufficie bu also ecessary. REMARK 3.2: The geeral heory preseed here assumes ha he subsample saisics are saioary, i.e., ha he sequece ˆ,,...,b4 b, is saioary. Noe ha his assumpio could be relaxed o accommodae local heeroskedasiciy ador chagig disribuios of he subsample saisics alog he lies of Poliis, Romao, ad Wolf Ž The ierval I i Ž iii. correspods o a oe-sided hybrid perceile ierval i he boosrap lieraure Že.g., Hall Ž A wo-sided equal-ailed cofidece ierval ca be obaied by formig he iersecio of wo oe-sided iervals. The wo-sided aalogue of I is ˆ Ž. ˆ I c 2, c Ž 2.. 2, b, b I2 is called equal-ailed because i has approximaely equal probabiliy i each ail: ˆ Prob c Ž 2. 4 ˆ Ž. 4 P, b Prob c 2 2, P, b

8 290 J. P. ROMANO AND M. WOLF where deoes equaliy up o a addiive ož. erm. As a aleraive approach, wo-sided symmeric cofidece iervals ca be cosruced. A wo-sided symmeric cofidece ierval is give by ˆ c, ˆ c ˆ ˆ, where ˆc is chose so ha Prob ˆ c4. Hall Ž 988. P ˆ showed ha symmeric boosrap cofidece iervals ejoy ehaced coverage ad, eve i asymmeric circumsaces, ca be shorer ha equal-ailed cofidece iervals. A aalogue for symmeric subsample cofidece iervals, for he applicaio of he sample mea, was provided by Poliis, Romao, ad Wolf Ž 999, Chaper 0.. To cosruc wo-sided symmeric subsamplig iervals i pracice, oe esimaes he wo-sided disribuio fucio 4 J Ž x, P. Prob ˆ x., P Ž. The subsamplig approximaio o J x, P is defied by, b Ž 5. L Ž x. ˆ ˆ, b, Ý ½ b b, x 5. b The asympoic validiy of wo-sided symmeric subsamplig iervals immediaely follows from Theorem 3. ad he coiuous mappig heorem. COROLLARY 3.: Make he same assumpios as i Theorem 3.. Deoe by J Ž P. he disribuio of Q, where Q is a radom ariable wih disribuio JŽ P.. Ž. i If x is a coiuiy poi of J Ž, P., he L Ž x. J Ž x, P., b, i probabiliy. Ž ii. If J Ž, P. is coiuous, he sup L Ž x. J Ž x, P. x, b, 0 i probabiliy. Ž iii. For Ž 0,., le c Ž. if x : L Ž x. 4, b,, b,. Correspod- igly, defie c Ž, P. if x : J Ž x, P. 4. If J Ž, P. is coiuous a c Ž, P., he ½ 5 Prob ˆ c Ž. as. P, b, Thus, he asympoic coerage probabiliy uder P of he ieral I Ž. ˆ c, c Ž. is he omial leel., b,, b, SYM ˆ The applicaio of Theorem 3. or Corollary 3. requires kowledge of he rae of covergece. I sadard cases, his is simply 2. I osadard cases, i may be aoher power of ; for example, see Subsecio 4.3. As log as he rae is kow, osadard cases do o pose a problem. O he oher had, for he parameer i is well-kow ha he rae of covergece of he OLS esimaor ˆ is give by 2 whe ad by whe, respecively. Hece, he applicaio of he basic subsamplig mehod would require he kowledge of wheher he ime series is red-saioary or has a ui roo! Foruaely, here is a way aroud his dilemma by cosiderig a sudeized saisic, amely he usual -saisic for ˆ. Ideed,

9 AUTOREGRESSIVE MODELS 29 his saisic has a proper limiig disribuio o maer wha he value of. The ex subsecio will provide he ecessary heory o apply he subsamplig idea i a sudeized seig Subsamplig Sudeized Saisics The focus is ow o a sudeized saisic Ž ˆ. ˆ, where ˆ Ž Y,...,Y. ˆ is some posiive esimae of scale. Noe ha he appropriae ormalizig cosa may be differe from is aalogue i he osu- deized case. Defie J Ž P. o be he samplig disribuio of Ž ˆ. b b b, ˆ b, based o he subsample Y,...,Y b, assumig ha his disribuio is ide- pede of. Also, defie he correspodig cumulaive disribuio fucio J Ž x, P. Prob ˆ x. ˆ ½ Ž. 5 b P b b, b, The subsamplig mehod is modified o he sudeized case i he obvious Ž. way. Aalogous o 4, defie Ž 6. b L Ž x. ˆ ˆ, b bž b, / ˆ b, x. b Ý ½ 5 L Ž x. he represes he subsamplig approximaio o J Ž x, P., b. The esseial assumpio eeded o cosruc asympoically valid cofidece regios for ow becomes more ivolved ha for he osudeized case. Ž. ASSUMPTION 3.2: J P coerges weakly o a odegeerae limi law J Ž P.. I addiio, here exis posiie sequeces a 4 ad d 4 such ha a d, a Ž ˆ. coerges weakly o a limi law VŽ P., ad dˆ coerges weakly o a limi law WŽ P. wihou posiie mass a zero. THEOREM 3.2: Assume Assumpio 3.2, aba0, b 0, b0, ad b as. Also assume ha X 4 is ear epoch depede of size q, for some q 2, o a basis process V 4 whose -mixig coefficies saisfy lim Ž. Ýh V h 2 r for some r0. Ž. i If x is a coiuiy poi of J Ž, P., he L Ž x. J Ž x, P., b i probabiliy. Ž. ii If J Ž, P. is coiuous, he sup L J Ž x, P. x, b 0 i probabiliy. Ž iii. For Ž 0,., le c Ž. if x : L Ž x. 4, b, b. Correspodigly, defie c Ž, P. if x : J Ž x, P. 4. If J Ž, P. is coiuous a c Ž, P. he ½ Ž. 5 Prob ˆ ˆ c. as. Ž P, b Thus, he asympoic coerage probabiliy uder P of he ieral I Ž. c Ž.,. is he omial leel. ˆ, b ˆ

10 292 J. P. ROMANO AND M. WOLF The issue of symmeric cofidece iervals applies as well o sudeized saisics. Le J Ž P. be he samplig disribuio of ˆ ˆ. Defie, Ž 7. b L Ž x. ˆ ˆ, b, b b, ˆ b, x. b Ý ½ 5 L Ž x. he represes he subsamplig approximaio o J Ž x., b,,. Theorem 3.2 ad he coiuous mappig heorem immediaely imply he followig corollary. COROLLARY 3.2: Make he same assumpios as i Theorem 3.2. Deoe by J Ž P. he disribuio of U, where U is a radom ariable wih disribuio J Ž P.. Ž. i If x is a coiuiy poi of J Ž, P., he L Ž x. J Ž x, P., b, i probabiliy. Ž. ii If J Ž, P. is coiuous, he sup L J Ž x, P. x, b, 0 i probabiliy. Ž iii. For Ž 0,., le c Ž. if x : L Ž x. 4, b,, b,. Correspod- igly, defie c Ž, P. if x : J Ž x, P. 4. If J Ž, P. is coiuous a c Ž, P., he Prob ˆ c Ž. P½ ˆ, b, 5 as. Thus, he asympoic coerage probabiliy uder P of he ieral I ˆ Ž. c Ž., ˆ Ž. c Ž. SYM ˆ, b, ˆ, b, is he omial leel. 4. SUBSAMPLING INFERENCE 4.. Cofidece Ierals for i he Full Model We will ow demosrae ha he subsamplig approach of Subsecio 3.2 ca be applied o cosruc asympoically valid cofidece iervals for he parameer. Hece, will play he role of he geeral parameer of he previous secio. The esimaor ˆ is he OLS esimaor for based o he Dickey-Fuller regressio form Ž. 2 ; oe ha i would be umerically equivale o compue ˆ p as Ý where he are he OLS esimaors of model Ž. iˆi, ˆi, 3. Cosequely, is he OLS esimaor for based o he block of daa Y,...,Y 4 ˆb, b. Deoe he correspodig OLS sadard errors by SE Ž. ad SE Ž. OLS ˆ OLS ˆb,. 2 For reasos o become appare shorly, defie ˆ SE Ž. OLS ˆ ad ˆ b, 2 b SE Ž. OLS ˆb,. To apply he mehodology of Subsecio 3.2, i is lef o specify he appropriae ormalizig cosa. Wih he defiiio of ˆ above, his urs ou o be 2 o maer wha he value of ; see he proof of he followig heorem. Deoe he mixig coefficies correspodig o sequece Y 4, which is saioary whe, by Y * ad he mixig coefficies correspodig o sequece Y 4, which is saioary whe, by. Y *

11 AUTOREGRESSIVE MODELS 293 THEOREM 4.: Assume ha b ad b0 as ad ha he saioary sequece 4 is a marigale differece sequece wih E 4 for some 0. 2 Ž4. Whe assume Ý Ž h. Ž h. h Y *. Whe assume ha Y is srog mixig. ˆ 2 Ž. 2 Le, ˆ, ˆ SE ˆ, ad. The, coclusios Ž i. Ž iii. OLS of Theorem 3.2 ad Corollary 3.2 hold. REMARK 4.: Noe ha ulike he Ž coveioal. boosrap, he subsamplig mehod ca hadle discoiuiies of he limiig disribuio of esimaors as a fucio of he uderlyig model parameers. The iuiio is ha he subsamplig approximaio of he samplig disribuio of a esimaor is based o subsample saisics compued from smaller blocks of he observed daa. The subsample saisics are herefore always geeraed from he rue model. The boosrap, o he oher had, bases is approximaio o pseudo saisics compued from pseudo daa accordig o a boosrap disribuio, which was esimaed from he observed ime series. The boosrap daa come from a model close o he ruh, bu o exacly he ruh ad his ca cause he boosrap o fail. A case i poi is he parameer i ARŽ p. models, where he Žcove- ioal. boosrap is icosise; see Basawa e al. Ž 99.. However, if oe is willig o assume i.i.d. residuals, i is possible o iver boosrap ess for o cosruc asympoically valid cofidece iervals; see Hase Ž REMARK 4.2: We have preseed a resul ha allows cosrucio of asympoically valid cofidece iervals for ay fixed Ž,. Sricly speakig, his problem is already solved by a prees mehod, a leas whe he residuals are assumed i.i.d. Žwe would like o hak a aoymous referee for poiig his ou.. The idea is o es for a ui roousig a sigificace level edig o zero wih he sample sizead o base he cofidece ierval o he ormal approximaio, whe he es rejecs he ull, or o se i equal o he sigleo uiy, oherwise. However, i is well-kow ha his mehod has errible fiie sample properies; his is oe of he reasos for cosiderig local-o-uiy asympoics such as i Sock Ž 99.. The problem wih he prees mehod is see by he fac ha i applies oe of wo iherely differe ypes of cofidece iervalsormal ierval or sigleo uiydepedig o he oucome of a es wih low power i fiie samples. Hece, quie ofe he false ierval will be used, resulig i poor coverage. O he oher had, he subsamplig mehod avoids his pifall, sice i employs oe uique cosrucio ha works boh whe ad whe. The iuiio ha subsamplig should herefore lead o good fiie sample properies is cofirmed by some simulaio sudies i Secio 6. REMARK 4.3: A impora byproduc of he proof of Theorem 4. is he fac ha, whe, he -saisic for ˆ has a limiig ormal disribuio wih mea 0 bu wih variace ha ca be arbirarily differe from if he

12 294 J. P. ROMANO AND M. WOLF iovaios are allowed o be a marigale differece sequece Ž m.d.s.. raher ha i.i.d.; oe ha m.d.s.-ype iovaios cao be rasformed o i.i.d. iovaios by icreasig he order of he ARŽ p. model. This is a impora resul, sice iferece for a leas whe i is kow ha is ofe based o sadard OLS oupu. However, his iferece ca be arbirarily misleadig, uless he iovaios are kow o be i.i.d. I he same way, ay oher iferece mehod for ha assumes i.i.d. iovaiossuch as Adrews ad Che Ž 994. or Hase Ž 999. is equally affeced i he red-saioary case. Subsamplig, o he oher had, offers a safey e agais iovaios ha are a m.d.s. Noe ha whe aeio is resriced o he case he assumpio of a m.d.s. could be relaxed o a ucorrelaed iovaio sequece. REMARK 4.4: The assumpio of saioariy of he iovaio process 4 is made o esure he saioariy of he subsample saisics ˆb,,,..., b 4. However, by exedig he geeral heory alog he lies of Poliis, Romao, ad Wolf Ž 997., his assumpio could be relaxed o allow for heeroskedasiciy ador chagig disribuios of he ; see Remark Cofidece Ierals for i Models wihou Time Tred Someimes i may be kow a priori ha *0 i model Ž.. I is he desirable o icorporae his kowledge i makig iferece o. Bu, he above resricio implies 0 i model Ž. 2. Therefore, he kowledge ca be icorporaed hrough compuig he resriced versio of ˆ by applyig OLS o model Ž. 2 excludig he ime red. Deoe he resriced versio of ˆ by 0 Ž 0 ˆ. Also, deoe he correspodig OLS sadard error by SE. OLS ˆ. The applicaio of he subsamplig mehod is aalogous o he geeral model ad is based o compuig he resriced versio ˆb 0 o all he subsamples of size b. The followig corollary shows ha he esuig cofidece iervals also have asympoically correc coverage probabiliy. COROLLARY 4.: Make he same assumpios as i Theorem 4.. I addiio, Ž. ˆ 0 2 Ž 0 assume ha *0 i model. Le,, SE. ˆ ˆ OLS ˆ, 2 ad. The, coclusios Ž. i Ž iii. of Theorem 3.2 ad Corollary 3.2 hold. REMARK 4.5: Cosider he case where a resriced model holds, ha is, where he ime red is equal o zero. Oe ca he base he esimaio of o he resriced model or o he full model. The proof of he above corollary shows ha whe he asympoic disribuio of ˆ 0 is equal o ha of ˆ. Therefore, here is o asympoic efficiecy loss for uecessarily icludig he ime red i he esimaio process; i sads o reaso, hough, ha fiie sample performace will be affeced. O he oher had, whe he asympoic disribuio of ˆ 0 does differ from ha of ˆ. Therefore, eve asympoically i is beeficial o exclude he ime red from he esimaio procedure if i is equal o zero ideed.

13 AUTOREGRESSIVE MODELS Cofidece Ierals for Oher Parameers The geeral resuls of Secio 3 also allow for he cosrucio of cofidece iervals for parameers of ieres oher ha ad he deails are sraighforward ad lef o he reader. To give oly oe example, cosider cofidece iervals for ay regressio coefficie of model Ž. 3 i he red-saioary case. The commo iferece is based o he limiig sadard ormaliy of he -saisic of he correspodig OLS esimaor. However, as he proof of Proposiio A. i he Appedix shows, if he iovaios are a m.d.s., his iferece is agai misleadig because he limiig variace of he -saisic is he i geeral o equal o. O he oher had, subsamplig is robus i his respec. Sice he rae of covergece is well-kow for all parameersgive by 2 for he i ad ad by 32 for, respecivelyhe basic subsamplig approach of Subsecio 3. ca be employed. Aleraively, he sudeized approach of Subsecio 3.2 is available as well. 5. CHOICE OF THE BLOCK SIZE A pracical issue i cosrucig subsamplig iervals is he choice of he block size b ad i ca be compared o he problem of choosig he badwidh for kerel mehods. I his secio, we propose wo mehods o selec b i pracice. The firs oe is very geeral ad ca be used wheever subsamplig applies. The secod oe ries o exploi he semi-parameric srucure of he ARŽ p. model wih liear ime red. 5.. Miimizig Cofidece Ieral Volailiy This geeral approach is of a heurisic aure ad we do o claim ay opimaliy properies. I is based o he fac ha, i order for he subsamplig mehod o be cosise, he block size b eeds o ed o ifiiy wih he sample size, bu a a smaller rae saisfyig b0. Ideed, for b oo close o all subsample saisics ˆ will almos equal o ˆ b,, resulig i he subsamplig disribuio beig oo igh ad i udercoverage of subsamplig cofidece iervals. If b is oo small, he iervals ca udercover or overcover depedig o he sae of aure Že.g., Poliis, Romao, ad Wolf Ž This leaves a umber of b values i he righ rage where we would expec almos correc resuls, a leas for big sample sizes. This idea is exploied by compuig subsamplig iervals for a umber of block sizes b ad he lookig for a regio where he iervals do o chage very much. Wihi his regio, a ierval is he picked accordig o some arbirary crierio. This idea is illusraed i Figure. For wo daa ses, symmeric subsamplig iervals are compued for a wide rage of block sizes b. The righ rages exed from b0 o abou b30 for he firs daa se ad from b0 o abou b80 for he secod daa se. The fac ha for very large block sizes he

14 296 J. P. ROMANO AND M. WOLF cofidece iervals will shrik owards he sigleo ˆ is a cosequece of he 2 fac ha he subsamplig approximaio of he samplig disribuio of Žˆ. ˆ collapses o a poi mass a zero as he block size b eds o. While his mehod ca be carried ou by visual ispecio, i is desirable o also have some auomaic selecio procedure, especially whe simulaio sudies are o be carried ou. The procedure we propose is based o miimizig a ruig sadard deviaio. Assume we compue subsamplig iervals for block sizes b i he rage of bsmall o b big. The edpois of he cofidece iervals will chage i a smooh fashio as b chages. A ruig sadard deviaio applied o he edpois he deermies he volailiy aroud a specific b value. We choose he value of b wih he smalles volailiy. Here is a more formal descripio of he algorihm. FIGURE.Cofidece iervals as fucio of he block size b for wo ARŽ. daa ses; he x-axis shows he block size b while he y-axis shows he upper ad lower cofidece ierval edpois. The daa were geeraed accordig o model Ž. wih * *0, 0.99, 200, ad i.i.d. sadard ormal iovaios. The iervals are omial 95% wo-sided symmeric iervals based o he sudeized approach of Subsecio 3.2 ad he block size rages from b0 o b80.

15 AUTOREGRESSIVE MODELS 297 Ž. ALGORITHM 5. Miimizig Cofidece Ieral Volailiy :. For bbsmall o bbbig compue a subsamplig ierval for a he desired cofidece level, resulig i edpois I ad I. b, low b, up 2. For each b compue a volailiy idex VIb as he sadard deviaio of he ierval edpois i a eighborhood of b. More specifically, for a small ieger k, le VI be equal o he sadard deviaio of I,..., I 4 b bk, low bk, low plus he sadard deviaio of I,...,I 4 bk, up bk, up. 3. Pick he value b* wih he smalles volailiy idex ad repor I, I b*, low b*, up as he fial cofidece ierval. Some remarks cocerig he implemeaio of his algorihm are i order. REMARK 5.: The rage of b values, deermied by bsmall ad b big, which is icluded i he miimizaio algorihm, is o of crucial imporace. O he oher had, o keep he compuaioal cos dow as well as o eforce he requiremes b ad b0 as, i is sesible o choose bsmallc ad bbigc2 for cosas 0cc2 ad 0. We recommed c 0.5,, c 2, 3, ad REMARK 5.2: The algorihm coais a model parameer k. Simulaio sudies have show ha he algorihm is very isesiive o is choice. We recommed k2 ork3. We ow illusrae how he algorihm works wih he help of wo simulaed daa ses. Firs, we geeraed a ime series of size accordig o model Ž. wih * *0, 0.95, 200, ad i.i.d. sadard ormal iovaios. The rage of b values was chose as bsmall0 ad bbig40. The miimizaio of he volailiy i Sep 2 was doe usig k2. The resuls are show a he op of Figure 2. The lef plo correspods o equal-ailed cofidece iervals while he righ plo correspods o symmeric cofidece iervals. The block sizes b chose by he algorihm are highlighed by a sar. The resulig fial cofidece iervals are icluded i he plos ogeher wih he poi esimae ˆ. This exercise was repeaed for aoher daa se accordig o model Ž. wih * *0,, 500, ad i.i.d. sadard ormal iovaios. The rage of b here was chose as bsmall5 ad bbig60. The resuls are show a he boom of Figure 2. The plos show ha symmeric iervals are somewha more sable, ha is, he edpois chage less as b is varied. This behavior is ypical ad was observed for may oher simulaios as well Choosig b accordig o a Esimaed Model The idea uderlyig he secod approach is ha he opimal fiie sample block size for a specific omial coverage probabiliy could be calculaed, or a leas simulaed, if he rue daa geeraig mechaism was kow. Usig

16 298 J. P. ROMANO AND M. WOLF FIGURE 2.Illusraio of he Miimizig Cofidece Ierval Volailiy Algorihm for wo daa ses. The plos o he lef correspod o equal-ailed cofidece iervals, while he plos o he righ correspod o symmeric cofidece iervals; boh ierval ypes are based o he sudeized approach of Subsecio 3.2. The block sizes seleced by he algorihm are highlighed by a sar. The fial cofidece iervals appear wihi he plos ogeher wih he poi esimaes. he simulaio mehod, oe would geerae a large umber K, say, of ime series accordig o he rue mechaism Žwih he same sample size as he observed series., cosruc subsamplig iervals usig a umber of differe block sizes for each geerae series, ad compue he esimaed coverage probabiliy for each block size as he fracio of he correspodig B iervals ha coai he rue parameer. Oe he would use he block size whose esimaed coverage probabiliy is closes o. Of course, his mehod is o feasible, sice he rue daa geeraig process is geerally ukow. However, i is reasoable o hope ha a feasible varia of his mehod will sill yield useful resuls i case he rue daa geeraig mechaism ca be cosisely esimaed. I ha case oe would use he above algorihm wih he

17 AUTOREGRESSIVE MODELS 299 esimaed process i place of he rue process. For a compleely oparameric applicaio, i is i geeral o clear how o cosisely esimae he uderlyig mechaism. Our applicaio, o he oher had, is of semi-parameric aure depedig o p2 real-valued parameers, each of which ca be cosisely esimaed by OLS, say, ad he probabiliy mechaism of he whie oise iovaio sequece, which ca be cosisely esimaed by applyig a ime series boosrap o he esimaed iovaios, say he movig blocks boosrap ŽKusch Ž 989., Liu ad Sigh Ž or he saioary boosrap ŽPoliis ad Romao Ž 994b... While i is well-kow ha his residual based boosrap yields icosise resuls whe used direcly, ha is, o approximae he samplig disribuio of Že.g. Basawa e al. Ž 99.. ˆ, i yields cosise resuls whe used idirecly, ha is, o esimae he opimal block size of he subsamplig mehod. This is a simple cosequece of he fac ha ay mehod of pickig oe of several compeig block sizes, eve coi-ossig, would yield cosise resuls as log as he block sizes icluded i he coes saisfy he regulariy codiios b ad b0 as. The poi is ha whe usig a esimaed model i pickig he block size, oe should expec beer fiie sample properies as compared o coi ossig. To provide a somewha more formal descripio, iroduce he oio of a calibraio fucio h: b ha expresses he rue coverage probabiliy of a omial cofidece ierval as a fucio of he block size b ha is used i cosrucig he ierval. If hž. was kow, oe could cosruc a ierval wih perfec coverage by employig a block size b wih hb Ž. Žprovided ha such a soluio exiss.. While he rue hž. is ukow, we ca approximae i as previously suggesed. The esimaed daa geeraig mechaism is based o OLS esimaio of model Ž. 3 ad he resulig esimaes, ˆ, ˆ ˆ,..., ˆp, ˆ p,..., ˆ where he subscrip correspodig o esimaio based o daa pois has bee suppressed. To geerae a correspodig esimaed or pseudo sequece, we sar by applyig a ime series boosrap o he esimaed iovaios o obai pseudo iovaios,..., p. The pseudo sequece is he defied by he recursive relaio Ž 8. Y Y Ž,..., p., ˆ Y ˆ ˆ Y ˆ Y Ž p,...,.. p p The followig he is he algorihm correspodig o he above calibraio idea. I is saed for a geeral parameer ad is correspodig esimae ˆ.Of course, we are maily ieresed i ad ˆ, bu he algorihm equally applies o ay oher parameers of ieres; see Subsecio 4.3. ALGORITHM 5.2 Ž Block Size Calibraio.:. Geerae K pseudo sequeces Y,...,Y, accordig o Ž., k, k 8. For each sequece k,..., K: a. Compue a level cofidece ierval CI b k j, for a grid of block sizes bmibjb max.

18 300 J. P. ROMANO AND M. WOLF ˆŽ. ˆ k 2. For each b compue hb CI 4 j j b j K. 3. Fid he value of bj wih ˆŽ hbj. closes o. 4. Cosruc a cofidece ierval usig he block size b j. REMARK 5.3: Algorihm 5.2 is relaed o adjusig he omial level of a cofidece ierval so ha is acual level beer maches he desired level i fiie samples, a idea ha daes back o Loh Ž However, o his ed he Ž sadardized. samplig disribuio of ˆ uder he esimaed mechaism mus be a cosise approximaio of he Ž sadardized. samplig disribuio of ˆ uder he rue disribuio for he resulig cofidece iervals o have asympoically correc coverage probabiliy. As meioed before, his codiio is violaed i our applicaio. REMARK 5.4: I is clear ha he grid of Ž subsamplig. block sizes o be used i Algorihm 5.2 should be as fie as possible wihi he limiaios bmi ad bmax0 as. Moreover, bmi ad bmax play roles aalogous o bsmall ad bbig i Algorihm 5. ad ca be picked i a similar fashio. However, a leas for simulaio sudies, icludig every ieger umber bewee bmi ad bmax migh be compuaioally oo expesive, i which case a appropriae subse ca be seleced. REMARK 5.5: As far as he choice of he ime series boosrap is cocered, we prefer he saioary boosrap, sice i is well-kow o be less sesiive o he choice of he Ž boosrap. block size ha he movig blocks boosrap ad also because i does o suffer from he ed effecs of he laer; see Poliis ad Romao Ž 994b.. 6. SMALL SAMPLE PERFORMANCE The purpose of his secio is o examie he small sample performace of he subsamplig cofidece iervals via some simulaio sudies. Performace is maily measured by coverage probabiliy of wo-sided omial 95% iervals for he parameer. We also look a media legh. The approach of Subsecio 3.2, subsamplig a sudeized saisic, is employed, usig boh equal-ailed ad symmeric iervals Ž deoed by idices ET ad SYM.. Moreover, he wo mehods of Secio 5 for choosig he block size are employed Ždeoed by idices MV ad CA sadig for Miimum Volailiy ad CAlibraio.. The four resulig ierval ypes are labeled Sub MV, ET, Sub MV, SYM, Sub CA, ET, ad Sub CA, SYM. For compariso, he ormal mehod, which bases he cofidece ierval o asympoic sadard ormaliy of he -saisic for ˆ, ad he mehod of Sock Ž 99. are also icluded. These wo iervals are labeled CLT ad Sock, respecively. Some brief remarks cocerig Sock s iervals are i order. Firs, his iervals are for he larges roo isead of he sum of he ARŽ p. coefficies. These wo parameers oly coicide whe p. Hece, Sock s iervals are

19 AUTOREGRESSIVE MODELS 30 oly icluded i he simulaios for he ARŽ. case. Nex, Table A. of Sock Ž 99. allows oe, up o some mior ierpolaio, o check wheher a paricular value of is coaied i he cofidece iervals i a way ha ca be auomaed for simulaios. However, he compuaio of he acual iervalsad hus heir leghrequires a graphical device ŽFigures ad 2 of Sock Ž 99.. he auomizaio of which seems very cumbersome. For his reaso, oly coverage, bu o media legh, is repored. 6.. ARŽ. Model We sar wih he mos simple model, amely ARŽ.. The daa are geeraed accordig o model Ž. wih * *0 ad oe of he followig:, 0.99, 0.95, 0.9, or 0.6; oe ha he value 0.6 is oo far away from o be hadled by Sock s iervals. The saisic ˆ is he OLS esimaor ˆ, based o model Ž. 3. The iovaios are eiher i.i.d. sadard ormal or of he form ZZ wih he Z i.i.d. sadard ormal. I he laer case, he iovaios are a marigale differece sequece bu depede. The sample sizes cosidered are 20 ad 240. The rage of b values used for Algorihm 5. is deermied by b 5 ad b 25 whe 20 ad by b 0 ad small big small bbig40 whe 240, respecively. The grid of block sizes bj for Algorihm 5.2 is 5, 8, 2, 8, 26, 40 whe 20 ad 0, 5, 20, 30, 40, 60 whe 240. The reaso for o employig a equally spaced grid is ha some prior simulaios Ž wih fier grids. showed ha he esimaed calibraio fucio ˆh Ž. chages more rapidly for smaller block sizes. The resuls are preseed i Table I. I is see ha equal-ailed subsamplig iervals perform worse ha symmeric iervals i geeral. The wo mehods for choosig he block size are comparable. Nex, oe oes he well-kow fac ha he CLT approach does o work whe ad ha i works poorly whe is close o. Sock s iervals, o he oher had, have raher accurae coverage whe hey apply. If depede iovaios of he form ZZ are employed, he CLT breaks dow. For 0.6, he coverage drops o abou 80%. I fac i ca be show ha, for iovaios of he form ZZ Z k, he coverage of CLT iervals will ed o 0 as k eds o ifiiy; see Romao ad Thombs Ž Also, i appears ha Sock s iervals are somewha less robus agais depedece as compared o subsamplig iervals Žespecially for 20.. The differece i empirical coverage bewee equal-ailed ad symmeric subsamplig iervals is oeworhy. A possible explaaio is ha he equalailed ierval is based o esimaig a 2.5% ad a 97.5% quaile while he symmeric ierval is based o esimaig a Ž sigle. 95% quaile, ad i is coceivable ha he laer ca be esimaed wih higher precisio. Oe way o examie his issue would be o redo he above able for a umber of differe cofidece levels, such as 90% ad 80%. Isead, we op for cosiderig all levels simulaeously by exploiig he dualiy bewee cofidece iervals

20 302 J. P. ROMANO AND M. WOLF TABLE I ESTIMATED COVERAGE PROBABILITIES OF VARIOUS NOMINAL 95% CONFIDENCE INTERVALS BASED ON 000 REPLICATIONS FOR EACH SCENARIO ARŽ. Model, 20, Z SubMV, ET SubMV, SYM SubCA, ET SubC A, SYM CLT Sock NA ARŽ. Model, 240, Z SubMV, ET SubMV, SYM SubCA, ET SubC A, SYM CLT Sock NA ARŽ. Model, 20, ZZ SubMV, ET SubMV, SYM SubCA, ET SubC A, SYM CLT Sock NA ARŽ. Model, 240, ZZ SubMV, ET SubMV, SYM SubCA, ET SubC A, SYM CLT Sock NA The daa were geeraed accordig o model Ž. wih * * 0ad p. The iovaios are eiher i.i.d. sadard ormal Ž Z. or of he form ZZ. All iervals were obaied by icludig a ime red i he fied model. ad hypohesis ess o calculae correspodig P-values. Hece, he P-value is give by mius he cofidece level of he ierval ha barely excludes a hypohesized value 0. Foruaely, his umber ca be direcly compued. For example, he symmeric sudeized P-value is give by 2 Ž. 2 b ˆ ˆ ˆ Ž ˆ. ˆ 4 b, b, 0 P-value Ž., b 0 b ad he remaiig P-values are compued aalogously. I is well kow ha if 0 is equal o he rue parameer, he he disribuio of a P-value correspodig o a exac hypohesis es Žor, equivalely, o a exac cofidece ierval. is give by Uiform0,, provided ha his disribuio is coiuous. Hece, oe ca judge he accuracy of hypohesis es Žor he correspodig cofidece iervals. by a Q-Q-plo of a large umber of simu-

21 AUTOREGRESSIVE MODELS 303 laed P-values Ž for he rue parameer. agais he Uiform0, disribuio. We do his i Figure 3 for he wo subsamplig ierval ypes ad he CLT ierval; Sock s mehod is excluded, sice he ables i Sock Ž 99. do o allow compuaio of P-values. To geerae he daa, we use i.i.d. sadard ormal iovaios, 240, ad he hree values, 0.95, ad 0.6. A fixed block size of b25 is used for all subsamplig iervals; his is somewha subopimal, sice he bes fixed block size i geeral depeds o he approach used as well as o he rue uderlyig parameer Žad possibly eve o he omial level of he ierval.. The plos show ha he wo subsamplig ierval ypes are qualiaively raher differe. The equal-ailed sudeized iervals work well whe bu geerally udercover whe. O he oher had, symmeric sudeized iervals are relaively accurae a large cofidece levels hroughou, bu a smaller cofidece level Ž 70%, say. udercover whe ad overcover whe. I addiio, oce more we observe he well-kow fac ha he CLT iervals work well for far away from, bu udercover icreasigly as ges closer o oe. As discussed i Subsecio 4.2, whe i is kow a priori ha *0 i model Ž. his kowledge should ad ca be icorporaed i cosrucig cofidece iervals for. I is of ieres o compare he gai i efficiecy, ha is, i ierval legh i hose isaces. We do his by compuig media legh of he 000 cofidece iervals i each sceario for he wo mehods of usig he full model ad of usig he resriced model wihou ime red Žhe laer is appropriae, sice we employ * *0 whe geeraig he daa.. Of course, i is also of ieres o compare he media legh of he various ierval ypes. The resuls are preseed i Table II. The gai from excludig he ime red whe i is ideed o eeded is subsaial whe is equal o or close o, bu i geerally decreases as ges furher away from. This is o surprisig because asympoically here is a gai i righfully omiig he ime red from he model whe bu here is oe whe ; see Remark 4.5. Moreover, symmeric sudeized subsamplig iervals are abou as good as he CLT iervals whe boh have approximaely correc coverage, ha is, whe 0.6. Noe ha we also compued empirical coverage for cofidece iervals cosruced wihou ime red. The resuls were similar o hose of Table I ad hus are o repored ARŽ. 2 Model The daa geeraig mechaism is ow model Ž., wih p2, oe of he values, 0.99, 0.95, 0.9, ad 0.6, ad equal o eiher 0.4 or o 0.4. The saisic ˆ is compued as ˆ, ˆ2, where he OLS esimaio is based o model Ž. 3 agai. Everyhig else is as i Subsecio 6.. Tables III ad IV provide empirical coverage for omial 95% cofidece iervals. The resuls are qualiaively comparable o he ARŽ. case.

22 304 J. P. ROMANO AND M. WOLF FIGURE 3.Q-Q-plos of 000 empirical P-values agais Uiform0,. The daa were geeraed accordig o model Ž. wih * *0, 240, ad i.i.d. sadard ormal iovaios. The hree values for, from op o boom, are, 0.95, ad 0.6. A sraigh lie hrough he origi wih slope is icluded i all plos.

23 AUTOREGRESSIVE MODELS 305 TABLE II MEDIAN LENGTH OF VARIOUS NOMINAL 95% CONFIDENCE INTERVALS BASED ON 000 REPLICATIONS FOR EACH SCENARIO ARŽ. Model icludig Time Tred, 20, Z SubMV, ET SubMV, SYM SubCA, ET SubC A, SYM CLT ARŽ. Model excludig Time Tred, 20, Z SubMV, ET SubMV, SYM SubCA, ET SubC A, SYM CLT ARŽ. Model icludig Time Tred, 240, Z SubMV, ET SubMV, SYM SubCA, ET SubC A, SYM CLT ARŽ. Model excludig Time Tred, 240, Z SubMV, ET SubMV, SYM SubCA, ET SubC A, SYM CLT Ž. The daa were geeraed accordig o model wih * * 0ad p. The iovaios are i.i.d. sadard ormal. 7. SUMMARY I his paper, we proposed a ew way of cosrucig cofidece iervals i ARŽ p. models wih liear ime red. While he focus was o he parameer, he sum of he ARŽ p. coefficies, he mehod is geeral eough o cover esseially ay oher parameer of ieres as well. The crux of he ew approach is o recompue a esimaor o smaller blocks of he observed daa o approximae he samplig disribuio of he esimaor compued from he eire sequece. This is he geeral idea of he subsamplig mehod of Poliis ad Romao Ž 994a.. The subsamplig mehod overcomes he oorious difficuly i he cosrucio of cofidece iervals for, amely ha he limiig disribuio, as well as he rae of covergece, of he OLS esimaor ˆ deped i a discoiuous way upo wheher or o. Some exesios of previous heory were ecessary o hadle he ukow covergece rae. The approach suggesed is

24 306 J. P. ROMANO AND M. WOLF TABLE III ESTIMATED COVERAGE PROBABILITIES OF VARIOUS NOMINAL 95% CONFIDENCE INTERVALS BASED ON 000 REPLICATIONS FOR EACH SCENARIO ARŽ. 2 Model, 0.4, 20, Z SubMV, ET SubMV, SYM SubCA, ET SubC A, SYM CLT ARŽ. 2 Model, 0.4, 240, Z SubMV, ET SubMV, SYM SubCA, ET SubC A, SYM CLT ARŽ. 2 Model, 0.4, 20, ZZ SubMV, ET SubMV, SYM SubCA, ET SubC A, SYM CLT ARŽ. 2 Model, 0.4, 240, ZZ SubMV, ET SubMV, SYM SubCA, ET SubC A, SYM CLT The daa were geeraed accordig o model Ž. wih * * 0ad p 2. The iovaios are eiher i.i.d. sadard ormal Ž Z. or of he form ZZ. All iervals were obaied by icludig a ime red i he fied model. based o subsamplig he -saisic for ˆ, which has a odegeerae limiig disribuio o maer wha he value of. The heory is flexible eough o allow iovaios of he ARŽ p. series o be a saioary marigale differece sequece raher ha a i.i.d. sequece Žbu eve he assumpio of saioariy could be relaxed.. This flexibiliy was see o be of pracical relevace, sice i he red-saioary case, he sadard iferece o bu as well o oher parameers of ieresca be arbirarily misleadig whe he iovaios are o idepede. Fiie sample performace was examied hrough some simulaio sudies ad was see o be saisfacory, a leas whe symmeric subsamplig iervals are used. The resuls were mos favorable i he case of depede iovaios, sice i his case he CLT iervals break dow Ž eve for far away from. ad Sock s Ž 99. iervals, which are asympoically valid as well, seem somewha more affeced ha subsamplig iervals i small samples. Fially, i should be poied ou ha subsamplig is a very geeral ad powerful echique ad o resriced o iferece i ARŽ p. models. Basically,