Robustness of Multistep Forecasts and Predictive Regressions at Intermediate and Long Horizons.

Transcription

1 Robustness of Multistep Foreasts and Preditive Regressions at Intermediate and Long Horizons. Guillaume Cevillon o ite tis version: Guillaume Cevillon. Robustness of Multistep Foreasts and Preditive Regressions at Intermediate and Long Horizons.. ESSEC Working paper. Doument de Reere ESSEC / Centre de reere de l ESSEC. ISSN : WP <al > HAL Id: al ttps://al-esse.arives-ouvertes.fr/al Submitted on 21 Aug 217 HAL is a multi-disiplinary open aess arive for te deposit and dissemination of sientifi resear douments, weter tey are publised or not. e douments may ome from teaing and resear institutions in Frane or abroad, or from publi or private resear enters. L arive ouverte pluridisiplinaire HAL, est destinée au dépôt et à la diffusion de douments sientifiques de niveau reere, publiés ou non, émanant des établissements d enseignement et de reere français ou étrangers, des laboratoires publis ou privés.

2 ROBUSNESS OF MULISEP FORECASS AND PREDICIVE REGRESSIONS A INERMEDIAE AND LONG HORIZONS RESEARCH CENER GUILLAUME CHEVILLON ESSEC WORKING PAPER 171 JULY 217

3 Robustness of Multistep Foreasts and Preditive Regressions at Intermediate and Long Horizons. Guillaume Cevillon ESSEC Business Sool & CRES July 23, 217 Abstrat is paper studies te properties of multi-step projetions, and foreasts tat are obtained using eiter iterated or diret metods. e models onsidered are loal asymptoti: tey allow for a near unit root and a loal to zero drift. We treat sort, intermediate and long term foreasting by onsidering te orizon in relation to te observable sample size. We sow te impliation of our results for models of preditive regressions used in te finanial literature. We sow ere tat diret projetion metods at intermediate and long orizons are robust to te potential misspeifiation of te serial orrelation of te regression errors. We terefore reommend, for better global power in preditive regressions, a ombination of test statistis wit and witout autoorrelation orretion. Keywords: Multi-step Foreasting, Preditive Regressions, Loal Asymptotis, Dynami Misspeifiation, Finite Samples, Long Horizons. JEL Classifiation: C22, C52, C53. evillon@esse.edu 1

4 1 Introdution and overview wo parallel literatures ave developed or aelerated reently tat aim to estimate relationsips over a so-alled multi-step orizon. On te one and, tere as been a renewed interest in assessing te relative merits of two foreasting metods: tose of iterated and diret multi-step foreasts denoted IMS and DMS. e former tenique onstitutes te standard in eonometris and onsists in estimating a one-step aead model relating, say, y t to y t 1 in a sample of observations and using it to foreast y +1 using y and extrapolating te relation to generate a foreast for y +2 using te foreast for y +1 tat as previously been obtained. Diret multi-step foreasting by ontrast will aim to develop a distint model for ea foreast orizon 1: relating, in-sample, y t to y t so tat a foreast for y + an be obtained diretly using y. e relative performanes of tese foreasts was first derived in general settings by Weiss 1991, but it as been been a ontinuous interest, sine, as in e.g., Clements and Hendry 1996, Ing 23, 24, Marellino, Stok, and Watson 26, and Sorfeide 25, most reently, Carriero, Clark, and Marellino 215 Cevillon 216, MElroy and MCraken 217 and Hendry and Martinez 217. On te oter and, te seminal work by Fama and Fren 1988, Campbell and Siller 1988 and Stambaug 1999 as spurred a wole literature witin finane of autors wo aim to assess te preditive power a variable x t as on anoter, say z t over some orizon. e prototypial preditive or long orizon regression will take te form of a regression of z t+ on x t, but z t+i or x t+i also appear as regressand and regressor in te literature see e.g. Lanne, 22, orous et al., 24, Valkanov 23, Boudouk et al., 28, Hjalmarsson, 211, Pillips and Lee, 213, Pillips, 215, and te referenes terein. e long orizon regression literature sares wit tat on diret multi-step foreasting tree key features: i te model wi is estimated is not a priori tat wi would most effiiently ose i.e. te one-step aead model but one tat indues te errors in te regression to be serially orrelated; te osen multi-step tenique works for te estimated model beause ii tis model is potentially misspeified as te errors are serially orrelated see Ferson et al., 23, and Pástor and Stambaug, 29 and iii te variables tat are being used are non-stationary or nearly so in addition to te papers above, see inter alia Stambaug, 1999, Lettau and Ludvigson, 21. In tis paper, we propose a loal-asymptoti model tat builds on te work of Kemp 1999, Valkanov 23, orous et al. 24, Cevillon and Hendry 25 and Hjalmarsson 211. We prove a new key property of diret multi-step estimators, namely teir robustness to misspeifiation of te serial orrelation of te error proess. We ten sow ow tis property also applies in te ase of long-orizon regressions and tat it provides a new justifiation for wy tey ave proved so suessful empirially. We sow tat te bias tat was found by Hjalmarsson relies on is assumption tat te orizon is small ompared to te observed sample, = o, but tat it vanises wen onsidering = O, as suggested by Corane 26 and seds ligt on is results. Our analyti results lead us to reommending at long orizon te ombination of a standard test and tat wit Heteroskedastiity and Autoorrelation orretion HAC. We sow by simulations tat te ombination aieves better global power. 2

5 is paper is organized as follows. Setion 2 presents te foreasting and preditive regression models tat we onsider and te way tey are related. We ten derive te distributions of iterated and direted multi-step estimators and foreasts in Setion 3. e same setion applies tese results to preditive regressions. A Monte Carlo assessment follows in Setion 4. In te paper, row vetors are denoted as x 1 : x 2 and olumn vetors as x 1, x 2. rougout, we also use te following notations: λ {} i= λi for 1, denotes weak onvergene of te assoiated probability measure, W r is a standard Brownian motion on C, 1, and w denotes te integer part of w for any real salar w. 2 e models and loal-asymptoti assumptions We introdue ere te literatures on multistep foreasting and long-orizon regressions. ese literatures present similarities wi ave not always been stressed. rougout te paper, we are onsidering te simple autoregressive model for te proess {y t } y t = τ + ρy t 1 + ɛ t 1 for t 1, were y as a finite distribution and te error ɛ t is assumed to satisfy te following ondition. Condition P. i E ɛ t = for all t Z; A sequene {ɛ t } satisfies Condition P if and only if ii sup t E ɛ t +η < for some > 2 and η > ; iii {ɛ t } is weakly stationary wit ovariane funtion series {ξ ɛ i} i= su tat i= iξ ɛ i <. Condition P allows to derive general results for general distributions of te errors. Here we restrit our attention to weakly stationary ɛ t as it allows to derive more expliit results. Yet, our results old replaing iii above wit te less restritive assumption tat ɛ t is strongly mixing wit mixing oeffiients α m su tat m=1 α1 2/ m < and tat lim 1 Var j=1 ɛ j = ξ ɛ < as in Pillips, In te following we will be led to restriting te serial dependene of ɛ t and onsider te ases were it is a wite noise proess or follows a moving average or order one, an MA1. Also, were use te notation ξ u = i= ξ u i for te long run variane of any proess u t satisfying te ondition. In te speifi ase of ɛ t in 1 we write σ 2 = ξ ɛ. In te time series foreasting literature, te standard multistep foreasting tenique onsists in estimating te parameters τ, ρ of model 1 and ten to use te estimators τ, ρ to ompute foreasts reursively at all orizons 1 : ŷ t+ t = τ + ρŷ t+ t = ρ {} τ + ρ y t 2 were we let ŷ t t = y t and ρ {} = i= ρi. is onstitutes te plug-in or iterated multi-step IMS tenique. 3

6 Diret multistep DMS as often been proposed as an alternative: it wi onsists in estimating te parameters τ, ρ of te projetion of y t+ on y t, y t = τ + ρ y t + w,t, for 1, 3 wit τ, ρ = ρ {} τ, ρ and w,t = i= ρi ɛ t i. e DMS foreasts are obtained from estimators τ, ρ as ỹ t+ t = τ + ρ y t. 4 o aieve robustness to misspeifiation, te literature as often onsidered τ, ρ and τ, ρ to be te ordinary least squares OLS estimators and we follow tis approa ere. e rationale for DMS lies in tat wen ɛ t in 1 is serially orrelated, IMS foreasts are biased and DMS an prove more aurate in terms of mean-square foreast error MSFE. e preditive regression literature sine Fama and Swert, 1977, and Rozeff, 1984, see Stambaug, 1999 as onsidered testing te null of not preditability in a bivariate setting: a standard model see e.g. Valkanov, 23 lets, for t =,..., zt+1 α zt εt+1 = + + τ ρ y t+1 y t were bot ε t and ɛ t are assumed to satisfy Condition P. 1 For instane, in Pástor and Stambaug 29, z t denotes te return on an asset, y t an imperfet preditor tereof, and te null ypotesis ɛ t+1 is H : =. Model 5 is often expressed, for 1, as z t = α + y t + ω,t 6 wit α, = α + τρ {}, ρ, ω,t+ = ρ i ɛ t +i + ε t ; or using Z t +1 = z t +i as a regressand see Valkanov, 23, and referenes terein. In Expression 6, te ypotesis of interest is H : =. e empirial literature as sown tat wereas H =1 often does not rejet, tis is not te ase wen onsidering large, in wi ase H may rejet and y t appears elpful in prediting z t. e question of ow large sould be is an empirial one: Hjalmarsson 211 studies te ase were is fixed, wereas Valkanov 23, orous et al. 24 and Hjalmarsson 211 ave onsidered letting te orizon grow wit te sample size as respetively = O and = o. In teir setting, orous et al. and Hjalmarsson allowed in addition for te error ε t+1, ɛ t+1 to exibit autoorrelation. is ompliates te derivation of te distributions of te estimators and test statistis but it yields insigt regarding te role played by. Indeed, Hjalmarsson sows tat te estimators of te regression oeffiient of Z t +1 on y t suffers from seond-order bias generated by te orrelation between ε t and ɛ t. is result is similar to tat of Banerjee et al in te ontext of a omparison between iterated and diret multistep foreasting. Here te main insigt we gain about preditive regressions from multistep foreasting ours under preditability, ene our results allow to devise tests wit inreased global power. 1 is preludes assumptions su as in Deng 213 were ɛ t exibits a moving average loal unit root. 5 4

7 In te following, we assume tat te parameters of 6 are estimated using OLS. is oie assumes tat te errors ε t are martingale differene sequenes MDS and is ommon in empirial work, see e.g. Stambaug In reality, tis assumption may be wrong and ε t may be autoorrelated as sown by Pástor and Stambaug 29 were tey follow an MA1. Altoug te literature as also onsidered variane estimators wi are eteroskedastiity and autoorrelation onsistent HAC, we do not study tem speifially ere. Indeed, tese do not orret for te bias in te autoregression. Also, by taking into aount te serial orrelation in te estimation of te varianes of w,t and ω,t tey sould benefit multistep or long orizon estimators and only strengten our argument. In tis paper, we aim to apture tree key issues tat arise bot te in preditive regression and multistep foreasting frameworks: i te interation between te orizon, te available sample size, ii te persistene in te time series y t and z t and iii te serial dependene in ɛ t and ε t. For tis we onsider te loal-asymptoti framework tat is now ommon in te eonometri literature. First, we follow te now standard assumptions tat ρ is lose to unity: we follow autors su as Pillips 1987 or Campbell and Yogo 26 and model tem as loal to unity ρ = exp φ/ = 1 + φ/ + O 2, 7 Expression 7 implies tat y t is near integrated and tat τ ats as a near drift. is latter issue as generally been avoided in te early literature by imposing α = τ = wi orresponds to using demeaned variables, but not in te some seminal artiles e.g. Campbell and Yogo, 26. Owing to te near non-stationary nature of te variables, demeaning may not provide more aurate estimates. In partiular if τ is indeed nonzero but very small so tat a near linear trend is mistaken for a non zero mean see for instane Pástor and Stambaug, 29, were τ is low wen ρ is lose to unity. Cevillon and Hendry 25 ave sown tat small nonzero drifts an ave a signifiant influene on te multistep foreasts wen dealing wit non-stationary variables. Also, te literature on returns foreasting as aknowledged te importane of slowly drifting expeted returns see e.g. Lettau and Van Nieuwerburg, 27. For tese reasons we allow for te parameter τ in 1 to be nonzero but assume tat it is small and model it via loal-asymptotis as a Pitman drift: τ = ψ. Su a loal drift would be of low magnitude, justifying te loal-asymptoti assumption. Loalto-zero drifts ave been used inter alia in Monte Carlo simulations of unit root tests in Vogelsang 1998, Rossi 25a and Busetti and Harvey 28; tey ave been studied analytially by Haldrup and Hylleberg 1995 and Stok and Watson Parameterizing te drift as 8 indues a nonlinear sine ρ < 1 deterministi trend of order O. In te paper, we denote by y t te triangular array tat is generated by te non-onstant parameters τ, ρ. Seond, we onsider eiter old te orizon onstant, or letting it grow as a onstant fration of te sample size as in te following definition. 8 5

8 Definition 1 Let 1 denote te orizon of interest. We refer to te orizon begin long wit respet to te sample size if tere exists a onstant, 1 su tat / as ; te orizon is sort if is onstant irrespetive of ; and te orizon is said intermediate in a sequential asymptoti setting were / as and ten we let. Long run foreasting as been studied by Stok 1996, Pillips 1998, Kemp 1999 and in long-run preditive regression by Valkanov 23, orous et al. 24, urner 24 and Elliott 26. Altoug different, te sequential asymptoti intermediate orizon framework relates te setting of Hjalmarsson 211 were /. Finally, te problem of misspeifiation may arise even for = 1 if ε t or ɛ t exibits serial orrelation and ross orrelation. We define teir joint autoovariane funtion εt εt k ξε k ξ ε,ɛ k E = Ξ k = ξ ɛ,ε k ξ ɛ k ɛ t ɛ t k wit Ξ = + k= Ξ k and denote ς 2 = + k= ξ ε k, ϱ 2 = + k= ξ ε,ɛ k, wit as before σ 2 = + k= ξ ɛ k. 3 Estimators and Foreasts is setion provides our main results. First, we onsider te asymptoti distribution of te OLS estimators τ, ρ and τ, ρ under various assumptions on te orizon. en we derive te impliations of our results for foreasting. 3.1 Distributions of empirial moments Under Condition P, 1/2 r ɛ i σw r, as, were W r denotes a Wiener proess. We define te Vasiček proess 2 K ψ,φ r = ψf φ r + r eφr s dw s,for r, 1, were te funtional 3 f : R C, 1 satisfies f φ : r e φr 1 /φ for φ R\ {} and f r = r. By extension, for a given σ >, denote by K ψ,φ r te funtional K ψ,φ r = σj ψ/σ,φ r solution to te linear stoasti differential equation. dk ψ,φ r = ψ + φk ψ,φ r dr + σdw r 9 wit initial ondition K ψ,φ =. K ψ,φ r is a Gaussian proess for fixed r wit expetation ψf φ r and variane σ 2 f 2φ r. For ψ =, it redues to an Ornstein-Ulenbek OU J φ r = K,φ r. 2 It is standard in te literature to parametrize instead te proess imposing ψ = λφ for some λ. 3 We denote by D, 1 te spae of real-valued funtions on te interval, 1 wi are rigt ontinuous and ave finite left limits àdlàg. C, 1 is te subspae of D, 1 of ontinuous funtions. We will straigtforwardly extend tis definition below to allow for r > 1 in foreasting. 6

9 First, olding onstant, te variane of te fixed orizon multi-step disturbane w,t admits te variane σ 2 w = lim 1 w,t 2 = σɛ i ξ ɛ i and let its long-run variane σ 2 = lim 1 Var r i= w,i = 2 σ 2. en, simply, for r, 1, 1/2 r i= w,i σ W r = σw r, as. Wen letting be a fration of te sample size : = O, w,t beomes a non-stationary series exibiting a stoasti trend and te usual saling fators for integrated proesses no longer old. We define te operator δφ wi, for any diffusion proess Z r defined on C, η, η lets δφ Z on C, η be su tat: δ φ Z r = Z r eφ Z r, for r η and δφ Z r = Z r, for r <. A proposition follows tat provides all te asymptoti onvergene properties tat we require in te paper. Proposition 1 Let y t be generated as 1 under Condition P wit loal asymptoti parameters 7 and 8. en, te following olds as, under sort orizon, 1, is onstant, a 1/2 y r K ψ,φ, 3/2 y t K ψ,φ, 2 y2 t b 1 y t w,t σ K ψ,φdw σ 2 σw 2 ; 1/2 r i= w,i σw r, 1 w2,t σ2 w. under long orizon /, 1, as, a 1/2 y r K ψ,φ r, 3/2 y t K ψ,φ, b 2 y t w,t K ψ,φ r δ φ J φ r dr 1 2 ψ2 e φ f φ 2 1/2 w, r δ φ J φ r, 3/2 w,t δ φ J φ r dr, K2 ψ,φ ; 2 y2 t K2 ψ,φ ; 2 w2,t o allow for a omparison between sort and long orizons in Proposition 1, te following orollary onsiders te intermediate orizon setting. e notation A L = B means tat as +, for all x R, Pr A < x / Pr B < x 1. Corollary 2 Intermediate Horizon Under te assumption of 1, te asymptoti distributions under long orizon settings satisfy as : a K ψ,φ K ψ,φ, K2 ψ,φ K2 ψ,φ ; L = L = b K ψ,φ r δφ J φ r dr 1 2 ψ2 λ f φ 2 = L σ K ψ,φ r dw r δφ J φ r + δ 2φ J φ 2 r dr = L σ 2. L = σ W r + W r, δ φ J φ r + dr L = σw 1, and One of te key impliations of te different beaviors under sort and long orizons relates to te sample ovariane between te regressors and te disturbanes in expressions 1 and 3. Under sort orizons: 1 1 y t w,t y t 1 ɛ t i ξ ɛ i. 1 t=1 δ φ J φ r 2 dr. 7

10 so differenes between multi-step and saled one-step moments only arise asymptotially wen te error ɛ t is autoorrelated. Expressions under long orizons are more involved, but intermediate orizons allow for an easy omparison: lim 2 L y t w,t = 1/2 σ K ψ,φ r dw r, 2 t=1 y t 1ɛ t σ K ψ,φdw σ 2 σɛ is latter expression sow tat in OLS estimation, tere will be a trade-off of bias and effiieny between one-step aead and multistep projetions. Indeed te expetation of lim 2 y t w,t is nonzero but o weter or not te error is autoorrelated. 4 By ontrast te orresponding expetation of 2 t=1 y t 1ɛ t is zero in te absene of misspeifiation but O oterwise. In terms of variane, te order is reversed: te multistep moments ave asymptoti variane in te O and te saled one-step in O 2 for. e previous analysis sow tat weter te orizon is sort or long will ave a signifiant impat on te estimators. Sort orizons multistep estimation will be affeted by misspeifiation, and tis may be benefiial or detrimental. By ontrast, long orizon multi-step estimation will be mostly unaffeted by te misspeifiation. is is due to te fat tat as, te multi-step error w,t beomes an integrated proess wose autoovariane funtion is onstant; in oter terms, te orretion σ 2 σ 2 w in Proposition 1-b. 3.2 OLS Estimators o empasize te different beaviors, define te saled deviations of OLS slope and interept estimators from te parameters as: 5 γ = ρ 1 γ, and π = 1/2 τ τ π. 12 We define, for notational ease, te following stoasti matrix 1 D = K ψ,φ K ψ,φ K2 ψ,φ. e one-step OLS estimator is ten araterized by π = D 1 σw 1 γ φ σ K ψ,φdw + σ2 σ 2 ɛ 2 13 e presene of a loal-to-zero drift implies tat te stoasti and te deterministi trends ave idential asymptoti orders of magnitude, bot O p 1/2. e unit-root estimator is superonsistent but te orresponding error is of order O p 1 and not te O p 3/2 observed 4 e term 1 2 ψ2 e φ f φ 2 tat arises in proposition 1-b is O 3. It is zero in te absene of a drift, i.e. wen ψ =. 5 We define γ as deviation for ρ for unity rater tan from ρ for ease of notation in te long orizon setting. 8

11 in te presene of a true linear trend. Wen ɛ t is wite noise, σ = σ ɛ and we denote te estimators of a true AR1 as π, γ = σd 1 W 1, 1 K ψ,φdw. In te presene of dependent errors, σ 2 σɛ 2 /2 = ξ ɛ i and tis is te annel troug wi misspeifiation of te innovations affets te estimators. e orrelation between π and γ is ten a positive funtion of K ψ,φdr and te latter s expetation as a sign opposite tat of ψ. Now, te previous results may be used for omputing IMS and DMS estimators of te multistep parameters τ,, ρ, = ρ {} τ, ρ e IMS estimators are naturally defined as. τ{},, ρ {}, = ρ {} τ, ρ. e DMS estimators τ,, ρ, are omputed via OLS of 1 over a sample of size. We denote te asymptoti limits as follows. Under sort orizon fixed, IMS : τ{}, τ,, ρ 1 π {}, γ {}, DMS : τ, τ,, ρ, 1 π, γ, and under long orizon, as /, IMS : 1/2 τ {}, τ,, ρ 1 π {}, γ {}, DMS : 1/2 τ, τ,, ρ, 1 π, γ. Using te results above, te following Proposition relates te distribution of te multi-step estimators to tose of te one-step. Proposition 3 Let y t be generated as 1 under Condition P wit loal asymptoti parameters 7 and 8. en te following olds as, under sort orizon, 1, is onstant, and te limits are, IMS : DMS : π{} γ {} π γ = = π π γ γ, σ 2 ɛ σ 2 w D 1 under long orizon, /, 1, te limits are π{} fγ f φ ψ + f γ π IMS : =, γ {} f γ γ π DMS : = + D 1 δ φ J φ r dr γ φf φ K ψ,φ r δφ J φ r dr 1 2 ψ2 e φ f φ 2. Proposition 3 allows for a omparison of IMS and DMS estimation auray. Bot estimators are onsistent for te multistep parameters at sort but not at long orizons. 1 ; Indeed for te latter, te estimators must be saled by an additional or to ensure tey weakly onverge. At sort orizons, IMS and DMS yield idential asymptoti distributions wen ɛ t follows a wite noise and tese are simply times te one-step. By ontrast, serial orrelation in ɛ t implies tat 9

12 DMS distributions wi are not times tat of te one-step model. o see te impat of te autoorrelation of ɛ t, onsider te differenes between π {}, γ {} and te orresponding random variables π {}, γ {} wen ɛ t is wite noise we define π, γ similarly: ten π {} π {}, γ {} γ {} = π π, γ γ π π, γ γ = σ 2 σ 2 w σ 2 σ 2 ɛ π π, γ γ were σ2 σ 2 w σ 2 σ = ɛ 2 ξ ɛ i 1 i ξ ɛ i. In partiular, if ɛ t follows an MAq, ten for > q, σ 2 σw 2 = qσ 2 σw 2 q. is sows tat if ɛ t follows a moving average as in e.g. Pástor and Stambaug, 29, were it is an MA1 ten te impat of te serial orrelation in ɛ t is inreasing linearly in te orizon for IMS but bounded by tat at orizon q for DMS. Banerjee, Hendry, and Mizon 1996 find a similar result. Now te atual distribution of π π, γ γ depends on te parameters of te DGP but its expetation as te sign of ψ, 1 σ 2 σɛ 2 = ψ, 1 i ξ ɛ i. Sine, in general, te bias in autoregressive parameter estimators is negative an AR1 wit a near unit root, tis implies tat E γ <. Hene if ɛt is negatively autoorrelated ten te probability E γ γ < so te distribution of γ is sifted to te left, i.e ρ furter away from unity, wit a larger absolute bias tan wen ɛ t is wite noise. As te orizon grows, ten IMS ompounds te bias but tat of DMS remains bounded if ɛ t follows a moving average. Given te negative expeted orrelation between te interept and slope estimators, positive ψ will ave te same effet on te bias of te multistep interept estimator. is is wat Cevillon and Hendry 25 found in teir simulations. Proposition 3 also allows for a omparison of te estimators at long orizons, but nonlinearities render te analysis of analytial results diffiult. For tis reason, te following orollary onsiders intermediate orizons. Corollary 4 Intermediate Horizon Under te assumptions of Proposition 3,te asymptoti distributions under long orizon settings satisfy as : π {} L = γ {} φ π π L and = γ φ γ φ Corollary 4 onfirms te analysis tat was made previously tat intermediate orizon DMS estimators are robust to serial orrelation of ɛ t sine teir distribution is a proportional to te unbiased π, γ. is is not te ase for IMS wi are biased. Yet, as, is of iger magnitude tan, so DMS suffers from iger variane tan IMS. γ π φ 3.3 Foreasting We now derive te distributions of te foreast errors. Parameter estimates are used to foreast te series steps aead from an end-of-sample foreast origin y using te expressions of Setion 2. Define te IMS foreast errors under sort orizon as ê = y + ŷ + and under long orizons as ê, = /2 ê. Denote te orresponding DMS foreast errors as ẽ and ẽ,. 1

13 In sort-orizon foreasting, onsisteny of te estimators imply tat te asymptoti limit of te foreast error is simply ê j= ɛ p + j and similarly for ẽ. Hene, for a omparison we derive te sort orizon distributions as deviations from j= ɛ + j. For te long orizon ase, we need to extend te definition of K ψ,φ r to over r, 1 + l for some l, 1. e following proposition provides asymptoti distributions of te foreast errors. Proposition 5 Let y t be generated as 1 under Condition P wit loal asymptoti parameters 7 and 8. en te following olds as, under sort orizons 1,, 1/2 ê j= ρj ɛ + j 1/2 ẽ j= ρj ɛ + j and under long orizons /, 1, π {} γ {} φ K ψ,φ 1 π γ φ K ψ,φ 1 ê, π {} + γ {} φf φ K ψ,φ 1 + δ φ J φ 1 +, ẽ, π + γ φf φ K ψ,φ 1 + δ φ J φ 1 +. e key to foreast auray is ere te orrelation between te slope estimator and te demeaned foreast origin. Indeed, wereas for stationary proesses it as been ustomary to assume tat te orrelation between te foreast origin and te estimators as little impat, tis assumption does not old in te presene of trending beavior see Ing, 24. In sort orizon foreasting, te proposition implies tat 1/2 K µ ψ,φ ê ẽ 1 i/ ξ ɛ i K µ ψ,φ were K µ ψ,φ = K ψ,φ r K ψ,φ u du. is expression sows tat for ɛ t MA q, wiever metod is more preise at orizon q + 1 will tend also to be so for q + 1, and te differene in foreast errors is lose to being linear in. Wen ɛ t is wite noise and te orizon sort, bot metods are asymptotially equivalent. Expression 14 also sows tat if E K µ ψ,φ 1 / 2 K µ ψ,φ >, su as wen ψ >, ten negatively autoorrelated ɛ t imply tat E ê ẽ >. In partiular, if ɛ t follows an MA1, ten sign E ê ẽ = sign ξɛ 1 ψ. 15 Heuristially, if E ê and E ẽ ave te sign of ψ, ten ξɛ 1 < implies tat foreast biases favor DMS: E ê > E ẽ. Next, we onsider intermediate and long orizon settings. For low, te foreast errors from eiter metod do not beave omparably wit respet to te orizon: 11

14 Corollary 6 Intermediate Horizon Under te assumptions of Proposition 5, te limiting distributions as satisfy: ê, ẽ, L = γ φ K ψ,φ 1 π + σ L = γ φ K ψ,φ 1 π + σ W 1 + W 1 W 1 + W 1 16a 16b e orollary sows te insigt we drew from te estimators arry over to te foreasts: i sine DMS estimator biases are not affeted by serial orrelation of ɛ t at intermediate orizons nor are te foreast; yet ii DMS foreasts ave iger variane. ere terefore exists a trade-off between DMS robustness to dynami misspeifiation and te ompounded variane due to te orizon effet. e orollary sows toug tat at intermediate orizons in te presene of serial orrelation of ɛ t, biases differ by an order of magnitude: E ê, /E ẽ, = O, but varianes are omparable Var ê, = O 1, Var ẽ, = O 1. If γ φ K ψ,φ 1 as zero expetation, ten DMS is unbiased but not IMS so te ontribution of te IMS bias to te MSFE is of order. 3.4 Preditive Regressions e results tat were derived in te multi-step autoregression an be used to obtain te distributions of te estimators in te preditive regression of z t on y t. Define te bivariate Brownian motion H r su tat 1/2 r t=1 ε t, ɛ t H r = H r, σw r were we write H = ςu + σδw. In Expression 6, α 1 = 1 ρ α + τ ene sine 1 ρ = O 1, only τ needs to be onsidered loal asymptoti. o mat te results from Proposition 1, we let G φ r = r eφr s dh s = G φ, J φ. A proposition follows. Proposition 7 Let {z t, y t } generated by 5, were ɛ t and ε t satisfy Condition P and wit loal asymptoti parameters 7 and 8. en te following olds, as, if under sort orizon, for 1, onstant, tere exist 6 ϖ R su tat a 1/2 ω,t H 1 + σ 1 W r b 1 y t ω,t K ψ,φd H + σ 1 W + ϖ under long orizon, for /, 1, 6 e definition of ϖ is ϖ = 1 1 ξ ε + ξ ε + ρ 2 1 ξ ε ξ ε 1 2 ρ 1 ξ ε ξ ε 1 ξε ξ ε,ɛ i ξ ε,ɛ 1 i + ρ j 1 1 ξ ε,ɛ ξ ε,ɛ j ρ 1 ξ ε,ɛ ξ ε,ɛ j 1 j=1 + ρ j+k 2 1 ξ ɛ ξ ɛ k j 2 j=1 k=1 12

15 a 3/2 ω,t δ φ J φds b 2 y t ω,t K ψ,φ r δφ J φ r dr 1 2 ψ2 e φ f φ 2 Corollary 8 Under te assumptions of Proposition 7, if = te results simplify to a 1/2 ω,t H 1, b 1 y t ω,t K ψ,φdh + i= ξ ε,ɛ i a 1/2 ω,t H 1 H 1, b 1 y t ω,t K ψ,φ s dh s Elements b and b sow tat only long orizons are robust to te ross-orrelation i= ξ ε,ɛ i. Proposition 7 sows tat te results tat were derived for multi-step foreasting an be used for te analysis of te preditive regression. In partiular, te saled empirial moments onverge to distributions tat are very lose to tose of DMS. ey sare te similar properties tat wen = misspeifiation of te regression errors as a negligible impat. By ontrast, if a modeler ad attempted to foreast using a one-step preditive regression, se would ave been subjet to errors omparable to tose found in IMS foreasting. ξ ε,ɛ Indeed onsider te OLS estimator of, in te regression z t = α + y t + ω,t. Let = i= ξ ε,ɛ i, ten a straigtforward appliation of Proposition 7 yields te following proposition. Proposition 9 Under te assumptions of Proposition 7, te following olds: first if, 2 1 K µ 1 ϕ,φ dr Kµ ϕ,φ d ςu + σδw + ϖ def 1 = λ K µ ϕ,φ e φ and if =,, 2 dr 1 Kµ ϕ,φ d ςu + σ δ + 1 W + ϖ K µ ϕ,φ Kµ ϕ,φ dh+ξi ε,ɛ K ϕ,φ µ 2 dr def = λ,{} 2 dr 1 Kµ ϕ,φ r δ φ J φ r dr 1 2 ψ2 e φ f φ 2 def = λ,, 1 Kµ ϕ,φ dh+ξ ε,ɛ, K ϕ,φ µ 2 dr Kµ ψ,φ s dhs K µ ϕ,φ 2 dr Corollary 1 Under te assumptions of Proposition 9, at intermediate orizons, L σ 1 λ, = Kµ ϕ,φ dw L if, and λ K ϕ,φ µ 2, = Kµ ϕ,φ dh dr 1 K ϕ,φ oterwise. µ 2 dr Corollary 11 Consider te regression of 1 k=1 z t+k on a onstant and x t, te estimator of te oeffiient of x t admits te following distribution: if is fixed, {} 1 k=1 λ,{k}; if /, f φ λ,sds. Proposition 9 sows tat intermediate and long orizon preditive regressions are robust to dynami misspeifiation yet not to ontemporary orrelation of te errors. As, te beavior of 1/2 λ, is lose to tat of λ provided tat all Ξ k, are diagonal. e main differene 13

16 is tat te former involves te stoasti integral of K µ ϕ,φ wit respet to inrements in W wereas tat of λ involves te inrements of H. Hene, wen, λ, is immune at intermediate orizons to te long run endogeneity and serial orrelation of te errors in te preditive regression. 4 Monte Carlo In order to illustrate te teoretial results presented above, we perform some simple simulations. We first ompare te distributions of IMS and DMS foreast error under dynami speifiation to tose under orret speifiation. For tis, we simulate an ARMA1, 1 data generating proess DGP y t = τ + ρy t 1 + ɛ t + θɛ t 1 were ɛ t i.i.d N, 1 as well as an AR1 wit te same long run variane y t = τ + ρy t θ ɛ t. Parameters vary as follows: and ρ {±.99, ±.95, ±.6, }, θ {±.9, ±.4} and {1, 25} wit an initialization of 2 observations wit ranging from 1 to /3. For ea DGP, we ompute 5, repliations of te IMS and DMS foreast errors based on an AR1 model. We report te p-values of a Kolmogorov-Smirnov test for te null of equal distributions of te foreast errors under te ARMA1, 1 and AR1 DGP. Non-rejetion of te null is interpreted as evidene tat for te DGP and orizon onsidered, te foreasting metod is robust to te dynami misspeifiation onsidered. Figures 1 and 2 report te p-values of te Kolmogorov-Smirov test as a funtion of te orizon and for, respetively, = 1 and 25 observations. e simulations all onfirm tat te p-values rejet equal distributions of te foreast errors and ene robustness to dynami misspeifiation at very low orizons in te presene of severe misspeifiation large θ. Yet te p-values inrease rapidly wit wen ρ is positive. is is espeially true of DMS; tis is less so for IMS: for instane, wen ρ =.99 and θ =.4, te test rejets at te 1% level for.15. e figures also report ases were ρ < and we see tat te foreasts ten tend to be less robust, in partiular wen θ > and ρ is lose to 1. o assess ow te results on multistep foreasting arry over to preditive regressions, we simulate Model 5 were α = τ = and = 1. Under dynami misspeifiation ε t follows an MA1 proess wit parameter θ and standard Gaussian wite noise innovations, wereas under i.i.d orret speifiation ε t N, 1 + θ 2. We onsider bot te ase of Corrε t, ɛ t = no endogeneity and of Corrε t, ɛ t = 1/ 2.7 endogenous ase. We let ɛ t i.i.d N, 1 and onsider various values of ρ and as before. We only reord te ase of = 25. Results are reported in Figures 3 and 4, respetively for te exogenous and endogenous situations: te graps present te p-values of te Kolmogorov-Smirnov test for te null tat te standardized i.e. divided by teir estimated standard error, witout autoorrelation orretion ave idential distributions for ε t MA 1 or ε t iid. e figures report patterns similar to tose observed under multi-step foreasting. Finally, we assess te impliations of te results above for te test of te null H : = at te 1% signifiane level in te preditive regression model wit ρ =.99. For tis, we onsider te simple situation were ritial values of te test statisti is obtained by parametri bootstrap over 14

17 1..5 ρ=-.99 ρ=-.95 ρ=-.6 ρ= ρ=.6 ρ=.95 ρ=.99 IMS, θ= DMS, θ= IMS, θ= IMS, θ= IMS, θ= DMS, θ= DMS, θ= DMS, θ= Figure 1: e figure reports p-values of te Kolmogorov-Smirnov test tat te distributions of foreast errors IMS, left and DMS, rigt are te same in te models wit misspefiied and orretly speified error dynamis. e orizontal axis is te orizon. e sample size is = 1 observations ρ=-.99 ρ=-.95 ρ=-.6 ρ= ρ=.6 ρ=.95 ρ=.99 IMS, θ= DMS, θ= IMS, θ= IMS, θ= IMS, θ= DMS, θ= DMS, θ= DMS, θ= Figure 2: e figure reports p-values of te Kolmogorov-Smirnov test tat te distributions of foreast errors IMS, left and DMS, rigt are te same in te models wit misspefiied and orretly speified error dynamis. e orizontal axis is te orizon. e sample size is = 25 observations. 15

18 no endogeneity, ^/s.e. ρ=-.6 ρ= ρ=.6 ρ=.95 ρ= θ= θ= θ= θ= Figure 3: e figure reports p-values of te Kolmogorov-Smirnov test tat te distributions of standardized are te same in te preditive regression models wit misspeified and orretly speified error dynamis witout long run endogeneity. e orizontal axis is te orizon. e sample size is = 25 observations. endogenous, ^/s.e. ρ=-.6 ρ= ρ=.6 ρ=.95 ρ= θ= θ= θ= θ= Figure 4: e figure reports p-values of te Kolmogorov-Smirnov test tat te distributions of standardized are te same in te preditive regression models wit misspefiied and orretly speified error dynamis wit long run endogeneity. e orizontal axis is te orizon. e sample size is = 25 observations. 16

19 a sample of = 25 observations under te assumption tat ε t is iid and normal. 7 Figures 5-8 report te rejetion probabilities of four statistis: t is obtained as as simple t-test were te DGP sows no serial orrelation in te errors, t, HAC is omputed wit a New-West HAC orretion in a DGP wit no serial orrelation, t is te statisti were ε t follows an MA1 wit parameter θ were θ =-.4 in Figures 5 and 6, θ =-.9 in Figures 7 and 8 and t HAC is te statisti wit Newey-West HAC orretion were ε t MA 1. In all DGPs onsidered te long run variane of ε t is 1 + θ 2, and we onsider bot te exogenous ase Corrε t, ɛ t = in Figures 5 and 7 and te presene of endogeneity Corrε t, ɛ t = 1/ 2 in Figures 6 and 8. e figures sow tat misspeifying te dynamis of ε t yields very low loal power for te standard t statisti lose to te null = at all orizons wen Corrε t, ɛ t =. In te exogenous ase, t HAC is sligtly undersized but sows better loal power tan t. As te orizon grows toug, HAC orretions lower te power, weter or not ε t is serially orrelated. By ontrast, standard t test do not suffer from tis upper limit and te power tends to unity as gets larger. Hene a ombined test tat rejets if eiter t or t HAC rejets will yield better loal and global power at all orizons. Wen θ =-.9 so te degree of misspeifiation is large, te loal power remains low toug. Similar results old for te endogenous ase were Corrε t, ɛ t = 1/ 2. e main differene is tat bot t and t HAC are loally biased and skewed at low. Bot are unreliable ere wen = 1 t HAC beome very liberal. Overall, our simulations sow tat te robustness of long orizon projetions to dynami misspeifiation advoates te use of te non HAC orreted statisti. o ensure better power, tis statisti sould be ombined wit its HAC version wi te empirial literature as usually onsidered: te ombined test rejets ours if eiter statisti does. 5 Conlusions In tis paper, we ave studied te properties of iterated and diret multi-step foreasts in te presene of model misspeifiation and non-stationarity bot stoasti and deterministi trends. We ave sown tat in tis framework, most general random walk estimation results apply wen standard Brownian motions are replaed wit trending Ornstein-Ulenbek proesses. is allowed us to araterize te non-linear patterns exibited by bot estimators and foreasts. In partiular, by letting te foreast orizon grow wit te sample size, we were able to sow ow mu IMS and DMS differ in terms of long range foreasting. A Monte Carlo simulation illustrated te analytial results tat were derived from te weak trend framework. Namely, tat DMS is exibits robustness to dynami misspeifiation at intermediate orizons, and, tese an be possibly very sort in finite samples. 7 is ritial value is unobtainable in pratie sine we ompute it under a known ρ so tis onstitutes an unfeasible bound were we do not need to resort to te orretions onsidered in te literature, e.g., Bonferroni as in Rossi 25 or te IVX of Pillips and Magdalinos 29 and Kostakis et al

20 exogenous ase, θ=-.4 Pr. Rej. t 1. =1 t, HAC t t HAC 1. = =1 1. = =3 1. = Figure 5: e figure reports te rejetion probabilities of four test statistis t, t,hac, t and t HAC for te null tat =. e sample size is = 25 observations, θ =.4 and tere is no endogeneity. endogenous 1. =1 ase, θ=-.4 Pr. Rej. t t, HAC t t HAC 1. = =1 1. = =3 1. = Figure 6: e figure reports te rejetion probabilities of four test statistis t, t,hac, t and t HAC for te null tat =. e sample size is = 25 observations, θ =.4 and tere is ontemporaneous endogeneity: Corrε t, ɛ t = 1/ 2. 18

21 exogenous 1. =1 ase, θ= =2 Pr. Rej. t t, HAC t t HAC =1 1. = =3 1. = Figure 7: e figure reports te rejetion probabilities of four test statistis t, t,hac, t and t HAC for te null tat =. e sample size is = 25 observations, θ =.9 and tere is no endogeneity. endogenous 1. =1 ase, θ=-.9 Pr. Rej. t t, HAC t t HAC 1. = =1 1. = = = Figure 8: e figure reports te rejetion probabilities of four test statistis t, t,hac, t and t HAC for te null tat =. e sample size is = 25 observations, θ =.9 and tere is ontemporaneous endogeneity: Corrε t, ɛ t = 1/ 2. 19

22 e reommendations tat we were able to derive are as follow. A foreaster wo is onfident tat er model is well-speified ougt to use iterated multi-step foreasts wen te orizon is small ompared to te sample size. If se must obtain long orizon foreasts using te available data, se sould ten resort to DMS. By ontrast, sould se suspet tat er model migt be misspeified, ten DMS ougt to be used at all orizons. e Diret Multi-Step Foreasting framework as also been sow to be useful for te analysis of preditive regressions as found in te literature. It follows tat long-orizon regressions an be understood to work well wen te model is misspeified for te serial orrelation of te regression errors. Using simple simulations, we were able to sow tat, at intermediate or long orizons, a ombination of te HAC test often onsidered in te empirial literature wit te non-hac version of te statisti aieves better global power tan eiter separately. e literature as also onsidered alternative test, optimal under Gaussianity in te ase of Jansson and Moreira, 26 or, e.g., Campbell and Yogo, 25 or finite sample distributional adjustments MCloskey, 212. Altoug we do not expliitly study tem ere, our teoretial analysis seem to indiate tat similar results are likely to old. Referenes Andrews, D. W Laws of large numbers for dependent non-identially distributed random variables. Eonometri teory 4 3, Banerjee, A., D. F. Hendry, and G. E. Mizon e eonometri analysis of eonomi poliy. Oxford Bulletin of Eonomis and Statistis 58, Boudouk, J., M. Riardson, and R. F. Witelaw 28. e myt of long-orizon preditability. e Review of Finanial Studies 21 4, Busetti, F. and A. Harvey 28. esting for trend. Eonometri eory 24 1, Campbell, J. and R. Siller e dividend-prie ratio and expetations of future dividends and disount fators. Review of Finanial Studies 1, Campbell, J. Y. and M. Yogo 26. Finanial Eonomis 81 1, Effiient tests of stok return preditability. Journal of Carriero, A.,. E. Clark, and M. Marellino 215. Bayesian vars: speifiation oies and foreast auray. Journal of Applied Eonometris 3 1, Cevillon, G Multistep foreasting in te presene of loation sifts. International Journal of Foreasting 32, Cevillon, G. and D. F. Hendry 25. Non-parametri diret multi-step estimation for foreasting eonomi proesses. International Journal of Foreasting 21, Clements, M. P. and D. F. Hendry Multi-step estimation for foreasting. Oxford Bulletin of Eonomis and Statistis 58,

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25 Appendies A Proof of Proposition 1 A.1 Sort orizon For r, 1, we write te series as te sum of a moving average and a deterministi omponent: r 1 1/2 y r = 1/2 e φ r / y + e iφ/ ψ 1 + 1/2 r 1 e iφ/ ɛ r i. Hene, i= 1/2 y r = e φ r / y + ψf φ r 1 + O 1 + 1/2 r e r iφ/ ɛ i, were 1/2 r e r iφ/ ɛ i J φ r Pillips Proof of a follows. Now, we write te statisti b as a funtional on D, 1. We first square 1/2 y t : 1 y 2 t = 1 τ + ρ y t + w,t 2 i= Hene 2 = 2 e iφ/ ψ e 2φ/ yt w,t /2 e i+φ/ ψy t,. i= i= 1 e 2 1 e φ/ y t, w,t. = 1 yt, 2 e 2φ/ yt, 2 2 φ/ 2 ψ 2 1 e φ/ 1 w,t 2 2 3/2 φ/ 1 eφ/ e ψy 1 e φ/ t, 2 3/2 ψw,t. We notie tat, summing over t, 1 1 ene, as 1 yt, 2 e 2φ/ yt, 2 = 1 yt, 2 e 2φ/ yt, 2 = t= +1 t= +1 i= yt, 2 1 yt, 2 e 2φ/ 1 1 2φ i+2 i + 2! i yt, 2 e 2φ/ yt, 2 {K ψ,φ 1} 2 2φ Colleting te elements we find: 1 y t, w,t { {K ψ,φ 1} 2 2φ 2 = { {K ψ,φ 1} t= y 2 t, 2φ 2 1 y 2 t,, K ψ,φ r 2 dr 2 yt, 2 1 K ψ,φ r 2 dr 1 σ 2 w 2ψ y 2 t, y 2 t, } K ψ,φ r dr ψkψ,φ + φk 2 ψ,φ r dr 1 σ 2 w }. A.1 23

26 Now, using Itô s lemma: dk 2 ψ,φ r = 2K ψ,φ r ψ + φk ψ,φ r + σ 2 dr + 2σK ψ,φ r dw r, {K ψ,φ 1} 2 = σ ψkψ,φ r + φkψ,φ 2 r dr + 2σ K ψ,φ r dw r. A.2 A.3 wene te result, using A.1 and te definition of σ 2 w. A.2 Long Horizon Preliminary results Item a is lear using te Funtional Central Limit teorem FCL and te Continuous Mapping teorem CM respetively. As regard b, we first derive te asymptoti distribution of sample moments of te multi-step residuals w,t tis onstitutes te proof of. ey follow an MA 1 w,t = ρ i ɛ t i = i= t j=t +1 e t jφ/ ɛ j, wi, using U r = 1/2 r t=1 ɛ t, an be rewritten so tat we let appear a stoasti integral: 1/2 w, r = = r j= r +1 r j/ j= r +1 r σ r e φ r j/ j/ j 1/ j 1/ du s e φ r / s du s = r r e φr s dw s = σj φ r σe φ J φ r. We reognize te quasi-differene of an Ornstein-Ulenbek proess: e φ r / s du s 1/2 w, r δ φj φ r = δ J φ r φf φ J φ r. A.4 Using te ontinuous mapping teorem, we obtain te limit distributions of empirial moments of 1/2 w, r, first te sample mean: 3/2 w,t = 3/2 j= w, r 3/2 w,j σ j= δ φj φ r dr, A.5 and te sum of squares: 2 w 2,j σ δ φ J φ r 2 dr. A.6 j= A useful lemma 24

27 Lemma 12 Using te definition of δ θ and K ψ,φ r from te main text, ten for nonzero φ Proof. Develop δ θk ψ,φ r 2 dr = θ Kψ,φ 2 r dr θ 2 Kψ,φ 2 r dr φ K ψ,φ 1 δθk ψ,φ 1 θ 2 1 σ2 φ + θ ψ {δ φ θk ψ,φ r + 1 θ K ψ,φ r } dr + σθ δ φ θk ψ,φ r dw r + K ψ,φ r d δθw r δ θk ψ,φ r 2 = K 2 ψ,φ r + θ 2 K 2 ψ,φ r 2θ K ψ,φ r K ψ,φ r = K 2 ψ,φ r θ 2 K 2 ψ,φ r 2θ δ θk ψ,φ r K ψ,φ r A.7 Wen taking te integral over, 1 wit respet to r, we reognize te sum of δ2 θ K2 ψ,φ r dr and of 2θ δ θk r K r dr. We analyze tem in turn. First, Expression A.2 implies tat θ 2 dk 2 ψ,φ r = 2θ 2 K ψ,φ r ψ + φk ψ,φ r + θ 2 σ 2 dr+2σθ 2 K ψ,φ r dw r ene d Kψ,φ 2 r θ 2 Kψ,φ 2 r = 2 ψ { K ψ,φ r θ 2 K ψ,φ r } + φ { Kψ,φ 2 r θ 2 Kψ,φ 2 r } 2 dr + σ 1 θ 2 dr + 2σ K ψ,φ r dw r θ 2 K ψ,φ r dw r. Integrating over, 1 erefore 2φ d K 2 ψ,φ r θ 2 K 2 ψ,φ r = 1 θ 2 1 σ 2 + 2ψ + 2φ { K 2 ψ,φ r θ 2 K 2 ψ,φ r } dr = 2φ { K 2 ψ,φ r θ 2 K 2 ψ,φ r } dr + 2σ K ψ,φ r dw r θ 2 δ 2 θ K 2 ψ,φ r dr A.8 { Kψ,φ r θ 2 K ψ,φ r } dr K ψ,φ r dw r. = K 2 ψ,φ 1 θ 2 K 2 ψ,φ 1 K 2 ψ,φ 1 θ 2 1 σ 2 2ψ { Kψ,φ r θ 2 K ψ,φ r } dr 2σ K ψ,φ r dw r θ 2 K ψ,φ r dw r. 25

28 Now, for δ θ K ψ,φ r K ψ,φ r dr, using te formula for stoasti integration by parts, if : d K ψ,φ r K ψ,φ r = ψ {K ψ,φ r + K ψ,φ r } + 2φK ψ,φ r K ψ,φ r dr sine dw r and dw r are independent. + σk ψ,φ r dw r + σk ψ,φ r dw r, Combining A.2 and te previous expression, te differene d K ψ,φ r δ θk ψ,φ r = d K ψ,φ r K ψ,φ r θ dk 2 ψ,φ r is ten equal to { ψ {δ θ K ψ,φ r + 1 θ K ψ,φ r } + 2φK ψ,φ r δ θk ψ,φ r σ 2 θ } dr + σ {δ θk ψ,φ r dw r + K ψ,φ r d δ θw r}. We ten re-express 2φK ψ,φ r δ θ K ψ,φ r dr as d K ψ,φ r K ψ,φ r θ dk 2 ψ,φ r { ψ {δ θk ψ,φ r + 1 θ K ψ,φ r } σ 2 θ } dr σ {δ θk ψ,φ r dw r + K ψ,φ r dδ θw r}. e expression for 2φ δ θ K ψ,φ r K ψ,φ r dr is terefore K ψ,φ 1 K ψ,φ 1 θ K 2 ψ,φ 1 σ {δ θk ψ,φ r dw r + K ψ,φ r dδ θw r} and te result follows using A.7. Proof of b. ψ {δ θk ψ,φ r + 1 θ K ψ,φ r } dr σ 2 θ 1 We an now move to finding te expression for b. For nonzero φ, we square y t, and express it as te sum τ, + ρ, y t, + w,t 2, or: 1 e yt, 2 = ψ 2 φ/ 1 e φ/ 1 e φ/ + 2ψ 1 e φ/ e 2φ/ y 2 t, + w 2,t 1/2 e φ/ y t, + 2e φ/ y t, w,t + 2ψ Summing over t ranging from to and rearranging yields 1 e φ/ 1 e φ/ 1/2 w,t. 2e φ/ 2 y t, w,t = 2 yt, 2 e 2φ/ 2 yt, 2 2 ψ eφ/ 2 w 2 1 e φ/,t 2ψe φ/ 1 1 eφ/ 1 e φ/ 1 1 eφ/ 2ψ 1 e φ/ 3/2 y t, 3/2 w,t. A.9 26