Tests of Equal Forecast Accuracy and Encompassing for Nested Models. Todd E. Clark and Michael W. McCracken * April 1999 (current draft: April 7)

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1 Prelmnary and ncomplee Tess of Equal Forecas Accuracy and Encompassng for Nesed Models Todd E. Clark and Mchael W. McCracken Aprl 999 (curren draf: Aprl 7) Absrac Ths paper examnes he asympoc and fne-sample properes of ou-of-sample ess for equal accuracy and encompassng as appled o nesed models. Wh nesed models, hese ess can be vewed as Granger causaly ess. Appled o nesed models, however, he sandard asympoc crcal values for many ess of equal accuracy and encompassng are nvald. Sascs such as hose proposed by Debold and Marano (995) and Harvey, e. al. (998) fal o converge o he sandard normal dsrbuon when he models are nesed raher han nonnesed. Buldng on McCracken s (999) resuls for equal accuracy ess, hs paper derves he asympoc dsrbuons for a se of sandard encompassng ess and one new encompassng es. Numercal smulaons are used o generae he approprae asympoc crcal values. Mone Carlo smulaons are hen used o evaluae he sze and power of a baery of equal forecas accuracy and encompassng ess, as well as sandard F-ess of causaly. In hese expermens, forecass from an esmaed VAR model are compared o hose from a null esmaed AR model. The smulaon resuls ndcae ha McCracken s ou-of-sample F-ype es of equal accuracy and he encompassng es proposed n hs paper can be more powerful han sandard F-ess of causaly. The Mone Carlo smulaons also show ha usng nvald asympoc crcal values can produce msleadng nferences. JEL Nos.: C53, C, C5 Clark: Economc Research Dep., Federal Reserve Bank of Kansas Cy, Kansas Cy, MO 6498, clark@frbkc.org. McCracken: Dep. of Economcs, Lousana Sae Unversy, 07 CEBA Buldng, Baon Rouge, LA 70803, mmccrac@unx.sncc.lsu.edu. The research asssance of Hlary Croke and he helpful commens of Luz Klan and Ken Wes are graefully acknowledged. The vews expressed heren are solely hose of he auhors and do no necessarly reflec he vews of he Federal Reserve Bank of Kansas Cy or he Federal Reserve Sysem.

2 . INTRODUCTION As evden from recen sudes such as Amano and van Norden (995), Blomberg and Hess (997), Bram and Ludvgson (998), Krueger and Kuner (996), and Mark (995), neres ofen les n examnng wheher one varable helps predc anoher, boh n-sample or ou-of-sample. The sandard n-sample merc s a smple Granger causaly es. Ou-of-sample predcve ably s usually gauged by frs consrucng forecass from models ha nclude and exclude he varable ha may have predcve capacy and hen esng for equal accuracy or encompassng. Typcally, here s concern ha n-sample Granger causaly ess may lead o overfng of he model. Ou-of-sample forecas comparson s wdely vewed as a more srngen es of he relaonshp beween he varables. Moreover, Ashley, Granger, and Schmalensee (980) argue ha s more n he spr of her reasonable defnon of Granger causaly o employ pos-sample forecas ess han o employ he sandard n-sample ess of causaly. Accordng o Ashley, e. al., f forecass of y from a VAR n x and y are superor o forecass from an AR model for y, hen x carres nformaon abou y and hence x causes y. Ths paper examnes he ably of dfferen pos-sample forecas ess o deermne wheher one varable has predcve ably for anoher. Parcularly, hs paper examnes he asympoc and fne-sample properes of ess for equal accuracy and encompassng appled o nesed models. Used wh nesed forecass from models such as a VAR vs. an AR, he ess can be vewed as Granger causaly ess. However, many of he sandard ess of equal accuracy and encompassng are desgned for forecass from non-nesed, raher han nesed, models. Many of he sandard es sascs such as he Debold and Marano (995) equal accuracy and he Harvey, Leybourne, and Newbold (998) encompassng sascs fal o converge o he Debold and Marano (995) also sugges usng ou-of-sample forecas ess o examne Granger causaly.

3 sandard normal dsrbuon when he models are nesed raher han non-nesed. Therefore, he sandard asympoc crcal values are nvald wh nesed models. Ths sudy s analyss of he use of equal accuracy and encompassng ess for nesed forecass complemens he non-nesed forecas analyses of, among ohers, Corrad, Swanson, and Olve (998), Debold and Marano (995), Harvey, Leybourne, and Newbold (997, 998), Wes (996), Wes and McCracken (998) and McCracken (998). In anoher relaed analyss, Swanson, Ozyldrm, and Psu (996) examne he fne-sample performance prncpally, he sze of dfferen Granger causaly ess and Debold-Marano equal accuracy ess wh boh saonary and non-saonary daa. The equal accuracy ess hey consder, however, are all compared agans sandard asympoc crcal values ha are nvald because he models are nesed. The same problem apples o he Mone Carlo resuls of Corrad, e. al. (998) on how Debold and Marano ess perform when appled o models wh conegrang relaonshps. Buldng on McCracken s (999) resuls for equal accuracy ess, hs paper frs derves he asympoc dsrbuons for a se of sandard encompassng ess and one new encompassng es. The sandard encompassng ess for whch hs paper derves asympoc dsrbuons are he Harvey, e. al. (998) and Ercsson (99) sascs. The se of sandard sascs also ncludes he Chong and Hendry (986) es, whch remans asympocally normal when appled o nesed raher han non-nesed forecass. The new es proposed below s a varan of he Harvey, e. al. sasc. As n Corrad, e. al. (998), McCracken (999), Wes (996), and Wes and McCracken (998), he derved dsrbuons of he ess explcly accoun for he uncerany nroduced by model esmaon. In order o faclae he use of he lmng dsrbuons derved here, asympocally vald crcal values are generaed numercally and Mos of he oher avalable ess of forecas accuracy or encompassng such as he Mzrach (99) and Granger and Newbold (977) ess of equal accuracy are also asympocally nvald when appled o nesed forecass.

4 repored n appendx ables. The equal accuracy ess nclude an ou-of-sample F-ype es of equal mean squared error (MSE) developed n McCracken (999) and he Debold and Marano (995) es of equal MSE, sascs for whch McCracken develops he correc asympoc dsrbuons and provdes crcal values. In order o evaluae he fne-sample sze and sze-adused power of hese ess we conduc a seres of Mone Carlo smulaons based on VAR daa-generang processes. For comparson, he se of ess consdered also ncludes sandard F-ess of Granger causaly. In addon, n order o evaluae he exen o whch usng nvald crcal values can produce msleadng nferences, resuls are presened for Debold and Marano (995), Harvey, e. al. (998), and Ercsson (99) sascs compared o he dsrbuons ha would be approprae f he forecass were from non-nesed models. To furher llusrae how he dfferen ess perform n praccal sengs, he baery of ess s appled o deermnng wheher he unemploymen rae has predcve power for nflaon n quarerly U.S. daa. Resuls summary. Asympocs. Fne-sample. Mone Carlo smulaons show ha usng nvald asympoc crcal values can produce msleadng nferences n small samples. The smulaons also ndcae ha ou-of-sample F-ype and encompassng ess can be more powerful han sandard, n-sample F-ess of causaly. The remander of he paper wll proceed as follows. Secon wo nroduces he noaon and general envronmen under whch he forecass are generaed and he ess of forecas accuracy and encompassng are consruced. Secon hree, and s subsecons, nroduce he es sascs consdered and provde he asympoc resuls under he null. In secon four we presen a collecon of Mone Carlo expermens desgned o deermne he fne-sample sze and power properes of he es sascs. Secon fve conans an emprcal applcaon of he ess o he 3

5 problem of deermnng wheher he unemploymen rae has predcve power for nflaon n quarerly U.S. daa. Secon sx concludes. All proofs are conaned whn he Appendx.. General Envronmen In order o presen he ess consdered we frs provde some general noaon, descrbe he forecasng schemes, and presen he assumpons under whch he asympoc resuls are derved. The sample of daa {y T, x } = s dvded no n-sample and ou-of-sample porons. The n-sample observaons span o R. Leng P denoe he number of -sep ahead predcons consruced, he ou-of-sample observaons span R hrough R P. The oal number of observaons n he sample s hen R P = T. The larges number of observaons used o esmae he model under he forecas schemes consdered s T = R P. The scalar varable o be predced s y, = R,,T. Forecass of y are generaed usng paramerc models g (x, β ) g, ( β ) denoed by = and, each of whch s esmaed. Model s he unresrced model, whch ness he resrced model. Under he null hypohess, model ncludes k excess parameers. Whou loss of generaly le = β k, 0 k ) ( β (k k = k ) such ha for all, g, ( β ) = g, ( β ). Under he alernave hypohess, he k resrcons are no rue, and model s correc. Noe ha whle models and ake he form of AR and VAR models n he Mone Carlo analyss, he asympoc resuls perm he use of nonlnear models. Followng Wes and McCracken (998), hree forecas schemes are consdered. Under he recursve scheme, each models parameers, β =,, are esmaed wh added daa as forecasng moves forward. The frs predcon, g,r (ˆ β, R ), s creaed usng model parameer 4

6 esmaes ˆβ esmaed usng daa from o R, he second predcon ), R g,r (ˆ β,r s creaed ˆ usng model parameer esmaes β,r esmaed usng daa from o R, ec. In general, for = R,,T, he predcon of y, g, (ˆ β, ), from me s creaed usng model parameer esmaes ˆβ, esmaed usng daa from o. Under he rollng forecas scheme, he model s esmaed usng only he mos recen R observaons. The frs rollng predcon, g,r (ˆ β, R ), s creaed usng model parameer esmaes ˆβ esmaed usng daa from o R, he second predcon ), R g,r (ˆ β,r s creaed ˆ usng model parameer esmaes β,r esmaed usng daa from o R, ec. In general, for = R,,T, he predcon of y, g, (ˆ β, ), from me s creaed usng model parameer esmaes ˆβ, esmaed usng daa from R o. Noe ha under he rollng scheme he parameer esmaes ˆβ, should also be subscrped by R n order o reflec he sze of he sample wndow. To reduce noaon we leave ha subscrp mplc. Under he fxed scheme, all forecass are generaed usng models esmaed wh daa from o R. Hence for each predcon of y, g, (ˆ β, ) = g, (ˆ β, R ), from me = R,,T, he predcon s creaed usng he same model parameer esmae ˆβ, = daa from o R. As was he case for he rollng scheme, he parameer esmaes ˆβ, R esmaed usng ˆβ, under he fxed scheme should also be subscrped by R o reflec he sample wndow. To reduce noaon we also leave hs subscrp mplc. For each of he hree forecasng schemes, he -sep ahead forecas errors are = y g ( βˆ ) and u = y g ( βˆ ) for models and, respecvely. Usng uˆ,,, ˆ,,, 5

7 he wo sequences of P forecas errors he ou-of-sample ess of forecas accuracy and encompassng are consruced. In all cases he ou-of-sample sascs rely on sums of funcons of hese forecas errors. To smplfy noaon, for any varable z we wll le z denoe he T summaon = P T = u R, ˆ = P uˆ,. R z. For example, he mean squared error (MSE) for model s MSE β f, g, Before geng o he assumpons some fnal noaon s needed. Le g ( β ) ( β ), q, ( β ) = g,, β ( β )g,, β ( β ) (y g, ( β )) g, ( β ) β β,, β = f =,, ( β ) and f = f, for any funcon f, h, ( β ) = ( y g, ( β ))g,, β( β ), B = (Eq ),, W(s) s a (k ) vecor sandard Brownan Moon, and for any (m n) marx A wh column vecors a le vec(a) denoe he (mn ) vecor [ a,a,...,a. n ] Gven he defnons and he hree forecasng schemes descrbed above, he followng fve assumpons are hose used o derve he lmng dsrbuons of encompassng ess presened n Theorems 3.4, 3.5, and 3.6. The assumpons are also suffcen for he resuls of McCracken (999) when MSE s he measure of predcve ably. These assumpons are no nended o be necessary and suffcen, only suffcen. All proofs can be found whn he Appendx. Assumpon : The parameer esmaes where for β &, on he lne beween ˆβ, and ˆβ,, =,, = R,,T, sasfy β, B ()H () equals β ˆ β = B ()H (), 6

8 ( q ( β & = )) ( = h ), (R q ( )) ( h ) R β & = = R and,,, R R (R q ( β & = )) (R = h ) respecvely for he recursve, rollng and fxed schemes.,,,, Ths frs assumpon provdes us wh one prmary pece of nformaon. Analycally ells us ha he parameers are esmaed by OLS, NLLS, or maxmum lkelhood under normaly assumpons. In he case where a VAR s beng used, he sysem mus be exacly denfed and esmaed by mulvarae OLS. Ths ype of resrcon s mposed o nsure ha he sascs n Theorems are pvoal. As n McCracken (999), achevng a lmng dsrbuon ha does no depend upon he daa-generang process requres ha he loss funcon used o esmae he parameers be closely relaed o he loss funcon used o measure predcve ably. Each of he sascs n Theorems s n one way or anoher esng wheher he wo models have equal mean square errors. In order hen o acheve a pvoal sasc he parameers mus be esmaed usng mean square error as he loss funcon. Alhough hs assumpon s resrcve n how he parameers are esmaed, oherwse does no place any resrcons on he ype of model. Sngle and mulple equaon models as well as lnear and nonlnear models are permed.,, Assumpon : For =,, (a) β Θ, Θ compac, (b) E[y g, ( β s unquely mnmzed )] a β Θ wh Eq, nonsngular, (c) In some open neghborhood N around β, and wh probably one [ y β )] g, ( s wce connuously dfferenable, (d) In he open neghborhood N, and for all here exss a posve consan ϕ and a posve random varable m such ha ϕ q, ( β ) q, ( β ) m β β wh Em < and ϕ <. 7

9 Mos of Assumpon s mposed n order o nsure ha he parameers are denfed and are conssenly esmaed. I s drecly comparable o Theorem (.) of Newey and McFadden (994). The subsanve componens of hs assumpon are ha he predcve funcon, g, ( β ), s he condonal mean funcon and ha he condonal mean funcon s wce connuously dfferenable. Assumpon 3: Le U [h, vec(h h σ B ), vec(q B ) ]. (a) EU = 0, (b) U s unformly L 8 bounded, (c) For some 8 > d >, U s srong mxng wh coeffcens of sze 8d T, (d) lm T E = U U = Ω <. 8 d T Assumpon 4: (a) Eh h = Eq σ σ B, (b) E(h h,q, =,,...) = 0. Boh Assumpons 3 and 4 largely conss of echncal condons suffcen for he applcaon of an nvarance prncple. Moreover hey are suffcen for on weak convergence of paral sums and averages of hese paral sums o Brownan Moon and negrals of hese Brownan Moon. Assumpon 3 s drecly comparable o he assumpons n Hansen (99) and hence we are able o apply hs Theorems (.) and (3.). The reasons for mposng Assumpon 4 are much he same as Assumpon. In order o nsure ha he lmng dsrbuon does no depend upon he underlyng daa generang process we mus mpose some exra condons. Here we essenally requre ha he dsurbances form a condonally homoskedasc marngale dfference sequence. 8

10 Assumpon 5: lm P / R = π, 0 < π <, T λ ( π). Ths fnal assumpon nroduces he means by whch he asympocs are acheved. As n Hoffman and Pagan (989), Wes (996), and Whe (998) he lmng dsrbuon resuls are derved by mposng a slghly sronger condon han smply ha he sample sze, T, becomes arbrarly large. Here we mpose he addonal condon ha boh he number of n-sample (R) and ou-of-sample (P) observaons also become arbrarly large a he same rae. In hs way we nsure ha he parameers esmaed n-sample and ceran ou-of-sample averages are boh conssen esmaors of her populaon level analogs. I should be noed ha he assumpon ha π s bounded from above and below s no rval. Ceranly n pracce P/R wll be bounded bu wheher s near zero or much larger could affec how well he asympoc approxmaon behaves n fne samples. If P/R s small hen he parameers may be well esmaed bu, for example, he ou-of-sample MSE wll be esmaed by oo few observaons for he emprcal MSE o form a srong esmae of he populaon MSE. If P/R s large hen we canno expec he parameers o be well approxmaed, especally under he fxed scheme, and hus regardless of he ou-of-sample sze he emprcal MSE may form a poor esmae of he populaon MSE. Hence when choosng how o spl he sample no n-sample and ou-of-sample porons one should consder choosng a spl ha leaves a szable number of observaons n each of he n-sample and ou-of-sample porons. Unless oherwse noed, he noaon and assumpons presened n hs secon hold hroughou he remander of he paper. 3. TESTS 9

11 Whle Ashley, e. al. (980) specfcally advocae usng ess of equal forecas accuracy o examne causaly, gven her defnon of causaly, any es desgned o examne wheher x carres nformaon abou y could reasonably be used. Accordngly, hs paper consders he ably of smple Granger causaly ess, equal forecas accuracy ess, and forecas encompassng ess o deermne wheher one varable has predcve power for anoher. Snce a large number of ess for equal accuracy and encompassng already exs, for racably he se examned s lmed based on consderaons of compuaonal smplcy and performance n he non-nesed nvesgaons of Clark (999), Debold and Marano (995), and Harvey, Leybourne, and Newbold (997, 998). The se of ess ncludes: smple Granger causaly sascs; McCracken s (999) ou-of-sample F-ype es for equal MSE; Debold and Marano s sasc for equal MSE; he Harvey, e. al. (998) encompassng es; he Ercsson (99) encompassng sasc; a modfed, asympocally vald verson of he Harvey, e. al. es developed below; and he Chong and Hendry (986) encompassng sasc. In he resuls below, he ess are appled o -sep ahead forecass. When mul-sep forecass from nesed models are used, he asympoc dsrbuons of he ess appear o depend on he parameers of he daa-generang process. For praccal purposes, such dependence elmnaes he possbly of usng asympoc approxmaons o es for equal accuracy or encompassng. Lukepohl and Burda (997) noe smlar dffcules assocaed wh n-sample ess nvolvng mul-sep forecass. Our belef s ha researchers comforable wh assumng lnear models should be adequaely served by ess based on -sep ahead forecass. Wh lnear models, mul-sep forecass are smply lnear combnaons of -sep ahead forecass. Hence here does no appear o be any reason o expec ess based on mul-sep forecass o be beer a deermnng predcve power han ess based on -sep ahead forecass. 0

12 3. Smple Granger Causaly (GC) Tess The emphass of hs paper s on usng ex-ane forecass, raher han ex-pos predcons, o es for forecas accuracy and encompassng. Even so, we nclude smple F-ype Granger causaly ess n he Mone Carlo smulaons. We do so because hey are he mos commonly used sascs for esng for causaly. We consruc hese sascs usng boh n-sample and ou-of-sample daa, usng R observaons n he former case and P observaons n he laer. In he resuls of secon 4, he ess are compued as smple F-sascs for excluson resrcons. 3 When lag lenghs are se usng daa-based procedures, he ou-of-sample GC ess rely on he lag order deermned usng he n-sample daa. Whle sandard GC ess are rarely appled o ou-of-sample daa, hey are no less vald for he purpose of esng causaly n ou-ofsample daa han are forecas accuracy or encompassng ess. Lke ou-of-sample accuracy and encompassng ess, smple ou-of-sample GC ess may be less prone o spurous resuls due o overfng han are n-sample causaly ess. 3. The Ou-of-Sample F (OOS F) Tes McCracken (999) develops an ou-of-sample F-ype es of equal MSE, gven by OOS F MSE MSE = P MSE P = P uˆ P, P uˆ, uˆ,. () Ths sasc s comparable o he smple F-es form of he sandard n-sample GC es and offers he advanage of beng parcularly smple o compue f forecas summary sascs are already

13 avalable. Usng assumpons broadly smlar o hose used n hs paper, McCracken shows ha he OOS F sasc converges o a funcon of sochasc negrals of quadracs of Brownan moon. The lmng dsrbuon under he null, whch vares wh he forecasng scheme, s a funcon of he rao of pos-sample o n-sample observaons, π, and excess parameers, k, n model. In he resuls of secon 4, he es sasc s compared agans asympocally vald crcal values abulaed by McCracken. Snce he models are nesed, he null hypohess s MSE MSE, and he alernave s MSE MSE >. The alernave s one-sded because, f he resrcons mposed on model are no rue, here s no reason o expec forecass from model o be superor o hose from model. 3.3 The Debold-Marano (DM) Tes Defne = uˆ, uˆ, d and P d = MSE MSE. d = The Debold and Marano (995) es for equal MSE s formed as DM = MSE MSE d d = = vˆ ar( MSE MSE ) vˆar( d ) P ( d ) d. () Whle he DM sasc s asympocally sandard normal when appled o non-nesed forecass (see Debold and Marano (995) and Wes (996)), he asympoc dsrbuon s non-normal when he forecass are nesed under he null hypohess. The roo of he problem s ha, under he null, g, ( β ) = g, ( β ) and hus boh u, = y g, ( β ) u and u, = y g, ( β ) = y g, ( β ) u. Hence, a 3 Compung he ess as Wald sascs and comparng hem agans he ch-square dsrbuon would make he ess robus o non-normally dsrbued daa. However, usng he Wald sascs produces resuls very smlar o hose

14 leas heurscally, asympocally he dfference n squared forecas errors s exacly 0, wh 0 varance. The usual DM es n whch he sasc s compared agans he sandard normal dsrbuon s herefore asympocally nvald. McCracken (999) shows ha, for forecass from nesed models, he DM es sasc converges o a funcon of sochasc negrals of quadracs of Brownan moon. Ths lmng dsrbuon depends on he forecasng scheme, π, and k. In he resuls of secon 4, he es sasc s compared agans asympocally vald crcal values abulaed by McCracken. Because models and are nesed raher han nonnesed, he alernave hypohess s one-sded nsead of wo-sded. Under he null, MSE MSE ; under he alernave, MSE > MSE. To evaluae how usng nvald asympoc crcal values would affec nference, resuls are also repored for a verson of he es comparng he DM sasc agans he P dsrbuon. Whle he DM sasc s asympocally sandard normal when he forecass are non-nesed, Harvey, e. al. (997) fnd ha comparng he DM sasc agans he P dsrbuon yelds beer small-sample properes. In he repored resuls, he nvald verson of he es (bu no he asympocally vald verson) also ncorporaes an adusmen, developed n Harvey, e. al (997), o correc for bas n he esmaed varance of d. Ths adusmen akes he form of mulplyng he es sasc () by ( P ) P. 3.4 The Harvey, e. al. (HLN) Tes Harvey, e. al. (998) use he basc mehodology of Debold and Marano (995) o develop a es of forecas encompassng drawn from he condonal effcency framework of repored, wh he dfference ha he emprcal sze s slghly hgher. 3

15 Granger and Newbold (973) and Nelson (97). 4 Condonal effcency s esed wh he OLS-based regresson uˆ, =,, λ ( uˆ uˆ ) η. (3) If λ = 0, forecas carres no useful nformaon no already n forecas, so forecas s condonally effcen. If λ > 0, forecass from model do carry nformaon no already n forecass from model. Harvey, e. al. (998) propose esng encompassng wh a -sasc for he covarance beween u ˆ, and u ˆ, uˆ, raher han wh a -sasc for he regresson coeffcen ˆλ. Le = uˆ, ( uˆ, uˆ, ) = uˆ, uˆ, uˆ, c and = P c. The Harvey, e. al. encompassng c es s formed as HLN = uˆ ˆ ˆ, P u, u, { }. ( uˆ uˆ uˆ ) c c P = = P vˆ ar( c) P ( c c) P,,, c (4) Under he null ha model forecas encompasses model, he covarance beween u, and u, u, wll be less han or equal o 0, whle under he alernave ha model conans added nformaon, he covarance should be posve. The es s one-sded when appled o eher nesed or non-nesed forecass. Whle he HLN sasc s asympocally sandard normal when appled o non-nesed forecass, he asympoc dsrbuon s non-normal when he forecass are nesed under he null. The acual lmng dsrbuon s provded n Theorem Condon effcency n urn draws from he forecas combnaon leraure, sared by Baes and Granger (969). The regresson (5) can be used o deermne he opmal combnng wegh. The lnear combnaon of forecass 4

16 χ Theorem 3.4: For HLN defned n (4), HLN d where χ 0. 5 equals (χ ) λ s W (s)dw(s) for he recursve scheme, λ {W() W( λ)} W( λ) for he fxed scheme, and χ equals λ λ { W(s) W(s λ)} dw(s) for he rollng scheme, λ s W (s)w(s)ds πλ W ( λ)w( λ ) for he recursve scheme, for he fxed scheme, λ λ { W(s) W(s λ)} {W(s) W(s λ)}ds for he rollng scheme. There are a couple hngs o noce abou Theorem 3.4. The frs s ha for each forecasng scheme he sasc s pvoal. Ths fac s no parcularly useful f asympoc crcal values, assocaed wh he lmng dsrbuons, are used o consruc asympocally vald ess. If he boosrap s used, as n Ashley (998), hen we know from Hall (99) ha he boosrap provdes refnemens o frs order asympocs and hence n fne samples may provde more accurae nference. Though he null lmng dsrbuons do no depend upon he daa generang process self, he dsrbuons are dependen upon wo parameers. The frs s he number of excess parameers k. I arses snce he vecor Brownan Moon, W(s), s (k ). The second parameer, π, also affecs he null lmng dsrbuon. I affecs he lmng dsrbuon n wo and wll have a smaller MSE han forecas unless he covarance beween ê, and ê, ê, and, equvalenly, he coeffcen λ, are 0. 5

17 ways. I drecly affecs he weghs on each of he componens of he sascs (recall ha λ = ( π) ). I also affecs he range of negraon on each of he sochasc negrals hrough λ. Snce he lmng dsrbuon of he HLN sasc s nonsandard (.e. neher normal nor ch-square) we provde asympocally vald crcal values n Tables A-A3. These were generaed numercally usng he lmng dsrbuon n Theorem 3.4 and hence can be consdered esmaes of he rue asympoc crcal values. The repored crcal values are he 90 h, 95 h and 99 h χ percenles of 5000 ndependen draws from he dsrbuon of 0. 5 (χ ) for a gven value of k and π. Generang hese draws proceeded as follows. Weghs ha depend upon π were esmaed n he obvous way usng π ˆ = P / R. The necessary k Brownan Moons were smulaed as random walks each usng an ndependen sequence of 0,000..d. N(0,T -0.5 ) ncremens. The negrals were emulaed by summng he relevan weghed quadracs of he random walks from he R s observaon o he T h observaon. The random number generaor was seeded so ha all k and π pars and all samplng schemes use he same 5000 draws of k sequences of 0,000..d. N(0,T -0.5 ) ncremens. A bref lsng of crcal values s provded n Tables A, A, and A3. Each of he ables corresponds o eher he recursve, rollng, or fxed forecasng scheme. Whn each able here are 330 crcal values. Each of hese correspond o one permuaon of hree parameers: k = {,, 3,, 9, 0}, π = {0., 0., 0.4,,.0,.,,.0} and nomnal sze of he es = {0.0, 0.05, 0.0}. Tables ha allow for larger values of boh k and π are avalable upon reques. In he resuls of secon 4, he HLN sasc s compared agans he asympocally vald crcal values abulaed n Tables A-A3. To evaluae how usng nvald asympoc crcal values would affec nference, resuls are also repored for a verson of he es comparng he 6

18 HLN sasc agans he P dsrbuon. For non-nesed models, Harvey, e. al. (998) fnd ha comparng he HLN sasc agans he P dsrbuon, raher han he sandard normal, yelds beer small-sample properes. In he repored resuls, he nvald verson of he es (bu no he asympocally vald verson) also ncorporaes an adusmen, developed n Harvey, e. al (997), o correc for bas n he esmaed varance of d. Ths adusmen akes he form of mulplyng he es sasc (4) by ( P ) P. 3.5 A New Encompassng (CM) Tes As dscussed below, Mone Carlo smulaons sugges ha he denomnaors of ess lke he DM sasc () and he HLN sasc (4) adversely affec he small-sample properes of he ess. The denomnaor of he HLN sasc, for example, s he sample varance of c (normalzed by P), whch s asympocally equal o 0. In parallel o he OOS F es, hs paper proposes a varan of he HLN sasc n whch c s scaled by he varance of one of he forecas errors raher han an esmae of he varance of c. Ths sasc, whch we wll refer o as he CM sasc, akes he form CM = P c MSE P uˆ, = P P uˆ, P uˆ, uˆ,. (5) The numeraor s he obec of neres n he HLN es he covarance beween u, and. The denomnaor, MSE, serves as a scale correcon. As was he case for he HLN u, u, sasc, he lmng dsrbuon s non-normal when he forecass are nesed under he null. The acual lmng dsrbuon s provded n Theorem

19 Theorem 3.5: For CM defned n (5) and χ defned n Theorem 3.4, CM χ. d Gven Theorem 3.4, hs resul s no surprsng. The sole dfference beween he HLN and CM sascs s he denomnaor. Hence we expec her lmng dsrbuons o be somewha relaed. As was he case for he HLN sasc, he lmng dsrbuon s pvoal and reles upon he parameers k and π. In he resuls of secon 4, he CM sasc s compared agans asympocally vald crcal values abulaed n Tables A4-A6. As was done for Tables A-A3, hese were generaed numercally usng he lmng dsrbuon n Theorem 3.5 and hence can be consdered esmaes of he rue asympoc crcal values. The numercal mehods used o consruc 5000 ndependen draws from he dsrbuon of χ were dencal o hose used o consruc Tables A-A3. Moreover he random number generaor was seeded so ha he same χ values were used n he consrucon of boh Tables A-A3 and A4-A6. Tables A4-A6 conan he same 330 permuaons of k, π and nomnal sze ha are used n Tables A-A3. Tables ha allow for larger values of boh k and π are avalable upon reques. 3.6 The Ercsson (ERIC) Tes Ercsson s (99) forecas-dfferenal encompassng es akes he same form as he condonal effcency regresson presened above: 5 uˆ, =,, λ ( uˆ uˆ ) η. (6) 5 As presened n Ercsson (99) and used by ohers, he es ofen s expressed as a regresson of he error from model on he dfference n forecass raher han forecas errors. Bu ha regresson s equvalen o (6), wh he approprae sgn change. 8

20 The es sasc s smply he -sasc for he OLS-based regresson coeffcen λˆ, whch can be expressed as ERIC P a / 0,T = (7) 0.5 [a,ta,t a 0,T ] where a ˆ ( ˆ ˆ 0, T = P u, u, u, ), a, T = P ( uˆ, uˆ, ) and a T = P uˆ,,. Under he null ha model forecas encompasses model, he covarance beween u, and u, u, wll be less han or equal o 0, whle under he alernave ha model conans added nformaon, he covarance should be posve. The es s one-sded when appled o eher nesed or non-nesed forecass. Once agan he ERIC sasc s asympocally sandard normal when appled o nonnesed forecass bu he asympoc dsrbuon s non-normal when he forecass are nesed under he null. The acual lmng dsrbuon s provded n Theorem 3.6. χ Theorem 3.6: For ERIC defned n (7) and 0. 5 (χ ) χ defned n Theorem 3.4, ERIC d (χ ) In Theorem 3.6 we fnd ha he ERIC and HLN sascs have he same lmng dsrbuon under he null. 6 Hence we can use Tables A-A3 o consruc asympocally vald ess of forecas encompassng when he ERIC sasc s used. I should be menoned however, ha hs does no mply ha he wo sascs wll have smlar fne sample properes. I s for hs reason ha we nclude boh he HLN and ERIC sascs n he Mone Carlo expermens of secon 4. 9

21 In he resuls of secon 4, he ERIC sasc s compared agans asympocally vald crcal values abulaed n Tables A-A3. To evaluae how usng nvald asympoc crcal values would affec nference, resuls are also repored for a verson of he es comparng he ERIC sasc agans he sandard normal dsrbuon. 3.7 The Chong-Hendry (CH) Tes Under he null ha he resrcons on model are correc, model forecas encompasses model. The Chong and Hendry (986) es of encompassng s formed as he -sasc on αˆ from he OLS-based regresson α ˆ, (8) uˆ, = g, ( β, ) n where g, (ˆ β, ) denoes he model forecas. 7 I follows from Wes and McCracken (998) ha he CH sasc s asympocally sandard normal even when he models consdered are nesed. Accordngly, he -sasc on αˆ s compared agans he sandard normal dsrbuon. As wh he DM, HLN, and ERIC sascs, o allow for non-normal daa, he esmaed varance of αˆ s robus o heeroskedascy. Snce he nesng of he models has no clear mplcaons for he sgn of αˆ, he null α = 0 s esed agans a wo-sded alernave. 4. MONTE CARLO RESULTS Resuls on he small-sample properes of he ess descrbed n secon 3 are generaed usng a bvarae VAR daa generang process, wh forecass of he varable of neres from an 6 There s a parallel o hs n McCracken (999). There s shown ha he DM sasc has he same lmng dsrbuon as he regresson-based es for equal MSE by Granger and Newbold (977). 7 For he basc models consdered below a modfed CH es, ha akes he form of he covarance beween he LHS and RHS of (8) dvded by an esmae of he sandard error of he covarance, has modesly beer sze properes bu 0

22 esmaed AR model (model ) compared o forecass from a VAR (model ). The presened resuls are based on daa generaed from he normal dsrbuon. The resuls are essenally unchanged when daa are generaed from a heaver-aled dsrbuon, suggesed by Debold and Marano (995), n whch one forecas error follows a 6 dsrbuon and he oher s a lnear combnaon of 6 varables. 4. Expermen Desgn In he presened resuls (currenly), daa are generaed usng one arfcal VAR() model and one emprcal VAR() model. The arfcal VAR() akes he form y x = 0.3y = 0.5x bx u x,, u y, (9-0) where y s he varable o be forecas and x s an auxlary varable. The error erms are ndependen sandard normal varables. To evaluae sze n fne samples, he coeffcen b s se a 0. To evaluae power, b s se a 0., 0., and 0.4. Smulaons based on VAR() models akng a comparable form, and smulaons based on he rvarae saonary VAR() and VAR(3) models of Swanson, e. al. (996), produced resuls n lne wh hose from he bvarae VAR(). Alernave models. I. Models ha generae larger sze bases for n-sample GC ess due o pure pre-es bas effecs (models wh us much rcher dynamcs?). II. Msspecfed models: (a) A VAR() n whch he rue model s an AR() n y and x(-) s srongly correlaed wh y(- ). (b) Forecass from bvarae VAR and AR when he rue model s rvarae. (c) Forecass from VAR and AR when he rue model s a VARMA. modesly lower power. Ths form of he es sasc parallels he Harvey, e. al. (998) modfcaon of he Ercsson

23 The emprcal VAR s f from quarerly daa on he change n core CPI nflaon and he change n he prme-age male unemploymen rae, where he change n core nflaon s he varable o be forecas. Whle he negraon orders of hese daa are admedly debaable, he use of changes s based on he resuls of un roo ess and produces auoregressve roos well whn he un crcle. Conssen wh he emprcal applcaon consdered n secon 5, he models are f exclusvely wh n-sample daa ha span 958:Q3-987:Q. 8 Under he null ha unemploymen does no affec nflaon, he esmaed model used n a sze expermen s (-) Infl =.04 Unemp = Infl var(u ) =.795, var(v.057 Infl.37 Infl ) =.07,cov(u, v ) =.084. u.05 Infl.703 Unemp.8 Unemp v Wh unemploymen affecng nflaon, he esmaed model used n a power expermen s (3-4) Infl = Infl Unemp =.009 var(u ) =.59, var(v.057 Infl.66 Infl.05 Infl ) =.07,cov(u, v ) = Unemp.703 Unemp.37 Unemp.8 Unemp u v The lag lenghs of boh models were seleced o mnmze he Akake creron, allowng a maxmum of four lags. Leng R denoe he number of n-sample observaons and P represen he number of predcons, Mone Carlo mehods are used o generae a oal of R P 4 observaons. 9 The addonal four observaons generaed allow for daa-deermned lag lenghs n he forecasng models esmaed n each smulaon. Leng L denoe he lag lengh of he daa-generang regresson es. 8 Usng Klan s (998) boosrap mehod o adus he coeffcens of he models produces essenally he same Mone Carlo resuls.

24 process (DGP), he frs L observaons are generaed by drawng from he uncondonal normal dsrbuon mpled by he model parameerzaon. The remanng R P 4 L observaons are consruced usng he auoregressve model srucure and draws of he error erms from he normal dsrbuon. Wh observaons hrough 4 reserved as nal observaons necessary o allow for as many as four lags n he esmaed models, a VAR n y and x (or Infl and Unemp) and an AR model for y (or Infl) are f over he n-sample perod spannng observaons 5 hrough R 4. In he neres of brevy he lag lengh of each esmaed model was se a he "rue" order, L, of he daa generang process. 0 Resuls for forecass based on lags se a he order mnmzng he n-sample Akake creron for he VAR or he AR model are essenally he same. In one unsurprsng excepon, he n-sample GC es ends o have slghly greaer sze and lower power han he presened resuls when he lag lengh s se o mnmze he n-sample Akake creron for he VAR. Observaons R 5 hrough R P 4 are hen used o form P -sep ahead forecass. The frs forecas s for perod R 4 ; he las s for perod R P 4. For brevy, 9 For he emprcal model, usng boosrap mehods o generae arfcal daa produces resuls much lke hose repored for Mone Carlo-generaed daa. 0 In he case of he smple VAR() model, when b s non-zero, he rue VAR mples ha he rue unvarae model for y s an ARMA(,). For he models consdered here, however, nverng he MA componen o rewre he ARMA(,) n AR form yelds very small coeffcens on lags greaer han, so he rue (nfne order) AR model s very close o an AR(). Accordngly, n he Mone Carlos he seleced AR lag lengh s n abou 75 percen of he power smulaons. In compung power when he lags are daa-deermned, he es sasc n smulaon, for whch he seleced lag s, s compared agans he dsrbuon of es sascs from he se of smulaons under he null n whch he lag was seleced o be. For example, f lag was seleced n J of he 50,000 sze smulaons of a gven expermen, emprcal crcal values for lag were calculaed from us hose J smulaed es sascs. In a correspondng power expermen, for hose smulaons n whch he lag was seleced o be, he es sascs were compared agans hese crcal values. Snce longer lags end o be somewha nfrequenly reeced, 50,000 smulaons were used n he sze expermens o ensure he accuracy of he resuls wh daa-deermned lags. 3

25 resuls are only presened for recursve forecass, as he basc conclusons are essenally he same for rollng and fxed forecass. Resuls are repored for smple, emprcally relevan combnaons of P and R such ha π ˆ P R akes a value of 0., 0., 0.4, 0.6,.0, or.0. Specfcally, n one se of expermens based on he arfcal VAR(), R = 00 and P = 0, 0, 40, 60, 00, and 00. In anoher se usng he VAR(), R = 00 and P = 0, 40, 80, 0, 00, and 400. In expermens based on he nflaon-unemploymen model, as suggesed above P and R are chosen o be conssen wh he emprcal applcaon consdered n secon 5. Parcularly, R = 5 and P = 46, for whch π ˆ = Sze Resuls Table presens he emprcal szes of Granger causaly, equal accuracy, and encompassng ess for daa from he VAR() of equaons (9)-(0), usng a nomnal sze of 0%. The general resuls are he same a he nomnal sze of 5%. In hese sze expermens, daa are generaed mposng he null of b = 0, under whch he AR and VAR forecass of y have equal MSE (asympocally), and he AR forecas encompasses he VAR forecas. Four general resuls are evden from Table. Frs, OOS F and CM ess perform que well, sufferng only slgh sze dsorons. For example, when R = 00 and P = 0, he emprcal szes of he OOS F and CM ess are 0.7% and.0% respecvely. Gven R, any dsorons n he OOS F and CM ess generally declne as P rses. Second, he asympocally vald versons of he DM and ERIC sascs and he CH es are modesly overszed n fne Whle resuls for rollng forecass are very smlar o hose for recursve forecass, resuls for fxed forecass do dffer slghly. For example, wh fxed forecass, he sze dsorons of he asympocally vald DM es are smaller, whle he sze dsorons of he CM es are a b bgger. 4

26 samples. For example, when R = 00 and P = 0, he emprcal szes of he asympocally vald DM, ERIC, and CH ess are 3.5%, 4.%, and 5.5%, respecvely. The dsorons declne as P rses for a gven R. 3 Wh R = 00 and P = 00, for nsance, he ERIC es s essenally correcly szed, reecng he null n 0.4% of he smulaons. The hrd general resul s ha comparng he DM, HLN, and ERIC ess agans nvald asympoc crcal values wll generally lead o oo-nfrequen reecons, he DM es more so han he HLN es and he HLN es somewha more so han he ERIC es. In he cases of he DM and HLN sascs, usng he ncorrec asympoc dsrbuon causes he ess o be underszed for all sample szes, and he ess become more underszed as P rses gven R. For nsance, wh R = 00 and P = 0, he DM and HLN ess reec he null n, respecvely, 5.5% and 8.3% of he smulaons. Wh P ncreased o 40, he DM and HLN szes fall o 3.8% and 7.4%, respecvely. In he case of he ERIC sasc, he nvald verson of he es s overszed for very small samples bu underszed for larger samples. Wh R = 00, for nsance, he sze of he es falls from 3.5% when P = 0 o 6.6% when P = 00. Fnally, smple GC ess are somemes slghly overszed when he number of observaons s small, bu abou correcly szed oherwse. The ou-of-sample GC es used wh P = 0, for example, has an emprcal sze of.6%, and he n-sample GC es wh R = 00 has sze of essenally 0%. Whle no shown n he neres of brevy, when he model lags are chosen o mnmze he Akake creron for he n-sample VAR, he n-sample GC es suffers modes sze dsorons, whle he sze of he ou-of-sample GC es s abou he same as when he 3 Increasng P, however, does no necessarly make he emprcal dsrbuon of he es sasc quckly approach a normal dsrbuon. CH es reecons occur more frequenly n he lef al han n he rgh al. When a nomnal sze α% s used, rgh al reecons occur n less han α/% of he smulaons. As P rses, rgh al reecons occur even less frequenly and accoun for mos of he observed mprovemen n he emprcal sze of he CH es. 5

27 lag s fxed a he DGP order. 4 Wh R = 00 and P = 0, seng he lags of he AR and VAR models a he opmal VAR lag yelds an n-sample GC es sze of 3.7 percen, up from 0.4 percen n he fxed-lag resuls. The ou-of-sample GC es sze s.8% and.6% wh he opmal VAR lag and fxed lag, respecvely. Table presens sze resuls based on he resrced VAR() model for he changes n core nflaon and unemploymen, equaons () and (). In hs sze expermen, daa are generaed mposng he null ha unemploymen does no affec nflaon. Ths mples ha AR and VAR forecass of Infl have equal MSE (asympocally), and ha he AR forecas encompasses he VAR forecas. Usng hs emprcal model produces resuls he same as hose for he arfcal VAR(). Agan, he performance of he OOS F and CM ess s que good, wh boh each essenally correcly szed. The asympocally vald versons of he DM and ERIC sascs and he CH es are subec o modesly larger dsorons. Comparng he DM, HLN, and ERIC ess agans nvald asympoc crcal values connues o produce oo-nfrequen reecons, wh he DM es more underszed han he HLN or ERIC ess. Fnally, for he R and P combnaon used n hs expermen, sandard GC ess are abou correcly szed when he lag lengh s se a he rue order. In resuls no repored, however, he n-sample GC es s sll subec o modes dsorons when he lag lengh s deermned wh daa-based crera. Inerpreaon/explanaon of resuls. 4.3 Power Resuls Tables 3-5 presen resuls on he power of smple Granger causaly, equal accuracy, and encompassng ess for daa from he smple VAR(), equaons (9)-(0). In hese power 4 However, f he lags of he AR and VAR models are se a he order mnmzng he Akake creron for he nsample AR model, he n-sample GC es s essenally correcly szed, us as when he lag s se a he rue order. 6

28 expermens, daa are generaed usng b = 0., 0., and 0.4, so VAR forecass of y have lower MSE han AR forecass, and he AR forecas does no encompass he VAR forecas. Because he ess are, o varyng degrees, subec o sze dsorons, he repored power fgures are based on emprcal crcal values and herefore sze-adused. 5 Wh emprcal raher han asympoc crcal values used, here s no dsncon beween he vald and nvald versons of he DM, HLN, and ERIC ess. The sze of he ess s 0%; usng 5% produces essenally he same resuls. For he OOS F, DM, CM, HLN, and ERIC ess, whch are one-sded, he null s reeced f he es sasc s greaer han he 90% fracle of he sasc n he correspondng sze expermen (for he same R and P) wh b = 0. The same apples o he GC ess. For he CH es, whch s wo-sded, he null s reeced f he es sasc les ousde he 5% and 95% fracles of he emprcal dsrbuon generaed n he correspondng sze expermen. Several general resuls are evden n Tables 3-5. Frs, he powers of he ess perm some smple rankngs: () CM > OOS F > DM > CH; () OOS F > ou-of-sample GC; and (3) CM > ERIC HLN > DM. In he VAR expermen desgn, he CM es for encompassng s clearly he mos powerful ou-of-sample es of predcve power n fne samples. Boh he CM encompassng and OOS F accuracy ess domnae GC ess appled o us he ou-of-sample daa. For example, as shown n Table 3, wh b = 0., R = 00, and P = 40, he CM es reecs he null n 3.4% of he smulaons, compared o 8.0% for he OOS F es and 8.% for he ou-of-sample GC es. Moreover, when he explanaory power of he Granger-causal varable s weak, as when b = 0., he power of he CM es somemes exceeds ha of he n-sample GC es. Ths holds even hough he CM uses fewer observaons han does he n-sample GC es. In he precedng example, he n-sample GC es reecs he null n 30.9% of he smulaons. 5 For hose ess subec o small sze dsorons, parcularly he OOS F and CM ess, power based on asympoc crcal values s very smlar o he repored sze-adused power. 7

29 The power of he DM and CH ess s lower, ofen subsanally, han he oher equal accuracy and encompassng ess. The HLN and ERIC encompassng ess have power somewhere n beween he power of he CM and DM ess. The second general resul s ha, gven R, power rses wh P. For example, as shown n Table 4, wh b = 0. and R = 00, he power of he OOS F es ncreases from 48.4% when P = 0 o 58.% when P = 40. Thrd and fnally, power ncreases wh he coeffcen b, whch deermnes he explanaory power of he causal varable. Power s sysemacally hgher when b = 0.4 han when b = 0. and, n urn, han when b =0.. Table 6 presens sze-adused power based on he VAR() for he changes n core nflaon and unemploymen, equaons (3)-(4). In he DGP underlyng hs power expermen, unemploymen does affec nflaon, so VAR forecass of Infl have lower MSE han AR forecass (asympocally), and he AR forecas does no encompass he VAR forecas. The crcal values used o evaluae power are calculaed from he dsrbuon of sascs generaed n he Table sze expermen. Usng he emprcal model produces resuls he same as hose for he arfcal VAR(). The rankngs () CM > OOS F > DM > CH, () OOS F > ou-of-sample GC, and (3) CM > ERIC HLN > DM sll apply. In hs example, he power of he CM es s no only greaer han ha of an ou-of-sample GC es bu also essenally he same as an nsample GC es, despe he fac ha P < R. These resuls sugges ha, f he DGP were he rue model, sascs lke he CM es would be almos sure o correcly pck up he predcve power of unemploymen. Inerpreaons and explanaons. 5. EMPIRICAL EXAMPLE 8

30 To provde a sense of he ably of he dfferen ess o deermne predcve power n pracce, hs secon presens resuls on a queson ha many sudes have examned: wheher unemploymen has predcve power, n-sample and pos-sample, for nflaon. Recen examples of sudes addressng hs queson nclude Cecche (995) and Sager, Sock, and Wason (997). In hs analyss, ess on Granger causaly, equal forecas accuracy, and forecas encompassng are appled o daa on core CPI nflaon and he prme-age male unemploymen rae. In he specfcaon used here, boh nflaon and unemploymen are dfferenced, conssen wh he resuls of augmened Dckey-Fuller ess for un roos. The quarerly daa avalable from 957:Q-998:Q4 are dvded no an n-sample poron and an ou-of-sample poron of modes lengh, so as o produce a π ˆ =P R value for whch McCracken (999) repors asympoc crcal values. Allowng for daa dfferencng and a maxmum of four lags n deermnng he model s lag order, he n-sample perod spans 958:Q3-987:Q, a oal of R = 5 observaons. The ou-of-sample perod spans 987:Q-998:Q3, yeldng a oal of P = 46 -sep ahead predcons. For hs spl, π ˆ =0.4. Over he n-sample perod, he Akake creron for boh he AR and he VAR s mnmzed a wo lags. Table 7 presens n-sample esmaes of an AR() f o changes n core CPI nflaon and a VAR() f o changes n core CPI nflaon and prme-age male unemploymen. The GC ess repored n he able sugges ha, over he n-sample perod, unemploymen has R s and predcve power for nflaon. The ou-of-sample evdence, however, s weaker. 6 The CM encompassng es, shown above o be more powerful han all he oher pos-sample ess, ndcaes ha unemploymen has predcve power for nflaon. The ERIC es, whch has lower 9

31 power han he CM es bu hgher power han he DM es, also reecs he null when compared agans he emprcal crcal value. Surprsngly, he CH es, generally he weakes, also suggess unemploymen has predcve power. None of he oher ess reec he null of no predcve conen n unemploymen. These resuls mgh lead o he concluson ha unemploymen n fac has no predcve power for nflaon. In hs vew he n-sample resuls, and ou-of-sample es resuls suggesng oherwse, are spurous. The problem wh hs nerpreaon s ha he Mone Carlo expermens of secon 4 sugges ha any sze bases n n-sample Granger causaly ess and, n parcular, he ou-of-sample CM es, should be small, even wh daa-deermned lags. Accordngly, he n-sample GC and CM sascs srongly reec he null when compared agans boh asympoc and emprcal crcal values. A more reasonable nerpreaon of he resuls s ha unemploymen does have predcve power for nflaon, power perhaps weakened by model nsables. Boh an n-sample GC es and he CM es for forecas encompassng overwhelmngly ndcae ha unemploymen has predcve conen. The Mone Carlo resuls n secon 4 ndcae hese ess have he greaes power n fne samples. Noneheless, he falure of oher ou-of-sample ess o deec any predcve power, despe he srong n-sample predcve power of unemploymen, suggess here s a dfference beween he n-sample and ou-of-sample predcve conen of unemploymen. Ths dfference may reflec model nsables. Neher he AR model nor he VAR pass he 6 The forecass are slghly based. However, demeanng he errors pror o calculang he es sascs has lle effec on he resuls. 30

32 supremum Wald or exponenal Wald ess for sably developed n Andrews (993) and Andrews and Ploberger (994), respecvely CONCLUSIONS 7 The models do, however, pass he Nyblom (989) es for sably and Chow ess for a shf n he parameer esmaes beween 958:Q3-87:Q and 987:Q-97:Q3. Followng Debold and Chen (996), he sably es resuls are based on boosrap crcal values. 3

33 References Amano, Rober A., and Smon van Norden, Terms of Trade and Real Exchange Raes: The Canadan Evdence, Journal of Inernaonal Money and Fnance, v.4, no. (February 995), pp Andrews, Donald W.K., Tess for Parameer Insably and Srucural Change wh Unknown Change Pon, Economerca, v.6, no.4 (July 993), pp Andrews, Donald W.K., and Werner Ploberger, Opmal Tess When a Nusance Parameer Is Presen Only Under he Alernave, Economerca, v.6, no.6 (November 994), pp Ashley, Rchard, A New Technque for Possample Model Selecon and Valdaon, Journal of Economc Dynamcs and Conrol, v. (998), pp Ashley, R., C.W.J. Granger, and R. Schmalensee, Adversng and Aggregae Consumpon: An Analyss of Causaly, Economerca, v.48, no.5 (July 980), pp Baes, J.M. and C.W.J. Granger, The Combnaon of Forecass, Operaonal Research Quarerly, v.0 (969), pp Blomberg, S. Brock, and Gregory D. Hess, Polcs and Exchange Rae Forecass, Journal of Inernaonal Economcs, v.43, no.- (Augus 997), pp Bram, Jason, and Sydney Ludvgson, Does Consumer Confdence Forecas Household Expendure? A Senmen Horse Race, Economc Polcy Revew, Federal Reserve Bank of New York, June 998, pp Cecche, Sephen G., Inflaon Indcaors and Inflaon Polcy, NBER Macroeconomcs Annual, 995, pp Chong, Yock Y., and Davd F. Hendry, Economerc Evaluaon of Lnear Macroeconomc Models, Revew of Economc Sudes, v.53, no.4 (Augus 986), pp Clark, Todd E., Fne-Sample Properes of Tess for Equal Forecas Accuracy, forhcomng, Journal of Forecasng (999). Corrad, Valenna, Norman R. Swanson, and Clauda Olve, Predcve Ably Wh Conegraed Varables, manuscrp, Pennsylvana Sae Unversy, December 998. Debold, Francs X., and Cela Chen, Tesng Srucural Sably wh Endogenous Breakpon: A Sze Comparson of Analyc and Boosrap Procedures, Journal of Economercs, v.70, no. (January 996), pp