Improving the Power of the Diebold- Mariano Test for Least Squares Predictions

Transcription

1 Improving h Powr of h Dibold- Mariano Ts for Las Squars Prdicions Spmbr 20, 2016 Walr J. Mar* Dparmn of Economics Univrsi of Mississippi Univrsi MS, wmar@olmiss.du Fng Liu Dparmn of Economics Univrsi of Mississippi Univrsi MS, Xin Dang Dparmn of Mahmaics Univrsi of Mississippi Univrsi, MS xdang@olmiss.du * Corrsponding auhor.

2 Absrac W propos a mor powrful vrsion of h s of Dibold and Mariano (1995) and Ws (1996) for comparing las squars prdicors basd on non-nsd modls whn h sd paramr is h xpcd diffrnc bwn h squard prdicion rrors. Th proposd s improvs asmpoic powr using an simaor wih a smallr asmpoic varianc han ha usd b h sandard vrsion of h s. Th simaor usd b h sandard vrsion of h s dpnds on h individual prdicions and ralizaions onl hrough h obsrvaions on h prdicion rrors. Th sd paramr, howvr, can also b xprssd in rms of momns of h prdicors and prdicd variabl, som of which canno b idnifid sparal b h obsrvaions on h prdicion rrors alon. Paramrizing hs momns in a GMM framwork and drawing on hor from Ws (1996), w dvis mor powrful vrsions of h s b xploiing a rsricion rouinl mainaind undr h null hpohsis b Ws (1996, Assumpion 2b) and lar sudis. Th mainaind rsricion rquirs onl fini scond-ordr momns and covarianc saionari o nsur ha h populaion linar projcion xiss. Simulaion xprimns show ha h ponial gains in powr can b subsanial. Kwords: prdicion, hpohsis ss. 2

3 1. Inroducion Th s of ou- of- sampl prdiciv accurac proposd b Dibold and Mariano (1995) and Ws (1996) is widl rgardd as an imporan s for comparing wo prdicors. 1 Applicaions includ Mark (1995), Swanson and Whi (1997), Corradi, Swanson and Olivi (2001), Hong and L (2003), Hanson and Lund (2005) and Androu, Ghsls and Kourllos (2013). Formal asmpoic hor was firs prsnd in Ws (1996), and furhr dvlopd b Clark and McCrackn (2001, 2014), McCrackn (2007) and ohrs. Th prsn papr proposs a mor powrful vrsion of h s obaind b using a mor fficin simaor of h sd paramr han ha usd in h liraur. Th sd paramr is h xpcd diffrnc bwn funcions of h prdicion rrors. In h sandard vrsion of h s, i is simad b h sampl man of h diffrnc and, consqunl, h individual obsrvaions on h prdicions and ralizaions gnrall nr h s saisic onl hrough h prdicion rrors. Th sd paramr, howvr, can also b xprssd in rms of momns of h prdicors and prdicd variabl, som of which canno b idnifid sparal b h obsrvaions on h prdicion rrors alon. 2 This raiss h possibili of using h individual obsrvaions on h prdicors and prdicd variabl o consruc a mor fficin simaor of h sd paramr 3 and, in urn, ss wih grar asmpoic powr. 1 In conras o Dibold and Mariano (1995), Ws (1996) xplicil accouns for paramr simaion. 2 Dibold and Mariano (1995, p.254) no ha h s is no limid o sing funcions of h prdicion rror bu can b applid o funcions in which h ralizaion and prdicion nr sparal. Th sam argumn of cours applis if momns of h prdicd variabl and h prdicors canno b idnifid b h obsrvaions on such funcions. 3 Th ponial infficinc from rplacing a sampl of obsrvaions (hr h individual obsrvaions on h prdicors and prdicd variabl) b funcions of h obsrvaions (hr h prdicion rrors) can b quanifid in rms of informaion marics. Spcificall, if g(x) is a masurabl funcion of a random variabl X wih a dnsi ha is a funcion of a vcor θ hn i can b shown ha diffrnc bwn h informaion marics wih rspc o is non-ngaiv dfini (Rao 1973, pp ). θ, J J, ( ) X g X 3

4 W pursu his for h commonl sd hpohsis of qual man squard rrors in h cas of las squars prdicors. A nw vrsion of h s is proposd basd on a mor fficin simaor of h xpcd diffrnc bwn h squard prdicion rrors. Paramrizing h momns of h prdicors in a GMM framwork, w driv a mor fficin simaor b incorporaing a rsricion ha is par of wak rgulari condiions rouinl mainaind undr h null hpohsis b Ws (1996, Assumpion 2b) and lar sudis. In h conx of linar las squars prdicion, h mainaind rsricion rquirs onl fini scond-ordr momns and covarianc saionari. Whil jus sandard rsricions on h daa-gnraing procss, his rquirmn migh b considrd mor rsriciv han h sup in Dibold and Mariano (1995) in which h undrling prdicors ar compll unspcifid. In applicaions ha involv valuaing prdicions from survs, for xampl, i is clarl dsirabl o lav h undrling prdicors unspcifid and impos rsricions dircl on h prdicion rrors. On h ohr hand, h proposd s should b usful givn h larg numbr sudis ha valua prdicions undr h condiions of h prsn papr. Exampls of sudis ha compar h man squard rrors of las squars prdicors undr covarianc saionari includ Mark (1995), Clark and McCrackn (2006), Sock and Wason (2002, 2007) and ohrs. Th proposd s and h GMM simaor on which i is basd ar dscribd in Scion 2. Scion 3 rpors vidnc from simulaion xprimns on h fficinc of h simaor and siz and powr of h ss. Consisn wih h asmpoic fficinc of h GMM simaor, h simulaions confirm subsanial powr gains for h proposd s ovr h sandard vrsion of h Dibold-Mariano (DM) s. Proposiion 1 in Scion 2 formall sablishs asmpoic normali and rlaiv fficinc of h GMM simaor ovr h simaor on which h sandard DM s is basd. Proof of Proposiion 1 follows from an applicaion of Ws (1996, Thorm 4

5 4.1). Dvising a DM-p s basd on h GMM simaor is sraighforward for non-nsd rgrssion modls. 4 Howvr, as is h cas for h sandard DM s, crain chnical problms aris whn applid o nsd and ovrlapping modls. For h sandard vrsion of h DM s, hs problms hav bn sudid in a sris of paprs b Clark and McCrackn. 5 For h nw GMM vrsion proposd hr, h problms ak h form of duplicad momn condiions and singular covarianc marics. Ths problms ar brifl discussd in scion 2.3 and lf as dircions for fuur rsarch. I is wll-known from GMM hor ha incrasing h numbr of momns gnrall improvs asmpoic fficinc. Undr h assumpion of covarianc saionari, h populaion linar projcion rror is uncorrlad wih an linar combinaion of h prdicor variabls. In scion 2.4 w xploi his o xpand h momn funcions usd o dfin h GMM simaor in scion 2.2. For h simulaion xprimns in Scion 3, ss basd on h xpandd momn funcions ar found o hav grar powr for boh small and larg sampls bu wors siz in small sampls. In scion 4 w illusra h s wih a forcasing applicaion o monhl US indusrial producion. Scion 5 concluds. 2. GMM Vrsion of DM Ts 2.1 Null Hpohsis, Prdicors and Rsricions Givn daa availabl a im, w considr wo comping prdicors 1, ˆ + and ˆ 2, + of a variabl + a im + wih prdicion rrors, ˆ 1, + = + 1, + and ˆ 2, + = + 2, +. Th DM s can b usd o s h null hpohsis of qual prdiciv accurac for a wid rang of loss funcions and prdicors. An imporan spcial cas is qual man squard rrors: 4 Rcn sudis ha us DM ss o compar non-nsd modls includ Nas, R., J.A. Skjlorp and B.A. Ødgaard (2011) and Androu, Ghsls and Kourllos (2013). 5 S, for xampl, Clark and McCrackn (2001, 2014) and McCrackn (2007). 5

6 H : E ( ) = 0 (1) 2 2 o θ 1, + 2, + W assum ha (1) is o b sd using P of h τ -sp ahad prdicions. Th prdicors ar of h form ˆ ˆ j, + = X j, + β j, (j=1,2), whr ˆ β ˆ ˆ ˆ j = ( β j,0, β j,1, β j,2 ) is h las squars simaor compud from h rgrssion of on X j, = (1, xj,, x) using a minimum of R in-sampl obsrvaions. Following Ws (1996), w assum a rcursiv forcasing schm which also is commonl usd in pracic. 6 Thrfor, ( ) 1 ˆ j X 1 js, X js, X s s 1 js, = = s β = for =R,,R+P-1, and j=1 and 2. Givn h P ou-of-sampl prdicions, h sandard DM s of (1) is basd on h following simaor: P+ R 1 ˆ 1 θ = ( )( + ) (2) 1, + 2, + 1, + 2, + P = R whr h subscrip mphasizs ha (2) is compud from h obsrvaions on h prdicion rrors. Sinc θ is a funcion of h momns of ˆ 1,, obsrvaions on ˆ 1,, + ˆ 2, + ˆ 2, and, + + h individual and + + can b usd o dvis an asmpoicall mor fficin simaor han (2). Th basis for our approach is h following assumpion: Assumpion 1 For j=1,2: h squnc { X j, } is covarianc-saionar, rgodic for scond momns, and EX ( X ) is nonsingular. j, j, 6 This assumpion accommodas our applicaion of Ws (1996, Thorm 4.1) in scion 2.3. Ws (1996) assums a rcursiv forcasing schm bu considrs xnsions o fixd and rolling schms in an unpublishd working papr, Ws (1994). 6

7 Assumpion 1 imposs wak rgulari condiions ha ar rouinl mainaind undr h null hpohsis in h liraur. S, for xampl, Ws (1996, pp ) and Clark and McCrackn (2014, p.418). I nsurs ha h momns EX ( j, X j, ) and EX (, ) ar fini, j 1 j, X j, xiss and, hrfor, h linar projcion EX ( ) β [ EX ( X )] EX ( ) xiss for 1 j j, j, j, j=1,2. Undr Assumpion 1, a law of larg numbrs can also b applid o show ha ˆ β j convrgs in probabili (as ) o β. A wll-known propr of h linar projcion j cofficin is ha EX [ j, ( X j, β j)] = 0, which is a spcial cas of Assumpion 2b mainaind in Ws (1996). 7 This, in urn, implis h following propr which is usd o rsric θ blow: EX [ β ( X β )] = 0 (3) j, j j, j Th prdicors and prdicion rrors obviousl dpnd on h simaors, ˆ β j. To rflc his, w l ˆ β ˆ ˆ j = ( β jr,..., β j, R + P 1) and ˆ β = ( ˆ β ˆ 1, β2) and rwri ˆ θ and θ, rspcivl, as ˆ θ ( ˆ) β and θ( ˆ β ). As mphasizd b Clark and McCrackn (2013), hpohss lik (1) can b inrprd in rms of fini-sampl or populaion-lvl prdiciv accurac. Th formr concrns θ( ˆ β ) for fini R, whras h lar concrns θ( ˆ β ) wih ˆβ rplacd b β = ( β1, β2), 7 This is asil vrifid b subsiuing for β j ino h xprssion. Th quani X j, β jis known as h linar projcion of on X j, and is formall dfind as h bs linar prdicor of givn X j, (Hamilon 1994, p.74). Consisn simaion of linar projcion cofficins rquir much wakr assumpions han h cofficins of srucural or causal rgrssion modls which rquir assumpions abou h funcional forms of condiional mans and ohr quaniis. Th linar projcion xiss for an s of random variabls wih fini variancs. Furhr discussions can b found in Hamilon (1994, Chapr 4), Wooldridg (2010, Chapr 2) and Hansn (2016, Chapr 2). Applicaions of linar projcions includ Chambrlain (1982). 7

8 giving θ( β ). Following mos prvious work, w focus on populaion-lvl prdiciv accurac 8 and hus inrpr (1) as " θ( β ) = 0." Assumpion 1 provids a basis for fficinc gains ovr ˆ θ ( ˆ) β hrough rsricd GMM simaion of h sd paramr θ( β ). Subsiuing 1, 2, ˆ 2, ˆ + + = + 1, + and ˆ ˆ 1, + + 2, + = + 2, , + ino ˆ θ ( ˆ) β and θ( β ) ilds: P+ R 1 P+ R = P X2, X2, + 2 P X1, X1, + 1 = R = R ˆ θ ( ˆ β) ˆ β ( ˆ β ) ˆ β ( ˆ β ) P+ R 1 P+ R 1 1 ˆ 1 ˆ 2, + β2 + 1, + β1 + = R = R + P X P X (4) θ = EX [ β ( X β )] EX [ β ( X β )] 2, , + 2 1, , EX [ β ] EX [ β ] 2, , (5) Equaions (4) and (5) rval ha ˆ θ ( ˆ) β is quivaln o an simaor ha rplacs h populaion momns in (5) wih h corrsponding sampl momns. As such (4) is infficin bcaus i ignors rsricions on (5) implid b Assumpion 1, naml (3) which rsrics h firs wo rms on h righ hand sid of (5) o b zro. 9 A mor fficin rsricd simaor of θ ha incorporas (3) is dvlopd in h nx subscion. Th ss dvlopd in h nx scions mainain (3) undr h null hpohsis of qual man squard rrors: θ( β ) = 0. I is imporan o mphasis ha h onl rsricion for (3) o hold is Assumpion 1. Hnc, h null considrd in h prsn papr is h sam as ha considrd in ohr sudis ha mainain fini scond-ordr momns and covarianc saionari 8 Excpions includ Giacomini and Whi (2006) and Clark and McCrakn (2015). 9 Th sampl counrpars of (3) clarl hold for h R in-sampl obsrvaions sinc h las squars prdicors, ˆ1 and ˆ 2, ar compud wih hs obsrvaions. Howvr, h sampl counrpars of (3) do no gnrall hold for h P obsrvaions usd o compu ˆ ˆ θ ( ) β and, hus, ar no incorporad ino h simaion of θ( β ). 8

9 undr h null. I should also b nod ha w do no mak an assumpions abou h saisical propris of h prdicors in fini sampls and, hrfor, do no rquir h prdicors o b unbiasd or fficin. Such propris would nail imposing assumpions much mor rsriciv han Assumpion 1, assumpions ha h prsn approach avoids. Unbiasdnss, for xampl, which rquirs ha E( ˆ X j, β j ) = 0 holds for fini, would nail adding h assumpion ha X j, β jis h condiional man of givn X., In conras, Assumpion 1 and, consqunl, j (3) onl rquir covarianc saionari and h xisnc of crain momns. 10 Assumpion 1 dos no rquir ha ihr prdicor is basd a corrcl spcifid modl of h condiional man or an ohr saisical funcional. 2.2 Rsricd GMM Esimaion of θ( β ) W nx dvis a mor fficin rsricd simaor of θ( β ) using a GMM framwork in which h momns in (5) ar simad joinl subjc o (3). No ha (5) and (3) can b wrin as: θ = 2 EX ( β ) EX ( β ) 2 EX ( β ) + EX ( β ) (6) 2 2 2, , + 2 1, , ( j, + β j + ) ([ j, + β j] ) 1, 2 EX = E X j= (7) L µ dno h vcor of h momns in (6): µ = [ E( X β ), E( X β ), E([ X β ] ), E([ X β ] )] , , + 2 1, + 1 2, Th wakr assumpions imposd on h prdicors can also b sn as an advanag of focusing on populaionlvl prdiciv accurac as opposd o fini-sampl prdiciv accurac. Dvloping an analogous approach for h lar would rquir ha (3) holds wih β rplacd b ˆβ which, in urn, would rquir ha X jβ j is h condiional man. 9

10 In wha follows ach lmn of µ is rad as a paramr o b simad. Th orhogonali condiion for h GMM simaor of µ is Eg [ + ( µβ, )] = 0, whr g ( mβ, ) = m mand + + m X X X X = [ + 1, + β1, + 2, + β2, ( 1, + β1), ( 2, + β2) ] Th fasibl sampl analog of Eg [ + ( µβ, )] is hus g ˆ = P g ˆ whr 1 R+ P 1 ( µβ, ) ( µβ, ), = R + ˆ β = ( ˆ β, ˆ β ). Th wo rsricions in (7) can b xprssd as Qµ = 0, whr Q = Th rsricd GMM simaor of µ, µβ ( ˆ), solvs h problm: min g( mβ, ˆ) Wg ˆ ( mβ, ˆ) subjc o Qµ = 0 (8) m whr ˆ W is a wighing marix. Th soluion o (8) is: ˆ ˆ * ˆ µβ ( ) = A µ (9) whr ˆ ˆ ˆ * 1 = and ˆ m = P m ( ˆ + β) is h unrsricd GMM A I W Q[ QW Q] Q R+ P 1 simaor of µ. Th rsricd GMM simaor of θ( β ) is θ( ˆ β ) = c µβ ( ˆ) whr c = ( , ) whil ˆ ˆ * ˆ c = R θ ( β ) = µ is h unrsricd GMM simaor. Consqunl givn rgulari condiions, h asmpoic fficinc of θ ( ˆ β) rlaiv o ˆ θ ( ˆ) β can b sablishd using sandard rsuls on rsricd simaion. Undr h nx assumpions, h limiing disribuions of follow from an applicaion of Ws (1996, Thorm 4.1). Pθ ( ˆ β) and Pθ ˆ ( ˆ) β 10

11 Assumpion 2 ˆ plim P W W = whr W is posiiv dfini. Assumpion 3 a) L m = m / β, ε, j = X j, β j and * h Euclidan norm. Thn for j=1,2 and som d>1, sup E [ vc( m ), m, X ] j,, j 4d <. b) For j=1,2 and som d>1, [ vc( m E m),( m Em), X j,, j] is srong mixing wih mixing cofficins of siz -3d/(d-1). c) For j=1,2: [ vc( m), m, X j,, j] is covarianc saionar. d) L Vmm = Γmm( j) whr Γ ( j mm ) = E [ m Em )( m j Em ]. Thn Vmm is j= posiiv dfini. Assumpion 4 lim RP, P/ R= π whr 0 π. Assumpions 3 and 4 ar h sam as Assumpions 3 and 4 in Ws (1996, p.1073) wih diffrn noaion. Assumpion 3 rsrics srial corrlaion whil Assumpion 4 spcifis h asmpoics for h numbrs of in-sampl and ou-of-sampl obsrvaions. Proposiion 1 1 For j=1,2: l ( ) 1 B () = X X, B( ) = diag( B1( ), B2( )), B= plim B ( ), j s= 1 js, js, 1 ( s= 1,, ) H j() = X jsεjs, H () [ H1(), H2()], h = X ε X ε, = ( 1, 1, 2, 2, ) Γ mh( j) = E( m Em ) h j, ( ) [ ], Γ hh j = E hh j Vmh = Γmh( j), Vhh = Γhh( j), Vmm VmhB 1 Vβ = BVhhB, S =, Π= 1 π ln(1 + π) for 0 < π <, Π= 0 for π = 0, and BV mh V β Π= 1for π =. If S is posiiv dfini, hn undr Assumpions 1, 2, 3 and 4: 11 j= j=

12 (i) P( µ ( β ) µ ) N(0, Ω) whr * ˆ dis Ω= V +Π[( E m ) BV + V B ( E m ) ] + 2 Π( E m ) V ( E m ). mm mh mh β dis (ii) P( θ ( ˆ β) θ( β )) N(0, caωa c ) whr = [ ] A I W Q QW Q Q dis (iii) P( ˆ θ ( ˆ β) θ( β )) N(0, cω c ) (iv) L W = W 1. Thn c c Ω - caω A c = Ω Ω Ω 1 c Q[ Q Q] Q c Proof: S Appndix. Using Proposiion 1 o dvis DM ss is sraighforward whn h prdicors ar basd * on non-nsd modls. In his cas, h asmpoic varianc-covarianc of µ ( ˆ β ), Ω, is gnrall nonsingular and h ss ar asmpoicall sandard normal. Th original DM s saisic is: DM ( ˆ θ ( ˆ β)) = P ˆ θ ( ˆ β )/ cω ˆ c (10) whil h analog basd on θ ( ˆ β) is: DM ( θ ( ˆ β)) = P θ ( ˆ β )/ caˆω ˆ Aˆ c (11) whr ˆΩ is a consisn sima of Ω. I follows from pars (ii) and (iii) of Proposiion 1 ha (10) and (11) ar ach asmpoicall sandard normal undr h null hpohsis, θ( β ) = 0. I follows from par (iv) ha θ ( ˆ β) is asmpoicall fficin rlaiv o ˆ θ ( ˆ) β whn θ ( ˆ β) is basd on h opimal choic for h wigh marix, W= Ω Th asmpoic fficinc advanag suggss ha (11) ma hav grar powr han (10). Consisn simaion of h componns of Ω is discussd in Ws (1996, pp ). A Nw and Ws (1987) p simaor can b usd forv mm, Vmh and V β ; π can b consisnl simad b P/R, and B and E m ( β ) b hir sampl analogus. Th marix Ω is h sum of four rms. As nod b Ws 12 1.

13 (1996,p.1072), h firs rm, Vmm, rflcs h uncrain ha is prsn in h simaion of µ condiional on h valu of β, whil h rmaining hr rms rflc h uncrain du o h simaion of β. If π = 0, hn h lar rms do no conribu o asmpoic varianc of ihr ˆ θ ( ˆ ) β or θ ( ˆ β). If π 0, howvr, hn h uncrain du o h simaion of β conribus o h asmpoic varianc of θ ( ˆ β) bu no o h asmpoic varianc of ˆ θ ( ˆ). β This can b sn b noing ha X 1, 0 0 X 2, m ( β ) = 2X 0 1, β1x1, 0 2X2, β2x 2, and hrfor ce m ( β ) = 0, sinc β is dfind o b a projcion cofficin. For h opimal wighing marix, W= Ω 1, h xprssions for h asmpoic variancs of ˆ θ ( ˆ) β and θ ( ˆ β) hus simplif, rspcivl, o: cω c = cv c (12) mm Ω = Ω Ω Ω (13) 1 ca A c cvmmc c Q ( Q Q ) Q c whr cω= cv + cπv B ( E m ( β )). mm mh 2.3 Nsd and ovrlapping modls Th DM ss ar lss sraighforward for nsd and ovrlapping modls. For hs modls, X1β1 = X2 β2can characriz h null hpohsis which rsuls in duplicad momn condiions and, consqunl, singular Ω. Whn X1β, 1 = X2 β2 w hav cω c = 0, 13

14 P ˆ θ ( ˆ β ) = o (1) and, hrfor, h asmpoic disribuion of (10) is no obvious. 11 For h p spcial cas of on-sp ahad prdicion and sriall uncorrlad prdicion rrors, Clark and McCrakn (2001) and McCrakn (2007) show (10) has a non-sandard disribuion ha is a funcion of Brownian moion if π > 0, and is asmpoicall sandard normal if π = 0. Unlik ˆ θ ( ˆ), β h asmpoic disribuion of h rsricd simaor θ ( ˆ β) dpnds on h choic of W. On problm for nsd and ovrlapping modls is ha h sandard opimal 12 choic W 1 = W is obviousl no availabl if Ω is singular. In a diffrn conx, Pnaranda and Snana (2012 pp ) xnd Hanson s (1982) opimal GMM hor o addrss h problm of a singular asmpoic covarianc marix. 13 Th propos a wo-sp GMM simaor ha is asmpoicall quivaln o h infasibl opimal (asmpoicall fficin) GMM basd on h gnralizd invrs of h singular covarianc marix. For h prsn problm, his would nail rplacing Assumpion 2 wih h assumpion ha p lim P ˆ + + Ω =Ω for som simaor Ω ˆ + + of h gnralizd invrs Ω. I is no clar, howvr, how such an simaor could b consrucd for h prsn problm. Unlik h ordinar invrs, gnralizd invrss ar disconinuous. Consqunl, as nod b Andrws (1987), consisnc of ˆΩ for Ω dos no, in gnral, nsur h consisnc of Ω ˆ + + for Ω. Andrws (1987, Thorm 2) provids an addiional ncssar and sufficin condiion for h lar o hold. Th condiion is Prob[ rank( Ω ˆ ) = rank( Ω)] 1 as P. This condiion, howvr, is unlikl o b saisfid in 11 P ˆ θ ( ˆ β ) = o (1) follows from quaion (4.1) in Ws (1996) sinc cm ( β ) = 0. p 12 Hansn and Timmrmann (2015) dmonsra ha for nsd modls, (10) can b xprssd as h diffrnc bwn wo convnional Wald saisics sing h hpohsis ha h cofficin in h largr modl is zro, on saisic basd on h full sampl and h ohr basd on a subsampl. Th show ha his dilus local powr ovr h convnional full-sampl Wald s. Th hn argu ha his raiss srious qusions abou sing populaionlvl accurac for nsd modls using ou-of-sampl ss. 13 Also s Diz d los Rios (2015). 14

15 h prsn problm if ˆΩ is a sandard covarianc simaor. Th rason is ha ˆΩ will gnrall b nonsingular if X ˆ ˆ 1, + 1β1 X2, + 1β2which will hold wih probabili on for all fini P if componns of X 1, + 1 or X 2, + 1ar coninuousl disribud. W lav for fuur rsarch h + problm of consrucing consisn simaors for Ω. Y anohr approach would b a wo-sp procdur in which h firs sp consiss of sing h hpohsis ha h modls ar ovrlapping agains h alrnaiv ha h modls ar non-nsd. Such ss hav bn proposd b Vuong (1989), Marcllino and Rossi (2008) and Clark and McCrackn (2014). If h s fails o rjc h hpohsis of ovrlapping modls hn h procdur sops. If h hpohsis of ovrlapping modls is rjcd, hn, in h scond sp, h hpohsis θ( β ) = 0 is sd wih (11) using an sima of h opimal wigh marix for θ ( ˆ β). 2.4 Addiional Momns and Rsricions Undr Assumpion 1, EX ( j, + ( + X j, + β j)) = 0 which implis ha h produc X β and an linar combinaion of h prdicor variabls has zro xpcaion. + j, + j Consqunl, man diffrn spcificaions of µ and m + ( β ) ar possibl. Up o his poin our spcificaion consiss of h four cross-produc momns in h rsricion EX [ β ( X β )] = 0 j=1,2, giving: j, + j + j, + j µ = [ E( X β ), E( X β ), E([ X β ] ), E([ X β ] )] , , + 2 1, + 1 2, + 2 m = X X X X (14) [ + 1, + β1, + 2, + β2, ( 1, + β1), ( 2, + β2) ] 15

16 Six addiional rsricions implid b EX ( j, + ( + X j, + β j)) = 0 ar Ex [ + β j2 ( + Xi, + βi)] = 0 and E( + X j, + β j) = 0 i,j=1,2. This suggss using h following xpandd vcors: µ = [ E( X β ), E( X β ), E([ X β ] ), E([ X β ] ), E( ), , , + 2 1, + 1 2, EX ( β ), EX ( β ), Ex ( β ), Ex ( β ), Ex ( β X β ), 1, + 1 2, , + 1 Ex ( + β12 X2, + β2), Ex ( + β22 X2, + β2), Ex ( + β22 X1, + β1)] m = [ X β, X β, [ X β ], [ X β ],, X β, X β, , , + 2 1, + 1 2, , + 1 2, + 2 x β, x β, x β X β, x β X β, x β X β, x β X β ] , , , , + 1 (15) I is wll known from sandard GMM hor ha incorporaing addiional momns and rsricions gnrall improvs asmpoic fficinc. Consqunl, (15) offrs ponial fficinc gains ovr (14) a las asmpoicall. Exnding h hor of h scion 2.2 o accommoda (15) is sraighforward. I is a mar of rplacing m + ( β ) and µ in Proposiion 1 wih m + ( β ) and µ. Using (15) insad of (14) dos no chang ˆ θ ( ˆ) β sinc unrsricd GMM is quivaln o quaion b quaion OLS in h prsn sing. I dos, howvr, chang h asmpoic covarianc of Pθ and, consqunl, h opimal wighing marix. 3. Simulaion Exprimns 3.1 Dsign Simulaion xprimns wr conducd o xamin h fini-sampl propris of h various simaors and ss. Sampls wr drawn from h following modl: 16

17 = δ x + δ x + u (16) , 2 2, + 1 whr u and xi, ar indpndn, sriall uncorrlad, u N(0,10) and xi, N(0,0.5). Th ss wr valuad for on-sp ahad ( τ = 1) prdicions from h wo modls: = β + β x + β + u (17a) , 12 1, + 1 = β + β x + β + u (17b) , 22 2, + 1 Th modls wr simad b OLS and a rcursiv schm was usd o gnra prdicions. In all xprimns h ss ar conducd a h 10 prcn significanc lvl and for 5,000 rplicaions. Exprimns wr run for P=20, 50, 100, 200 wih R=jP, and j=2,3,4,5. Sampls wr gnrad from (16) using a non-nsd spcificaion in which w s δ 1 = 2 in all xprimns. Th xpcd valu of h squard prdicion rror in his cas quals δ 2 2 for (17a) and quals 12 for (17b). In h siz xprimns w s δ 2 = 2, and in h powr xprimnsδ 2 = 1. In wha follows, h DM s basd on a givn simaor is dnod b DM( simaor ). Th rsricd GMM simaor basd on h opimal wigh marix for j momns is dnod θ ˆ ( ), j=4,13. I is compud from h wo-sp GMM simaor µβ ( ˆ) j givn b (9). Th firs-sp simaor is OLS. Th wigh marix is h invrs of a consisn sima of h asmpoic covarianc. Th lar is srial-corrlaion robus, and uss a Barl krnl and h auomaic lag-slcion algorihm proposd b Nw and Ws (1994). 17

18 3.2 Rlaiv Efficincis of h GMM Esimaors W hav argud ha ss basd on θ ˆ (4) and θ ˆ (13) ma hav grar powr han ss basd on ˆ θ ( ˆ) β bcaus of h asmpoic fficinc of rsricd simaion. Bfor valuaing h ss, w firs xamin h fficinc issu b comparing h sampl mans and sandard dviaions of ˆ θ ( ˆ) β, θ ˆ (4) and θ ˆ (13) for h sampls usd o valua h ss. Undr h null hpohsis of qual man squard prdicion rrors, h sandard dviaions and mans ar rpord, rspcivl, in Tabls 1 and 2. In all cass h sampl mans ar much smallr han h sandard dviaions and, consqunl, biasdnss is no a significan sourc of h simaion rror. Consisn wih h asmpoic fficinc of rsricd simaion, h sandard dviaions of θ ˆ (13) and θ ˆ (4) ar smallr han hos of ˆ θ ( ˆ ) β for all P and R. Also as xpcd, h sandard dviaions of θ ˆ (4) ar (slighl) largr han hos of θ ˆ (13). Avraging h prcnags ovr R, h sandard dviaion of θ ˆ (13) is abou 61% of h sandard dviaion of ˆ θ ( ˆ) β for P=20, 59% for P=50 and P=100, and 57% for P=200. Th fficinc gains gnrall incras as P/R dcrass. Th prcnags rang from 64% o 67% for P/R=1/2, 57% o 61% for P/R=1/3, 52% o 59% for P/R=1/4, and 52% o 55% for P/R=1/5. Th fficinc gains for θ ˆ (4) ovr ˆ θ ( ˆ) β ar onl slighl lss Siz and Powr Rsuls Nx w valua h siz and powr of h ss. As Proposiion 1 shows, h asmpoic covarianc marix Ω dpnds on is h limi of P/R, π. Whn R is larg rlaiv o P on migh considr sing π =0 which gral simplifis h simad asmpoic covarianc. In iniial 18

19 xprimns, howvr, w found ha DM ( θ ˆ (4) ) and DM ( θ ˆ (13) ) wr gral ovrsizd for all P and R whn π =0 was imposd. For xampl, for h 10% nominal siz undr h null hpohsis, h rjcion ras for DM ( θ ˆ (4) ) wih π =0 imposd rangd from an avrag of 30% for P=200 o an avrag of 57% for P=20. Alhough ˆβ is asmpoicall irrlvan in h disribuions of θ ˆ (4) and θ ˆ (13) if h ru valu of π is zro, in fini sampls h disribuions gnrall dpnd on h disribuion of ˆ. β Imposing π =0 in fini sampls nglcs his dpndnc and, consqunl, h simad asmpoic varianc ma b a poor approximaion o h fini-sampl varianc. Consisn wih his, h siz of DM ( θ ˆ (4) ) and DM ( θ ( ˆ β)) improvd considrabl whn π =0 was no imposd. As discussd blow, h (13) rjcion ras of DM ( θ ˆ (4) ) and DM ( θ ˆ (13) ) ar picall wihin wo or hr prcn of h nominal siz whn π =0 is no imposd. In conras, h valu of π is no an issu for DM ( ˆ θ ( ˆ β )). As quaions (12) and (13) rval, unlik θ ˆ (4) and θ ˆ (13), ˆβ is asmpoicall irrlvan in h disribuion of ˆ θ ( ˆ) β for all valus of π. Tabl 3 rpors h rjcion ras undr h null hpohsis for a nominal siz of 10%. In lin wih prvious sudis, h sandard form of h DM s, DM ( ˆ θ ( ˆ β)), prforms wll in rms of siz. I is slighl ovrsizd in mos cass bu h rjcion ras ar wihin 2% of h nominal siz for all cass in which P 50 and wihin abou 3% for P=20. For similar sampl sizs, Clark and McCrackn (2014, pp ) also found DM ( ˆ θ ( ˆ β)) o b slighl ovrsizd. Th conjcur ha boh P and R migh hav o b largr for h asmpoics o kick in. Th rjcion ras of DM ( θ ˆ (13) ) ar, xcp wo cass, wihin abou 2 prcn of h nominal 19

20 siz for P 50 wih P/R 1/3. Th rjcion ras of DM ( θ ˆ (4)) ar somwha br as h ar wihin 2 prcn of h nominal siz for P 20 wih P/R 1/3 for 14 ou of 15 cass. Thrfor, DM ( θ ˆ (4)) prforms abou as wll as DM ( ˆ θ ( ˆ β )) in hs cass. On diffrnc, howvr, is ha whras DM ( ˆ θ ( ˆ β)) nds o b slighl ovrsizd, DM ( θ ˆ (4) ) and DM ( θ ( ˆ β)) nd o b undrsizd and, hrfor, ar mor consrvaiv. I should also b (13) nod ha h rjcion ra of DM ( ˆ θ ( ˆ β)) is gnrall lss snsiiv o P/R han DM ( θ ˆ (4) ) and DM θ(13) ( ( ˆ β)). Again, his migh b xplaind b h fac ha h asmpoic varianc of ˆ θ ( ˆ ) β dos no dpnd on π. Tabl 4 rpors h rjcion ras undr h alrnaiv hpohsis. In rms of powr, h rjcion ra of DM ( ˆ θ ( ˆ β)) is also lss snsiiv o h valu of P/R han h ohr ss. For h cass of P 50 wih P/R 1/3 in which h ss hav similar siz, DM ( θ ˆ (13) ) and DM ( θ ( ˆ β)) hav much grar powr han DM ( ˆ θ ( ˆ β )). For xampl wih P/R=1/5, h (4) rjcion ra of DM ( θ ˆ (13) ) is abou 2.3 ims grar han DM ( ˆ θ ( ˆ β )) for P=50, 2.1 ims grar for P=100 and 1.5 ims grar for P=200. As xpcd DM ( θ ˆ (4)) has lss powr han DM θ(13) ( ( ˆ β)) bu grar powr han DM ( ˆ θ ( ˆ β )). 4. Illusraiv Empirical Applicaion W nx illusra h ss using a forcasing applicaion o monhl US indusrial producion. Th daa wr downloadd from FRED wbsi of h Fdral Rsrv Bank of S. Louis. On forcasing modl is a rgrssion of h growh ra of h indusrial producion 20

21 indx on a consan, on lag of h growh ra and wo lags of h sprad bwn Mood s Aaa and Baa corpora bond ilds. Th ohr modl rplacs h crdi sprad wih h log of housing sars for prival ownd housing. Th daa ar monhl from 1959:03 o 2015:08. W appl h ss o on-monh ahad forcass for 2011:06 o 2015:08. Th modls wr simad rcursivl. Using 4,999 boosrap draws, w applid h boosrap s of Clark and McCrakn (2014, p.421) o drmin if h modls ar ovrlapping or non-nsd. Th null hpohsis ha h modls ar ovrlapping can b rjcd a h 5% lvl. For h null hpohsis of qual prdiciv accurac w obain DM ( ˆ θ ( ˆ β )) = 1.47 which is insignifican a h 10% lvl, whras DM ( θ ˆ (13) ( β )) = 3.025and DM ( θ ˆ (4) ( β )) = 2.162which ar significan, rspcivl, a h 1% and 5% lvls. 5. Conclusion Th s proposd b Dibold and Mariano (1995) is widl rgardd as an imporan s ha addrsss h nd o formall assss whhr diffrncs in h accurac of wo prdicors ar purl sampling rror. Using a GMM framwork, w proposd a mor powrful vrsion ha can b usd o compar h accurac of rgrssion prdicors basd on non-nsd modls. On advanag of h original DM s is ha i circumvns h problm of imposing assumpions on h prdicors b imposing assumpions dircl on h prdicion rrors. As w dmonsrad, howvr, his coms wih a cos in rms of powr. Spcificall, w showd ha mor powrful vrsions of h ss can b dvisd b xploiing propris of linar projcions ha rquir assuming onl h xisnc of crain momns and covarianc-saionari. Th simulaion xprimns illusra ha h ponial gains in powr can b considrabl. Dircions for fuur rsarch includ xnsions o hpohss of fini-sampl (as opposd o populaion-lvl) 21

22 prdiciv accurac, and h simaion of h opimal wigh marix whn h covarianc marix is singular for ss of ovrlapping and nsd modls. Appndix To prov par (i) of Proposiion 1, i suffics o show ha Assumpions 1 hrough 4 of Ws (1996) hold for m ( β ). Th rsul hn follows dircl from Thorm 4.1 of Ws (1996) which also assums a rcursiv forcasing schm. As nod abov, our Assumpions 3 and 4 ar Assumpions 3 and 4 in Ws (1996) in diffrn noaion. Clarl, m ( β ) is masurabl and wic coninuousl diffrniabl wih scond-ordr drivaivs ha do no dpnd on β. Undr our Assumpion 1, h scond-ordr drivaivs hav fini xpcaion. Thrfor, Assumpion 1 of Ws (1996) holds. Nx no ha undr our Assumpion 1: ˆ β β = BH () () whr B( ) = diag( B ( ), B ( )), s= 1 H () = [ H (), H ()] = h ( β ), B= plim B ( ), B has full s column rank, and Eh ( β ) = 0. Thrfor, Assumpion 2 of Ws (1996) holds. Consqunl, par (i) follows from Ws (1996, Thorm 4.1). Par (ii) follows from par (i) b noing ha  convrgs in probabili o A undr Assumpion 2 and ˆ ˆ * ˆ P( θ θ( β )) = ca P( µ ( β ) µ ) sinc Qµ = 0 undr Assumpion 1. Par (iii) follows from par (i) b noing ha ˆ ˆ * ˆ c P P( θ θ( β )) = ( µ ( β ) µ ). Par (iv) follows from subsiuing W 1 = W ino A and mulipling ou caω A c. 22

23 Rfrncs Androu, E., E. Ghsls and A. Kourllos (2013) Should Macroconomic Forcasrs Us Dail Financial Daa and How? Journal of Businss and Economic Saisics 31-1, Andrws, D.W.K. (1987) Asmpoic Rsuls for Gnralizd Wald Tss, Economric Thor 3, Chambrlain, G. (1982) Mulivaria Rgrssion Modls for Panl Daa, Journal of Economrics 18, Clark, T.E. and M.W. McCrackn (2001) Tss of Equal Forcas Accurac and Encompassing for Nsd Modls, Journal of Economrics 105, Clark, T.E. and M.W. McCrackn (2006) Th Prdiciv Conn of h Oupu Gap for Inflaion: Rsolving In-Sampl and Ou-of-Sampl Evidnc, Journal of Mon, Crdi and Banking 38-5, Clark, T.E. and M.W. McCrackn (2013) Advancs in Forcas Evaluaion Handbook of Economic Forcasing 2, Clark, T.E. and M.W. McCrackn (2014) Tss of Equal Forcas Accurac for Ovrlapping Modls, Journal of Applid Economrics 29, Clark, T.E. and M.W. McCrackn (2015) Nsd Forcas Modl Comparisons: A Nw Approach o Tsing Equal Accurac, Journal of Economrics 186(1), Corradi, V., N.R. Swanson and C. Oliva (2001) Prdiciv Abili wih Coingrad Variabls, Journal of Economrics 104, Dibold, F.X. and R.S. Mariano (1995) Comparing Prdiciv Accurac, Journal of Businss and Economic Saisics 13, Diz d los Rios, A. (2015) Opimal Asmpoic Las Squars Esimaion in a Singular S- Up, Economics Lrs 128,

24 Giacomini, R. and H. Whi (2006) Tss of Condiional Prdiciv Abili, Economrica 74-6, pp Hamilon, J. D. (1994) Tim Sris Analsis. Princon: Princon Univrsi Prss. Hanson, B.E. (2016) Economrics. Unpublishd xbook manuscrip. Univrsi of Wisconsin. Hanson, L.P. (1982) Larg Sampl Propris of Gnralizd Mhod of Momn Esimaors, Economrica 50, Hanson, P.R. and A. Timmrman (2015) Equivalnc bwn Ou-of-Sampl Forcas Comparisons and Wald Saisics, Economrica 83-6, Hanson, P.R. and A. Lund (2005) A Forcas Comparison of Volaili Modls: Dos Anhing Ba a GAR(1,1) Modl?, Journal of Applid Economrics 20: Hong, Y. and T.H. L (2003) Infrnc on Prdicabili of Forign Exchang Ras via Gnralizd Spcrum and Nonlinar Tim Sris Modls, Rviw of Economics and Saisics 85-4, Marcllino, M and B. Rossi (2008) Modl Slcion for Nsd and Ovrlapping Nonlinar, Dnamic and Possibl Misspcifid Modls, Oxford Bullin of Economics and Saisics 70, Mark, N.C. (1995) Exchang Ras and Fundamnals: Evidnc on Long-Horizon Prdicabili, Amrican Economic rviw 85, McCrackn, M. W. (2007) Asmpoics for ou of sampl ss of Grangr causali, Journal of Economrics 140-2, Nas, R., J.A. Skjlorp and B.A. Ødgaard (2011) Sock Mark Liquidi and Businss Ccl, Journal of Financ 66, Nw, W.K. and K.D. Ws (1987) A Simpl, Posiiv Smi-dfini Hroscdasici and auocorrlaion Consisn Covarianc Marix, Economrica 55,

25 Nw, W.K. and K.D. Ws (1994) Auomaic Lag Slcion in Covarianc Esimaion, Rviw of Economic Sudis 61, Pnaranda, F. and E. Snana (2012) Spanning Tss and Sochasic Discoun Man-Varianc Fronirs: A Unifing Approach, Journal of Economrics 170, Rao, C.R. (1973) Linar Saisical Infrnc and Is Applicaions (2 nd d.), Nw York: John Wil & Sons. Sock, J.H. and M.W. Wason (2002) Macroconomic Forcasing using Diffusion Indxs, Journal of Businss and Economic Saisics 20-2, Sock, J.H. and M.W. Wason (2007) Wh has U.S. Inflaion Bcom Hardr o Forcas?, Journal of Mon, Crdi and Banking 39-1, Swanson, N.R. and H. Whi (1997) A Modl Slcion Approach o Ral-im Macroconomic Forcasing using Linar Modls and Arificial Nural Nworks, Rviw of Economics and Saisics 79-4, Vuong, Q. (1989) Liklihood Raio Tss for Modl Slcion and Non-Nsd Hpohss, Economrica 57, Ws, K.D. (1994) Asmpoic Infrnc abou Prdiciv Abili, manuscrip, Univrsi of Wisconsin. Ws, K.D. (1996) Asmpoic Infrnc abou Prdiciv Abili, Economrica 64, Ws, K.D. (2006) Forcas Evaluaion, in Handbook of Economic Forcasing, Volum 1, did b G. Ellio, C.W.J. Grangr and A. Timmrman. Amsrdam: Norh Holland. Wooldridg, J.M. (2010) Economric Analsis of Cross Scion and Panl Daa, Scond Ediion, Cambridg, MA: MIT Prss. 25

26 Tabl 1 Sandard Dviaions undr h Null Hpohsis P R ˆ θ ( ˆ β θ ˆ (13) θ ˆ (4)

27 Tabl 2 Mans undr h Null Hpohsis P R ˆ θ ( ˆ β θ ˆ (13) θ ˆ (4)

28 Tabl 3 Rjcion Ras for nominal siz of 10% undr h Null Hpohsis P R DM ( ˆ θ ( ˆ β )) DM ( θ ˆ (13) ) DM ( θ ˆ (4) )

29 Tabl 4 Rjcion Ras for nominal siz of 10% undr h Alrnaiv Hpohsis P R DM ( ˆ θ ( ˆ β )) DM ( θ ˆ (13) ) DM ( θ ˆ (4) )