Comparing Possibly Misspeci ed Forecasts

Transcription

1 Supplemenl Appendx : Cmprng Pssbly Msspec ed Frecss Andrew J. Pn Duke Unversy 4 Augus Ts supplemenl ppendx cnns w prs. Appendx SA. cnns dervns used n e nlycl resuls presened n e pper. Appendx SA. cnns prfs f Prpsns 4 nd 6 presened n e mn pper. Appendx SA.: Dervns Te resuls belw drw n e fllwng lemm, wc summrzes sme useful resuls n mmens rse wen e d re Gussn nd e lss funcn s expnenl regmn. Lemm If X s N ; nd (; b) R, en () E [exp f + bxg] = exp + b + b () E [exp f + bxg X] = exp + b + b + b () E exp f + bxg X = exp + b + b + + b (v) E exp f + bxg X 3 = exp + b + b + b b Sme resuls belw re smpl ed f we cnsder e fllwng de nn: De nn A frecs ^Y s men unbsed f ^Y = E Y jf.s. y e lw f ered expecns, s mples E Y j ^Y = ^Y :s. Ne s des n requre F cnns ll relevn nfrmn fr frecsng Y ; nly ^Y pmlly uses ll nfrmn vlble n F : Appendx SA.: Dervns fr e AR(p) mdels n Secn. Te Gussn AR(5) spec cn n equn () mples: Y Y Y Y 3 Y 4 s N (5 ; ) (5)

2 were 5 s (5 ) vecr f nes, s e men f Y nd s e cvrnce mrx f e lef-nd sde vecr. Tese cn be bned usng sndrd meds frm me seres nlyss, see Hmln (994) fr exmple). Le [ ; ; :::; 5 ] (6) en = were F = ; nd vec () = (I 5 (F F )) vec (Q) I 4 4 5, Q = e e, e [; ; ; ; ] Ten ne e jn dsrbun f (Y ; Y ) s [Y ; Y ] s N ( ; ) (7) 3 were 4 5, nd j Cv [Y ; Y j ], fr j = ; ; ; ::: Dene j j = : Ten e cndnl dsrbun f Y jy s: Y jy s N ( ) + Y ; () nd s fr e prmeers used n e exmple we nd [ ; ] = [ ( ) ; ] = [:5; :5]. Smlr clculns fr e AR() mdel yeld: [Y ; Y ; Y ] s N ( 3 ; 3 ) (9) 3 were () 5 Y j (Y ; Y ) s N ( + Y + Y ; V AR ) = 4 ( ) 5, = ( ) 3 3 V AR = = + And fr e prmeers used n e exmple we nd [ ; ; ] = [:3; :76; :] : Nw we derve e expeced lss fr e AR(), AR() nd AR(5) frecss. Frs ne snce ll ree f ese frecss re men-unbsed, e expeced lss f ny regmn lss funcn ()

3 smpl es : E L Y ; ^Y ; = E [ (Y )] E ^Y E ^Y E Y j ^Y Fr expnenl regmn lss, were (Y ; ) = exp fy g, Lemm mples T bn E exp n ^Y n E [ (Y )] = E [exp fy g] = exp ( + ) ^Y = E [ (Y )] E ^Y : () (3) fr e AR(), AR() nd AR(5) frecss, we expl e fc fr s Gussn uregressn, ll f ese frecss re uncndnlly nrmlly dsrbued: ^Y ARk s N (; V ARk ), were [V AR ; V AR ; V AR5 ] = ; + + ; : Tus we nd E ^Y ARk = ARk E exp ^Y n = exp ( + V ARk) frm Lemm. Te expeced lss frm n AR (k) frecs s E L Y ; ARk ^Y ; n = exp ( + ) n exp ( + V ARk)! V ARk s! Ne we knw V AR V AR V AR5 nd s we mmedely see e rnkng under MSE (.e., expnenl regmn w! ) s E L Y ; ^Y AR E L Y ; ^Y AR E L Y ; : ^Y AR5 Appendx SA..: Dervns fr e ernull frecsers n Secn. Snce ^Y X nd ^Y W re b pml w respec er lmed nfrmn, ey re b exp fy g exp n ^Y : men unbsed nd s er expeced regmn lss smpl es E Fr s DGP we esly nd: E [exp fy g] = exp + ( L + C ) pq + exp + ( H + C ) ( p) q (4) + exp + ( L + M ) p ( q) + exp + ( H + M ) ( p) ( q) E exp ^Y X = exp f ( L + q C + ( q) M )g p + exp f ( H + q C + ( q) M )g ( p) E exp ^Y W = exp f (p L + ( p) H + C )g q + exp f (p L + ( p) H + M )g ( q) 3

4 Fgure nrmlzes e expeced lss frm frecs X nd W by e pml frecs, usng b sgnls: wc leds XW E exp ^Y ^Y XW = X L + ( X ) H + W C + ( W ) M (5) = exp f ( L + C )g pq + exp f ( H + C )g ( p) q (6) + exp f ( L + M )g p ( q) + exp f ( H + M )g ( p) ( q) Appendx SA..3: Dervns fr e lner mdel n Secn.3 Te rs-rder cndn fr e pml prmeer [; ] s: E [L (Y; m (X; ) ; = E (m (X; )) (E [Y jx] m (X; (X; = E exp f ( + X)g X X [; X] (7) S e w rs-rder cndns re: = E exp f ( + X)g X E [exp f ( + X)g] E [exp f ( + X)g X] () = E exp f ( + X)g X 3 E [exp f ( + X)g X] E exp f ( + X)g X (9) Usng Lemm bve we ve ec f e fur unque erms bve n clsed frm. Subsung ese n nd slvng fr slvng fr [; ] yelds e expressns gven n equn (5). 4

5 Appendx SA.: Addnl prfs Te prfs belw use e fllwng resuls n unfrm rndm vrble, nd rngulr rndm vrble w mde L nd PDF declnes lnerly zer U > L. X s Unf (L; U) Z s T r (L; U) () ; z < L ; z < L >< >< F x (x) = x L (U L) U L ; z [L; U] F z (z) = (U z) ; z [L; U] () (U L) >: ; z > U >: ; z > U < U L ; z [L; U] < (U x) ; z [L; U] (U L) f x (x) = f z (z) = (3) : else : else F x () = L + (U L), fr [; ] F z () = U (U L) p, fr [; ] (4) E [X] = (U + L) E [Z] = 3 (L + U) (5) E X = 3 L + U + LU E Z = 6 3L + LU + U (6) E X 3 = 4 L3 + U 3 + L U + LU E Z 3 = 4L3 + 3L U + LU + U 3 (7) M x Medn [X] = (U + L) M z Medn [Z] = U U L p () E [ fx < bg X] = b L (U L), fr b [L; U] E [ fz bg Z] = 3b U b 3 L (3U L) 3(U L), fr b [L; U] (9) E fx < bg X = b3 L 3 3(U L) E fz bg Z = 4b3 U 3b 4 L 3 (4U 3L) 6(U L) () E fx < bg X 3 = b4 L 4 4(U L) E fz bg Z 3 = 5b4 U 4b 5 L 4 (5U 4L) () (U L) E [ fx < M x g X] = 3L+U E [ fz M z g Z] = U(p p )+L(4 ) 6 () E fx < M x g X = 7L +4LU+U 4 E fz M z g Z = 9L +LU(7 4 E fx < M x g X 3 = (3L+U)(5L +LU+U ) 64 E fz M z g Z 3 = L3 ( p )+U ( p ) 4 (3) p )+3L U( p ) 4 (4) + LU (9 3 p )+U 3 ( p 3) 4 5

6 Te resuls n pr () belw use e dsrbun f Y = X + Z; were X s Unf (L; ) nd Z s Unf (; U) ; were L < < jlj < U: Ts vrble s e fllwng prperes: ; y < L y L (L y) LU ; y [L; ] >< LU y [L; ] >< F y (y) = y L U ; y [; L + U] U y [; L + U] nd f z (z) = U y LU (U y) LU y [L + U; U] LU y [L + U; U] >: >: ; else ; y > U L + p jlj U; L ; >< U Fz () = L + U; L U ; L+U U And en >: U p ( ) jlj U; L+U U ; (3) (3) E [Y ] = (L + U) (3) E Y = 6 L + 3LU + U (33) E Y 3 = 4 L3 + L U + LU + U 3 M y Medn [Y ] = (L + U) (34) E [ fy M y g Y ] = 6L L 4 U + 3U (35) E fy M y g Y = (L + U)3 L 3 4U E fy M y g Y 3 = 5 (L + U)4 6L 4 3U (36) (37) Anlgus e men cse, de ne n -qunle unbsed frecs s ne wc ss es: Ne fr n -qunle unbsed frecs we ve: E L Y; ^Y ; g E Y ^Y g ^Y n = E g ^Y Y ^Y = E g ^Y E Y ^Y j ^Y = E [g (Y )] E Y ^Y E Y ^Y j ^Y = (3) g (Y ) g (Y ) E Y ^Y E n Y ^Y g (Y ) g (Y ) + E [g (Y )] (39) 6

7 Hlzmnn nd Euler (4) presen d eren prf f pr () f Prpsn 4 belw. We presen e fllwng fr cmprbly w e cndnl men cse presened n Prpsn f e pper. Prf f Prpsn 4. () We wll sw under Assumpns () (3), LnLn LnLn A ) F F A ) E L Y ; ^Y E L Y ; ^Y A L L GP L ; were LnLn j E LnLn Y ; ^Y j fr j fa; g nd LnLn s e Ln-Ln lss funcn n equn (7). Frs: we re gven LnLn LnLn A ; nd ssume F A F : Ts mples E LnLn Y ; ^Y A jf E LnLn Y ; ^Y jf :s: ; snce ^Y A F A F, nd E LnLn Y ; ^Y A E LnLn Y ; ^Y by e LIE. Te nly wy s ls ssfy e ssumpn LnLn LnLn A LnLn s f E Y ; ^Y A jf = E LnLn Y ; ^Y jf :s:. Le Y c n^y = : = F ^Y, fr fa; g : Ts ccmmdes e fc we d n ssume F ; fr fa; g ; s srcly ncresng, nd s e -qunle s n necessrly unque. Te necessy nd su cency f GPL lss (wccludes LnLn lss) fr qunle esmn, mples s se cn lernvely be de ned s Y c = rg mn^y Y b E LnLn (Y ; ^y)j F : Tus E LnLn Y ; ^Y A jf = E LnLn Y ; ^Y jf :s: mples ^Y A Y c nd s c Y A \ c Y 6=? : Ts vles Assumpn, ledng cnrdcn. Tus LnLn LnLn A ) F F A : Nex: Le L j E L GP L Y ; ^Y j ; ; g, j fa; g were L GP L (; ; ; g) s GPL lss funcn de ned by g; nndecresng funcn. Under Assumpns ()-(3) we knw ^Y j s e slun mn^y E L GP L Y ; ^Y j ; ; g jf j : I s srgfrwrd sw ^Y j en ss es = E Y ^Y j jf j : Ts lds fr ll pssble (cndnl) dsrbuns f Y ; nd frm Serens () nd Gneng (b) we knw s mples (by e necessy f GPL lss fr pml qunle frecss) ^Y j ^Y j = rg mn ^y E ( fy ^yg ) (g (^y) g (Y )) jf j en mrever ss es fr ny nndecresng funcn g: If F F A en by e LIE we ve L (g) L A (g) fr ny nndecresng funcn g: (b)() We rs cnsder e cse f nn-nesed nfrmn ses (vlng Assumpn ). Cn- 7

8 sder e fllwng smple exmple: Y = X + Z (4) were X s Unf (; ), Z s T r (; ) ; X?Z Le = ; nd ssume frecs A cndns n X nd frecs cndns n Z: Ten: ^Y = X + Medn [Z] = X + :45, snce Medn [Z] = 6 p 3:5 (4) ^Y b = Z + Medn [X] = Z + :5, snce Medn [X] = 5 (4) Nex cnsder e GPL lss funcns genered by g (y) = y nd g (y) = y 3 : Nce b ^Y nd ^Y b re medn-unbsed frecss, wc smpl es e clculn f er expeced lss. L A (g ) E Y ^Y = ^Y = n E [Y ] E Y ^Y Y Y (43) were E [Y ] = E [X] + E [Z] (44) nd E Y ^Y Y = E [ fx + Z X + M z g (X + Z)] (45) = E [ fz M z g] E [X] + E [ fz M z g Z], snce X?Z = E [X] + E [ fz M zg Z], snce E [ fz M z g] = = We nd n nlgus expressn fr e er frecser: L (g ) = n E [Y ] E Y ^Y b Y = E [Y ] E [Z] + E [ fx M xg X] (46) Nex cnsder e lss GPL funcn bned wen g (y) = y 3 : n L A (g ) E Y ^Y 3 = ^Y = E Y 3 E Y 3 (47) ny ^Y Y 3 (X + Z) 3 = E X 3 + 3E X E [Z] + 3E [X] E Z + E Z 3 (4) E Y 3 = E E Y ^Y Y 3 = E [ fz M z g] E X 3 + 3E [ fz M z g Z] E X (49) +3E fz M z g Z E [X] + E fz M z g Z 3

9 Pullng ese erms geer nd usng e expressns fr ese mmens gven bve, we nd: L A (g ) = :7 < :5 = L (g ) (5) L A (g ) = 35:45 > 349:3 = L (g ) (5) Tus e rnkng s reversed dependng n e cce f funcn g. Ne wle e d erences n ese vlues my pper smll, ese re nlycl ppuln vlues, nd s ere s n smplng r smuln vrbly. () Nex we cnsder e cse b frecsers use crrecly spec ed mdels, gven er (nesed) nfrmn ses, bu ey re subjec esmn errr. Assume Y = X + Z (5) X s Unf ( ; ), Z s Unf (; ), X?Z Assume frecser A uses n cndnng nfrmn, nd s reprs er pml frecs s: ^Y = Medn [Y ] = (53) Frecser uses nfrmn n Z; bu expl mus esme Medn [X] : He res s n unknwn prmeer nd ssume e esmes usng n = bservn f X. Frecser s predcn wll en be ^Y b = X ~ + Z (54) were X ~ s relzn frm Unf (L; ) dsrbun, ndependen f (X; Z) : Ts desgn llws fr sgnl/nse rde- : In s desgn we nd : L A (g ) E Y ^Y = ^Y Y (55) = E [( fy M y g =) (M y Y )] = M y E [ fy M y g] E [ fy M y g Y ] = (M y E [Y ]) 9

10 Fr frecser we nd: L (g ) E Y ^Y b = ^Y b = E X + Z X ~ + Z = E X X ~ ~X X ~X Y ~X + Z X Z = = E ~X + Z = E X X ~ E X X ~ X = E E X X ~ j X ~ ~X E E X X ~ = E F x ~X ~X E [( F x (X)) X] Y, ne E ~X + Z jx X Y = (56) = E [F x (X) X] E [X], snce ~ X d = X And fr e secnd lss funcn we bn: n L A (g ) E Y ^Y 3 = ^Y = E ( fy M y g =) M 3 y Y 3 Y 3 = M 3 y E [ fy M y g] E fy M y g Y 3 = M 3 y E Y 3 (57) nd n L (g ) E Y ^Y b 3 = ^Y b n = E X + Z X ~ + Z n = E X X ~ 3 ~X + Z Y 3 3 = ~X + Z (X + Z) 3 = (X + Z) 3 3 E ~X + Z = E X X ~ ~X X ~ Z + 3 XZ ~ X 3 3X Z 3XZ = E X X ~ ~X 3 E X X ~ X 3 +3E [Z] E X X ~ ~X E X X ~ X +3E Z E X X ~ ~X E X X ~ X (5) E (X + Z) 3 Ten we use, fr p = ; ; 3 : E X X ~ ~X p = E E X X ~ j X ~ ~X p = E [F x (X) X p ], snce X ~ = d X (59) E X X ~ X p = E E X X ~ jx X p = E [( F x (X)) X p ] = E [X p ] E [F x (X) X p ] (6)

11 And s L (g ) = E F x (X) X 3 E X 3 (6) +3E [Z] E F x (X) X E X +3E Z (E [F x (X) X] E [X]) Fr X s Unf (L; U) we ve: E [F x (X) X] = L + U 6 E F x (X) X = L + LU + 3U E F x (X) X 3 = L3 + L U + 3LU + 4U 3 (6) (63) (64) Pullng ese erms geer, we nd L A (g ) = :5 > :67 = L (g ) (65) L A (g ) = 7:65 < 9 = L (g ) (66) Tus e rnkng s reversed dependng n e cce f funcn g. () Fnlly, we cnsder vln ssumpn 3, nd cnsder mdels re msspec ed. We wll smplfy e DGP, nd ssume Y = X s Unf (; ) (67) We wll ssume e w frecsers use msspec ed mdels, n ey use lner mdel w prmeers d er frm (; ): ^Y = + X (6) ^Y b = + X (69) Of curse ere we cnn use e smplfcn lds wen e frecss re medn unbsed. In s exmple, f we se ( ; ) = (:33; :67) nd ( ; ) = ( :5; :5) en b frecss use e sme nfrmn se, neer s esmn errr, bu b re bsed n msspec ed mdels.

12 In s cse we nd: L A (g ) E Y ^Y = ^Y Y (7) E [ f( ) X g] = E [ f( ) X g X p ] = = E [( fx + Xg =) ( + X X)] = E [ f( ) X g] + ( ) E [ f( ) X g X] F x >< ; < F x ; > >: >< >: f g ; = E X E E [X p ] E [X] X p ; < n X X p ; > f g E [X p ] ; = (7) (7) Te sme expressns cn be used fr L (g ) pluggng n ( ; ) fr ( ; ) : We use p = fr e rs GPL lss funcn bve, nd p = ; ; 3 fr e secnd, belw. Nex cnsder n L A (g ) E Y ^Y 3 = ^Y Y 3 = E f( ) X g ( + X) 3 X 3 =E ( + X) 3 X 3 (73) = 3 E [ f( ) X g] + 3 E [ f( ) X g X] +3 E f( ) X g X + 3 E f( ) X g X 3 = E [X] + 3 E X + 3 E X 3 Te sme expressns cn be used fr L (g ) pluggng n ( ; ) fr ( ; ) : Pullng ese erms geer, we nd L A (g ) = :6 > :5 = L (g ) (74) L A (g ) = 79:44 < :9 = L (g ) (75) Tus e rnkng s reversed dependng n e cce f funcn g. We ve us demnsred nlyclly e presence f ny f nn-nesed nfrmn ses, esmn errr, r mdel msspec cn cn led sensvy n e rnkng f w qunle frecss e cce f cnssen (GPL) lss funcn.

13 Prf f Prpsn 6. () We gn prve s resul by swng E L F A ; Y E L F ; Y fr sme L LPrper ) F F A ) E L F A ; Y E L F ; Y L L Prper : Frs: we re gven E L F A ; Y E L F ; Y ; nd ssume F A F : Under Assumpns ()-(3), s mples we cn ke F s e d generng prcess fr Y : Ten E L F ; Y jf = EF L F ; Y jf EF L F A ; Y jf e prprey f L: y e LIE s mples E L F ; Y E L F A ; Y ; wc cn nly ld f E L F ; Y jf = E L F A ; Y jf :s: ; bu snce L s srcly prper scrng rule s mples F A = F :s: wc vles Assumpn, ledng cnrdcn. Tus E L F A ; Y E L F ; Y fr sme L L Prper ) F F A : Nex, usng smlr lgc bve, gven F F A we ve E L F A ; Y E L F ; Y fr ny L LPrper ; cmpleng e prf. (b)() We rs cnsder e cse f nn-nesed nfrmn ses (vlng Assumpn ). Cnsder e fllwng exmple: Y = A ( A) + + ( ) (76) A s ernull (p) s ernull (q),?a > > Te ndcr, A revels weer e lef l wll be lng r sr, nd revels weer e rg l wll be lng r sr. Frecser A bserves e sgnl A nd frecser bserves sgnl ;.e., ec frecser nly ges nfrmn bu sngle l (lef r rg). Ten we nd: Z E [wcrp S (F A ; Y;!)] = pq ( q)! (z) dz + q ( p) ( q) E [wcrp S (F ; Y;!)] = pq ( p) Z! (z) dz + p ( p) ( q) Z Z! (z) dz (77)! (z) dz Te w prper scrng rules we cnsder (equn 33) plce d eren wegs n e lef vs. rg ls usng e lgsc funcn:! (z; ) = + exp f zg Wen > mre weg s plced n e rg l, nd wen < mre weg s plced n e (7) 3

14 lef l. We en cmpue e negrls, seng! R (z) =! (z; +) nd! L (z) =! (z; ) Z! R (z) dz = Z! R (z) dz = Z Z! L (z) dz = + lg lg (exp f g + exp f g) (79)! L (z) dz = lg ( + exp f g) W ese n nd, f we se (p; q; ; ) = (:5; :75; ; 5) we nd: E [wcrp S (F A ; Y ;! R )] = :5 > :5 = E [wcrp S (F ; Y ;! R )] () E [wcrp S (F A ; Y ;! L )] = :5 < :5 = E [wcrp S (F ; Y ;! L )] () And s e rnkng f ese w dsrbun frecss cn be reversed dependng n e cce f (prper) scrng rule. () Nex, we cnsder vln ssumpn 3, nd cnsder mdels re msspec ed. In s cse, cnsder e cse were frecser A uses e uncndnl dsrbun f e rge vrble, wle frecser cnnues use er sgnl, bu bsed n ~p 6= p: If we se (p; q; ; ; ~p) = (:5; :75; ; 5; :5) we nd E wcrp S F A ; Y ;! R E wcrp S F A ; Y ;! L = :6 > :33 = E wcrp S ~F ; Y ;! R = :6 < :67 = E wcrp S ~F ; Y ;! L () (3) And s e rnkng f ese w dsrbun frecss cn be reversed dependng n e cce f (prper) scrng rule. (Ne E wcrp S F A ; Y ;! R = E wcrp S FA ; Y ;! L s e dsrbun frecs FA s symmerc rund zer, nd e wegng funcns ssfy!r (z) =! L ( z).) () Fnlly, we cnsder e cse b frecsers use crrecly spec ed mdels, gven er (nesed) nfrmn ses, bu re subjec esmn errr. Cnsder e cse frecser A gn uses e uncndnl dsrbun f e rge vrble, wle frecser uses er sgnl, bu d s mus esme e prmeer p: Assume se des s bsed n n bservns f e sgnl A: (Ne snce frecser bserves e sgnl ; e vlue fr A cn be bcked u, ex ps, frm e relzed vlue f e rge vrble.) Ten n^p = nx A s nml (n; p) (4) = 4

15 In s cse, we ve: E wcrp S ^F (^p) ; Y ;! = X ~p E wcrp S ~F (~p) ; Y ;! Pr [^p = ~p] (5) nd we cn use e expressns frm pr () elp slve s prblem. Cnsder e cse n = 4; nd s ^p cn ke ne f ve vlues f; =4; =; 3=4; g : In s cse we nd E wcrp S E wcrp S F A ; Y ;! R F A ; Y ;! L = :6 > :3 = E wcrp S ^F (^p) ; Y ;! R = :6 < :6 = E wcrp S ^F (^p) ; Y ;! L (6) (7) And s e rnkng f ese w dsrbun frecss cn be reversed dependng n e cce f (prper) scrng rule. 5