CONSISTENT ESTIMATION OF THE NUMBER OF DYNAMIC FACTORS IN A LARGE N AND T PANEL. Detailed Appendix

Transcription

1 COSISE ESIMAIO OF HE UMBER OF DYAMIC FACORS I A LARGE AD PAEL Dealed Aendx July 005 hs verson: May 9, 006 Dane Amengual Dearmen of Economcs, Prnceon Unversy and Mar W Wason* Woodrow Wlson School and Dearmen of Economcs, Prnceon Unversy and he aonal Bureau of Economc Research

2 hs aendx conans dealed roofs for resuls saed n Amengual and Wason 005 o mae hs documen self-conaned begns wh a descron of he model and assumons before sang he resuls and roofs Model: Λ F + e, for,,, where and e are, F s r, and Λ s r F evolves as a VAR: F Φ F + ε, where ε Gη where G s r q wh full column ran and η s sequence of shocs wh mean zero and covarance marx Σ I Combnng he equaons yelds ηη q Y Γ η + e, 3 where Y ΛΦF and Γ Λ G ransosng and sacng he equaons yelds FΛ + e, 4 where s, F s r, Λ s r, and e s he h rows of, F and e are, F, and e ; he h row of Λ s λ ; he h elemen of s denoed, and smlarly for e, so ha λ F + e

3 Le F,, F F,,,, he defnon for Y are hen Φ Φ Φ Φ, and Π ΛΦ he VAR for F and F Φ F + Gη and Y ΠF Fnally, leng π denoe he h row of Π and γ denoe he h row of Γ, hen η γ + F π + e Assumons: Raes:, jonly equvalenly ha wh lm Le s mn, A E FF I r A E λλ Σ, where Σ ΛΛ s a dagonal marx wh elemens σ > σ > 0 for ΛΛ < j When Λ s deermnsc, Σ ΛΛ s nerreed as he lmng emrcal average jj A3 A4 FF Ir λλ ΣΛΛ e A5 e σ > 0

4 A6 For some neger m and for all negers j m, j j max {, } Erace ee O A7 λ s s E Fe O A8 λλ E e O A9 E Fe O A0 Le F,, F F, hen he sochasc rocess { F } s saonary and ergodc; E FF s non-sngular; and vec η F s a marngale dfference sequence wh fne second momens A F E e O Addonal oaon:, Λ λ V F arg mn V F, Λ mn Λ Λ Wh FF / I, V, Λ race FF R F race FF mn 3

5 F : Maxmzng R F yelds F wh columns gven by he normalzed egenvecors of corresondng he larges egenvalues; hese maxmze R F and mnmze V Λ F / Λ F / and λ F /, where s he h column of F denoes a marx and { / } I R F race F F * R race FΛΛ FF F denoes he se of ordered egenvecors of, normalzed as F F / I g, s a deermnsc sequence ha sasfes g, 0 and s g, m/m, where m s gven n assumon A5 for he larges-o-smalles ordered egenvalues of σ are ω, ω,, R RF ω PC, σ R, + g, ICP, ln σ R, + g, PC B argmn PC,, max 0 r ICP B argmn ICP, max 0 r AB For conformable marces A and B, Σ m AB, where m s he number of rows of A 4

6 Lemmas and heorem n Amengual and Wason 005: Lemma Ba-g: Under assumons A-A9, PC B r and ICP B r Follows from R30, R3, R34, and R36 below Lemma : Suose A-A9 are sasfed and + b where, hen PC b O s B r and ICP B r Follows from R4 below heorem: Consder he model -3 Suose ha sasfes A-A9, ha he analogous assumons are sasfed for 3, and ha A0 s sasfed hen a PC a B Y q and IPC a B Y q b In addon, suose ha A s sasfed hen PC b B Y q and IPC b B Y q a Follows from R48 and R55 below; b follows from R54 and R55 5

7 Dealed Resuls: R For j m, j j j+ race ee O s he resul follows from A6 and he defnon of s R λ s s Fe O he resul follows from A7 R3 Λ e O Λ λλ where he rae follows from A8 e e O R4 e O he resul follows mmedaely from R wh j R5 For all j, Fje O he resul follows mmedaely from A9 R6 m/ m su race F eef O s 6

8 su race F eef s equal o sum of he larges egenvalues of ee whch s less han or equal o µ, where µ denoes he larges egenvalue of ee Bu m m m µ s he larges egenvalue of ee, he larges egenvalue s bounded above by he race, so ha µ, where he las m m m m+ race ee O s equaly follows from R, and he resul follows drecly R7 / su race F F ef O Λ Le f m denoe he m h column of F and f m denoe he h elemen of f m hen Λ m Λ m m race F F ef f F ef f m f smf λes m s f m f sm F λes m s / f m f sm F λes m s s bu f / m f sm f m f m, and for all s, / I hus, / / Λ λ s s su, race F F ef F e O where he las equaly follows from R /, R8 and * / su RF R F O s * R F R F race F eef + race F FΛ ef 7

9 * su su RF R Λ race FeeF + Λ su, race F F ef where he wo erms on he rhs of he nequaly are R7, resecvely O s and / / O by R6 and R9 * / su RF su R F O s, * * / su RF su R F su RF R F O s where he frs nequaly follows by he defnon of he su and he convergence follows from R8 R0 Le * su mn r, R σ / / FF / FF / FF / denoe he Choles facorzaon of F F / Le F be reresened as + where VF 0 oe: / F F FF/ V F F / + VV /, so ha for all, I hus, we can wre * / / su R su : I race FF/ / FF/ F ΛΛ A drec calculaon shows ha he soluon s mn r, / / : I race FF ΛΛ FF σ su / / /, where σ s he h larges egenvalue of FF / / FF / / / ΛΛ oe, o derve hs, frs noe ha whou loss of generaly we can assume ha s dagonal, because osmullyng by an orhonormal marx does no change he value of he race Omzaon can hen be carred ou on each column of sequenally, and hs yelds he sandard egenvalue resul 8

10 Bu ΛΛ / Σ and FF / I by A3 and A4, so ha ΛΛ / / FF / / FF / ΛΛ ΛΛ Σ, and by connuy of egenvalues σ σ R su RF σ mn r, hs follows from R9 and R0 R mn r, * R F σ F arg su R, so he resul follows from R8 and R R3 Λ Fe F O FF Fe ΛF λfe s O, s where he nequaly follows from CS aled o he sum over mlc n Fe and he rae follows from R R4 Fee O s FF Fee ees O s s, where he nequaly follows from CS aled o he sum over mlc n Fe and he rae follows from R wh j 9

11 R5 Le f denoe he frs column of F and le S sgn f f meanng S f f f 0 and S f f f < 0 hen SfF / where, 0,, 0 For arcular values of and V, we can wre / f F FF / V + where VF 0 and oe ha s r Le ha Snce hus * R f C C ΛΛ C FF FF Σ + Σ / / / ΛΛ / / and noe * R f σ C ΛΛ λλ σ r C Σ ΛΛ + σ+ σ Σ and s bounded, he frs erm on he rgh hand sde of hs exresson s o hs resul ogeher wh R when mles σ + σ 0 r Snce σ > 0,,, r assumon A, hs mles ha and 0 for > oce, ha hs resul, ogeher wh f / f mes ha / VV 0 he resul hen follows from he assumon ha FF / I r assumon A3 R6 Suose ha he r marx F s formed as he r ordered egenvecors of normalzed as FF / Iwh he frs column corresondng he larges egenvalue, ec Le S dag sgn F F hen SF F / I he resul for he frs column of SF F / s gven n R5 he resuls for he oher columns mmc he argumen n R5 bu usng R when j and j o show R f σ 0 * j jj 0

12 R7 / ΛΛ Σ ΛΛ Σ Λ j j jj ΛΛ R f σ, where he convergence follows from R ΛΛ 0, for j by consrucon j R8 ΛΛ FF ΛΛ ΛΛ ΛΛ J Σ Σ Σ Σ SΣ J he resul follows from R6, R7, A4, and Slusy s heorem R9 J J he resul follows from R6 e S s full ran, A e heorem Σ ΛΛ s full ran and Slusy s R0 F F F ef / e eef / FF FF Σ Σ Σ + Λ Σ + ΛΣ Σ + Σ ΛΛ ΛΛ ΛΛ ΛΛ ΛΛ Because F are he egenvecors of he corresondng egenvalues on he dagonal, F F ΛΛ / he resul follows from FΛΛ F + FΛ e + eλ F + ee and ΛΛ / s a dagonal marx wh F F ΛΛ /, so ha R Le F denoe he ransose of he h row of F and F J F +Σ Fe Λ F +Σ Σ Λ e +Σ Fee ΛΛ ΛΛ FF ΛΛ I follows from drec calculaon from R0 Σ Σ Σ hen, J ΛΛ FF ΛΛ

13 R F J F O s he resul follows from R6 and R7 whch show ha Σ S and FF ΛΛ ΛΛ Σ Σ, R3 for he erm F eλ F, R3 for he erm Λ e, and R4 for he erm F ee R3 Le a λ J F J F, hen s s a a O s s s s λ λ s s a a F J F F J F J J F J F F J F J J O s s λ s s, λ where he nequaly uses CS, he equaly s a rearrangemen, and he rae follows from R aled o each of he frs erms, J J from R9 and Σ Σ A4 ΛΛ ΛΛ R4 Le a denoe a marx wh,j elemen a j, where a j s defned n R3 hen su race F aa F O s

14 m m m race F aa F f a af f m f smaas m s bu f f f f s f m f sm aas m s / f m f sm aas m s s m sm m m /, and for all, / I hus, / s s su race aa a a O s, where he las equaly follows from R3 /, R5 Suose WW O /, hen F JF W O s, F J F W F J F WW O s where he nequaly s CS, and he rae follows from R and he assumon of he resul R6 F J F e O s F J F e e F J F F J F e O s 3

15 where he nequaly follows from CS, he frs equaly s a rearrangemen and he rae follows from R and R4 R7 λ λ J O s From Λ F/ and FΛ + e, we have FF λ λ + F e, where e s he h column of e Wre Hence, F F FJ + FJ and use FF I o oban J J Fe + F F J F + F J F e λ λ λ J 9 J Fe + 9 F F J F + 9 F J F e 9 9 J Fe + F J F F J λ + 9 F JF e λ λ λ where he frs nequaly uses a+ b+ c 9 a + 9 b + 9 c, and he second nequaly uses CS hus ' λ J λ J Fe F J F F J λλ + 9 F J F e he frs erm n O by R9 and R5; he second erm s O fnal erm s O s by R6 s by R5 and A4; he 4

16 R8 For > r, wre u e ΛJ F J F hen r F F H ; le P H H H H, and ω r+ r R F R F u Pu R F s he sum of squares from he rojecon of ono F, and smlarly for F r Bu r P F P F + P P F H, where he wo erms on he rhs are orhogonal Wre Λ F + e Λ J F + e ΛJ F J F Λ J F + u he resul hen follows drecly R9 m/ m ω u Pu O s + O s r+ + 3 ΛJ F J F P ΛJ F J F r r 3su r race eef r u P u 3 e P e + 3su r r r race F aaf r m/ m, O s + O s where he frs nequaly uses c + d 3c + 3d, he nex nequaly relaxes he consran ha H s orhogonal o F r, and he rae uses R6 and R4 R30 For r, PC PC σ where PC PC R F + R F + g,, RF RF σ from R and g, 0 by assumon 5

17 R3 For > r R F R F O s + O s r, he resul follows from R8, R9 and he defnon of R3 For r [ PC r PC ] hus because >, Pr < 0 [ r PC r PC s R F R F ] r g, s g, [ r s R F R F ] Pr [ PC r PC < 0] Pr < r, s g, [ r s R F R F ] 0 Where he fnal resul follows because s g, r s R F R F O R3 and s g, by assumon R33 σ σ σ e r + σ λ λ e + F + F e e e σ from A5, r λ F σ from A3 and A4, and λfe 0 R34 For from R r σe + σ r, IC IC ln r σ e + σ + 6

18 σ RF IC IC ln g, and σ RF r σ RF σe + σ ln ln r σ RF σ e + σ + connuous mang heorem and R and R33, and he resul follows from g, 0 R35 For s σ ln σ r σ RF > r, s ln O σ RF r s R F R F, where R σ R r RF RF s beween R F and r R F R σe 0 σ > by R, R33 and A5, and r s R F R F O by R3 R36 For > r, Pr IC r IC < 0 IC r IC RF r σ RF s ln σ g, s g, r hus, r σ RF s ln σ RF Pr IC r IC 0 Pr r < < s g,, 7

19 because s r σ RF ln σ RF s g, 0 Where he fnal resul follows because s r σ RF ln O σ RF R35 and s g, by assumon For he followng resuls, le + b, or + b Le ω denoe he h larges egenvalue of Le R, ω, PC, R, g,, and ICP, ln R, g, R37 Le µ denoe he larges egenvalue of bb, hen / / ω + µ ω µ ω ω + µ + ω µ From Horn and Johnson σ A B σ A σ B + + +, where A and B are j j wo marces and σ denoes he h larges sngular value hus, and ω σ + b σ + σ b ω + µ / / / / / /, / / / / / / ω σ [ + b] σ + σ b ω + µ, whch ogeher yeld he resul R38 Suose b O s, hen µ s he larges egenvalue of b b, hus µ O s race b b b O s µ, and he resul follows mmedaely 8

20 R39 Suose µ o, hen ω ω o for,, r For r, ω σ R, and he resul follows drecly from R37 R40 Suose µ O s, hen ω ω O s for > r ω O s + / from R8; from R37 ω ω O s + O s, and he resul follows drecly R4 Resuls R30-R36 connue o hold n he model wh relacng hs follows from R39 and R40 R4 Le J I J, hen J I J J he resul follows mmedaely from R9 R43 + F J F O s F JF j j + j + F J F F J F O s where he nequaly follows from addng osve erms and he rae follows from R R44 Suose WW O /, hen W O s F J F 9

21 he roof mmcs R5 usng R43 n lace of R FF J FF J R45 E FF J FFJ F-J he frs erm converges n robably o J F FJ J F F -J F F - JF F -JF hree erms converge n robably o zero by R44 and R43 E FF J by R4 and A0, and he fnal R46 / η + F O s Fη J Fη + F JF η where he frs erm s R44 and A0 / O by R4 and A0, and he second erm s O s by / R47 / J O s Φ Φ J Φ FF FF, and usng F Φ F + Gη, + + Φ + η + Φ F J J F J G F J F J J F J F, so ha 0

22 ΦJ Φ J G + F J F J Φ J ηf F J F JF F FF + / O s where he rae follows from R9 and R4 whch mly ha R46, R5, and R44 whch show ha he erms + F J F F are O whch s nonsngular by A0 / + J J and J J, η F, F JF F, and + FF J FF J s, and R45 whch shows E R48 Le π Φ λ and π Φ λ, hen Wre λ J λ λ J λ π π J O s + and Φ J Φ + ΦJΦ J J J J, so ha J π J π J ΦJ λ λ + Φ Φ J λ J J λ J λ + Φ Φ and he resul follows from R9, R7, R4, and R47 R49 ππ Φ' ΣΛΛΦ π Φ λ, hen he resul follows drecly from A4

23 R50 ηγ F O Fηγ λλ F η G G O, where he equaly uses γ λ G, and he rae follows from A4 whch mles ha G λλ G G' ΣΛΛG and A0 whch mles ha / F η O ηγ R5 F J F O s F JF η γ λλ F JF η G G O s where he equaly uses γ λ G, and he rae follows from A4 whch mles ha G λλ G G' ΣΛΛG and R44 wh η W R5 ηγ F O s Fη γ Fη γ + F JF η γ J and he resul follows from R50 and R5 R53 F e O s

24 so ha F F + F JF e e e F 3 3 F + F JF e e e O + O s where he nequaly uses a+b 3a + 3b, and he rae follows from A and R6 usng F n lace of F R54 so ha Le π OLS + + FF F, hen OLS π π J O s e F π + ηγ + FJ π F J F J π + ηγ + e OLS J FF F F JF J + + π π π FF Fηγ FF F e and he resul follows from R9 and R4 whch mly ha J J and J J, R49, R44, R5, R53, and R45 whch shows A0 FF JE FF J whch s nonsngular by R55 Le π denoe an esmaor of π and b F π F π If π π J O s, hen O b s 3

25 Wre + and F J F F J F and π π + π π J J, so ha b FJ π J π + F J F J π + F J F π J π, where he frs erm n O π F J J π + F JF J π b s + F JF π J π from A0, R4 and he assumon of he resul; he second erm n O s from R4, R43, and R49; he fnal erm s O he assumon of he resul s from R43 and 4

26 References Amengual, D and MW Wason 005, Conssen Esmaon of he umber of Dynamc Facors n a Large and Panel, manuscr, Prnceon Unversy Ba, J 003, Inferenal heory for facor models of large dmensons, Economerca 7:35-7 Ba, J, and S g 00, Deermnng he number of facors n aroxmae facor models, Economerca 70:9- Ba, J and S g 005b, Confdence nervals for dffuson ndex forecass and nference for facor-augmened regressons, manuscr, Unversy of Mchgan Horn, RA and CR Johnson 99, ocs n Marx Analyss, Cambrdge Unversy Press, Cambrdge Soc, JH, and MW Wason 00, Forecasng usng rncal comonens from a large number of redcors, Journal of he Amercan Sascal Assocaon 97: