Solving Projection Problems Using Spectral Analysis

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Vincent Bradley
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1 Financ 50, Tim Sris Analysis Christiano Solving Projction Problms Using Spctral Analysis This not dscribs th us of th tools of spctral analysis to solv projction problms. Th four tools usd ar th Wold dcomposition thorm, th annihilation oprator, th cross spctrum and partial fractions xpansions.. Th Problm Th problm is to solv: P [y t x t,x t,x t,...] d j x t j, (.) whr y t and x t ar two purly indtrministic, covarianc stationary stochastic procsss. Th projction problm in (.) can aris in various ways. For xampl, this is a standard forcasting problm if y t x t+k,k>0. Anothr xampl is a classic from th arly days of rational xpctations, du to Muth (960). In Muth s problm an agnt is intrstd in tracking a variabl, y t, givn a signal, x t, which is corruptd by masurmnt rror: x t y t + ε t (.) y t ρy t + u t. (.3) Hr, u t is iid and orthogonal to y t s,s>0. Also, ε t is orthogonal to u t at all lads and lags. At first glanc, th solution to (.) sms intractabl. Th prsnc of an infinity of right hand xplanatory variabls in (.) rquirs, in ffct, invrting an infinit ordrd matrix to solv for th d j s. To s this, not that th orthogonality condition associatd with th projction problm rquirs c yx (k) d j c xx (k j), (.4) whr c yx (k) Ey t x t k,c xx (l) Ex t x t l. For illustration, suppos that th agnt is only intrstd in using a finit rcord of past obsrvations on x t,...,x t l to stimat y t. In this cas, th in (.4) is rplacd by l and ths quations ar rprsntd in matrix form as follows: c yx (0) c xx (0) c xx ( ) c xx ( l) d 0 c yx (). c xx () c xx (0) c xx ( l) d c yx (l) c xx (l) c xx (l ) c xx (0) d l To solv for th d j s vidntly rquirs invrting th l + l +matrix in squar brackts. Whn l as in (.) and (.4), this mans invrting an infinit dimnsional matrix. This dfinitly looks difficult!

2 . Gtting Hlp from th Wold Dcomposition Thorm Th problm with th matrix invrsion just dscribd is that autocorrlation among th x t s nsurs that th matrix to b invrtd grows non-trivially largr as l gts larg. If, for xampl, th matrix wr simply diagonal for all l, th invrsion would not b a problm, no mattr how larg l is. Th Wold dcomposition thorm in ffct allows us to rplac th projction (.) with a projction on non-autocorrlatd variabls and thus solv th matrix invrsion problm. In particular, according to th Wold dcomposition thorm any purly indtrministic procss can b linarly dcomposd in trms of its history of on-stp-ahad forcast rrors, which ar thmslvs not autocorrlatd ovr tim. Dnoting th on-stp-ahad forcast rror in x t by w t, it follows that any variabl that can b constructd by som linar function of [x t,x t,...] can also b rcovrd by a linar function of [w t,w t,...]. This is bcaus, according to th Wold dcomposition thorm, thr is a squar summabl squnc, {d 0,d,...}, such that x t d (L) w t. Th oprator, d (L), is asy to find for our xampl. Apply ( ρl) to both sids of (.) and tak into account (.3) to obtain: ( ρl) x t u t + ε t ρε t. Th varianc and lag-on autocovarianc of th trm on th right sid of th quality is σ u + +ρ σ ε, and ρσ ε, rspctivly. W obtain th Wold rprsntation for this trm by choosing λ and σ w to solv ρσ ε σ u +(+ρ ) σ λ ε +λ,σ w ρσ ε λ. Thn, d (L) λl ρl,ew t σ w. Not that if th masurmnt rrors, σ ε, ar larg compard with σ u thn ρ λ and σ w σ ε so that x t ε t, as w would xpct. In th othr xtrm, whn σ ε 0 w can show that x t y t and w t u t. Not that in our xampl, w t canbxprssdasasquarsummablsumofcurrnt and past x t s: w t ρl λl x t u t + ε t ρε t. (.) λl Th xprssion on th right of th quality shows how a ralization of currnt and past shocks, u t and ε t, maps into a ralization of Wold rrors. To gain intuition into (.), suppos that σ ε/σ u is larg, so that ρ ' λ. In this cas, (.) indicats that an ε t shock gnrats virtually no prsistnc in w t. This is not surprising sinc in this cas x t basically is ε t. It is also not surprising that whn σ ε/σ u is larg and thr is a positiv shock to u t, this gnrats a long squnc of positiv Wold forcast rrors, w t, whn ρ is larg. To undrstand this, not from (.) and (.3) that if x t jumps du to u t whn σ ε/σ u is larg, thn th optimal rspons is to trat this as a jump in ε t and not rvis up on s forcast of x t. But, whn u t jumps, y t rmains high for a long tim whn ρ is larg, and this translats into a prsistnt Itisausfulxrcistovrifythatw t is not autocorrlatd ovr tim. This may not b obvious at first sight, but rflcts that ρl and -λl roughly cancl. A simpl simulation will vrify that a prturbation in ε t gnrats a virtually compltly transitory rspons in w t whn ρ ' λ.

3 ris in x t. This prsistnt ris in x t will at first b assumd to b a prsistnt squnc of positiv ralizations in ε t. At som point th improbability of this assumption bcoms ovrwhlming and it is optimal to start thinking that th obsrvd ris in x t was du to u t and not ε t. It is convnint to considr th moving avrag rprsntation of th vctor procss, X t (y t,w t ) 0, as wll as its spctral dnsity. Using (.3) and (.): µ ut X t C (L), ε t " # µ ut Th spctral dnsity of X t is: µ yt w t ρl 0 λl ρl λl S X iω C iω VC iω 0,V σ u 0 0 σ ε As in rsult # in th class handout on spctral analysis, w hav S X iω c (0) + c () iω + c () iω c () 0 iω + c () 0 iω +..., ε t. whr and Similarly, S X iω c (j) EX t Xt j 0 cyy (j) c wy (j) c yw (j) c ww (j) c yw (j) Ey t w t j,c wy (j) Ew t y t j,c ww (j) Ew t w t j. Syy iω S yw iω S wy iω S ww iω σ u ( ρ iω )( ρ iω ) σ u ( λ iω )( ρ iω ), σ u ( ρ iω )( λ iω ) σ u ( ρ iω )( ρ iω ) + σ ε ( λ iω )( λ iω ) Considr th following projction: P [y t w t,w t,...] φ j w t j. Th orthogonality condition which is ncssary and sufficint to solv th projction problm is: E y t φ j w t j w t k 0,k0,,..., which is satisfid by stting th φ j s as follows: φ j Ey tw t j σ, for all j, σ w Ewt. w 3

4 Th z transform of th φ j s is P φ (z) φ j z j c yw (j) z j σ w [S yw (z)] + σ w σ u σ w ( ρz)( λz ) Hr, th + subscript indicats th annihilation oprator, which mans ignor trms in ngativ powrs of z. Our task now is to dvlop a simpl xprssion for [ ] Finishing Off th Problm Using th Annihilation Oprator and Partial Fractions Expansions It is convnint to dvlop th partial fraction xpansion of th trm insid th annihilation oprator. This is xprssd as follows: σ u S yw (z) ( ρz)( λz ) zσ u ( ρz)(z λ) A ρz + A z λ, whr A and A ar constants whos valus ar to b dtrmind. In th usual undtrmind cofficint styl, th valus of A and A ar dtrmind by th rquirmnt that th third quality in th last xprssion b satisfid. Aftr multiplying both sids of that quality by ( ρz)(z λ), w obtain: For th lft and right sids to b qual rquirs zσ u A (z λ)+a ( ρz) A A λ +(A ρa ) z. A A λ A ρa σ u, or, A σ u ρλ,a λσ u ρλ. W writ th partial fraction xpansion of th cross spctrum as follows: σ u S yw (z) ( ρz)( λz ) σ u ρλ Th two trms in th partial fractions xpansion ar: σ u ρλ ρz λσ u z ρλ λz σ u. + ρz + λσ u. (3.) ρλ λz +ρz + ρz +... z ρλ λσ u z + λz + λ z ρλ Bfor continuing, w paus to tak a closr look at som stps takn abov which may at first sm arbitrary. For xampl, w might hav considrd th following xpansion: σ u ( ρz)( λz ) Ã ρz + Ã λz. 4

5 It is asy to s that thr dos not xist an xpansion lik this, bcaus thr do not xist constants, Ã and Ã, that satisfy this quation. Multiply both sids by ( ρz) λz to obtain: σ u Ã λz + Ã ( ρz), and not that thr ar no valus for Ã and Ã so that th xprssion on th right is indpndnt of z. Anothr stp takn abov may also appar arbitrary at first glanc. In particular, w could hav adoptd th following xpansion: A z λ λ A λ z λ A λ λ z A λ +λ z + λ z + λ 3 z (3.) W did not adopt this xpansion bcaus, although it is tchnically valid, it is not rlvant for our purposs. W sk th xpansion of S yw (z), th z transform of a cross-covarianc function, whr th cross-covariancs ar squar summabl: S yw (z) τ c yw (τ) z τ. Although th polynomial rprsntation of S yw (z) tchnically has many xpansions in powrs of z, only on corrsponds to th on w ar intrstd in. Th xpansion that intrsts us is th on whos cofficints on powrs of z ar squar summabl. From this prspctiv, (3.) is not of intrst bcaus th cofficints on powrs of z ar xploding. It is our spcial intrst in th xpansion that involvs squar summabl cofficints that drov us to adopt th xpansion of A / (z λ) that is implicit in (3.). In sum, w hav xprssd S yw (z) in th form of (3.), with th undrstanding that th first polynomial rprsnts an xpansion in positiv powrs of z whil th scond polynomial rprsnts an xpansion in purly ngativ powrs of z. This rprsntation of S yw (z) puts us in a prfct position to valuat th annihilation oprator. Using th obvious fact that g (z) a (z) +b (z) implis [g (z)] + [a (z)] + +[b (z)] +, w obtain: [S yw (z)] + σ u ρλ ρz σ u ρλ ρz. + λσ + u ρλ z λz + W conclud that P [y t w t,w t,...] φ j w t j [S yw (z)] + σ w t w σ u/σ w ρλ ρz w t σ u/σ w ρλ λl x t, 5

6 whr th last trm maks us of (.). Thus, P [y t x t,x t,...] σ u/σ w ρλ λ j x t j. This is a prtty simpl solution to what at first may hav appard to b a formidabl projction problm. Th xprssion says that whn data, y t, ar corruptd by nois as in (.)-(.3), it maks sns to smooth th obsrvd data whn using thm to formulat a guss about y t. For a mor complt discussion of th matrial in this handout, s Sargnt (979, sction 7, pag 75) and Whittl (983, sction 3.7, pag 4). 4. Discussion Th tools dscribd hr ar not just usful for solving th projction problm in (.). Thy ar also usful in solving for th quilibrium in crtain conomic modls. In ths modls, th nois which corrupts obsrvations is idiosyncratic to individual agnts and th way thy do thir forcasting hlps dtrmin th structur of th stochastic procsss driving th variabls. Riccardo Masolo studis a simpl xampl of this in his thsis work. In his problm, thr is a procss dnotd l t, which is analogous to y t abov and which volvs according to an AR() : l t ρl t + η t. Hr, l t is an aggrgat variabl obsrvd with nois by individual agnts. Th nois is idiosyncratic across diffrnt agnts and agnt h obsrvs: s h t l t + ε h t. Hr, ε h t is a masurmnt rror, which avrags out to zro across th larg numbr of agnts in th conomy. Basd on obsrving th history of signals, th h th agnt maks a dcision, dnotd wt h according to following rul: h i h i wt h α P w t s h t,s h t,s h t,... + α P l t s h t,s h t,s h t,.... Hr, w t is th aggrgat of all agnt dcisions. Agnts do not s this aggrgat, sinc thy only s thir own history of s h t. Th actions of individual agnts dpnd on th tim sris rprsntation of w t. Atthsamtim,thtimsrisrprsntationofw t is dtrmind by thos sam actions. This fixd point problm can asily b solvd with an xtnsion of th mthods dscribd in this handout (this is a point that was particularly strssd by Kasa (000).) Rfrncs [] Kasa, Knnth, 000, Forcasting th Forcasts of Othrs in th Frquncy Domain, Rviw of Economic Dynamics 3, [] Muth, John F., 960, Optimal Proprtis of Exponntially Wightd Forcasts, Journal of th Amrican Statistical Association, 55,

7 [3] Sargnt, Thomas, 979, Macroconomic Thory. [4] Whittl, Ptr, 983, Prdiction and Rgulation by Linar Last-Squars Mthods, scond dition. 7

10. The Discrete-Time Fourier Transform (DTFT)

10. The Discrete-Time Fourier Transform (DTFT) Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w