Solving Non-Linear Rational Expectations Models: Approximations based on Taylor Expansions

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1 Work progress Solvg No-Lear Raoal Expecaos Models: Approxmaos based o Taylor Expasos Rober Kollma (*) Deparme of Ecoomcs, Uversy of Pars XII 6, Av. du Gééral de Gaulle; F-94 Créel Cedex; Frace rober_kollma@yahoo.com; Cere for Ecoomc Polcy Research, UK Frs draf: Ocober, Ths draf: February 8, Ths paper preses a algorhm ha compues Taylor seres expasos of order k=,, 4 of he polcy fuco of o-lear raoal expecaos models. Approxmaos of order k may be markedly more accurae ha lear approxmaos (k=) f he varably of he exogeous shocks s hgh ad/or he model exhbs srog curvaure JEL classfcao: C6, C68, C88. Keywords: Soluo mehods; No-lear raoal expecaos models; Taylor expasos (*) Tel.: ; Fax: ; E-mal: kollma@ww.u-bo.de hp:// I hak Ke Judd, Ala Suherlad ad parcpas a he November CFS-JEDC coferece (Frakfur) for useful suggesos/dscussos.

2 . Iroduco May wdely suded sochasc raoal expecaos models ca be expressed as: EG( ω, ω, ξ ε ) =, () m where E deoes he mahemacal expecao codoal upo complee formao abou m perods ad earler; G: R R s a fuco, ad ω s a vecor of varables kow a dae ; ξ s a scalar, ad ε = ( ε, ; ε, ;...; εm, ) s a m vecor of dae exogeous depede radom varables. The followg dscusso assumes ha ε has 4 bouded suppor ad hese momes: Eε, =, E( ε, ) =, E( ε, ) =, E( ε, ) =. The soluo of () s a "polcy fuco" ω = f ( ω, ξε, ξ), () ha sasfes he codo EG ( f ( ω, ξε, ξ), ω, ξε ) = ω, ξ. Whe G s lear, he he soluo f s lkewse lear ad ca easly be compued usg well-kow algorhms (e.g., Hase ad Sarge (98), Blachard ad Kah (98), Aderso ad Moore (985), Kle () ad Sms ()). However, mos ecoomc models (G) are o-lear. A wdely used approach (e.g., Kg, Plosser ad Rebelo, 988) cosss akg a lear approxmao of o-lear models, aroud a deermsc seady sae. A drawback of ha approach s ha does o allow o capure he effec of he volaly of exogeous shocks o he mea values of edogeous varables, as he learzed soluo exhbs ceray-equvalece; ha mehod s hus o sued for compug welfare or for he aalyss of rsk prema o facal asses. Judd ad Guu (99), Judd ad Gaspar (996) ad Judd (998) propose a geeral approach for approxmag he polcy fucos of couous me ad dscree me models usg Taylor expasos of order k>, aroud a seady sae. To dae, applcaos of ha approach o dscree me models have maly focused o quadrac approxmaos, k=. See, e.g., Sms (), Collard ad Jullard (), Schm-Grohé ad Urbe (4), ad Schaumburg () who have produced (ad made publcly avalable) compuer programmes for k=; several sudes have used hese programmes for he aalyss of medum scale macroecoomc models (e.g., Kollma (,, 4), Schm-Grohé ad Urbe ad Km ). Judd ad J (), J ad Jullard recely developed compuer code for dscree me approxmaos of order k> (I lear abou hese corbuos afer compleg mos of he work descrbed here). Ths paper preses a algorhm for compug approxmaos of order k=, ad 4, usg a approach ha dffers from ha used by he papers ha were jus ced (see dscusso Sec.. below). The compuaoal approach used here dffers from ha used by he papers ced above. MATLAB code ha mplemes he prese algorhm wll be made avalable o my web page. The frs o fourh momes of ε, correspod hus o hose of a sadard ormal radom varable. Noe ha E ( ε ) = holds for ay symmerc dsrbuo. The algorhm ca easly be adaped o allow for, 4, E ( ε ).

3 I appears ha approxmaos of order k may be markedly more accurae ha lear approxmaos (k=). A fourh order approxmao may be oceably more accurae ha a secod order approxmao f he varably of he exogeous shocks s hgh ad/or he model exhbs srog curvaure. Seco descrbes he algorhm (ad compares o prevous mehods). Seco apples o seleced models.. The mehod I beg by dscussg defos/oao. Throughou hs paper, he erm "seady sae" refers o he deermsc seady sae,.e. o a model soluo whch ω = ω = ω, ε = m, wh G( ω, ω, m) =, where m s a colum vecor of zeros ( m elemes). Seady sae values are deoed by varables whou me subscrps, ad dz = z z s he devao of a varable z from s seady sae value. R deoes a polyomal cossg of powers of order ad hgher of elemes of { dω τ ; ξε τ } τ. ( ) h deoes a -h order accurae approxmao of varable h, he followg ( ) {} s ( s) ( s ) sese: h h = R. Le h h h () () {} =, for s >. Thus, h = h h, () {} {} h = h h h ec. If a ad b are marces, he ( ab ; ) deoes he marx obaed by vercally cocaeag a ad b (provded a ad b have he same umber of colums), whle ( ab, ) deoes horzoal cocaeao. Le k be a colum vecor wh N elemes. P ( k ), for =,,... deoes a colum vecor cossg of all h order powers ad cross-producs of he elemes of k. I he compuer programs, hese powers/cross-producs are arraged he followg order: P( k) = k ad Ps ( k) = ( kps( k); kps(( k; k;..; kn));...; kn Ps(( kn ; kn)); knps( kn) ) for s >, where k s he h eleme of k... Frs-order approxmaos The algorhm for geerag secod (ad hgher) order approxmaos preseed here akes as s sarg po a frs-order accurae (lear) model soluo. As dscussed above, several soluo mehods for lear(zed) raoal expecaos models are avalable he leraure. Ay of hese mehods could be used o geerae hgher order accurae soluos. Here, I use Sms' () algorhm (ha ca be mplemeed usg Chrs Sms' compuer program gesys, avalable a www. prceo. edu / ~ sms ). Ths seco brefly revews Sms' () approach. Followg Sms () oe ha () mples G( ω, ω, ξε ) Π η =, wh Eη =, () where Π s a marx of sze p, where p equals he umber of model equaos ha clude dae expecaos of dae varables. η s fuco of ε (ha fuco s o kow a pror). Sms () shows ha he soluo of he model ca be wre as: y = F( y, ξε ), x = M( y ), wh ( y; x) = Z ω, where Z s a o-sgular marx. y ad x are colum vecors wh y ad x elemes, respecvely, wh = y x; F ad M are fucos.

4 Take a frs-order Taylor expaso of () aroud a seady sae. Ths gves: Gdω = Gdω Gξε Π η R, (4) where G, G ad G are marces/vecors of sze, ad, respecvely. Usg, we ca wre (4) as: K( dy ; x ) = K( dy ; x ) Gξε Π η R, (5) wh K= GZ, K= GZ. Sms () shows here exss a marx T wh he followg properes: I y H F H TK =, TK= J J ad y p T Π=, (6) Π where H ( ), J ( ), F ( ), H ( ), J ( ), Π ( p) are marces y x x x y y y x x x x (szes show pareheses). Premulplyg (5) by T hus gves a block-recursve sysem of equaos: dy Hdx = Fdy Hdx Fξε R, (7) Jdx Jdx Gξε = Π η R, (8) where F ad G are marces wh y ad x rows respecvely ( ( F; G) = T G ). The assumed saoary of he model soluo mples ha he egevalues of F ad ( J) J As dx Edx are sde he u crcle. Solvg forward (8) yelds: j= j = { (( ) ) ( ) ( ξε j Π η j) } j= dx J J J G R. (9) = holds, ad Eε j s = Eη j s = s, (9) mples: dx = R, ad hus: () dx =. () (9) ad (7) mply: () dy = F dy F ξε. ().. Hgher order approxmaos To h order accurae soluos ( ), ake a h order Taylor expaso of (): = Gdω = Gdω G ξε ΘP( dψ ) η Π R =, () where Ψ = ( ω ; ω; ξε ), whle Θ,..., Θ are marces. Usg, we ca rasform () o: = K( dy ; x ) = K( dy; x) G ξε ΩP( dλ ) η Π R =, wh Λ = ( y ; x ; y; x; ξε ), where Ω,..., Ω are marces. Premulplyg by T (see (6)) yelds: = dy Hdx = Fdy H dx F ξε Φ P( dλ ) R =, (4) = Jdx = J dx Gξε Π η Φ P( dλ ) R =, (5) where Φ ad Φ are marces: ( Φ; Φ ) = T Ω for =,...,. Solvg (5) forward ad akg codoal expecaos gves: I Chrs Sms' gesys program, T correspods o he produc of he marces ma ad q: T = ma q. Ω (for =,..., ) s a fuco of Θ ad of Ζ. Deermg Ω s smple bu edous. Ieresed readers may cosul he compuer code for he deals. Aalogous remarks apply o may of he coeffces he res of hs paper. 4

5 ( ) s = ( ) = (( ) ) ( ) Φ( ( Λ )) = s ( ). (6) dx J J J E { P d } Thus, o deerme dx we have o compue he pah of he d o horder powers ad ( ) cross-producs of he sae varables. Gve such a pah, ad a soluo for dx, he me ( ) pah for ( dy ) ca be deermed recursvely usg (4). A h order accurae soluo ( ) ( ) ( ) for ω ca he be compued usg : ( ) dω (( ) ;( ) = Z dy dx ). ( ) The algorhm descrbed below s based o he fac ha ha Pd ( ) Λ (for () () ( ) =,..., ) ca be deermed from ( d Λ ), ( d Λ ),., ( ) dλ. For example, a secod () order accurae model soluo requres kowledge of P ( dλ ). P ( dλ ) s a vecor cossg of he producs of he elemes of dλ. Le dk ad dq be wo elemes of dλ. Noe ha () () () ( dk dq) = dk dq. 4 (7) To geerae a hrd order accurae model soluo, we eed a hrd order accurae evaluao of producs of pars ad rples of elemes of he vecor dλ. Such a evaluao ca be obaed from frs- ad secod order accurae model soluo, as he produc of wo varables dk ad dq ca be expressed as: () () () {} {} () ( dk dq ) = ( dk) ( dq) ( dk) ( dq) ( dk) ( dq), (8) whle a hrd order accurae approxmao of he produc of hree varables dk, dq, dr s gve by: () () () ( dk dq dr) = ( dk) ( dq) ( dr). (9) A fourh order accurae model soluo requres a fourh order accurae evaluao of producs of pars, rples, ad quadruples of elemes of he vecor dλ. Ths ca be obaed from frs order, secod order ad hrd order accurae soluos. Noe ha (4) () () () {} {} () () {} ( dkdq) = ( dk) ( dq) ( dk) ( dq) ( dk) ( dq) ( dk) ( dq) {} () {} {} ( dk ) ( dq) ( dk ) ( dq ), () (4) () () () {} () () ( dk dq dr ) = ( dk ) ( dq ) ( dr ) ( dk ) ( dq ) ( dr ) () {} () () () {} ( dk ) ( dq ) ( dr ) ( dk ) ( dq ) ( dr ), () ( dk dq dr ds ) = ( dk ) ( dq ) ( dr ) ( ds ). () (4) () () () ()... Secod order accurae soluo For =, (4) ad (6) are gve by: dy Hdx = Fdy Hdx Fξε Φ P ( dλ ) R, () s () = (( ) ) ( ) Φ( ( Λ s )) dx J J J E P d. (4) Le Z ( ξ ; P (( dy ; ξε ))), (5) 4 Noe ha dk dq = ( dk R )( dq R ) = dk dq R. () () () () 5

6 h ξ. (6) ( ; P ( dy)) (7) mples ha ( Pd ( Λ )) = P(( dλ ) ). I follows from (),() ha ( Λ ) = ( ; ; ; ; ). (7) () () () d F dy F ξε dy x ξε x () Hece, ( Pd ( Λ )) s a lear fuco of he squares ad cross-producs of he vecor ( dy; ξε ), ad hus of he elemes of P(( dy; ξε )). Thus we ca wre () () Φ ( Pd ( Λ )) =Ψ Z, Φ ( Pd ( Λ )) =Ψ Z, (8) for some marces Ψ, Ψ. As dy ad ε are depede, EZ s a lear fuco of ξ ad of P ( dy ), ad hus: EZ = Ξ h, (9) for some marx Ξ. The logc ha uderles (8) also mples ha () ( h ) =Ψ Z, for some marx Ψ. Hece, () E( h ) =ΨΞ h () s (8)- mply ha Φ E( P( dλ s)) = Ψ Ξ ( Ψ Ξ ) h. Subsug hs o (4) gves: () s dx = Sh, wh S (( J) J) s ( J) Ψ Ξ ( ΨΞ). Usg (8), ad, we ca wre as: () ( dy ) = Fdy Fξε P Z, () for some marx P., () ca also be expressed as: () dx = Qξ Q P ( dy), (4) () ( dy ) = Fdy Fξε Fξ F4 P(( dy; ξε ), (5) where Q, Q, F ad F 4 are marces/vecors. (4),(5) have he same form as he secod order accurae soluos derved by Sms () ad by Schm-Grohé ad Urbe (4). Applcao of he secod-order accurae algorhm preseed here o several models yelded coeffces F, F, F, F 4 ad M, M ha are umercally dsgushable from coeffces mpled by he Sms () algorhm.... Thrd order accurae soluo For =, (4) ad (6) are gve by: dy Hdx = Fdy Hdx Fξε Φ P ( dλ ) Φ P ( dλ ) R, (6) Le 4 s = (( ) ) ( ) Φ( ( Λ s)) Φ( ( Λ s)) dx J J J E { P d P d }. (7) ( ξ ξ ) ad h ( ξ ; P ( dy ); ξ dy ; P ( dy )) Z ; P ( dy ; ξε ); ( dy ; ξε ); P (( dy ; ξε ), (8). (9) 6

7 ,, (4), (5) mply: {} ( dλ ) = Fξ F4 P(( dy; ξε ); S Ψ ( ξ ; P(( dy; ξε )); ; Q ξ Q P( y); ), (4) ( y m {} d Λ s a lear fuco of Z. (7), (8), (9), (7), (4) mply ha ( Pd ( )).e. ( ) Λ ca be expressed as lear fucos of Z. Thus: Φ P( dλ ) Φ P( dλ ) =Ψ Z, (4) Φ P( dλ ) Φ P( dλ ) =Ψ Z, (4) for some marces Ψ ad Ψ. As dy ad ε are depede, ad as he hrd momes of he elemes of he vecor ε are zero, EZ s a lear fuco of h : EZ = Ξ h, (4) for some marx Ξ. The logc ha uderles (4),(4) also mples ha ( h ) =Ψ Z, (44) for some marx Ψ. Hece, E( h ) =ΨΞ h. (45) s (4), (4), (45) mply E{ Φ P( dλ s) Φ P( dλ s) } =ΨΞ( ΨΞ ) h. Therefore, s dx = Sh, wh S (( J) J) ( J) Ψ s Ξ ( ΨΞ). (46) Usg (4), (44) ad (46), we ca wre (6) as: ( dy ) = Fdy Fξε PZ, (47) for some marx P. (46), (47) mply: dx = Qξ Q P( dy) Qξ dy Q4 P( dy), (48) ( dy ) = Fdy Fξε Fξ F4 P(( dy; ξε ) F5 ξ ( dy; ξε ) F6 P(( dy; ξε ), (49) where Q, Q, Q, Q4, F, F4, F 5 ad F 6 are marces/vecors. The coeffces of he frs order erms (49) (.e. F, F ) are, by cosruco decal o he correspodg coeffces he frs- ad secod order accurae soluos (), (5). I appears ha he coeffces of he secod order erms Q, Q, F, F 4 are also decal across he secod- ad hrd order accurae soluos (4)-(5) ad (48)-(49). A proof of hs s provded he Appedx.... Fourh order accurae soluo The dervao of a fourh order accurae soluo follows he same logc as he prevous dscussos. For =4, (4) ad (6) are gve by: dy Hdx = Fdy Hdx Fξε Φ P ( dλ ) Φ P ( dλ ) Φ4 P ( dλ ) R, (5) 4 5 (4) s (4) (4) (4) = (( ) ) ( ) { Φ( ( Λ s)) Φ( ( Λ s)) Φ4( 4( Λ s)) } dx J J J E P d P d P d (44), (46), (47) ad (48), mply: {} ( dλ ) = ( F5 ξ ( dy ; ξε ) F6 P (( dy ; ξε ); SΨ Z ;. (5) 7

8 ; Qξ dy 4 ( ); ). y Q P dy m (5) Le 4 Z4 ( ξ ; P( dy; ξε ); ξ ( dy; ξε ); P(( dy; ξε ) ; ξ ; ξ P(( y; ε )); P4(( y; ε ), (5) 4 h4 ( ξ ; P( dy); ξ dy; P( dy); ξ ; ξ P( dy); P4( dy) ). (5) Usg (7)-(), (7), (4) ad (4), we oba: (4) (4) (4) Φ( Pd ( Λ )) Φ( Pd ( Λ )) Φ4( Pd 4( Λ )) = Ψ 4 Z4, (4) (4) (4) Φ ( Pd ( Λ )) Φ ( Pd ( Λ )) Φ4( Pd 4( Λ )) = Ψ 4 Z4, for some marces Ψ 4 ad Ψ 4. Also, EZ 4 = Ξ 4h4, (4) (4) ad (4 h ) =Ψ 4Z4, E(4 h ) =Ψ4Ξ 4h4, for some marces Ξ 4 ad Ψ 4. We have (4) s ( dx) = S4h4, wh S4 (( J) J) ( J) Ψ s Ξ4 ( Ψ4Ξ4), (5) ad ( dy ) = Fdy Fξε P4 Z4, (5) for some marx P 4. (5), (5) ca be wre as: (4) 4 ( dx ) = Qξ Q P ( dy ) Qξ dy Q4 P ( dy ) Q5 Q6 ξ P ( dy ) Q7 P ( dy ), (54) 4 ξ ( dy ) = Fdy Fξε Fξ F4 P (( dy ; ξε ) F5 ξ ( dy ; ξε ) F6 P (( dy ; ξε ) (4) 4 F7ξ F8 ξ P(( y; ε )) F9 P4(( y; ε ). (55) where Q, Q, Q, Q4, Q4, Q6, Q7, F, F4, F5, F6, F7, F 8 ad F 9 are marces/vecors... Relaed approaches The compuaoal approach used here dffers from ha of Judd ad Guu (99), Judd ad Gaspar (996), Judd (998) (ad mos subseque papers ha compue Taylor expasos of he polcy fuco). These auhors oba he coeffces of a h order Taylor expaso of by compug he s o h order (cross-) paral dervaves of H( ω, ξ) EG ( f( ω, ξε, ξ), ω, ξε ) wh respec o ω ad ξ, a he seady sae. Noe ha hese dervaves all have o equal zero, as H ( ω, ξ ) = ω, ξ. Thus: H( ω, ξ) / ( ω, ξ) = for =,..,. (56) ω= ω, ξ= Ths gves a sysem of equaos he (ukow) s o h order (cross-) paral dervaves of he polcy fuco f. These paral dervaves ca be obaed sequeally: he ercep of f ( ω, ξε, ξ) s deermed by he codo G( f ( ω,,), ω, ) = ; frs-order dervaves ca be foud by cosderg (56) wh =. (56) wh = ps dow he secod order dervaves, ec. The basc dfferece bewee ha approach ad he approach used here ca be llusraed usg he followg smple sac model (see Judd (998), p.449): gxε (, ) =, (57) where x ad ε are a edogeous ad a exogeous varable, respecvely. The soluo of ha model s gve by a fuco x = f ( ε ) ha sasfes g( f( ε ), ε) = ε. (58) 8

9 We are eresed compug a Taylor expaso of f aroud bechmark value ε : dx = f ' dε f '' ( dε) 6 f ''' ( dε)..., wh dx = x x, dε = ε ε, gx (, ε ) =. (All dervaves are evaluaed a ε ). The Judd approach a deermg f ', f '', f '' ec. s based o hese codos: g( f( ε ), ε ) / ε = =,.., k. ε = ε For example: gf ' g =, whch mples ha f ' = g/ g; g( f ') g f ' g f '' g =. Subsug f ' = g/ g o hs expresso allows o deerme f '' : f '' = [ g( g/ g) g( g/ g) g]. The approach adoped here, by coras, compues a secod-order approxmao usg a (slghly) dffere procedure: amely a secod-order approxmaos of he squared erms (of secod-order Taylor expaso) s compued usg he frs-order soluo. A frs-order Taylor expaso of (57) gves gdx gdε R =, whch mples ha () ( dx) = ( g / g ) dε. A secod-order Taylor expaso gves: ε ε ε () () gdx g d g ( dx) g dxd g ( d ) R =. As () () (( dx) ) = (( dx) ) ad ( dx dε ) = ( dx) dε, we have: () () () ( dx) = ( g) [ g dε g(( dx) ) g ( dx) dε g ( dε) ]. Ths mples ha () ( dx) = ( g / g ) dε [ g ( g / g ) g ( g / g ) g ]( dε ). Thus, he mpled frs ad secod dervaves of he polcy fuco are decal o hose obaed usg he Judd approach. The approach here s closely relaed o work by,.a., Km ad Km (999), Woodford (999), ad Woodford ad Bego who have show ha a secod-order accurae evaluao of codoal ad ucodoal expeced values of ω ( a model of ype ()) ca be acheved usg a frs-order accurae model soluo. These mehods explo he fac ha a frs-order accurae (.e. lear) approxmao of he polcy fuco perms a secod order accurae evaluao of he squares ad cross-producs of he sae varables (ad hus of he secod momes of hese varables). 5 However, he mehods preseed by hese auhors do o readly perm o compue smulaed me seres { ω s } ha are secod order accurae. Suherlad () uses a lear approxmao of he model o provde a secod-order accurae evaluao of he codoal expeced value of he me pah { Eω s } s, gve he sae of he ecoomy a some dae =. The paper here adaps ad geeralzes Suherlad's () approach o compue k-h-order accurae smulaed pahs { ω s }. 6. Applyg he mehod 5 These papers focus o he compuao of welfare. As he vecor ω ca be specfed such a maer ha oe of s elemes cludes he uly level of he ages assumed he model, ha approach s suffce for compug expeced welfare. 6 Afer he research here was compleed, papers by Schaumburg () ad Lombardo ad Suherlad (4) were brough o my aeo ha prese secod-order accurae soluos based o he same dea. 9

10 Refereces Aderso ad Moore (985) Blachard ad Kah (98) Collard ad Jullard () Km ad Km (999) Kle () Sms () J Judd ad Guu (99) Judd ad Gaspar (996) Judd (998) Jullard Km ad Km (999) Km Kg, Plosser ad Rebelo (988) Kollma (,, 4) Lombardo ad Suherlad (4) Schaumburg () Schm-Grohé ad Urbe (4) Sms () Sms () Suherlad () Woodford (999) Woodford ad Bego