Optimal Control Theory for Inflation Targeting

Transcription

1 Unversdade Federal de Sana Caarna From he SelecedWorks of Sergo Da Slva Aprl, 28 Opmal Conrol Theory for Inflaon Targeng Thago Veloso Robero Meurer, Federal Unversy of Sana Caarna Sergo Da Slva, Federal Unversy of Sana Caarna Avalable a: hps://works.bepress.com/sergodaslva/9/

2 Opmal conrol heory for nflaon argeng Thago Veloso Naonal Agency for Elecrc Energy Robero Meurer Deparmen of Economcs, Federal Unversy of Sana Caarna Sergo Da Slva Deparmen of Economcs, Federal Unversy of Sana Caarna Absrac We make a case for he usefulness of an opmal conrol approach for he cenral banks choce of neres raes n nflaon arge regmes. We llusrae wh daa from seleced developed and emergng counres wh longes experence of nflaon argeng. TV acknowledges fnancal suppor from he Brazlan agency CNPq, and SDS acknowledges suppor from CNPq and Capes-Procad. Caon: Veloso, Thago, Robero Meurer, and Sergo Da Slva, (28) "Opmal conrol heory for nflaon argeng." Economcs Bullen, Vol. 3, No. 24 pp. -4 Submed: February 2, 28. Acceped: Aprl 2, 28. URL: hp://economcsbullen.vanderbl.edu/28/volume3/eb-8c6a.pdf

3 l. Inroducon Inflaon argeng s now a new gold sandard for cenral banks. The regme s beleved o perform beer han, for nsance, he alernave of conrollng money for clampng down on nflaon by gvng moneary polcy more ransparency and hus credbly (Svensson 997, Mshkn and Schmd-Hebbel 27). Insead of ryng o mee moneary arges, cenral banks use her own money o deermne shor-erm neres raes and hus conrol nflaon drecly. Teherng nflaonary expecaons s val under hs regme. If agens beleve ha he nflaon arge wll be h, hen nflaonary shocks wll be absorbed. The 99s were favorable o low nflaon regardless of nflaon argeng (Masson e al. 997). And he case for nflaon argeng s no ha sraghforward for emergng marke counres. Ths s so because of her fragle nsuons (Echengreen 22, Calvo and Renhar 2, Mshkn 24), excess lables n foregn currency, and hgh degree of passhrough (Echengreen 22). Exchange rae deprecaon sze also maers n such counres (Echengreen 22) n ha mgh cause a nonlnear mpac on oupu, as n Aghon e al. (999) and Krugman (23). e by 25 egh developed markes and hreen emergng counres had adoped nflaon argeng; concdenally or no, nflaon was amed n such counres (Mshkn and Schmd- Hebbel 27). Anoher appeal of nflaon argeng s s conssency wh he Taylor rule and hus s supposed advanage over a fxed exchange rae anchor o moneary polcy (Echengreen 22, Masson e al. 997). e mxng nflaon argeng wh flexble exchange raes s no always feasble (Calvo and Renhar 2). Inflaon argeng has also been lnked o a more favorable nflaon-unemploymen radeoff (Clfon e al. 2, Clarda e al. 999). Bu he regme can also creae more nomnal rgdy and hus worsen he nflaon-unemploymen radeoff n he presence of low nflaon and longer-erm conracs (Posen 998, Huchson and Walsh 998). Ths paper wll make a case for he usefulness of opmal conrol analyss for he cenral banks choce of neres raes n nflaon arge regmes. We wll employ a cenral bank reacon funcon consderng he Taylor rule whn a framework of opmal conrol (Chow 975). The model wll selec he nflaon-argeng neres rae as a soluon o he mnmzaon of he cenral bank s loss funcon subjec o he behavor of oupu, nflaon, and exchange rae changes. The res of he paper s organzed as follows. Secon 2 wll presen our model. Secon 3 wll show daa. Secon 4 wll perform analyss. And secon 5 wll conclude. 2. Theorecal model Now we presen an opmal conrol model ha bulds on he Taylor rule model of Echengreen (22). Echengreen s model racks he major feaures of open emergng markes, and can be descrbed by equaons () (3). π = π + β ( ) + β ( e e ) + ε () 2 + = α ( ) α ( ) + α e + η (2) Ee ( + ) e = + v, (3)

4 where π and π are nflaon rae and nflaon rae arge respecvely, s oupu devaon from s naural level, e s he nomnal exchange rae (dollar prce of a counry s currency),,, and are domesc, foregn, and neural neres rae respecvely, ε and η are dsurbance erms, and ν s a fnancal dsurbance (Calvo s shock). Equaon () s he expecaonal Phllps curve, and equaon (2) s he aggregae demand for an open economy. The neres rae mpac on oupu s capured by parameer α 2 (and ndrecly hrough α 3 ). Equaon (3) s uncovered neres pary, where Ee ( + ) s assumed o be consan when dervng he Taylor rule. Hgh degree of passhrough s racked by boh a bg β 2 and a small α 3 because hese values mean ha exchange rae deprecaon causes rapd ncrease n domesc and radable prces, decreased compeveness, and hen low effec on oupu. Excess lables n foregn currency can also be represened by a small α 3. If α 3 s small (and posve) he cenral bank has less fear of floang. e a bg deprecaon means a negave α 3, and hs ncreases he fear of floang. Our opmal conrol approach o nflaon argeng can be exended o oher formulaons of he Phllps curve and aggregae demand by changng he assumpons abou he forward or backward lookng feaures of he model, as dscussed by e.g. Mara-Dolores and Vazquez (26). The soluon o he model above s an neres rae reacon funcon. We sugges ha such a reacon funcon wll resul from he mnmzaon of he cenral bank s loss funcon. We arbrarly choose o mnmze he loss funcon (4) over en perods subjec o a sysem of equaons represenng he behavor of oupu, nflaon, and exchange rae changes,.e. 2 2 = µ, ( ) + µ 2, ( π π ) = EW E (4) s.. = α + α + α + ε (5) 2 π = β + βπ + β ( e e ) + β + η (6) 2 3 e e = γ + γ ( e e ) + ν, (7) 2 where EW s he loss funcon (gven he values of oupu, nflaon, and exchange rae changes a = ), and µ, and µ 2, are he coss of no reachng he desred oupu level and he nflaon arge respecvely. Equaons (5) (7) are smlar o Echengreen model (equaons () (3)). The dfferences are as follows. Equaon (5) shows oupu pah as a funcon of he neres rae. Boh equaons (5) and (6) are que sandard (e.g. Svensson 997) bu consder desred oupu raher han he devaon from naural oupu (Romer 2). I s preferable o use oupu raher han oupu gap because of he dffculy nvolved n fndng a sascally sgnfcan coeffcen for he relaon oupu gap-nflaon n quarerly models (Gal and Gerler 999). Equaon (7) comes

5 from uncovered neres pary and a frs-dfference auoregressve model (Munhos e f al. 22). By rewrng E( e+ ) e = δ + δ( ) + u n frs dfferences and f consderng [ E( e+ ) ] = E e+ one ges E e+ e = δ ( ). Inserng he rule f for expecaons formaon E e+ = γ e + γ2( π π ) no he frs-dfference f f equaon produces e = γ e δ ( ) + γ2( π π ) + ε. The laer can hen be furher smplfed o generae equaon (7). Noe ha expecaons play a role n monorng nflaon and hs can affec he model s reduced-form coeffcens,.e. he model s subjec o he Lucas crque. A more rgorous approach would be o frsly esmae he srucural parameers assocaed wh a new Keynesan-ype, open economy model (as n Del Negro e al. (25) and Mara-Dolores and Vazquez (26)), and hen solve he opmal conrol problem faced by he cenral bank subjec o he resrcons mposed by he model. However, for our purposes n here suffces o ake a nonsrucural model. The soluon o he problem s he neres rae reacon funcon (8),.e. π = G + g, (8) e [ ] where G = θ, θ2, θ3, θ4,. Reacon funcons for each of he en perods oban afer reckonng G, G9,..., G and g, g9,..., g by dfferenang he Lagrangean 2 λ = =, (9) L = ( y a ) K ( y a ) ( y Ay Cx b) where y π =, e K µ, = µ 2,, a π = e, x = [ ]. () Marces A and C, and vecor b are he parameers of equaons (5), (6), and (7) n her reduced form, and are assumed o be consan. Equaon (9) refers o he mnmzaon of loss funcon W = 2 ( y a) K( y a) subjec o equaons (5), (6), and (7) rewren as a frs-order dfference equaon sysem,.e. y = Ay + Cx + b. And W s loss funcon (4) n marx noaon for K and a. By dfferenang (9) wh respec o x, y, and λ, and consderng only he deermnsc par of (5), (6), and (7), one can ge G, G9,..., G and g, g9,..., g usng (Chow 975) G = ( CH C) CH A ()

6 and g = CH C C H b h, (2) ( ) ( ) where H = K and h = K a for =. One advanage for a cenral bank o employ reacon funcon (8) s ha can choose he bes neres rae by consderng s effecs n several subsequen perods. Anoher advanage of he opmal conrol approach s o allow one o calbrae he heorecal model wh economerc esmaes of he parameers n A, C, and b. 3. Daa We consdered a sample of developed and emergng counres wh longes experence of nflaon argeng. They are he Uned Kngdom (UK), Canada (CAN), New Zealand (NZL), and Sweden (SWE). The emergng counres are Chle (CHI), Poland (POL), Czech Republc (CZE), and Korea (KOR). The quarerly daa for he varables n equaons (5) (7) as well as were aken from he IMF s Inernaonal Fnancal Sascs. The daa range s from 99 Q o 25 Q. Our am s o esmae such equaons and parameers, and hen ge he neres rae reacon funcon. We consdered real GDP o represen oupu. We used he GDP mplc prce deflaor and made 2:Q = n order o ge real GDP from nomnal GDP. For nflaon we ook changes n he producer prce ndex, apar from Chle (where he consumer prce ndex was aken). For exchange rae changes we consdered he closng quoes. For neres rae we consdered he money marke rae, apar from he UK, Sweden, and Chle. For he UK and Sweden we ook he governmen bond yeld, and for Chle we consdered he dscoun rae. 4. Analyss Tables 3 show he esmaes for equaons (5) (7) usng ordnary leas squares. (There are oher esmaon echnques, such as a Bayesan approach; hs has been employed, for nsance, n a dynamc sochasc general equlbrum framework by Del Negro e al. 25.) Our esmaon choce can be jusfed on he bass ha here s no nerdependence beween he endogenous varables; pu anoher way, each equaon presens a one-way causal relaonshp. Dsurbances ε, η, and ν were found conemporaneously unrelaed. We also checked for auocorrelaon n resduals employng Breusch-Godfrey s LM es. The presence of auocorrelaon was correced by Cochrane-Orcu esmaon. Coeffcen α 3 s absen from Table,.e. exchange rae changes were no sascally sgnfcan. The oupu response o was sronger for he UK, Sweden, and Poland. And he coeffcen values n Tables 3 show ha makes no dfference wheher a counry s developed or no. The coeffcen of passhrough (.e. ha of π ) s hgher for he emergng counres (Table 2), bu even for hs se of counres he values vary a grea deal. Table 4 and 5 show he cenral banks reacon funcons reckoned by Chow (975) mehodology. The coeffcens of marx G were que smlar for he en

7 perods, and hen we dsplay hose for wo perods only. The values of G were no nfluenced by eher oupu arge, nflaon arge, or he nal condons. To calculae marces g, g9,..., g we se boh oupu and nflaon arge a.5 percen per quarer (~ 2 percen a year); hs fgure s based on he raonale presened n Fscher (996). For he nal condons we ook he endogenous varables values a. We also assumed ha he cenral banks do no change he penales for oupu and nflaon devaon from he arge, whch means assumng µ, and µ 2, consan for he en quarers. The F es n Table 4 shows ha he counres are smlar regardng he sensbly of he opmal neres rae o nflaon and exchange rae. The observed F s less han he abulaed value of 5.99 (5 percen sgnfcan). The resuls n Table 4 also depend crcally on α 2. Havng found he reacon funcons, we hen appled opmal conrol analyss (and loss funcon (4)) o ge he pahs of oupu and nflaon devaon. The pahs allow one o assess he performance of a counry regardng he chosen arge. Chow mehodology suggess decomposng (4) no one deermnsc and one sochasc par. For convenence, here we consder he deermnsc par only. The deermnsc loss funcon can be found by rewrng (4) as 2 2 µ, ( ) µ 2, ( π π ), (4 ) = W = + where and π are mean ha we dropped ε and η from () and (2). Table 5 shows he oal of devaons for he nal condons =, π = π, and e = e. The emergng counres were found o devae more from he arge. (Fgures and 2 show he pahs for oupu and nflaon afer opmzaon a =.) The arges were no h n Fgures and 2 because we negleced he sochasc par n he loss funcon. Targes are only h when he number of varables n he loss funcon maches he number of nsrumenal varables. Ths canno occur n our model of wo varables (oupu and nflaon) and only one nsrumenal varable (neres rae). Calbrang he nflaon wegh n he loss funcon (.e. makng µ 2, = 2) shows ha he counres can approach more closely he nflaon arge a he expense of he oupu arge. 5. Concluson Ths paper sraghforwardly shows how an opmal conrol analyss can be employed by cenral banks n her choce of neres raes under nflaon arge regmes. Daa from seleced developed and emergng counres wh longes experence of nflaon argeng were aken o llusrae. We ncdenally found ha he emergng counres devae more from he arge afer opmzaon.

8 UK GDP Targe GDP (µ2=) GDP (µ2=2) Inflaon Targe Inflaon (µ2=) Inflaon (µ2=2) CAN GDP Targe GDP (µ2 = ) GDP (µ2 = 2) Inflaon Targe Inflaon (µ2 = ) Inflaon (µ2 = 2) NZL GDP Targe GDP (µ2 = ) GDP (µ2 = 2) Inflaon Targe Inflaon (µ2 = ) Inflaon (µ2 = 2) SWE GDP Targe GDP (µ2 = ) GDP (µ2 = 2) Inflaon Targe Inflaon (µ2 = ) Inflaon (µ2 = 2) Fgure. Developed counres opmal pah for GDP and nflaon

9 CHI GDP Targe GDP (µ2 = ) GDP (µ2 = 2) Inflaon Targe.2 Inflaon (µ2 = ). Inflaon (µ2 = 2) POL GDP Targe GDP (µ2 = ) GDP (µ2 = 2) Inflaon Targe.2 Inflaon (µ2 = ). Inflaon (µ2 = 2) CZE GDP Targe GDP (µ2 = ) GDP (µ2 = 2) Inflaon Targe Inflaon (µ2 = ) Inflaon (µ2 = 2) KOR GDP Targe GDP (µ2 = ) GDP (µ2 = 2) Inflaon Targe Inflaon (µ2 = ) Inflaon (µ2 = 2) Fgure 2. Emergng counres opmal pah for GDP and nflaon

10 Table. GDP behavor (equaon (5)) Dependen varable: Coeffcen (-sasc) UK CAN NZL SWE CHI POL CZE KOR 99:Q 99:Q 99:Q 99:Q 996:Q 994:Q4 994:Q 99:Q Inercep (2.622) (.79) (.728) (4.98) (.737) (4.53) (2.829) (3.78) (7.844) (98.686) (94.45) (3.398) (2.927) (3.) (9.379) (7.7) ( 2.89) ( 2.88) ( 2.92) ( 3.95) ( 2.59) ( 3.3) (.694) ( 2.766) R squared Adjused R LM es lag p = p = 7 p =.265 p =.97 p =.775 p =.35 p =.592 p =.37 2 lags p =. p =.939 p =.53 p =. p =.3 p =.67 p =. p =.5 ARCH lag p =.83 p =.86 p =.85 p =.7 p =.5 p =.29 p =.7 p =.723 Whe p =.255 p =.2 p =.774 p =.27 p =.962 p =.69 p =.36 p =.9 sgnfcan a %, sgnfcan a 5%, sgnfcan a %

11 Table 2. Inflaon behavor (equaon (6)) Dependen varable: π Coeffcen (-sasc) UK CAN NZL SWE CHI POL CZE KOR 99:Q 99:Q 99:Q 99:Q 996:Q 994:Q4 994:Q 99:Q π (4.967) (2.626) (2.6) (3.288) (2.725) (8.259) (4.28) (3.66) e (.87) (5.928) (4.63) (3.74) (.63) (4.426) (.969) (8.48) (2.562) (2.32) (3.434) (2.266) (2.666) (2.286) (2.634) (2.483) LM es lag p =.469 p =.39 p =.938 p =.82 p =.73 p = 877 p =.47 p = lags p =.63 p =.43 p =.783 p =.39 p =.83 p =.32 p =.73 p =.649 ARCH lag p =.45 p =.683 p =.37 p =.44 p =.254 p =.49 p =.657 p =.366 Whe p =.464 p = 8 p =. p =.656 p =.487 p =.42 p =.973 p =. sgnfcan a %, sgnfcan a 5%, sgnfcan a % Table 3. Exchange rae changes behavor (equaon (7)) Dependen varable: e Coeffcen (-sasc) UK CAN NZL SWE CHI POL CZE KOR 99:Q 99:Q 99:Q 99:Q 996:Q 994:Q4 994:Q 99:Q e (.38) (.852) (3.35) (.832) (.763) (.622) (2.24) (2.4) LM es lag p =.29 p = p =.79 p = p =.545 p =.23 p = p =.34 2 lags p =. p =.44 p =.38 p =.9 p =.44 p =.273 p =.926 p =.94 ARCH lag p =.839 p =.344 p =.93 p =.7 p =.24 p = 6 p =.79 p =. Whe p =.38 p =.285 p =.94 p =.25 p =.437 p =.543 p =.649 p =. sgnfcan a %, sgnfcan a 5%, sgnfcan a %

12 Table 4. Opmal neres rae s reacon funcon Counry/Tme Conrol G coeffcens g coeffcens perod varable UK = = π e = = π +.29 e CAN = = π e = +.69 π e NZL = = π e = π e SWE = = π e = π e.287 Average ( = ) Average ( = ) CHI = = π e = π e POL = = π e = π e.3456 CZE = = π e = π e KOR = = π e = π e Average ( = ) Average ( = ) F value ( = ) F value ( = )

13 Table 5. Devaons from he arge of 2 percen annual growh for boh GDP and nflaon ( ) ( π π ) = = µ, =, µ 2, = µ, =, µ 2, = UK.96. CAN NZL SWE.63.8 Average CHI POL CZE KOR Average F value

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15 Munhos, M.K., S.A.L. Alves, and G. Rella (22) Modelo Esruural com Seor Exerno: Endogenzacao do Premo de Rsco e do Cambo Brazlan Cenral Bank workng paper 42. Posen, A. (998) Cenral bank ndependence and dsnflaonary credbly: a mssng lnk? Oxford Economc Papers 5, Romer, D. (2) Advanced Macroeconomcs, 2 nd ed., McGraw-Hll: New ork. Svensson, L.E.O. (997) Inflaon forecas argeng: mplemenng and monorng nflaon arges European Economc Revew 4, 46.