Estimating nonlinear DSGE models with moments based methods

Transcription

1 Dyare Workig Papers Series hp:// simaig oliear DSG models wih momes based mehods Ivashcheko Sergey Workig Paper o. 32 Jauary , rue du hevalere 7503 Paris Frace hp://

2 simaig oliear DSG models wih momes based mehods y Ivashcheko Sergey bsrac This aricle suggess he ew approach o a approximaio of oliear DSG models momes. This approach is fas ad accurae eough o use i for a esimaio of oliear DSG models. The small fiacial DSG model is repeaedly esimaed by several modificaios of suggesed approach. pproximaios of momes are close o he resuls of large sample Moe arlo esimaio. Qualiy of parameers esimaio wih suggesed approach is close o he eral Differece Kalma Filer he DKF based. he same ime suggesed approach is much faser. Keywords: DSG DSG-VR, GMM, oliear esimaio Jel-codes: S. Peersburg Isiue for coomics ad Mahemaics Russia cademy of Scieces Tchaikovsky Sr., S. Peersburg, 987 RUSSI glucke_ru@pisem.e el:

3 . Iroducio Moder macroecoomics seeks o explai he aggregae ecoomy usig heories based o srog microecoomic foudaios. The advaage of such a approach is a descripio of models i erms of deep srucural parameers ha are o iflueced by ecoomic policy [Wickes 2008]. Neverheless, hese parameers should be esimaed for usage of DSG models. There are differe ecoomeric echiques for models esimaio bu empirical sudies have coceraed heir aeio o a esimaio of firs-order liearized DSG models [Tovar 2008]. No-liear approximaios of DSG models have several impora advaages, i paricular, hey allow uceraiy o ifluece ecoomic choices [Ruge-Murcia 202]. Parameers drifig ad sochasic volailiy are examples of impora elemes ha are esseially worhless wih liear approximaios [Feradez-Villaverde, Guerro e al. 200]. Liear approximaio of DSG models behavior may sigificaly differ from higher order more accurae approximaios [ollard ad Juillard 200]. Secod order approximaio makes differece bewee he models ad he approximaio behavior much smaller. There are a few mehods for o-liear approximaios of DSG models, bu he perurbaio mehod is he mos widely used [Schmi-Grohe ad Uribe 2004]. Thus advaages of o-liear approximaios are kow ad echiques for approximaios are developed. However, hese advaages ca o be fully implemeed due o lack of esimaio echiques for o-liear approximaios exisig echiques are oo compuaioally expesive for usage of hem wih medium-scale DSG models. There are wo mai approaches for a esimaio of DSG models: momes based ad likelihood based [DeJog ad Dave 2007, aova 2007]. The likelihood based approaches use o-liear filers for he cosrucio of he likelihood fucio. The firs of hem is he paricle filer. This ool could produce all advaages of oliear approximaio, icludig sharper he likelihood fucio ad smaller variace of parameers esimaor [ ad 2

4 Schorfheide 2006, Feradez-Villaverde, Guerro e al. 200]. However, he paricle filers have some disadvaages: he likelihood evaluaio wih he paricle filer is a radom variable each evaluaio of he likelihood for he same model ad observaios produces differe value. Thus usual maximizaio algorihms ca be used ad Markov hai Moe arlo iefficiecy grealy icreases - Pi, Silva e al. 202 show ha umber of draws should be 0 from 5 o 400 depedig o he umber of paricles imes higher for he same accuracy of MM. There are aleraive filers ha ca be used for he likelihood calculaio. For example, drease 2008 has show a advaage of he eral Differece Kalma Filer over a few versios of paricle filer i erms of he qualiy ad compuig ime 00 imes faser. The Quadraic Kalma Filer has a advaage i he qualiy wih some loss i compuig ime over he Usceed Kalma filer Julier ad Uhlma 997 ad he eral Differece Kalma Filer Ivashcheko 203. Momes based approaches for esimaio of DSG models are more robus [Ruge- Murcia 2007, reel ad Krisese 20]. The firs of hem is he isrumeal variables approach which is a special case of he geeralized mehod of momes [aova 2007]. ll variables of he DSG model should be observed for usage of his approach. I ca be rue oly for small-scale DSG models. oher versio of he GMM is used for a liearized model because a wide rage of empirical arges ca be calculaed aalyically [DeJog ad Dave 2007]. The simulaed mehod of momes has almos he same saisical efficie as he GMM, bu i is more compuaioal demadig [Ruge-Murcia 2007]. The SMM ca be implemeed for o-liear DSG models [Ruge-Murcia 202, Kim ad Ruge-Murcia 2009]. u i s oo slow for a esimaio of medium-scale or large-scale models ha are used for policymakig. oher momes based approach is he idirec iferece. Theoreically i s more efficie ha he GMM [reel ad Krisese 20]. The usage of he idirec iferece is more complicaed ha he GMM, because i requires kowledge of he momes disribuio fucio. I ca be calculaed easily oly for a arrow rage of momes usually i is parameers esimaio of a 3

5 4 ecoomeric model, example is DSG-VR model. So, usually his approach is described differely [DeJog ad Dave 2007]. This aricle suggess a approach for a fas calculaio of o-liear DSG model s momes. This momes calculaios are compared wih aleraive approaches. small oliear DSG model is esimaed wih momes based approaches. The qualiy of such esimaios is compared wih liear maximum likelihood esimaio ad he DKF based quasimaximum likelihood esimaio. 2. The approach for momes calculaio The equaio describes he daa geeraig process for sae variables which is approximaio of a raioal expecaio model wih perurbaio mehod. xogeous shocks have ormal disribuio wih covariace marix ad mea equal zero. The measureme equaio 2 describes he depedece of observed variables Y o sae variables ad measureme errors u ha have ormal disribuio wih zero mea ad covariace marix u. x xx 0 u D S Y 2 The measureme equaio 2 is liear, herefore depedece bewee momes of observed variables Y ad sae variables is sadard. quaios for firs ad for secod momes of sae variables 3-5 are more complicaed. xx 3 U U 4

6 5 s xx s s I I I I 5 where is he fourh mome of vecor, I is ideiy marix of he same size as sae variables vecor, U _ ad U _ are permuaio marixes ha describe depedece bewee vecors idicaed i he subscrip equaio 6 is a example: U _ 6 The equaio 7 shows marix formula for calculaio of he fourh mome for : vec vec vec vec vec vec vec vec 7 where is he umber of elemes i he vecor, vecmm is he vecorizaio fucio ha rasforms marix M o marix wih rows ad m colums. The soluio of he equaio 3 requires kowledge of vecor s secod mome. The equaio 4 for he secod mome requires kowledge of he hird ad he fourh momes. So, i is impossible o solve hese equaios direcly. There are some approaches for approximaio of equaio 3-5 soluio. The key quesio is wha o do wih higher order momes. The firs way is o approximae he higher order momes ormal disribuio would be used for his approximaio. The secod way is o assume ha he higher order momes are equal o zero. quaios 8-9 are he formulas for he hird ad he fourh momes of ormal disribuio: U _ 8 U U U U 9 where is he firs mome of a vecor, is he secod ceral mome of a vecor, is he fourh ceral mome of a vecor. quaios 3-4 wih ormal approximaio of he higher order momes are oliear i momes. alyical soluio of a large o-liear sysem is problemaic. Thus umeric soluio is required. The simple umeric mehod is used: k xx k k,,, 0

7 , k, k, k, k, k, k, k, k, k, k, k U _ U, k, k _, k, k, k where,k,,k,,k,,k are he firs, he secod, he hird ad he fourh momes of a vecor afer ieraio k. I should be oed ha oly a few ieraios ca be calculaed due o compuaioal coss. I meas ha geeral properies of he covergece are less impora. The umerical comparisos of momes ad esimaio based o hem are made isead of i. The equaio 5 ca be solved wihou umerical approximaio jus wih ormal approximaio of hird mome of vecor [ -s ]. This approach would be called he ormal approximaio of he higher ha 2 d momes he NHM2. leraive approach where he higher order momes are zeros is also used i his aricle. Due o lower compuaioal coss he hird mome of ca be calculaed large formula 2. 6

8 7,k,k,k,k _,k,k,k,k,k,k,k _,k,k,k,k,k _,k _,k _,k _,k _,k _,k _,k _,k,k,k _,k _,k,k _,k,k _,k _,k,k,k,k,k,k,k,k _,k,k,k,k,k U U U U U U U U U U U U U U U U 2

9 2 U U U,k U U,k U U U U U,k,k _,k,k,k,k _,k,k,k,k,k,k,k,k,k,k,k,k,k U U,k U,k U,k U U _ U,k,k,k,k _,k,k,k,k,k _,k The equaio for he hird mome wih zero higher momes is liear i he momes of sae variables, bu i is a large scale equaio. quaio for he secod momes is Lyapuov equaio ad here are special algorihms for solvig i. The equaio for he hird momes is large ad here are o special algorihms which use specific srucure of marixes for solvig i. This is why he same ieraio algorihm is used for momes calculaio. This approach would be called he zero approximaio of he higher ha 3 momes he ZHM3. The case wih zero approximaio of he higher ha he secod mome is calculaed oo he ZHM2. I should be oed ha differece bewee he NHM2 ad he ZHM2 is 3 of 20 sum compoes i he dispersio formula. The suggesed approaches use properies of exogeous shocks ormal disribuio for calculaio of exogeous shocks higher momes. I meas ha hese approaches ca be easily,k,k,k,k 8

10 modified for oher disribuio of exogeous shocks. However, a approximaio of a raioal expecaio model wih he perurbaio mehod ca have ifiie ucodiioal momes if shocks have heavier ails such as -disribuio. The reaso is followig: -disribuio has ifiie higher momes depedig o degrees of freedom formula meas ha deped o 2, - 4, -2 8 ad so o. Thus, he suggesed approaches ca be modified for o-ormal disribuio of exogeous shocks, if all momes of his disribuio are fiie. 3. ompariso echiques Fiace is oe of hose areas where liear approximaio of DSG models is usuiable. Therefore a fiace model is used here for comparig differe esimaio approaches. The same model as [Ivashcheko 203] is used bu wih addiioal observed variables. Households maximize he expeced uiliy fucio 3 wih budge cosrai 4. There are 3 ypes of a expediure: cosumpio wih exogeous price Z P,, oe period bods, ad socks, he price of which is S. There are 3 sources of icome: exogeous icome S Z I,, bods wih ieres ha were bough oe period ago R - -, ad socks wih divided ha were bough oe period ago - S +D. 0 max 0 S D SZI Z P, S R, 4 This model suggess ha divided growh is exogeous 5, he bod amou is se by goverme 6, ad amou of socks is equal o 7. D Z D, D 5 Z, S 6 7 The model 3-7 is rasformed io 8-22 where saioarized variables are used. Table shows he relaio bewee he origial ad saioarized variables. 3 9

11 TL. The DSG model variables Variable Descripio Saioary variable Value of bods bough by households a period b / S osumpio a ime c l Z P, / S D Divideds a ime d ld / S R Ieres rae a ime r lr S Price of socks a ime s ls / S mou of socks bough by households a period x Lagrage muliplier correspodig o budge resricio of households a period xogeous process correspodig o earraioaliy of households wih heir bod posiio z,, Z,, xogeous process correspodig o earraioaliy of households wih heir cosumpio z,, Z,, xogeous process correspodig o earraioaliy of households wih heir socks posiio z,, Z,, Z, xogeous process correspodig o bod amou z, Z, sold by he goverme Z D, xogeous process correspodig o growh of z D, Z D, divideds Z I, xogeous process correspodig o icome of z T, ZT, households z Z Z Z,, Z,, Z,S, Z P, xogeous process correspodig o price level P, l P, P, s z, c i P i e e c 0 i0 0 e max c b x b x e b x d e zi, r s d d s zd, b z, 2 x 22 The opimal codiios of 8-9 problems wih addiioal exogeous processes z,s,, z,,, z,, are he followig: s zp, d e z, S, l e e 23 z,, r s l s zp, e e 24 c c z,, 25 addiioal exogeous process z,s,, z,,, z,, ca be ierpreed as ear-raioal households hese processes have zero mea. oher ierpreaio is a compesaio of approximaio errors hese processes allows he use of a liear approximaio for parameer esimaio. ll he exogeous processes are R wih he followig parameerizaio: 0

12 z *, 0,*,,*,,*, z*, *, 26 The model parameers are esimaed wih idirec iferece approach DSG-VR wih 4 lags ad 5 ieraios wih zero higher ha secod momes. Mohly daa verage rae o - moh cerificaes of deposi MSI US price reur MSI US gross reur persoal cosumpio expediures compesaio of employees from Jauary 975 ill December 202 is used. simaed values are used for geeraig observaios. The followig wo approaches are used for compariso of he momes esimaio echiques. The firs of hem is a compariso of momes approximaios wih a usual simulaio based approach for mome s calculaio i should be more accurae i case of a large sample simulaio. The secod approach is a compariso of esimaio error geeraio of he observed variables by he DSG model ad he esimaio of models wih differe momes based approaches. The DSG model is simulaed for observaios. The momes are calculaed. This procedure is repeaed 0 imes. Mea ad sadard deviaio of he momes are repored. Deviaios from his mome s mea errors are repored for each approach. The followig procedure is used for compariso of he esimaio resuls:. Geeraio of 400 observaios 600 ad drop of he firs 200 observaios from he secodorder approximaio of model 2. Parameers esimaio by differe approaches liear maximum likelihood he DKF based maximum likelihood he idirec iferece maximum likelihood ad he GMM wih differe mome s calculaio approaches. The rue values of parameers are used as he iiial values. 3. Seps -2 are repeaed 00 imes. The idirec iferece maximum likelihood is DSG-VR wih ifiie weigh of prior parameer maximum likelihood esimaio Del Negro ad Schorfheide The Newey- Wes esimaor wih widow equal 74T/00 2/9 is used for calculaio of momes variace for he GMM [Hamilo 994].

13 The DKF is used as a bechmark oliear esimaio echique because i has advaage over oher approaches i erms of speed wih appropriae qualiy ha are well kow i DSG lieraure [drease 2008, Ivashcheko 203]. 4. The resuls Table 2 shows he resuls of mome s esimaio. ll approaches have very high errors afer he firs ieraio, bu hese errors grealy decrease afer he secod ieraio o he same level for each approach. The errors afer 5 ad 0 ieraios are very close ha idicaes fas covergece. I should be oed ha he errors of all approaches are close o errors of mea over sample for he mos of he momes errors are smaller ha sadard deviaio. This idicaes a very high qualiy of such simple mome approximaios. Thus suggesed approaches produce more accurae esimaio of he momes ha exisig approach simulaio ad require less compuer ime 4 sec. for simulaio ad less ha secod for he NHM2 ad he ZHM2. The qualiy of he NHM2 ad he ZHM3 is almos he same. The qualiy of he ZHM2 is 3-4 imes worse afer 5-0 ieraios. I should be oed ha he RMS of all momes for he ZHM2 wih 0 ieraios is a lile bi worse ha oes wih 5 ieraios. more deailed view shows ha RMS of he firs momes are much smaller for 0 ieraios case, bu RMS of secod momes is a lile bi higher for 0 ieraios case. There are oly 5 firs momes ad 5 secod momes icludig all 4 lags herefore ifluece of he secod momes is higher for he RMS of all momes. This differece ca be more impora for he GMM approach due o smaller variace of he secod momes which meas higher weigh of secod momes. This effec ca be impora for he NHM2 ad he ZHM3 because here are a few momes ha have higher errors i 0 ieraios case ha i 5 ieraios case i ca make score fucio of he GMM wih 5 ieraios beer ha wih 0 ieraios for some weigh marixes. 2

14 TL 2. The DSG model momes esimaio Mea Sd NHM2 ZHM2 ieraio umber obs_pp obs_pg obs_r obs_c obs_i obs_pp obs_pg obs_r obs_c obs_i obs_pp Firs Secod Secod lagged obs_pg obs_r obs_c obs_i RMS of firs momes RMS of secod momes RMS of secod lagged momes lag RMS of all secod momes 4 lags RMS of all momes TL 2 coiued. The DSG model momes esimaio Mea Sd ZHM2 ZHM3 Firs Secod Secod lagged ieraio umber obs_pp obs_pg obs_r obs_c obs_i obs_pp obs_pg obs_r obs_c obs_i obs_pp obs_pg obs_r obs_c obs_i RMS of firs momes RMS of secod momes RMS of secod lagged momes lag RMS of all secod momes 4 lags RMS of all momes RMS of parameers esimaio by he idirec iferece DSG-VR wih 4 lags wih differe mome s calculaio echiques preseed a he able 3. RMS for he GMM 3

15 approach preseed a he able 4. I should be oed ha he idirec iferece wih momes calculaed by he NHM2 or he ZHM2 wih 2 ieraios produce errors covariace marix which is o posiive-defiie. Thus, RMS of he idirec iferece preseed oly for 5 ad 0 ieraios. The resuls for he ZHM3 are o preseed due o compuaioal expese of his approach he ZHM3 wih 5 ieraios requires abou 2 secod i requires abou secods for 2 ieraios wha is much higher ha he DKF. TL 3. The RMS of parameers esimaio he liear likelihood he DKF DSG-VR4 he NHM2, 0 ier. DSG-VR4 he NHM2, 5 ier. DSG-VR4 he ZHM2, 0 ier. DSG-VR4 he ZHM2, 5 ier. Sd of, Sd of, Sd of,s Sd of Sd of D Sd of I Sd of P l , ,D ,I ,P , , ,S , ,D ,I ,P Sum of RMS Sum of MS Time for likelihood calculaio sec* *P used: Iel core 2 Duo GHz, Gb RM, Widows P. I should be oed ha a few parameers have much higher RMS for each approach which mea ha is ifluece is criical for such measure as sum of MS. The idirec iferece wih he ZHM2 wih 5 ieraios produces exremely high qualiy of esimaio. The reaso is a high amou of local exremums may parameers values produce errors covariace marix 4

16 which is o posiive-defiie. bou half of cases produce low value of log-likelihood fucio local exremums which are close o he iiial values rue values, produce low RMS. The NHM2 wih 0 ieraios produces he bes qualiy accordig o sum of RMS. However, he ZHM2 wih 0 ieraios produces he bes qualiy accordig o sum of MS. The DKF is he secod bes for he boh measures of qualiy. I should be oed ha he NHM2 wih 0 ieraios has almos he same sum of MS as he DKF. Uexpeced resul is ha he ZHM2 is beer ha he NHM2 for esimaio purpose GMM wih 2 ad 5 ieraios despie worse qualiy of momes calculaio. The NHM2 is beer ha he ZHM2 GMM wih 0 ieraios accordig o he oe of qualiy measures. advaage of he idirec iferece over he GMM is expecable [reel ad Krisese 20]. TL 4. The RMS of parameers esimaio GMM GMM he NHM2 0 ier GMM he NHM2 5 ier GMM he NHM2 2 ier GMM he ZHM2 0 ier GMM he ZHM2 5 ier GMM he ZHM2 2 ier Sd of, Sd of, Sd of,s Sd of Sd of D Sd of I Sd of P l , ,D ,I ,P , , ,S , ,D ,I ,P Sum of RMS Sum of MS Time for likelihood calculaiosec* *P used: Iel core 2 Duo GHz, Gb RM, Widows P. 5

17 The speed of likelihood fucios calculaios for ormal approximaios of higher momes he NHM2 is bewee liear-likelihood ad he DKF-likelihood. For he ZHM2 he speed is almos he same as for liear-likelihood. 5. oclusios This aricle suggess he ew approach o approximaio of oliear DSG models momes. These approximaios are fas ad accurae eough o use hem for a esimaio of parameers of oliear DSG models i produces a more accurae esimaio of momes ha simulaio of sample. xisig mehod of approximaio of oliear DSG models momes Moe-arlo simulaios is 32 o 2 imes slower ha he suggesed approaches depedig o versio. The suggesed approaches are sec. or sec. imes faser depedig o versio ha he DKF 3.95 sec.. ombiaio of he suggesed approaches wih he GMM does produce high qualiy esimaio, bu combiaio of he suggesed approaches wih idirec iferece has almos he same qualiy as he DKF. Oe of qualiy measure sum of RMS is 7.5% 3.79 worse or 5.0% 2.75 beer depedig o versio ha he DKF oher measure sum of MS is 0.7% 2.39 worse or 9.4% 2.5 beer depedig o versio ha he DKF Thus, he suggesed approaches are close i he erms of esimaio qualiy bu much faser ha oe of he bes exisig oliear esimaio approaches he DKF. Lieraure S. ad Schorfheide F., ayesia aalysis of DSG models. Workig Papers from Federal Reserve ak of Philadelphia. No drease M.M., No-liear DSG models, he eral Differece Kalma Filer, ad he Mea Shifed Paricle Filer. RTS Research Paper vailable a SSRN: hp://ssr.com/absrac=

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19 Ruge-Murcia F.J simaig oliear DSG models by he simulaed mehod of momes: Wih a applicaio o busiess cycles. Joural of coomic Dyamics ad orol, 36 6: Ruge-Murcia F.J Mehods o esimae dyamic sochasic geeral equilibrium models. Joural of coomic Dyamics ad orol, 3 8: Schmi-Grohe S. ad Uribe M Solvig dyamic geeral equilibrium models usig a secod-order approximaio o he policy fucio // Joural of coomic Dyamics ad orol, 2004, vol. 28, issue 4, pages Tovar DSG models ad ceral baks. IS Workig Papers from ak for Ieraioal Selemes. No Wickes M.R MROONOMI THORY DYNMI GNRL QUILIRIUM PPROH. Priceo Uiversiy Press, Priceo. 8