IV. Sign restrictions and Bayesian VAR

Transcription

1 PhD Course: Srucural VAR models IV. Sign resricions and Bayesian VAR Hilde C. Bjørnland Norwegian School of Managemen (BI)

2 Lecure noe IV: Sign resricions and Bayesian VAR Conen 1. Inroducion 2. Bayesian versus classical inference 3. Priors for VARs 4. Comarisons of DSGE and SVAR models 5. Sign resricions Evaluaion

3 1. Inroducion In he SVAR lieraure, researchers exerimen wih model secificaions unil he emirical resuls sar o look reasonable. For insance in sudies of moneary olicy, reasonable behavior is usually aken o mean ha moneary conracion should a leas on imac raise ineres raes, i should lower rices and some moneary aggregaes in he long run, and a leas emorary lower ouu. The search for models ha roduce such resuls is acually ar of he esimaion rocess, and he reasonableness crieria are informal (over-) idenifying resricions. The esimaion sraegy can be characerized as secify model, idenify shocks and imulses. If he model is reasonable one can so here, or if no declare a uzzle and sar over again. Bayesian analysis is abou how o formally incororae hese yes of beliefs ino a rior on he model arameers. Bayesian analysis can herefore also hel in idenificaion.

4 Assume again he linear dynamic srucure of he -h order vecor auoregression: y μ + e (1) = + A1y 1 + A2 y A y The reresenaion in (1) included n 2 unknown arameers in he A j marices and anoher n(n+1)/2 in he covariance marix E ( e e ') = Σ. However, by he order condiions, his would leave use wih (n-1)n/2 fewer arameers in he reduced form (seing =1) han in he srucural model. Since he roeries of he daa are deermined by he reduced form arameers, any aem o deermine he srucural arameers from roeries of he daa will face indeerminacy, unless we can find (n-1)n/2 idenifying resricions. As n increases, need o search for oher mehods for how o develo arsimonious reresenaions ha caure rich dynamic relaionshis. Bayesian analysis using for insance informaive Bayesian riors in an mehodology ha has araced lo of research. The generalizaion of Dynamic facor models is anoher aroach; see for insance Forni e al. (2000).

5 2. Bayesian versus classical inference Assume again he reduced form VAR in (1). Le δ denoe he vecor conaining he unknown elemens of μ A,..., A, Σ and le y = [ y',..., y', y ] denoe he vecor of observaions a all daes., 1 T 2 ' 1 A classical economerician forms an esimae of δ such as he maximum likelihood esimae (δˆ), and ask wha he disribuion of (δˆ) would be if one reea he inference on a large number of samle. The arameers are reaed as fixed unknown quaniies. Unbiased esimaors are imoran, as he average value of samle esimaor converges o he rue value wih he law of large numbers. For he Bayesian economerician, robabiliy measures he degree of beliefs ha a researcher has in an even. Parameers δ are random variables wih a robabiliy disribuion (δ ). For insance, here could be a 50% robabiliy ha δ exceeds one.

6 There is no oin in esing he arameers in reeaing samles since beliefs are no necessary relaed o he relaively frequency of an even. The beliefs of he Bayesian economerician summarized by he robabiliy densiy is called he rior. The rior is formed before seeing he daa. The rior could be weak or srong. To moivae he Bayesian aroach, consider wo random variables A and B. The rules of robabiliy imlies ha ( A, B) = ( A B) ( B) or ( B A) ( A) where ( A, B) is he join robabiliy of A and B occurring, ( A B) is he robabiliy of A occurring condiional on B having occurred (condiional robabiliy) and (B) is he marginal robabiliy of B. Equaing he wo exressions above and rearranging, give us he Bayes rule ( A B) ( B) ( B A) = (2) ( A)

7 Les urn o he classical regression model. The goal of he Bayesian economerician is o use he observed daa, y, o calculae he oserior densiy g( δ y), ha is a summary of wha we believe now ha we have seen he daa. We can find he oserior densiy using Bayes Law: f ( yδ ) ( δ ) f ( yδ ) ( δ ) g ( δ y) = = f ( yδ ) ( δ ) = L( δ y) ( δ ) ( δ ) f ( yδ ) dδ f ( y) (3) where f ( y δ ) = L( δ y) is he likelihood funcion and he inegral reresens a definie inegral over all ossible values of δ. We can rea f(y) as consan, so ha equaion (3) saes ha he oserior will be roorional o he likelihood imes he rior. The roblem wih (3) is ha he inegral is no known analyically. However, develomens of algorihms have allowed us o simulae draws from he disribuion g( δ y) direcly wihou needing o calculae he disribuion iself (imorance samling, Gibbs samler).

8 3. Priors for VARs The rior could be eiher diffuse or igh, where a igh rior allow for only a minimum of variance away from he rior. Examle: Consider he linear regression, y = x ' α + e, e ~ iidn(0,1) A rior could ake he simle form on he disribuion of he coefficien α, 2 α~ N(0, τ I ), kx1 kxk ex 1 2τ = ( α ) (2πτ 2 2 ) k / 2 α ' α where Large τ means diffuse rior (variance large) Small τ means igh rior (variance small)

9 Bayesian riors The Bayesian aroach is o secify resricions on he coefficiens ha are fuzzy, raher han hard exclusion resricions. Shrinkage esimaors have long been suggesed for dealing wih mulicollineariy and similar roblems. The usual aroach in linear regressions o deal wih degrees of freedom roblems is o reduce he numbers of regressors, which in auoregressive models is o exclude lags, hereby forcing he coefficien o zero. Bayesian VARs insead suggess ha coefficiens on long lags are more likely o be close o zero han coefficiens on shorer lags. However, daa may override his assumion if he evidence of a coefficien is srong. Use a normal disribuion wih a mean of zero and small sandard deviaions. In VARs, he riors can ake many forms, A general shrinkage of all coeffiens owards zero Shrinkage owards secific yical dynamic aerns as in Sims and Zha (1998) Shrinkage owards fully secified dynamic sochasic general equilibrium models as in Del Negro and Schorfheide (2004).

10 Dummy Observaion Priors The basic idea of dummy observaion riors for VAR s is he same as ha for dummy observaion riors in ordinary regression models. Inuiively, one adds exra daa o he samle ha exress rior beliefs abou he arameers. The rior akes he form of he likelihood funcion for he dummy observaions. The difference from single equaion regression models is ha he added observaions are used in all he equaions of he sysem, no jus in one. Minnesoa Prior In VARs, one mus concern iself no only wih lags of he deenden variable, bu also wih lags on he oher endogenous variables. Furher, economic variables are ofen non-saionary so suiable ransformaions has o be aken before invering he VAR. The Minnesoa (or Lierman) rior deals wih hese issues. The rior suggess ha one should se he rior mean of he firs lagged value of he deenden variables o one, while all he oher coefficiens in he model should be se, wih varying degrees of uncerainy, close o zero. The varying degrees of uncerainy are indicaed by he sandard deviaions calculaed from benchmark ou-of-samle forecass made wih simle univariae models, and he variaion in he degree of uncerainy is assumed o decrease as he lengh of he ime-lags increases.

11 The arameers in he rior marix are calculaed from hese sandard deviaions and from "hyerarameer" facors ha vary along a coninuum ha indicaes how likely he coefficiens on he lagged values of he oher endogenous variables are o deviae from a rior mean of zero. Circumsances can arise where one has beliefs abou arameers in some equaions bu no ohers, or has differen beliefs abou differen equaions; see Sims and Zha (1998). Training Samle Prior Relace dummy observaions by acual observaions from a re- or raining samle. DSGE model Prior Use arificial observaions generaed by a DSGE model.

12 4. Comarisons of DSGEs wih SVARs Comare DSGE model dynamics o idenified VARs. Two lines of research: 1. Imrove VAR esimaes by resricing is arameer esimaes. General equilibrium models are used o derive riors o imrove he forecas erformance of VAR models. Synheic daa are simulaed for a DSGE model, an combined wih acual daa o esimae Bayesian VARs. The weigh on he rior is deermined by he amoun of synheic daa relaive o acual daa, see Del Negro and Schorfheide (2004). 2. Imrove DSGE models by relaxing is resricions. The VAR is used o minimize he disance beween a funcion of he arameers in he heoreical model and a funcion of daa. The lieraure has focused on a se of arameers o minimize he disance beween he model imulse resonses and hose from an emirical VAR. The simulaed mehods of momens esimaor (SMM) solves θ = arg min θ [ ˆ ]' D D( θ ) V [ Dˆ D( θ )] ˆ 1 (4) Where D(θ) is a vecor of imulse resonses given θ from he heoreical model and emirical counerar derived from he VAR. V is a diagonal weighing marix. Dˆ is he

13 Noe however ha he sae-sace reresenaion of he linearized DSGE model may no always be aroximaed by a finie order VAR, see Fernandez-Villaverde er a. (2005). Wih regard o he lieraure on forecasing, one can comue draws from he oserior disribuion and comare resonses. DSGE-VARs imroves. Careful consrucion of VAR riors is crucial, for insance a Minnesoa syle rior (Sims and Zha, 1998) or when using riors for DSGE-VARs, see Del Negro and Schorfheide (2004)

14 5. Sign resricions Make imlicily held assumions exlici. The sign resricion mehodology daes back o Faus (1998), Uhlig (1999) and Canova and de Nicolo (2000). They use he convenional view (=rior) ha conracionary olicy shocks lead o a rise in ineres raes and declines in money, rices and ouu, direcly for idenifying moneary olicy shocks. The idenifying resricions are herefore suored by consrucion. (However, noe ha Uhlig (1999) has shown ha moneary olicy shocks, idenified via he effecs of raising ineres raes, lowering money and inflaion, have no clear effecs on ouu, see also Sims and Zha (1998)). Sign resricions requires ha we define he whole sace of resonses, since we will be searching among all ossible yes of resonses (bu kee only hose we have a rior for).

15 Consider again he reduced form VAR (ignoring any consan) y + e = A1 y 1 Tha has a MA reresenaion y = ( I A L) B( L) e 1 1 e = Le e Ψ 1 and assume = E[ '] I =, so ha he VAR can be exressed in srucural form as y = Be j B L 1 = ( ) Ψ 1 1 where ( e e ') = Ψ Ψ ' = Σ, since j= 0 E E[ '] I. = Recall ha we could decomose he CVM ino he lower riangular marices (i.e. he Cholesky decomosiion), Σ = PP'.

16 Le Λ be an orhonormal marix, ha is a marix wih he roery Λ Λ ' = Λ' Λ = I. The relaionshi beween he reduced from residuals and he srucural relaionshi can hen be described as follows e = PΛ (5) Since E( e e ') = E[ PΛ ' Λ' P' ] = PΛΛ' P' = PP' = Σ I is edious o characerize he sace of orhonormal marices. For n=2, i can be shown (see Schorfheide 2006) o be cos λ sin λ Λ( λ) = sin λ cos λ (6) where λ [ π, π ]. Idenificaion requires resricions on λ.

17 Before roceeding, see how his work for he shor run resricions. Consider a reduced form VAR in inflaion and ouu growh. + Δ Δ = Δ Δ e e x a a a a x 2, 1, , We assume ha he variables are driven by wo srucural shocks (moneary olicy and roduciviy shocks) = r m,, Le he idenifying resricion be ha moneary olicy can no affec ouu growh emorarily. This can be found by seing λ=0, in P e Λ =, so ha we ge he lower riangular Choleski facor (cos0=1, sin0=0) as before = = r m r m e,, ,,

18 Sign resricions are less binding. Le he idenifying assumion be ha a conracionary moneary olicy reduces boh inflaion and ouu on imac (in conras o he recursive VAR assumions ha assume a zero resonse on imac). Noe ha as e = PΛ and y = B( L) e y = B( L) PΛ Δ Δx = B( L) cos λ sin λ sin λ cos λ m, r, I can be verified ha Δ y MP, 11 cos λb11,1 + ( 21 cos λ + 22 sin λ) b12,1 = = Δ MP, x 11 cos λb21,1 + ( 21 cos λ + 22 sin λ) b 22,1 (7) MP,

19 The imlicaion for our resricion is hen 11 cos λb 11,1 + ( 21 cos λ + 22 sin λ) b 12, cos λb 21,1 + ( 21 cos λ + 22 sin λ) b 22,1 0 Which resrics λ o be in a cerain subse of [ π,π ]. The resricions will no give you one imulse resonse, bu a whole range of resonses.

20 Esimaion sraegy Consider he VAR of he form y + = A1 y 1 e, e PΛ(λ) =, A = ' A 1 A Bayesian vecor auoregression is fied o he daa. Form a rior for he reduced form VAR (i.e. he Normal-Wishar rior, which is a very weak rior ha ermis saionary, uni and exlosive roos). Using he daa, form he oserior disribuion of ( A, Σ). Take a draw from he oserior disribuion for he VAR coefficiens using samling echniques. Le P = chol(σ). Comue moving average reresenaion j =0 y = B ( A) e j For sign resricions, condiional on A and P, assign a rior disribuion o he se of λ s for which he sign resricions are saisfied. Generae a draw λ from his rior. Since he samle has no informaion on λ given A and P, he rior equals oserior.

21 Wih λ deermined, comue imulse resonses and variance decomosiion. Take as many draws on he sace of ossible imulse vecors. If he range of imulse resonse is comaible wih he sign resricion, kee i, oherwise discard i. Hence, kee he draws ha saisfy he sign resricions while discarding he ones ha do no. This leaves you wih many ossible draws. Calculae summary saisics like he mean, median, robabiliy bands ec.

22 Evaluaion Minimal sign resricions do no in down he resonse o shocks. There is an idenificaion roblem For he sign resricions o be informaive, a large number of sign resricions has o be imosed May conflic wih he desire o secify robus sign resricions.

23 References Canova, Fabio and Gianni de Nicolo, (2000) Moneary disurbances maer for business cycle flucuaions in he G-7, Journal of Moneary Economics, 49, Del Negro, Marco and Frank Schorfheide, (2004). Priors from General Equilibrium Models for VARS, Inernaional Economic Review, Dearmen of Economics, Universiy of Pennsylvania and Osaka Universiy Insiue of Social and Economic Research Associaion, 45, Fernandez-Villaverde, J., J.Rubio-Ramirez, and T. Sargen, (2005), A, B, C s (and D s) for Undersanding VAR s, Naional Bureau of Economic Research Technical Working Paer no. 308 and forhcoming, American Economic Review. Faus, Jon, (1998) The Robusness of Idenified VAR Conclusions abou Money, Carnegie-Rocheser Conference Series on Public Policy; 49, Schorfheide, Frank (2006), Bayesian Mehods for Macroeconomerics, mimeo, Universiy of Pennsylvania. Sims, Chrisoher A. and Tao Zha (1998), Bayesian Mehods for Dynamic Mulivariae Models, Inernaional Economic Review, 39, Uhlig, Harald (1999), Wha are he resuls of moneary olicy shocks on ouu? Resuls from an agnosic idenificaion rocedure, CEPR Discussion Paer 2137 (forhcoming in Journal of Moneary Economics).