A Test of Identification for Government Spending Shocks.

Transcription

1 A Tes of Idenificaion for Governmen Spending Shocks. Anna Kormilisina December 14, 2015 Absrac The response of consumpion o an increase in governmen spending in SVAR models may be influenced by he shock idenificaion sraegy. The popular idenificaion sraegy in Blanchard and Peroi (2002) (BP) is based on he assumpion ha any unpredicable adjusmen in governmen spending is due o he governmen spending shock. This implies ha he governmen spending shock can be idenified in an SVAR model by placing he governmen spending variable firs and using Cholesky facorizaion of error covariance marix o idenify he governmen spending shock. I urns ou o be imporan for he quanifying he macroeconomic impac of he governmen spending shocks. I firs show in his paper ha while BP idenificaion scheme resuls in a posiive consumpion response, his resul does no hold for alernaive idenificaion sraegy. Namely, if governmen spending is ordered second raher han firs, hen consumpion response o governmen spending is miigaed: I is negaive, alhough no significanly differen from zero. While his idenifying assumpion is cenral in SVAR models, i has never been formally esed. In his paper, I use an innovaive approach o es if governmen spending can indeed be reaed as predeermined relaive o all sources on uncerainy no relaed o he governmen spending policy. Wih his purpose, I modify a sandard DSGE model wih informaional iming resricions following Kormilisina (2013). In paricular, I firs assume ha each period can be divided ino wo subperiods, and all shocks have realizaions in he second subperiod, excep for he shock o governmen spending. Also, he amoun of governmen purchases mus be deermined before all shocks are realized. This model seup mirrors he idenificaion sraegy of BP. In an alernaive seup, I assume ha governmen spending can, 0 Deparmen of Economics, Souhern Mehodis Universiy, annak@smu.edu 1

2 a leas parially, reac o conemporaneous realizaions of oher macroeconomic shocks. I esimae his conemporaneous effec on governmen spending, which allows o evaluae he validiy of he BP idenificaion sraegy. RESULTS TO FOLLOW. Keywords:governmen spending shocks, DSGE model esimaion, iming, informaional subperiods 2

3 1 Inroducion 2 SVAR model 1) Simple 3 variable SVAR model, where G is ordered firs, like in Blanchard and Peroi 2) Simple 3 variable SVAR model where G is ordered second, and Y is ordered firs 3) Fully-fledged SVAR model wih and wihou Blanchard-Peroi idenificaion 3 Model Firs, assume a simple New-Keynesian model. The model can be reduced o four equaions: ineremporal Euler equaion, he resource consrain, and moneary policy rule. There are hree shocks ha govern he economy: governmen spending shock, echnology shock, moneary policy, and preference shock. 4 Fully fledged DSGE model The economy is populaed by a coninuum of infiniely-lived households. Each household paricipaes in he following aciviies. I consumes, supplies differeniaed labor services o he labor packer, accumulaes capial by means of invesing, rens capial services o firms, pays axes and receives dividends from ownership in firms Households Each household derives uiliy from a consumpion measure X, he exac definiion of which differs across he hree models, and homogenous labor h. The life-ime expeced uiliy of households is defined as E 0 =0 β d U(C bc 1, h ), where E 0 denoes expecaions based on period zero informaion se, β is he discoun facor, and d is he preference shock, evolving according o an AR(1) process: ( ) ( ) d+1 d log = ρ d log + ϵ d d d +1, (1) 3

4 where 0 < ρ d < 1, and ϵ d i.i.d.(0, σ 2 d ), wih σ d > 0, is he preference shock. The inraemporal uiliy funcion follows King, Plosser, and Rebelo (1988) in ha i is nonseparable in leisure and consumpion and consisen wih long-run balanced growh: U(C bc 1, h ) (C bc 1 ) 1 σ (1 h ) ζ, (2) 1 σ where he inverse of σ > 0 is he inverse of he ineremporal elasiciy of subsiuion in consumpion, and ζ is he elasiciy of he demand for leisure. 1 Homogenous labor h is a Dixi-Sigliz aggregae of differeniaed labor services h j, for j [0, 1] supplied by households o a labor packer: ( 1 ) 1 h = (h j ) η w η 1 dj w. 0 Here, η w is he elasiciy of subsiuion across differen ypes of labor, and he upper scrip j helps o disinguish beween differen ypes of labor. The homogenous labor h is supplied o firms a a real rae W. Households possess monopolisic power over heir wages, and have he abiliy o se he labor specific wage rae; however, hey are required o saisfy he demand for labor a his wage rae. Changes in he wage rae are subjec o quadraic adjusmen cos, deermined as Ψ ( W j W j 1 ) = α w 2 ( 2 W j W j µ z π), 1 per (real) dollar of he wage bill. In his formula, α w > 0 is he wage adjusmen cos parameer, W j is he individual real wage rae, π is he inflaion rae along he balanced growh pah, and µ z is he rae of growh of he economy (oupu, consumpion, and wages) along he balances growh pah. The households own physical capial, K. Capial is accumulaed hrough he process of invesing, and he oal sock of capial depreciaes a a variable rae depending on how inensively i is used. Moreover, invesmen adjusmen is cosly, wih he capial loss of S( ) per uni of invesmen. The dynamics of capial is herefore: ( )) I K +1 = (1 δ(u ))K + I (1 S, (3) 1 This uiliy funcion assumes he exisence of wealh effec from he governmen spending shock on labor supply, as opposed o he GHH ype preferences, where his wealh effec is absen. Monacelli and Peroi (2008) show ha absen he wealh effec, a sandard new-keynesian model wih price rigidiies will produce a posiive response of consumpion o a governmen spending shock. We choose o avoid he possibiliy of auomaically generaing he posiive response wihou eliminaing his possibiliy, allowing he esimaion procedure o deermine he size of he wealh effec. 4 I 1

5 where u deermines he inensiy of capial uilizaion as a fracion of capial being used in producion, and δ(u ) is he depreciaion funcion, parameerized as follows: δ(u ) = δ 0 + δ 1 (u u) + δ 2 2 (u u) 2, (4) where δ 0, δ 1, δ 2 0, and u is he seady sae rae of capial uilizaion. In Equaion (3), he cos of invesmen S( ) is he quadraic funcion: ( ) I S = κ ( ) 2 I µ I, 2 I 1 I 1 where κ > 0, and µ I is he seady-sae growh rae of capial and invesmen. Following Fisher (2003), invesmen goods I are obained from consumpion using a sochasic linear echnology, according o which a each dae, one uni of consumpion can produce Υ unis of invesmen. We call Υ he invesmen specific echnology. Denoing µ Υ, Υ /Υ 1, he gross growh rae of Υ, he dynamics for µ Υ, is ( ) ( ) µυ,+1 µυ, log = ρ Υ log + ϵ Υ +1, (5) µ Υ where ϵ Υ i.i.d.(0, συ 2 ), wih σ Υ > 0, and µ Υ is he growh rae of he invesmen specific echnology along he balanced growh pah. Capial services u K are rened ou o firms a a real renal rae R k. Households own shares in firms, and receive dividends wih he real value Φ. They pay a disorionary income ax, a he rae τ, and receive lump-sum ransfers in he amoun T r in erms of consumpion. Complee se of one-period sae-coningen asses, as well as he risk-free governmen bonds are raded in financial markes. If households have access o financial markes, 2 hen he budge consrain can be wrien in real erms as 3 E r,+1 L +1 + C + Υ 1 I + B +1 R = L π + (1 τ )R k u K + µ Υ ( ( )) W j 1 τ Ψ W j W j h j dj + Φ + B + T r, π 1 where L is he payoff in period of sae-coningen securiies raded in period 1, r,+1 is he price of a sae coningen securiy raded a dae for a claim on consumpion delivered in period + 1, C is real consumpion, τ is he income ax rae, and B is he real value of non-sae coningen governmen bonds in possession of households. The new bonds are purchased a a price 1/R. 2 This is he case in all models excep for he model wih rule-of-humb consumers. 3 To simplify noaion, we omi he household specific superscrip j when i is possible. 5

6 4.0.2 Firms A coninuum of monopolisically compeiive firms of measure 1 produce differeniaed inermediae goods. For producion, each firm uses capial and labor services, u K and h according o he following echnology F (u K, Z h ) q (u K ) θ (Z h ) 1 θ Z ϑ, (6) where 0 < θ < 1, variable q is model specific, inroduced in Secion??, Z ϑ represens he fixed coss of operaing a firm in each period, 4 Z is he sochasic labor-augmening produciviy process, growing a a rae of µ z,, µ z, Z /Z 1, which evolves according o and AR(1) process: ( ) ( ) µz,+1 µz, log = ρ z log + ϵ z +1. (7) µ z Here, µ z is he growh rae along he balanced growh pah, 0 < ρ z < 1, and ϵ z i.i.d.(0, σz), 2 wih σ z > 0 is he neural echnology shock. Each firm i [0, 1] maximizes he presen discouned value of dividend paymens, given by E s=0 µ z r,+s P i +sφ i +s, where r,+s s k=1 r +k 1,+k, for s 1, wih r, 1, and period dividend paymens in real erms are Φ i = P ( ) i P a i R k u i P K W h i i Ω, P 1 i where a i is he demand for he firm i s oupu, Ω( ) is he quadraic cos of price changes, which is proporional o he sochasic rend Z : ( ) P i Ω = α pz P 1 i 2 0 ( P i 2 π), P 1 i wih α p > 0, denoing he degree of price sickiness. Monopolisically compeiive firms mus saisfy heir demands a he posed price. The final good is he aggregae of differeniaed goods produced by monopolisically compeiive firms using a Dixi-Sigliz echnology: ( 1 ) 1 (Y i ) η ηp 1 p, di. where η p is he elasiciy of subsiuion beween individual good varieies. 4 Z is he sochasic rend for he economy, which is combinaion of he invesmen specific and labor-augmening echnologies. 6

7 4.0.3 Fiscal and moneary policy The fiscal auhoriy levies axes, provides lump-sum ransfers and develops public projecs wih real cos of G. We assume ha each period, he governmen saisfies a balanced budge. To ensure he model has a well-defined balanced growh pah, we assume ha governmen expendiures G evolve along he same sochasic rend as oupu and consumpion. Wih his purpose, we assume he raio ς g = G /Y is an AR(1) process: 5 ( ς g log +1 ς g ) = ρ g log ( ς g ) + ϵ g ς +1, (8) g where 0 < ρ g < 1, and ϵ g i.i.d.(0, σg), 2 wih σ g > 0, is he governmen spending shock. We assume ha moneary policy is described by a generalized Taylor ype rule wih he ineres rae smoohing and response o inflaion and oupu growh, as follows: log ( R R ) = α R log ( R 1 R ) + α π log ( π ) + α Y log π ( Y ) + ϵ r Y 1 µ, (9) z where Y is aggregae real oupu, α R, α π, α Y are Taylor rule parameers, and ϵ r i.i.d.(0, σ 2 r) is he moneary policy shock, wih σ r > Modeling Informaional Subperiods. Conrary o models wihou informaional subperiods, iming resricions may be imposed on agen decisions and realizaions of shocks. As a resul, no all realizaions of sochasic processes occur in he beginning of a period, and decisions of differen conrol variables mus be made before or afer hese shocks realize. Thus, any period can be divided ino one or several subperiods, each saring wih realizaions of paricular shocks, and choices for endogenous variables mus be made in differen subperiods. The SVAR model wih B-P idenificaion assumes ha no shock (excep for he governmen spending shock) can have a conemporaneous effec on governmen spending. This idenificaion sraegy can be embedded in a heoreical model wih informaional ime periods described in Kormilisina (2013). The iming resricion would specify ha in a given period, while he governmen spending shock has realizaions in he beginning 5 Such modeling assumpion is moivaed by he fac ha planned governmen expendiures are decided upon prior o he year of implemenaion, and herefore curren public expendiures are predeermined wih respec o curren oupu. To implemen his idea, we define ς g as he raio of G and he previous, raher han curren period oupu, Y 1. We find ha his approach o modeling governmen spending shock improves he marginal likelihood of all models. Alernaively, wihin-period iming resricions could be imposed, as in Kormilisina (2013). 7

8 of he period, all oher shocks occur in he middle of each period, and afer ha, decisions for all endogenous macroeconomic variables are made. As a resul, in a given period,s governmen spending is predeermined relaive o all oher endogenous variables. I also assume ha governmen spending is parially predeermined, which means ha G may respond o shocks a leas o some exen. The parial adjusmen is parameerized wih row vecor J, he elemens of which I esimae wih Bayesian mehods. Noice ha imposing specific iming consrains does no aler he sysem of equaions f(y, X, Y, X). To ake ino accoun he iming srucure in he soluion mechanism, i is convenien o pariion he conrol and sae vecors as follows: X = [x; θ], (10) Y = [y; z], (11) where n x 1 vecor x consiss of endogenous predeermined variables, and exogenous variables wih realizaion in he beginning of he firs subperiod, θ conains n θ exogenous variables wih realizaions in he second subperiod, y is he n y 1 vecor of full informaion conrol variables. These are endogenous variables he decision for which is made in he beginning of second subperiod, when realizaions of all shocks are known. Finally, z is he n z 1 vecor of parial conrol variables. These endogenous variables mus be decided in he firs subperiod, when realizaions of only a subse of shocks are known. Suppose equaions in f in Formula (??) are arranged as follows f = [f 0 ; f 1 ; f θ ]. (12) The se of equaions in f 0 includes n z equaions deermining he choice of parially endogenous variables in z, while f 1 includes n y equaions ha deermine fully endogenous variables y, and n x equaions deermining he dynamics of he sae variables in vecor x. The se of equaions in f θ describes he evoluion of exogenous shocks in θ, which presumably can be represened as AR(1) processes: θ = P θ + σϵ θ, (13) where P is he n θ n θ marix of auoregressive coefficiens, and ϵ θ is an n θ 1 vecor of i.i.d. shocks wih mean 0, and variance Σ. Because we derive he soluion o his problem using he soluion o is version wihou informaional subperiods, we assume ha sochasic processes in θ are saionary o guaranee exisence of ḠX and H X. Therefore, all eigenvalues of P are smaller han one in absolue value. 8

9 Denoing E he expecaions operaor ha akes ino accoun his iming srucure, Sysem (??) becomes E[f(Y, X, Y, X)] = 0. (14) Given he sochasic process for θ, he soluion o his sysem of expecaional difference equaions can be presened in general form as y = g(x, θ, θ 1, σ) z = j(x, θ, θ 1, σ) (15) x = h(x, θ, θ 1, σ) + σϵ x, where ϵ x is he op subvecor of innovaions e wih he lengh of n x, and subscrip 1 denoes previous period value. This sysem implies ha parially endogenous variables in vecor z canno respond o he curren realizaion of shocks in θ because informaion abou θ is no available ye a he ime when decision on z is made. Thus, he bes opion for a consrained decision maker is o rely on he condiional forecas of θ ha is deermined by θ 1 due o he auoregressive srucure of he shock process. When choosing fully endogenous variables in y in he second subperiod, z is reaed as a sae variable. As a resul, hrough heir response o z, fully endogenous variables in y will implicily ake ino accoun previous period s realizaion of he shock θ 1. Finally, endogenous predeermined variables in x will respond o θ 1 for he same reason. I assume j θ = J, where J is a marix of size n z n θ. The firs-order approximaion of equaions in Sysem (15) is given by y = g x x + g θ θ + g θ 1 θ 1, z = j x x + J θ + j θ 1 θ 1, x = h x x + h θ θ + h θ 1 θ 1 + σϵ x, where g x, g θ, g θ 1, j x, j θ 1, h x, h θ, and h θ 1 are marices of coefficiens o be deermined. 6 These marices have sizes n y n x, n y n θ, n y n θ, n z n x, n z n θ, n x n x, n x n θ, and n x n θ respecively. The firs-order approximae soluion can be presened in a more compac way as follows Y = G X x θ θ 1, X = H X x θ θ 1 + σϵ, (16) 6 In Appendix, I show ha same as in he model wihou informaional subperiods, he size of uncerainy does no influence he firs order approximae policy funcions in models wih iming resricions. 9

10 where in which G x = [ H x = G X = [G x, G θ, G θ 1 ], H X = [H x, H θ, H θ 1 ] (17) [ gx j x h x 0 nθ n θ ] [, G θ = ], H θ = g θ 0 nz n θ [ hθ P ] [ gθ 1, G θ 1 = ], H θ 1 = j θ 1 [ hθ 1 0 nθ n θ Suppose in he unresriced problem, he soluion can be represened wih marices Ḡ X and H X. For convenience, I pariion hese marices as follows: ], ]. Ḡ X = [ n x {}}{ n θ {}}{ ] ḡ Ḡ θ }ny, x j θ }n z H X = [ n x n θ {}}{{}}{ ] h x hθ }nx 0 P }n θ. Kormilisina (2013) shows ha he soluions o he model wih wo informaional subperiods is relaed o he soluion of he model wihou subperiods as follows: h x = h x, G x = Ḡx, and g θ 1 + g θ P = ḡ θ P, h θ 1 + h θ P = h θ P, j θ 1 = j θ P, (18) whereas h x, Ḡ x, ḡ θ, j θ, h θ, and P are submarices of HX and ḠX as defined above, and h θ 1 and g θ 1 solve he linear sysem of equaions [ ] (f 1 hθ 1 ) [x,y] = fz 1 j g θ 1, (19) θ 1 in which (f 1 ) [x,y] = [f 1 Y G x + f 1 x, f 1 y ] (20) is he jacobian of he sysem of equaions f 1 wih respec o vecor [x, y], and fy 1, f x 1, fy 1, and fz 1 are derivaive marices of a vecor f 1 wih respec o vecors Y, x, y and z correspondingly, evaluaed a a seady sae. 10

11 Kormilisina (2013) suggess ha he soluion o a model wih iming consrains can be obained from he soluion o he he model wihou informaional subperiods using simple linear ransformaions. One implicaion of Proposiion?? is ha he model wih informaional subperiods will inheri he propery of equilibrium uniqueness or indeerminacy (mulipliciy of equilibria) from he counerpar model wihou informaional subperiods. 7 Anoher imporan implicaion of Proposiion?? is ha he soluion o he model wih informaional subperiods can be obained easily once marices H X and ḠX are available, no maer wha mehod was used o recover H X and ḠX. 5 Esimaion Model-implied dynamics of variables of ineres depends on paramerizaion of he model, as well as on he responsiveness of governmen spending o curren sae, given by he elemens of he row vecor J. I rely on he Bayesian mehods o esimae hese parameers. I use governmen spending, oupu, and consumpion as observable variables. I assume he each observable variable is relaed wih he corresponding model implied variable as follows: log(y /Y 1 ) = log(y /y 1 ) + µ z,, where µ z, is he growh rae of he echnology process, µ z, = log(z /Z 1s ). The firs column of Table?? summarizes he prior disribuion for he esimaed parameers. The prior disribuions are cenered around he values common in he lieraure. DISCUSSION The prior disribuion for he values of row-vecor J are all cenered around zero, o reflec he prior beliefs ha he idenificaion is correc. Alernaively, we could cener he priors for hese parameers around he values implied by he unresriced model (These values, however, implicily are deermined by oher esimaed parameers). I rely on Bayesian mehods o esimae he model, where he likelihood funcion is esimaed using he Kalman filer, and combined wih prior disribuions for model parameers. The daa y is he 5 1 vecor of observable variables defined as follows y = { (log(g ) (log(c )), (log(i )), 4 (log(p Y, )), R }, where G, C, and I, are governmen spending, consumpion and invesmen expendiures, which appear as growh raes o be consisen wih heir model implied nonsaionariy propery. P Y, is GDP deflaor, and herefore 4 (log(p Y, )) measures annualized inflaion 7? derives his resul more formally in a slighly differen framework. 11

12 rae based on he GDP deflaor. R is he nominal ineres rae, measured by he effecive (annualized) Federal Funds rae.all he daa in vecor y are expressed as deviaions from heir means, and appear in quarerly frequency, spanning from 1954:3 o 2010:4. The vecor of esimaed model parameers is defined as θ = { J, b, α p, α w, κ, δ 2 /δ 1, σ, α R, α π, α Y, ρ g, ρ z, ρ Υ, ρ d, σ g, σ z, σ Υ, σ d, σ r }. Parameers presened in Table?? are calibraed, eiher because i is convenional in he lieraure, or because esimaing hese parameers is problemaic due o idenificaion issues. The parameer governing he seady sae share of capial is se a θ = 0.3. Following Alig, Chrisiano, Eichenbaum, and Linde (2011), he seady sae growh rae of oupu, µ z, is calibraed a , while he growh rae of he embodied echnology is se a The seady sae gross rae of inflaion is calibraed as π = , o mach he average yearly rae of inflaion of 3.5 percen. The ineremporal discoun facor β = This relaively high value for β ensures he seady sae nominal ineres rae is below 6 percen, because smaller values for β implies unrealisically large seady sae nominal ineres raes. The seady sae rae of capial uilizaion is u = 1, while he seady sae depreciaion rae is fixed a a convenional value δ 0 = The acual average share of governmen expendiures in GDP, G/Y = 0.2, is used o calibrae he seady sae share of governmen expendiures in he model. Finally, we fix he elasiciy of subsiuion for inermediae goods and labor ypes, because esimaing hese parameers is problemaic. We se η p a 6 and η w a 21, which imply he seady sae price and wage markups of 20 and 5 percen correspondingly. 6 Resuls 7 Conclusion References Alig, D., L. Chrisiano, M. Eichenbaum, and J. Linde (2011): Firm-Specific Capial, Nominal Rigidiies and he Business Cycle, Review of Economic Dynamics, 14(2), Blanchard, O., and R. Peroi (2002): An Empirical Characerizaion Of The Dynamic Effecs Of Changes In Governmen Spending And Taxes On Oupu, The Quarerly Journal of Economics, 117(4),

13 Fisher, J. (2003): Technology Shocks Maer, FRB of Chicago w.p # , December. King, R. G., C. I. Plosser, and S. T. Rebelo (1988): Producion, Growh and Business Cycles : The Basic Neoclassical Model, Journal of Moneary Economics, 21(2-3), Kormilisina, A. (2013): Solving Raional Expecaions Models wih Informaional Subperiods: A Perurbaion Approach, Compuaional Economics, 41(4), Monacelli, T., and R. Peroi (2008): Fiscal Policy, Wealh Effecs, and Markups, NBER Working Papers 14584, Naional Bureau of Economic Research, Inc. 13