Robert Kollmann. 6 September 2017

Transcription

1 Appendix: Supplemenary maerial for Tracable Likelihood-Based Esimaion of Non- Linear DSGE Models Economics Leers (available online 6 Sepember 207) hp://dx.doi.org/0.06/j.econle Rober Kollmann ECARES Universié Libre de Bruxelles & CEPR 50 Avenue Roosevel 050-Brussels Belgium rober_kollmann@yahoo.com 6 Sepember 207 Par A. of his Appendix provides supplemenary informaion for he second-order approximaed RBC el discussed in he paper published in Economics Leers (Kollmann (207)). Par B. The echnique described in Kollmann (207) can also be used for likelihood esimaion of DSGE els ha are approximaed o an order ha is higher han he second order. Par B of his Appendix shows how hird-order approximaed DSGE els can be esimaed using observaion equaion inversion.

2 A. Supplemenary informaion for he second-order approximaed RBC el considered in Kollmann (207) Comparison beween decision rule (4) and ified decision rule (5) Table a documens ha he decision rule (4) and he ified decision rule (5) discussed in Kollmann (207) are (essenially) indisinguishable. An idenical sequence of random exogenous innovaions of lengh T= was fed ino (4) and ino (5). Table a shows ha he resuling ime series of endogenous variables are almos perfecly correlaed across (4) and (5) and ha hey have (essenially) he same sandard deviaion. This holds boh for levels and for firs differences of logged simulaed endogenous variables. Sandard deviaions of firs- and second-order approximaed els Table a2 repors prediced sandard deviaions of firs- and second-order approximaed variables (log levels and log firs differences). The Table documens ha each of he four ypes of exogenous shocks accouns for a sizable share of he variance of GDP (see Panel (a) Col. ()). In he small shocks el varian he firs- and second-order approximaed els produce almos idenical sandard deviaions of endogenous variables (see Panel (a)). In he big shocks el varian by conras he second-order approximaed variables are more volaile han he firs-order approximaed variables; his is especially he case for GDP invesmen and hours worked (see Panel (b)). 2

3 Table a. Second-order approximaed RBC el: correlaions across ime series generaed by decision rule (4) [ω] and ime series generaed by he ified decision rule (5) [ ω ] Y C I N K () (2) (3) (4) (5) (a) Model varian wih small shocks ( σθ= σg= σψ= % σλ= 0.025%) Correlaions beween ω and ω Levels Firs differences Relaive sandard deviaions: sd(ω )/sd( ω ) Levels Firs differences Relaive sandard deviaion of difference beween decision rules: sd(ω -ω Levels Firs differences (b) Model varian wih big shocks ( σθ= σg= σψ= 5% σλ= 0.25%) Correlaions beween ω and ω Levels Firs differences Relaive sandard deviaions: sd(ω )/sd( ω ) Levels Firs differences Relaive sandard deviaion of difference beween decision rules: sd(ω -ω Levels Firs differences )/sd( ω ) )/sd( ω ) Noe: Correlaions of simulaed ime series (of variables lised above Cols. ()-(5)) generaed by he decision rule (4) and by he ified decision rule (5) are repored as well as he relaive sandard deviaion of hese wo ses of ime series. These saisics are repored for variables in log levels and for variables in log firs differences. Y: GDP; C: consumpion; I: gross invesmen; N: hours worked; K: capial sock. Correlaions greaer han are repored as Repored saisics are based on one sequence of T= random exogenous innovaions ha was fed ino (4) and (5). 3

4 Table a2. RBC el: prediced sandard deviaions (in %). Comparison beween s order and 2 nd order accurae el soluions Y C I N K () (2) (3) (4) (5) (a) Model varian wih small shocks ( σ = σ = σ = % σ = 0.025%) θ G ψ λ Variables in levels s order all shocks s order jus θ shock s order jus G shock s order jus ψ shock s order jus λ shock nd order all shocks Firs-differenced variables s order all shocks nd order all shocks (b) Model varian wih big shocks ( σθ= σg= σψ= 5% σλ= 0.25%) Variables in levels s order all shocks nd order all shocks Firs-differenced variables s order all shocks nd order all shocks Noe: Sandard deviaions (in %) of simulaed variables (lised above Cols. ()-(5)) are shown for he RBC el. Rows labeled s order and 2nd order show sandard deviaions prediced by he firs- and secondorder accurae el soluions respecively. The saisics are repored for variables in log levels and for variables in log firs differences. Y: GDP; C: consumpion; I: gross invesmen; N: hours worked; K: capial sock. All saisics are compued using one simulaion run of periods. 4

5 B. Tracable Likelihood-Based Esimaion of Third-Order Approximaed DSGE Models The echnique described in Kollmann (207) can also be used for likelihood esimaion of DSGE els ha are approximaed o an order ha is higher han he second order. This is illusraed here for hird-order approximaed els. The hird-order accurae el soluion of he DSGE el () is given by: ω = Fξ + ( F+ F ξ ) x + ( F + F ξ ) ε + F x x + F x ε + F ε ε ξ 2 2ξ F x x x + F x x + F x + F wih x =Λ ω. (B.) 2 ε+ 22 ε+ ε+ 222ε+ ε+ ε+ F F F F F F ξ 2ξ are marices ha are funcions of he srucural el parameers ( F0 F F2 F F2 F 22 are idenical o he corresponding coefficiens in he second-order accurae el soluion; see (2) in Kollmann (207)). Pruning is also essenial for applied work based on hird-order approximaed els-- he un-pruned sysem (B.) can exhibi explosive dynamics in response o big shocks (see discussion in Kollmann (207)). To apply he logic of pruning o equaion (B.) noe ha he following condiions hold up o hird-order accuracy: ξ 2 2 x () = ξ x x x= x x + x ( x x ) (2) () (2) (2) () x (2) ε+ = x ε+ x x x x x x () () () = x x x x () () ε + = ε + x = (B.2) () ε+ ε+ x ε+ ε+ where he superscrip (i) denoes variables solved o i h accuracy and () i () i x =Λω. The Dynare oolbox (Adjemian e al. (204)) implemens a pruned version of he hird-order soluion in which produc erms in equaion (B.) are replaced by heir hird-order accurae equivalens saed in (B.2): ω = Fξ + Fx + F ξ x + ( F + F ξ ) ε + F { x x + x ( x x )} + F x ε + F ε ε () 2 (2) () () (2) () (2) + 0 ξ 2 2ξ F x x x + F x x + F x + F (B.3) () () () () () (). 2 ε+ 22 ε+ ε+ 222ε+ ε+ ε+ (This pruned hird-order soluion was also proposed by Kollmann (2004).) The dynamics of he firs- and second-order approximaed quaniies is governed by (3) and (4) in Kollmann (207) resaed here for convenience: () (2) (2) For variable a we can wrie a= a + R and (3) ( n) a= a + R where R conains erms of order n or higher in deviaions from he seady sae. The produc ab can hus be expressed as () (2) () (3) () (2) () (3) () (2) (2) () () (4) (3) () (2) (2) () () ab= ( a + a a + R )( b + b b + R ) = a b + ( a a ) b + R ; hence ( ab) = a b + ( a a ) b. (Noe ha (2) () (2) (2) () (2) () (4) = and hence ( a a )( b b ) R.) a a R (3) () () () = The same logic shows ha ( abc) = a b c. 5

6 ω () () + = Fx + F2ε + ω = Fξ + Fx + Fε + F x x + F x ε + F ε ε. (B.4) (2) 2 (2) () () () The moving average represenaion of he hird-order pruned soluion (B.3) depends on firs- second and hird-order erms in exogenous innovaions (ε) bu no on higher-order erms. The hird-order pruned sysem (B.3) is saionary if he firs-order sysem is saionary. To allow observaion equaion inversion I replace squares and cubes of ε + in (B.3) by heir expeced values. This gives he ified decision rule ω = Fξ + Fx + F ξ x + ( F + F ξ ) ε + F { x x + x ( x x )} + F x ε + F E( ε ε ) () 2 (2) () () (2) () (2) + 0 ξ 2 2ξ F x x x + F x x ε + F x E( ε ε ). (B.5) () () () () () () Noe ha E( ε+ ε+ ε+ ) = 0 because ε + is normally disribued. The subsequen discussion assumes ha (B.5) is he rue daa generaing process. Assume ha he economerician observes a vecor z + comprising m elemens of he vecor ω + (recall ha m is he number of exogenous innovaions). Thus he observaion equaion is z = Qω where Q is an mxn selecion marix. Subsiuion of equaion (B.5) ino + + he observaion equaion gives z+ = γ + λε + where 2 2 () (2) () () (2) () () () () () γ Q [ F0ξ + Fx + F ξξ x + F{ x x + x ( x x )} + F22E( ε + ε + ) + Fx x x + F22x E( ε + ε+ )] 2 (2) () () and λ is an mxm marix such ha λ ε Q [( F+ F ξ ) ε + F x ε + F x x ε ]. Provided λ is non-singular we hus have: + 2 2ξ z+ ε = λ ( γ ). (B.6) Given he iniial saes x x x and daa { z } T = one can recursively exrac he () (2) innovaions { ε } T = using (B.4)(B.5) and (B.6). The log likelihood of he daa (condiional on x x x ) is: () (2) ln L({ z} x x x ) ln(2 ) ln ε { ' ( ε) ln }. (B.7) T () (2) 2 T mt T 2 = = 2 π 2 ξ Σ 2 ε ξ Σ ε λ = Srucural el parameers (and he iniial saes) can be esimaed by maximizing his funcion. Illusraion: RBC el approximaed o hird-order I compue a hird-order approximaion of he RBC el described in Kollmann (207). Boh he small shocks varian of ha el and he big shocks varian are considered. Table b documens ha decision rule (B.3) and he ified decision rule (B.5) are (essenially) 6

7 indisinguishable. An idenical sequence of random exogenous innovaions of lengh T= was fed ino (B.3) and ino (B.5). Table b shows ha he resuling ime series of endogenous variables are almos perfecly correlaed across (B.3) and (B.5) and ha hey have (essenially) he same sandard deviaions. This holds boh for levels and for firs differences of logged simulaed endogenous variables. Table b2 repors prediced sandard deviaions of firs- second- and hird-order approximaed variables (log levels and log firs differences). In he big shocks RBC el varian GDP invesmen and capial are noiceably more volaile under a hird-order approximaion han under firs- or second-order approximaions (see Panel (b)). Finally I esimae he el parameers using simulaed ime series by maximizing he likelihood funcion (B.7). As for he Mone Carlo described in Kollmann (207) I generaed 30 simulaion runs of 00 periods each. 2 In compuing he sample likelihood I assume ha he iniial saes x x x equal heir uncondiional mean. The firs 0 periods in each simulaion () (2) run are used as a raining sample. Table b3 repors he median mean and sandard deviaion of he esimaed el parameers across he 30 simulaion runs for he small shocks el varian (Columns ()-(3)) and for he big shocks varian (Cols. (4)-(6)). As for he second-order accurae el discussed in Kollmann (207) mos el parameers are ighly esimaed. References Adjemian S. H. Basani M. Juillard F. Mihoubi G. Perendia J. Pfeifer M. Rao S. Villemo 204. Dynare: reference manual Version Working Paper CEPREMAP. Kollmann R Solving Non-Linear Raional Expecaions Models: Approximaions Based on Taylor Expansions WP Universiy of Paris XII. Kollmann R Tracable Likelihood-Based Esimaion of Non-Linear DSGE Models. Economics Leers (available online 6 Sepember 207) hp://dx.doi.org/0.06/j.econle To eliminae he influence of iniial condiions he el was simulaed over 500 periods; he firs 5000 periods were discarded. 7

8 Table b. Third-order approximaed RBC el: correlaions across ime series generaed by decision rule (B.3) [ω] and ime series generaed by ified decision rule (B.5) [ ω ] Y C I N K () (2) (3) (4) (5) (a) Model varian wih small shocks ( σθ= σg= σψ= % σλ= 0.025%) non Correlaions beween ω and ω Levels Firs differences Relaive sandard deviaions: sd(ω)/sd( ω ) Levels Firs differences Relaive sandard deviaion of difference beween decision rules: sd(ω -ω Levels Firs differences (b) Model varian wih big shocks ( σθ= σg= σψ= 5% σλ= 0.25%) non Correlaions beween ω and ω Levels Firs differences Relaive sandard deviaions: sd(ω)/sd( ω ) Levels Firs differences Relaive sandard deviaion of difference beween decision rules: sd(ω -ω Levels Firs differences )/sd( ω ) )/sd( ω ) Noe: Correlaions of simulaed ime series (of variables lised above Cols. ()-(5)) generaed by decision rule (B.3) and by he ified decision rule (B.5) are repored as well as he relaive sandard deviaion of hese wo ses of ime series. The saisics are repored for log levels and for log firs differences of endogenous variables. Y: GDP; C: consumpion; I: gross invesmen; N: hours worked; K: capial sock. Correlaions greaer han are repored as Repored saisics are based on one simulaion run of periods. 8

9 Table b2. RBC el: prediced sandard deviaions (in %). Comparison beween s order 2 nd order and 3 rd order accurae el soluions Y C I N K () (2) (3) (4) (5) σ = σ = σ = σ = (a) Model varian wih small shocks ( % 0.025%) θ G ψ λ Variables in levels s order nd order rd order Firs-differenced variables s order nd order rd order (b) Model varian wih big shocks ( σθ= σg= σψ= 5% σλ= 0.25%) Variables in levels s order nd order rd order Firs-differenced variables s order nd order rd order Noe: Sandard deviaions (in %) of simulaed variables (lised above Cols. ()-(5)) are shown for he RBC el. Rows labeled s order 2 nd order and 3 rd order show sandard deviaions prediced by he firs- second- and hird- order accurae el soluions respecively. The saisics are repored for log levels and for log firs differences of endogenous variables. Y: GDP; C: consumpion; I: gross invesmen; N: hours worked; K: capial sock. All saisics are compued using one simulaion run of periods. 9

10 Table b3. Mone Carlo: parameer esimaes for hird-order approximaed RBC el Model varian Model varian Parameer wih small shocks wih big shocks () (2) (3) (4) (5) (6) Median Mean Sd Median Mean Sd σ η ρ θ ρ G ρ ψ ρ λ s θ (%) s G (%) s ψ (%) s λ (%) Noe: The Table summarizes parameers esimaes across 30 simulaion runs of 00 periods. Cols. labelled Median Mean and Sd repor he median mean and sandard deviaion of esimaed parameers (lised in lef-mos column) across he 30 runs. Cols. labelled ()-(3): small shocks el varian. Cols. (4)-(6): big shocks el varian. The rue parameer values are: σ = 0 η= 0.25 ρθ = ρg= ρψ= ρλ= True sandard deviaions of exogenous innovaions in small shocks el varian: sθ= sg= sψ= % sλ= 0.025%. Big shocks varian: s= s = s = 5% σ = 0.25%. θ G ψ λ 0