Approximating time varying structural models with time invariant structures. May 2015

Transcription

1 Approximating time varying structural models with time invariant structures Fabio Canova EUI and CEPR Filippo Ferroni, Banque de France and University of Surrey Christian Matthes, Federal Reserve Bank of Richmond May 25

2 Introduction Typical to assume that DSGE are structural, i.e. preference, technology parameters are invariant to policy interventions (Hurwicz, 962). Mounting evidence (Dueker et al., 27, Fernandez and Rubio, 27, Canova, 29, Rios and Santaeularia, 2, Liu et al., 2, Galvao, et al., 24, Vavra, 24, Seoane, 24, Meier and Sprengler, forthcoming) that DSGE parameters are not time invariant. Parameter variations may be due to misspecication of a time invariant structure (Cogley and Yagihashi, 2; Chang, et al., 23, Basile and Carvalho, 25). Parameter variations may be needed to insure the existence of stationary equilibrium (see e.g. Schmitt Grohe and Uribe, 23).

3 Standard approach to modelling parameter variations follows VAR literature: parameters are exogenously drifting as a RW with small variance. Justication comes from Stock and Watson (996): economic relationship display small but persistent variations. Many questions suggest that time variations may be endogenous: - Is it reasonable to assume that the Federal Reserve reacts with the same coecients to ination in an expansion or in a contraction? (Davig and Leeper (26): state dependent rule). - Are households as risk averse when they are wealthy or poor? Or as impatient when the capital stock is high or low? - Does the propagation of shocks depends on the state of private and government nances? Or on inequality? (e.g. Brinca et al, 24).

4 Goals of the paper Characterize dierences in decisions rules, impulse responses, etc. when parameters variations are exogenous or state dependent. In the latter case, examine externalized or internalized variations. Provide diagnostics to detect misspecication driven by parameter variations and to distinguish exogenous from state dependent variations. Study the consequences of considering time invariant models when parameters are time varying in terms of identication, estimation, inference. Compare Likelihood and SVAR based estimates of structural dynamics. Apply the technology to Gertler-Karadi (2) model.

5 Related literature - Seoane (24): Use patterns of time variations as model respecication tool. - Kulish and Pagan (22): Likelihood estimation in models with structural breaks; Galvao et al. (24): time varying structural estimation. - Magnusson and Mavroedis (24, Econometrica): time variations helps identication of time invariant parameters in GMM. Does it apply to likelihood based estimation? - Huang (24): Moderate (exogenous) variations in weakly identied DSGE parameters do not make ML asymptotics and standard break test wrong. Moderate variations in strongly identied parameters do. - Ireland (27, JMCB): trend ination reects exogenous structural shocks. Ascari and Sbordone (24, JEL): trend ination reects monetary choices.

6 Results If parameter variations are exogenous, structural dynamics are the same as in a time invariant model. Little inferential loss if structural disturbances are correctly identied (variance/ historical decompositions distorted). If parameter variations are endogenous, structural dynamics may be altered. Extent of the dierences depends on the specication. Identication and inferential problems may arise when parameter variations are neglected. SVAR methods competitive with likelihood based methods. Financial (moral hazard) friction in Gertler-Karadi model is time varying and endogenous.

7 Plan of the talk. Some analytical results and an example. 2. Diagnostics for time varying misspecication; and for exogenous vs. endogenous time variations. 3. Identication and misspecication. 4. Some likelihood based and semi-structural (SVAR) based MC evidence. 5. An application.

8 The setup E t [f(x t+ ; X t ; X t ; Z t+ ; Z t ; t+ ; t )] = () X t : n x vector of the endogenous variables; Z t : n z vector of the exogenous variables; t : n vector of possibly time varying (TV) structural parameters; f is continuous and dierentiable up to order q. Z t+ = (Z t ; z t+ ) (2) is continuous and dierentiable up to order q; z t+ iid(; I) a n e vector, n z n e ; an auxiliary scalar; a known n e n e matrix.

9 Evolution of t = [ t ; 2t ]; where 2t : n n x vector: t+ = (; X t ; U t+ ) (3) is continuous and dierentiable up to order q, is a vector of constants, U t : n u vector of disturbances, n = n ( + n x ) n u : U t+ = (U t ; u u t+ ) (4) is continuous and dierentiable up to order q; u t iid(; I) is n u vector, uncorrelated with z t+, u is a known n u n u matrix. Let t+ = [ z t+ ; u t+ ] ; = diag[ z ; u ]. Decision rule: X t = h(x t ; Z t ; U t ; t ) (5) Time variations only aect structural parameters (see e.g. Andreasan, 22 for TV in auxiliary parameters ).

10 First order approximations The linear approximation of () is = E t [F x t+ + Gx t + Hx t + Lz t+ + Mz t + N t+ + O t ] (6) where F t+, G t, H t, L t+, M t, N t+ ; O t. The linear approximation to the decision rule: x t = P x t + Qz t + Ru t (7) where P t, Q t ; R t.

11 Proposition The matrices P; Q; R are obtained as follows: P solves F P 2 + (G + N x )P + (H + O x ) = Given P, V Q = vec(l z +M) and V = z F +I n z (F P +G+N x ) where vec denotes the columnwise vectorization Given P, W R = vec(n u! u + O u ) and W =! u F + I n (F P + G + N x ) where u t+, x t, z =@Z t ;! u t ;where! u is a n u n u matrix and z is a n z n z matrix, both with all eigenvalues strictly less than one in absolute value.

12 Corollary 2 If x =, the dynamics in response to the structural shocks z t are identical to those obtained when parameters are time invariant. Variations in the j-th parameter have instantaneous impact on the endogenous variables x t, if and only if the j th column of N u! u + O u 6=. Corollary 3 If u = and the matrices N x and O x are zero, variations in the j-th parameter have no dynamic eects on the endogenous variables x t.

13 Proposition : have time varying parameters equivalent to having additional shocks to the model. Corollary 2: with exogenous parameters variations, contemporaneous and lagged dynamics to structural shocks are the same as in a time invariant model. Can we recover structural shocks with a time invariant model? Corollary 3: with endogenous parameter variations, contemporaneous and lagged dynamics to structural shocks may be aected. Need to know whether N x or O x are zero or not.

14 Higher order approximations Do the conclusions change? Let W t = [Z t ; U t ], Y t+ = [X t+ ; X t ], Using the decision rule is X t = h(x t ; W t ; t+ ), () is = [F (X t ; W t ; t+ ; )] (8) - The second order approximation of (8) is [(F x x t + F w w t + F ) + :5(F xx (x t x t ) + F ww (w t w t ) + F 2 ) +F xw (x t w t ) + F x x t + F w w t ] = (9) The second order expansion of the decision rule is x t = h x x t h w w t + :5(h xx (x t x t ) + h ww (w t w t ) + h 2 ) + h xw (x t w t ) + h x x t + h w w t ()

15 Proposition 4 Time variations in the parameters aect h xx ; h xw ; h ww if only if h x and h w are aected In higher order approximations there are terms of the form F x 6= ; F w 6= :These require a correction of the linear terms in the decision rule to account for uncertainty. The dynamics induced by structural shocks in xed coecient and TV coecient models will be dierent (some shocks are omitted in xed coecient models).

16 Discussion - Time variations are assumed to be continuous. - Can accommodate once-and-for-all breaks (at known dates) with smooth transition (e.g. t+ = ( ) + t + a exp(t)=(b + exp(t)), t = T ; : : : ; ; ; ; : : : T 2 ; exogenous; t+ = ( )+ t +a exp( (K t K + U ;t+ ))=(b + exp( (K t K + U ;t+ )); endogenous). - Can not allow for Markov switching variations in the parameters or nonsmooth transition e.g. t+ = ( I(K t > K)) +I((K t > K)) (see Davig and Leeper (26)).

17 Structural Break Kulish and Pagan (24) have a solution method with abrupt structural breaks and learning - valid for rst order approximations only. We use nal form to derive results (no distinction between states and controls). - Are time varying parameters CKMcG wedges? No: across equation restrictions; less time varying parameters than optimality conditions.

18 A RBC example X max E t ( C t t= A N t + + ) () Y t ( g t ) = C t + K t ( t )K t (2) Y t = t Kt N t (3) Y t is output, C t consumption, K t the stock of capital, N t hours worked and g t = G t Y t the share of government expenditure. Exogenous disturbances: ln t = ( ) ln + ln t + e z t (4) ln g t = ( g ) ln g + g ln g t + e g t (5)

19 - 2 parameters: structural (,,, A, t t ), auxiliary ( ; g; ; g ; ; g ). - t and t allowed to be time varying (Meier and Sprengler, forthcoming; Dueker, et al., 27, Liu et al., 2). Note: Ireland (24): and weakly identied. Canova and Sala (29): partially identied. The optimality conditions: AC t N t = ( )( g t )Y t =N t (6) C t+ t = E t C t+ t+)y t+ + t+ ) t t+ + E t u(c t+ ; N t+ ) K t ) t ( g t )Y t = C t + K t ( t )K t (8) Y t = t Kt N t (9)

20 Two eects of parameter variations on optimality conditions: - direct eect in the Euler equation and in the resource constraint when t and t are time varying. - If agents take into account that their decisions aects parameter variations, there is an additional (indirect) eect due to the derivatives of t+ and t+ with respect to the endogenous states (the capital stock).

21 Model A: Constant coecients. Let t = t and t =. The optimality conditions are ACt N t + = ( )( g t )Y t Ct = E t C t+ (( g t+)y t+ =K t + ) ( g t )Y t = C t + K t ( )K t Y t = t Kt N t In the steady state: K Y ( g) = + = ; C Y = K Y g; N Y = K Y ; Y = " A ( )( g) # + C N Y Y (2)

22 Model B: Exogenous parameter variation Let d t = t+ = t ; t+ (d t+ ( ); t+ ( )) = U t+ u ;t+ = d u ;t + e ;t+ (2) u ;t+ = u ;t + e ;t+ (22) t+ =@k t t+ =@k t =, the optimality conditions are E t E t [f(x t+ ; X t ; X t ; Z t+ ; Z t ; t+ ; t )] = AC t N + t ( )( g t )Y t d t C t+ =C t (( g t+ )Y t+ =K t + t+ ) C ( g t )Y t C t K t + ( t )K t A = (23) Y t t Kt N t with X t = (K t ; Y t ; C t ; N t ), Z t = ( t ; g t ) ; t = t.

23 - Since E(d t ) = ; E( t ) =, the steady states are as in model A. - Since x =, P and Q are as in model A. - Is R 6=? Check whether the columns of N u! u + O u are zero. N u! u + O u = = = k C A 6= (24) Note: if d t+ = t+ = t (d t is a fast moving variable), time variations in d t matter only if d 6=

24 θ θ Model C: State dependent parameter variations, no internalization t+ = [ u ( u l )e (K t K) ] + [ u ( u l )e 2(K t K) ] + u ;t+ (25) ; 2 ; u ; l are vectors. Let u ;t+ be zero mean, iid shocks K t K K t K If d l = =2 and l = =2, the steady states are as model A.

25 - Assume that agents treat K t appearing in (25) as an aggregate t+ =@K t t+ =@K t =. The optimality conditions are as in (23). - Are the P and Q matrices aected? N x = O x = = C A = k (d u =2)( 2 ) ( u =2)( 3 4 ) C A! (d u =2)( 2 ) ( u =2)( 3 4 )! (26) (27) P and Q dier from those of model A if 6= 2 or 3 6= 4

26 - Do time variation in t aect X t? N u! u +O u = =(d u =2)( + 2 ) k( u =2)( ) (28) C A - R 6= if 6= 2 or 3 6= 4 :

27 Model D: State dependent parameter variations, internalization The relevant derivatives are d t+ =@K t = (d u =2)[ e (K t K) + 2 e 2(K t K) ] (29) t+ =@K t = ( u =2)[ 3 e 3(K t K) + 4 e 4(K t K) ] (3) If = 2 = ; 3 = 4 = 3, the steady states are as in model A. Here E t E t [f(x t+ ; X t ; X t ; Z t+ ; Z t ; t+ ; t )] = AC t N + t ( )( g t )Y t d t u(c t+; N t+ )=C t d t C t+ =C t (( g t+ )Y t+ =K t+ + ( g t )Y t C t K t + ( t )K t t+ + t+ K t) Y t t Kt N t C A =

28 where X t = (K t ; Y t ; C t ; N t ), Z t = ( t ; g t ) but now t = d t+ t+ d t+ t+ C A = (; K t ; u t+ ) = 2d u (d u =2)[e (K t K+u ) ;t+ + e (K t K+u ) ;t+ ] 2 u ( u =2)[e 3(K t K+u ) ;t+ + e 3(K t K+u ) ;t+ ] (d u =2) [ e (K t K) + e (K t K) ] ( u =2) 3 [ e 3(K t K) + e 3(K t K) ] The relevant matrices are: C A (3)

29 t+ = t = t = u = t+ = = u(c; N)=C K = K C A 2(d u =2) 2 2( u =2) 2 3 C A 2(d u =2) 2 2( u =2) 2 3 Then N x 6=, O x = and N u! u + O u = since! u = 22. Thus, P is aected even when = 2 and 3 = 4. C A C A

30 Why are the structural dynamics in models C and D dierent? Common parameters: = :3, = :99, = :25, = 2, = 2, A = 4:5, = ; z = :9, z = :72, g = :8; g = :5 and g = :52. Model B : = :985; = :95; =.2; = :7. Model C : = :; 2 = :3; = :2; 2 = :; = = :5; d u = :999; u = :25. Model D : = :; 2 = :6; = :2; 2 = :; =., = :; d u = :999; u = :25.

31 Output Consumption Capital Hours x 3 Techology shocks x 3 G expenditure shocks x x B C D A Income and substitution eects are dierent!

32 Punchline Having exogenous parameter variations, is like adding shocks to the original model without altering the existing dynamics. Approximating an exogenous TVC model with a constant coecient model may be less costly than approximating a endogenous TVC model. Dynamics in models with endogenous time variations may dier from those of exogenous and time invariant models because income and substitution eects are altered.

33 Characterizing time invariant misspecication Two ways: "Wedges" (see Chari et al., 28). Forecast errors.

34 Wedges CC Model: = [F (X t ; W t ; z z t+ ; )] = (32) X t = h(x t ; W t ; z z t+ ; ) (33) TVC Model: = [F (X t ; W t; t+ ; )] = (34) X t = h (X t ; W t; t+ ; ) (35) Wedge: ) [F (X t ; W t; z z t+ ; )] 6= ( since z z t+ 6= t+; h 6= h ). 2) [F (X t ; W t; z z t+ ; )] predictable using past X t

35 In rst order system the wedge is (F (P P ) 2 + G(P P ))x t + (F (Q Q) z + G(Q Q) + F (P P )(G G))z t + (F (P P )R + GR + F R! u )u t (36) If P = P; Q = Q wedge is (GR + F R! u )u t (37) Dierent from zero if R 6= and predictable if! u 6= If P 6= P; Q 6= Q wedge is non zero (even if R = ) and predictable using past x t even if! u =.

36 Forecast errors - Linearized decision rule in time invariant model: x t = P x t + Qz t - Linearized decision rule in TVC model: x t = P x t + Q z t + R u t. Let vt be the forecast error in predicting x t constant coecient model and TVC data using the decision rules of the v t x t P x t = Q z t + R u t + (P P )x t (38) Forecast error is function of the lags of the observables x t : True when P 6= P but also if P if u t are serially correlated (they aect x t ).

37 Misspecication diagnostic: RBC example Euler wedge. DGP c t r t B (.4) (.4) C.6.8 (.2) (.2) D (.9) (.7)

38 Forecast error: hour equation. DGP n t y t c t Ftest, P-value B (.4) (.7) (.2) C (.2) (.3) (.29) D (.6) (.2) (.2) - Counterfactual: Constant coecients but one period time to build model. First lag real rate -.3, rst lag of consumption growth -.6, both insignicant.

39 Puchline Can detect time invariant misspecication by estimating a time invariant model and a bunch of regressions. Much less costly than, e.g. rolling estimation or estimating dierent versions of the model (with and without TV coecients). Exploit the structure of optimality conditions and of forecast errors.

40 How to detect exogenous vs. endogenous time variations? Use a DSGE-VAR (Del Negro and Schorfheide, 24) setup. Idea: - Simulate T data point for each model. Add it to actual T data points. - If simulated data come from the DGP, precision of estimates improve, ML increase. If simulated data does not come to DGP, noise is added, precision decrease ( bias may be generated), ML decrease. - Compare ML of adding T data from the exogenous model to the ML of adding T data from the endogenous model.

41 RBC example: T=5 Log Marginal likelihood T =5 T =75 DGP Model BModel CModel DModel BModel CModel D Simulated from B Simulated from C Simulated from D

42 Inferential distortions : parameter identication Can time invariant parameters be identied from a potentially misspecied likelihood function? - Canova and Sala (29): DSGE models have POPULATION identication problems. - Can identication problems arise because parameter variations are neglected? What should we expect to happen to the likelihood function when time invariant parameters are assumed?

43 log likeli log likeli True RBC B Estimated with RBC B True RBC C Estimated with RBC C True RBC D Estimated with RBC D γ.5.5 η γ.5.5 η γ.5.5 η True RBC B Estimated with RBC A True RBC C Estimated with RBC A True RBC D Estimated with RBC A γ.5.5 η γ.5.5 η γ.5.5 η 2 2.5

44 log likeli log likeli True RBC B Estimated with RBC B True RBC C Estimated with RBC C True RBC D Estimated with RBC D α.25.8 ρ ζ α.25.8 ρ ζ α.25.8 ρ ζ.9 True RBC B Estimated with RBC A True RBC C Estimated with RBC A True RBC D Estimated with RBC A α.25.8 ρ ζ α.25.8 ρ ζ α.25.8 ρ ζ.9

45 Koop, Pesaran, Smith identication diagnostic Parameter T=5 T=3 T=5 T=75 T= T=5 T=25 DGP Model B, Estimated model A z.8e+4 2.6e+4 4.2e+4 4.2e+4 4.5e+4 4.9e e+4 g e+4.7e+4 2.4e+4 2.3e+4 2.5e A DGP Model C, Estimated model A z g e+5 4.4e+5 4.3e+5 4.e+5 3.8e+5 4.4e+5 4.3e+5.8e+4.e+4.4e+4.2e+4.e+4.6e+4.5e+4 A DGP Model D, Estimated model A z g e+5 6.7e+5 6.5e+5 6.e+5 5.7e+5 5.8e+5 5.7e+5.e+4 2.5e+4 2.4e+4.9e+4 2.e+4 2.e+4 2.e+4 A

46 - Curvature of the correct likelihood OK. Maximum is at = 2; = 2; = :3; z = :9 for all models. - When the decision rules of model A are used and the truth are models B-C-D, distortions are large and some partial identication problem exist (notice ridges in contours). - Misspecication of P and Q distorts the likelihood. Neglecting existence of other shocks tilts the likelihood without changing location. - KPS statistic agrees: weak identication only for g with some DGPs. - Results consistent with Huang (24).

47 Inferential distortions 2: ML based parameter and response estimates Use RBC models B, C, D; N=5; T= or T=. Two DGPs: one where parameter variations explain little of output variability (less than 5 percent); one where parameter variations explain a sizable portion of output variability (2 percent). Estimate a time invariant model and the correct model (to control for possible identication and numerical problems). Compare parameter estimates, impulse responses and variance decompositions.

48 Small time variations True Correct Time invariant Time invariant Parameter Mean Mean 5th percentile 95th percentile Mean 5th percentile 95th percentile T=5 T=5 T= DGP Model B = 2: = 2: z = : g = : = : = : A = 4: DGP Model C = 2: = 2: z = : g = : = : = : A = 4: DGP Model D = 2: = 2: z = : g = : = : = : A = 4:

49 Model D Model C Model B η γ ρ z ρ g δ α A True T= Incorrect T= Incorrect T=

50 Consumption Output Capital Hours x 3 Technology 3 x G.expenditure x 3 Technology 3 x G.expenditure x 3 Technology 3 x G.expenditure x x x 3 x 3 x 3 x 3 x x x 4 2 x x x x Model B A A84 B5 A Model C A A84 C5 A Model D A A6 A84 D5 Impulse responses, DGP

51 Variance decomposition DGP: Small time variations Variable Technology Government Technology Government Model B Time invariant Y C N K Model C Time invariant Y C N K Model D Time invariant Y C N K

52 Large time variations True Correct Time invariant Time invariant Parameter Mean Mean 5th percentile 95th percentile Mean 5th percentile 95th percentile T=5 T=5 T= DGP Model B = 2: = 2: z = : g = : = : = : A = 4: DGP Model C = 2: = 2: z = : g = : = : = : A = 4: DGP Model D = 2: = 2: z = : g = : = : = : A = 4:

53 Model D Model C Model B η γ ρ z ρ g δ α A True T= Incorrect T= Incorrect T=

54 Consumption Output Capital Hours x 3 Model B A x 3 Model C A x Model 3 D A x 3 Model B A x 3 Model C A x 3 Model D A x x x x x x x x x x x x A84 A6 B5 A84 A6 C5 A84 A6 D5 Impulse responses, DGP2

55 Variance decomposition DGP:Large time variations Variable Technology Government Technology Government Model B Time invariant Y C N K Model C Time invariant Y C N K Model D Time invariant Y C N K

56 Punchline - Large distortions in and 's, i.e. the parameter controlling income and substitution eects - Results with T= do not improve - If the DGP has more important time variations, results worsen.

57 SVAR based inference - Can less structural methods be useful to recover structural dynamics? - Canova and Paustian (2): robust SVAR methods can be used if theoretical model is misspecied in certain way. - Is the conclusion still true if we neglect time variations in the structural parameters? - Use the DGP where parameter variations explain less than 5 percent of the variance of output.

58 - Simulate (long) data using the decisions rules of a TVC model. - Construct VAR errors using the dynamics (matrix P) of a constant coecient model. - Rotate VAR errors until they satisfy robust sign restrictions (common to all models): Technology shocks: Y, H, K up; Government expenditure shocks: Y, H, C and K down. - For each replication l = ; : : : ; L store the median of the true and incorrectly specied model. Report the 68 percent interval for the median of the incorrectly specied model and the median value of median for the correctly specied model.

59 Consumption Output Capital Hours x Technology 4 shocks 5 x G 4 expenditure shocks x Technology 4 shocks 5 x G 4 expenditure shocks x Technology 4 shocks 5 x G 4 expenditure shocks x x x x x x x 3 x 3 x 3 x 3 x 3 x Models B A upper 84 lower 6 mean true Models C A upper 84 lower 6 mean true Models D A upper 84 lower 6 mean true

60 - Performance good also with models C and D. Only consumption responses not well captured - entries of P matrix o primarily in one row. - Performance is good since shock misaggregation is minor.

61 Estimation of Gertler and Karadi (2) model Three goals: i) Estimate the parameters specic to the model ( : the share of projects that bankers can steal,! : the fraction of wealth given to new bankers, : the lifetime horizon of a banker). ii) Apply diagnostics for TV misspecication iii) Estimate TVC models. Compare responses to a capital quality shock to those in a xed coecient model.

62 Parameter estimates Parameter Mode Gertler- Karadi (.82)!.5.2 (.8) (.98) SS Leverage Lifetime bank <y y

63 Forecast error regressions diagnostic Equation T-stat F-stat Y t C t Credit t Leverage t Spread t Y C Credit Leverage Spread Wedge diagnostic Mean C t IY t Euler wedge (.3)(.)(.3)

64 Estimates with TVC models Parameter Time Invariant Exogenous TVC Endogenous TVC Function of net worth h.43 (.6).9 (.3).9 (.2).24 (.).37 (.3).55 (.3)!. (.8).2 (.2). (.8).46 (.9).54 (.).52 (.2).99 (.4).2 (.2).3 (.3) u.98 (.8).2 (.7) 2.5 (.9) Log ML

65 Leverage Output Investments Net worth Inflation Spread x 3 x x x C onstant E xogenous E ndogenous Capital quality shock

66 A remaining question - How do you distinguish a model with m shocks from a model with m shocks and m 2 time varying parameters,m + m 2 = m? Or with m shocks and m 2 measurement errors? Solution with m shocks and m 2 time varying parameters: x t = P x t + Q z t + R u t = P x t + Qz t (39) Q=[Q ; R ], z t = [zt ; u t ]. Solution with m shocks and m 2 measurement errors v t is x t = P x t + Q z t + v t = P x t + Qz t (4) Q=[Q ; I], z t = [zt v t ]. If v t Ru t very hard to distinguish the two unless u t makes P; Q dierent. Solution with m shocks is x t = P x t + Qz t (4) Q full matrix.