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					  QUANTITATIVE ANALYSIS OF DYNAMIC
STOCHASTIC GENERAL EQUILIBRIUM MODEL

                   Tonner J., Polanský J., Vašíček 0.

                     Faculty of Economics and Administration
                                Masaryk University


   Econometric Day, November 2008, Brno, Czech Republic.



This work was supported by Czech Science Foundation grant 402/05/2172,
MŠMT project Research centre 1M0524, and funding of specific research at
                               ESF MU.


           Tonner J., Polanský J., Vašíček 0.   Quantitive Analysis of DSGE Model
Outline




      Introduction
      Model formulation
      Estimation
      Conclusions




            Tonner J., Polanský J., Vašíček 0.   Quantitive Analysis of DSGE Model
Introduction


  Our quantitative analysis is explicitly focused on
       time - variant ’structural’ parameters estimation (”structural’
       in the sense of Hurwitz(1962): they are invariant to
       interventions, including shocks by nature’, Villaverde, 2007)
  in
       Dynamic Stochastic General Equilibrium Model with Rational
       Expectation (’Since parameters are fully interpretable from the
       perspective of economic theory and invariant to policy
       interventions, DSGE models avoid the Lucas critique. . .,
       Villaverde, 2007).
  The aim is to explore a relation between changes in the conduct of
  monetary policy and changes in the ’structural’ parameters.



              Tonner J., Polanský J., Vašíček 0.   Quantitive Analysis of DSGE Model
Introduction



  Controversy: Why to estimate time - variant parameters?
      ’It is hard to believe that the ’structural parameters’ of DSGE
      models are realy structural. . . .’(Villaverde & Rubio-Ramírez,
      June , ’How structural are structural parameters?’, 2007)
      ’A key assumption underlying the policy analysis with DSGE
      models is that the parameters characterizing preferences and
      technologies as well as the law of motion of aggregate shocks
      are invariant to the policy changes studied with the DSGE
      model.’(Schorfheide, F., Commment on ’How structural are
      structural are structural parameters?’, May 2007)




             Tonner J., Polanský J., Vašíček 0.   Quantitive Analysis of DSGE Model
Introduction

  Empirical findings (Villaverde, 2007): ’. . ., we report a move by the
  Fed toward a much more aggressive stand against raising prices in
  the late 1970s. Also, we find that changes in the inflation target
  account, at most, for 40-50 percent of the increase in inflation in
  the 1970s.’




              Tonner J., Polanský J., Vašíček 0.   Quantitive Analysis of DSGE Model
Introduction




  Empirical findings (Schorfheide, 2007): ’I will contrast the authors’
  empirical findings with estimates obtained from a
  constant-parameter DSGE model that is fitted to three different
  post-war periods. Although posterior distributions for some of the
  model parameters have shifted, there is no much evidence that the
  transmission of monetary policy shocks and the inflation-output
  trade-off have significantly changed.’




              Tonner J., Polanský J., Vašíček 0.   Quantitive Analysis of DSGE Model
Introduction



  If we admit Villaverde’s hypothesis, there are at least three reasons
  for parameters drifting (Villaverde, 2007):
      Pure econometric approach or capricious nature.
      Parameters drifting as a character of economic environment.
      Parameters drifting as a telltale of model misspecification.
  We would like to identify changes of structural parameters as
  consequence of the monetary regime change in 1997 or as
  consequence of the Czech integration with european structures. So
  our approach coincides with the second way.




              Tonner J., Polanský J., Vašíček 0.   Quantitive Analysis of DSGE Model
Introduction



  Our contribution embodies in:
      constructing a functional tool of time - variant parameters
      estimation (our first article was adopted in May 2007,
      Villaverde’s in June 2007),
      BECAUSE time - variant parameters cause the model to be
      overcomplete. This is a general mathematical problem and we
      think it must be solved properly, because Villaverde only
      marked ’the information in the sample is limited, and it is
      difficult to obtain stable estimates otherwise’.




             Tonner J., Polanský J., Vašíček 0.   Quantitive Analysis of DSGE Model
Model Formulation




  A baseline New Keynesian business cycle model was chosen as an
  example of our analysis, because:
      it is paradigmatic representative of the DSGE economies
      estimated by practitioners,
      it is the most favourite one for policy - making institutions.




             Tonner J., Polanský J., Vašíček 0.   Quantitive Analysis of DSGE Model
Model Formulation


                            HOUSEHOLDS PROBLEM
            ∞                                                                     1+γ
                                                                    mjt          ljt
   max E0         β t dt    log(cjt − hcjt−1 ) + v log(                 ) − ϕt ψ
                                                                    pt           1+γ
            t=0

  under five conditions:
      log dt = ρd log dt−1 + σd              d,t   where    d,t   ∼ N(0, 1),
      log ϕt = ρϕ log ϕt−1 + σϕ      ϕ,t where ϕ,t ∼ N(0, 1),
                      mjt  bjt+1
      cjt + xjt +     pt+ pt + qjt+1,tajt+1 dωj,t+1,t =
                                            m          b
      wjt ljt +(rt ujt −µ−1 Φ[ujt ])kjt−1 + jt−1 +Rt−1 pjt +ajt +Tt +Ft ,
                         t                    pt         t
                                                         xjt
      kjt = (1 − δ)kjt−1 + µt 1 − V                     xjt−1       xjt ,
      µt = µt−1 eΛµ +zµ,t where zµ,t = σµ                   µ,t   and    µ,t   ∼ N(0, 1).


              Tonner J., Polanský J., Vašíček 0.    Quantitive Analysis of DSGE Model
Model Formulation



                                    LABOUR SUPPLIER
                                                          1
                                             d
                                  maxljt wt lt −              wjt ljt dj
                                                      0
  under condition:
                                     η
                       η−1          η−1
        d          1 η
       lt   =     0 ljt      dj           .

                                                                    wjt    −η
  By the zero profit condition, we get ljt =                                      d
                                                                                lt and finally
                                                                    wt
        w d
  lt = vt lt .




                 Tonner J., Polanský J., Vašíček 0.   Quantitive Analysis of DSGE Model
Model Formulation


                            CALVO’S WAGES SETTING

  The relevant part of the lagrangian for the household is then:
                                                                            τ
              ∞
                                                ljt+τ 1+γ                       Πχw
                                                                                 t+s−1
  maxwjt Et          (βθw )τ      −dt ϕt ψ                + λjt+τ                      wjt ljt+τ
                                                  1+γ                            Πt+s
              τ =0                                                        s=1

  subject to:
                        τ  Πχw
                            t+s−1 wjt
                                                     −η
                                                           d
       ljt+τ =          s=1 Πt+s wt+τ                     lt+τ .
  Because of complete markets, we can drop jth index.
  Consequently, the real price index evolves:
                             1−η
                                        wt−1              1−η
                     Πχw                zt−1    zt−1
       1 = θw         t−1
                      Πt                 wt
                                                 zt             + (1 − θw )(Π∗w )1−η
                                                                             t
                                         zt




                Tonner J., Polanský J., Vašíček 0.     Quantitive Analysis of DSGE Model
Model Formulation




                     THE FINAL GOOD PRODUCER
                                                       1
                                       d
                            maxyit pt yt −                 pit yit di
                                                   0
  under condition:
                      −1
       d        1                 −1
      yt =     0 yit       di          .
                                                                        −
                                                                  pit        d
  By the zero profit condition, we get yit =                       pt        yt .




             Tonner J., Polanský J., Vašíček 0.   Quantitive Analysis of DSGE Model
Model Formulation

       INTERMEDIATE GOOD PRODUCERS TWO-STAGES
                      PROBLEM
                                                 d
                                minld ,kit−1 wt lit + rt kit−1
                                        it

  subject to their supply curve:
      yit = At kit−1 (lit )1−α − φzt with
                α      d

      At = At−1 eΛA +zA,t where zA,t = σA                     A,t and A,t ∼ N(0, 1),
                                                       zA,t +αzµ,t         Λ +αΛ
      zt = zt−1      eΛz +zz,t     where zz,t =           1−α      , Λz,t = A,t1−α µ,t .
  So the firms rent inputs to satisfy their static minimization
  problem:
       ut kt−1        α wt
           d
          lt
                 =   1−α rt ,
                            1−α            1−α α
                      1               1 α wt    rt
      mct =          1−α              α      At    .

                 Tonner J., Polanský J., Vašíček 0.   Quantitive Analysis of DSGE Model
Model Formulation



                            CALVO’S WAGES SETTING
                ∞                                    τ
                                 λt+τ                                   pit
    maxpit Et          (βθp )τ                           Πχ
                                                          t+s−1             − mct+τ              yit+τ
                                  λt                                   pt+τ
                τ =0                            s=1

  subject to:
                                                         −
                         τ    χ      pit                       d
       yit+τ =           s=1 Πt+s−1 pt+τ                      yt+τ .
  Because of complete markets, we can drop ith index.
  Consequently, the real wage index evolves:
                  Πχ        1−
                   t−1
       1 = θp      Πt              + (1 − θp )Π∗1−
                                               t




                Tonner J., Polanský J., Vašíček 0.           Quantitive Analysis of DSGE Model
Model Formulation




                                          MARKETS

  Markets clear
       d
                              d
             At (ut kt−1 )α (lt )1−α −φzt
      yt =                vtp             ,
       d
      yt =   ct + xt + µ−1 a[ut ]kt−1 .




               Tonner J., Polanský J., Vašíček 0.   Quantitive Analysis of DSGE Model
Model Formulation




                              GOVERNMENT PROBLEM
                                             ytd     1−γR
                    γR                        yd
  Rt       Rt−1                Πt γΠ
  R    =    R
                         
                               Π
                                              t−1
                                              Λy d
                                                                emt

  with
         m t = σm      m,t   where        m,t   ∼ N(0, 1).




                  Tonner J., Polanský J., Vašíček 0.    Quantitive Analysis of DSGE Model
Model analysis



                             GDP growth, %, QoQ                                                                MU growth, %, QoQ
   5                                                                               3
                                                                    GDP growth                                                              rel.price I w.r.t. C growth
                                                                    Mean                                                                    Mean
   4
                                                                                   2

   3


                                                                                   1
   2


   1
                                                                                   0

   0

                                                                                  −1
  −1


  −2
                                                                                  −2


  −3

                                                                                  −3
  −4


  −5                                                                              −4
  1948:1   1958:1   1968:1        1978:1          1988:1   1998:1          2008:1 1947:1     1957:1   1967:1       1977:1          1987:1      1997:1            2007:1




                       Tonner J., Polanský J., Vašíček 0.                              Quantitive Analysis of DSGE Model
Model analysis



                              hours worked, %, QoQ                                                                              FF growth, %, YoY
  0.37                                                                             18
                                                                   hours worked                                                                                                funds rate

  0.36                                                                             16


  0.35                                                                             14


  0.34                                                                             12


  0.33                                                                             10


  0.32                                                                              8


  0.31                                                                              6


   0.3                                                                              4


  0.29                                                                              2


                                                                                    0
   1948:1   1958:1   1968:1         1978:1           1988:1   1998:1        2008:1 1954:1   1959:1   1964:1   1969:1   1974:1    1979:1   1984:1    1989:1   1994:1   1999:1   2004:1




                       Tonner J., Polanský J., Vašíček 0.                           Quantitive Analysis of DSGE Model
Preliminary estimation
                                                   M:1      M:2
                           beta                 0.9999   0.9999
                           h                    0.8773   0.8773
                           psi                  8.9420   8.9420
                           gamma                1.3586   1.3586
                           eta                  7.9650   7.9650
                           kappa                7.6790   7.6790
                           chi w                0.8500   1.0000
                           theta w              0.4500   0.0000
                           delta                0.0149   0.0149
                           u                    1.0000   1.0000
                           gamma2               0.0010   0.0010
                           alpha                0.2550   0.2550
                           epsilon              7.9570   7.9570
                           theta p              0.9067   0.0000
                           chi p                0.1505   1.0000
                           phi                  0.0000   0.0000
                           lambda mu            0.0100   0.0100
                           lambda A             0.0005   0.0005
                           lambda z             0.0041   0.0041
                           gamma R              0.7900   0.7900
                           gamma y              0.1904   0.1904
                           gamma pi             1.2596   1.2596
                           target pi            1.0078   1.0078
                           rho d                0.9506   0.9506
                           rho varphi           0.9420   0.9420
                           std eps mu           0.1010   0.1010
                           std eps d            0.0600   0.0600
                           std eps A            0.0072   0.0072
                           std eps m            0.0030   0.0030
                           std eps varphi       0.0700   0.0700


           Tonner J., Polanský J., Vašíček 0.   Quantitive Analysis of DSGE Model
Impulse response - Monetary policy shock


                  yd                     c                  x                         R                  k                        w
       1                        1                   1                                            1                    1
                                                                     1.002
                            0.9995               0.995
                                                                                              0.998
                                                                     1.001                                         0.995
    0.995                                         0.99
                             0.999                                        1
              09:1 11:1              09:1 11:1           09:1 11:1              09:1 11:1             09:1 11:1               09:1 11:1
                  ld                     u                 dot_z                    pi                   pi_st                    mc
       1                        1                   2                     1                      1                    1

                             0.995                  1                0.995
                                                                                              0.995
    0.995                     0.99                                    0.99                                         0.995
                                                    0
              09:1 11:1              09:1 11:1           09:1 11:1              09:1 11:1             09:1 11:1               09:1 11:1
                 w_st                 pi_w_st             lambda                    q                     Q                       pK
       1                                                                  1                                           1
                                1                                                            1.0005
                                                  1.01
                             0.999                                   0.995                                        0.9998
    0.995                                        1.005
                             0.998                                    0.99                       1
                                                    1                                                             0.9996
              09:1 11:1              09:1 11:1           09:1 11:1              09:1 11:1             09:1 11:1               09:1 11:1
                 v_p                   dot_yd             mu_obs                 YD_obs                 R_obs                   pi_obs
       1                                            1                     0                                           0
                                1                                                                1
    0.999                                                            −0.2
                             0.999                  0                                                             −0.005
                                                                                                0.5
                                                                     −0.4
    0.998                    0.998
                                                                     −0.6                        0                 −0.01
                                                   −1
               09:1 11:1             09:1 11:1           09:1 11:1              09:1 11:1             09:1 11:1               09:1 11:1
                −3                                                               varphi                                        −3 m
            x 10 l_obs                dot_mu                 d                                          dot_A              x 10
       0                        2                   2                 2                          2                    3
                                                                      1                                               2
      −1                        1                   1                                            1
                                                                      0                                               1
      −2                                                                       09:1   11:1
                                0                   0                     SS           M1        0
                                                                                                M2                    0
              09:1 11:1              09:1 11:1           09:1 11:1                                    09:1 11:1               09:1 11:1




                           Tonner J., Polanský J., Vašíček 0.             Quantitive Analysis of DSGE Model
Closing




               Thank you for your attention
                                jtonner@tiscali.cz




          Tonner J., Polanský J., Vašíček 0.   Quantitive Analysis of DSGE Model

				
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