433 lines
22 KiB
Modula-2
433 lines
22 KiB
Modula-2
% this is the Smets and Wouters (2007) model for which Komunjer and Ng (2011)
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% derived the minimal state space system. In Dynare, however, we use more
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% powerful minreal function
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/*
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* This file provides replication files for
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* Smets, Frank and Wouters, Rafael (2007): "Shocks and Frictions in US Business Cycles: A Bayesian
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* DSGE Approach", American Economic Review, 97(3), 586-606, that are compatible with Dynare 4.5 onwards
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*
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* To replicate the full results, you have to get back to the original replication files available at
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* https://www.aeaweb.org/articles.php?doi=10.1257/aer.97.3.586 and include the respective estimation commands and mode-files.
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*
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* Notes:
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* - The consumption Euler equation in the paper, equation (2), premultiplies the risk premium process \varepsilon_t^b,
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* denoted by b in this code, by the coefficient c_3. In the code this prefactor is omitted by setting the
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* coefficient to 1. As a consequence, b in this code actually is b:=c_3*\varepsilon_t^b. As a consequence, in
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* the arbitrage equation for the value of capital in the paper, equation (4), the term 1*\varepsilon_t^b
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* is replaced by 1/c_3*b, which is equal to \varepsilon_t^b given the above redefinition. This rescaling also explains why the
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* standard deviation of the risk premium shock in the AR(1)-process for b has a different standard deviation than reported
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* in the paper. However, the results are unaffected by this scaling factor (except for the fact that the posterior distribution
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* reported in the paper cannot be directly translated to the present mod-file due to parameter correlation in the posterior.
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* - As pointed out in Del Negro/Schorfheide (2012): "Notes on New-Keynesian Models"
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* in the code implementation of equation (8) for both the flex price and the sticky price/wage economy,
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* there is a (1/(1+cbetabar*cgamma)) missing in the i_2 in front of q_t (denoted qs in the code).
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* Equation (8) in the paper reads:
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* (1-(1-delta)/gamma)*(1+beta*gamma^(1-sigma))*gamma^2*varphi
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* which translates to the code snippet:
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* (1-(1-ctou)/cgamma)*(1+cbetabar*cgamma)*cgamma^2*csadjcost
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* But the code implements
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* (1-(1-ctou)/cgamma)*cgamma^2*csadjcost
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* which corresponds to an equation reading
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* (1-(1-delta)/gamma)*gamma^2*varphi
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* - Chib/Ramamurthy (2010): "Tailored randomized block MCMC methods with application to DSGE models", Journal of Econometrics, 155, pp. 19-38
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* have pointed out that the mode reported in the original Smets/Wouters (2007) paper is not actually the mode. \bar \pi (constepinf) is estimated lower
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* while \bar \l (constelab) is higher.
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* - Note that at the prior mean, [cmap,crhopinf] and [cmaw,crhow] are pairwise collinear. Thus, running identification at the prior
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* mean will return a warning. But this is only a local issue. These parameters are only indistinguishable at the prior mean, but not
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* at different points.
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* - In the prior Table 1A in the paper, the
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* - habit parameter $\lambda$ is erroneously labeled h
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* - the fixed cost parameter $\phi_p$ is labeled $\Phi$
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* - Table 1B claims that $\rho_{ga}$ follows a beta prior with B(0.5,0.2^2), but the code shows that it actually
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* follows a normal distribution with N(0.5,0.25^2)
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*
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* This file was originally written by Frank Smets and Rafeal Wouters and has been updated by
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* Johannes Pfeifer.
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*
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* Please note that the following copyright notice only applies to this Dynare
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* implementation of the model
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*/
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/*
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* Copyright (C) 2007-2013 Frank Smets and Raf Wouters
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* Copyright (C) 2013-15 Johannes Pfeifer
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* Copyright (C) 2020 Dynare Team
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*
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* This is free software: you can redistribute it and/or modify
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* it under the terms of the GNU General Public License as published by
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* the Free Software Foundation, either version 3 of the License, or
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* (at your option) any later version.
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*
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* This file is distributed in the hope that it will be useful,
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* but WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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* GNU General Public License for more details.
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*
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* You can receive a copy of the GNU General Public License
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* at <http://www.gnu.org/licenses/>.
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*/
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var labobs ${lHOURS}$ (long_name='log hours worked')
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robs ${FEDFUNDS}$ (long_name='Federal funds rate')
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pinfobs ${dlP}$ (long_name='Inflation')
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dy ${dlGDP}$ (long_name='Output growth rate')
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dc ${dlCONS}$ (long_name='Consumption growth rate')
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dinve ${dlINV}$ (long_name='Investment growth rate')
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dw ${dlWAG}$ (long_name='Wage growth rate')
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ewma ${\eta^{w,aux}}$ (long_name='Auxiliary wage markup moving average variable')
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epinfma ${\eta^{p,aux}}$ (long_name='Auxiliary price markup moving average variable')
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zcapf ${z^{flex}}$ (long_name='Capital utilization rate flex price economy')
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rkf ${r^{k,flex}}$ (long_name='rental rate of capital flex price economy')
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kf ${k^{s,flex}}$ (long_name='Capital services flex price economy')
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pkf ${q^{flex}}$ (long_name='real value of existing capital stock flex price economy')
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cf ${c^{flex}}$ (long_name='Consumption flex price economy')
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invef ${i^{flex}}$ (long_name='Investment flex price economy')
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yf ${y^{flex}}$ (long_name='Output flex price economy')
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labf ${l^{flex}}$ (long_name='hours worked flex price economy')
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wf ${w^{flex}}$ (long_name='real wage flex price economy')
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rrf ${r^{flex}}$ (long_name='real interest rate flex price economy')
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mc ${\mu_p}$ (long_name='gross price markup')
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zcap ${z}$ (long_name='Capital utilization rate')
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rk ${r^{k}}$ (long_name='rental rate of capital')
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k ${k^{s}}$ (long_name='Capital services')
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pk ${q}$ (long_name='real value of existing capital stock')
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c ${c}$ (long_name='Consumption')
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inve ${i}$ (long_name='Investment')
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y ${y}$ (long_name='Output')
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lab ${l}$ (long_name='hours worked')
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pinf ${\pi}$ (long_name='Inflation')
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w ${w}$ (long_name='real wage')
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r ${r}$ (long_name='nominal interest rate')
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a ${\varepsilon_a}$ (long_name='productivity process')
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b ${c_2*\varepsilon_t^b}$ (long_name='Scaled risk premium shock')
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g ${\varepsilon^g}$ (long_name='Exogenous spending')
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qs ${\varepsilon^i}$ (long_name='Investment-specific technology')
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ms ${\varepsilon^r}$ (long_name='Monetary policy shock process')
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spinf ${\varepsilon^p}$ (long_name='Price markup shock process')
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sw ${\varepsilon^w}$ (long_name='Wage markup shock process')
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kpf ${k^{flex}}$ (long_name='Capital stock flex price economy')
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kp ${k}$ (long_name='Capital stock')
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;
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varexo ea ${\eta^a}$ (long_name='productivity shock')
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eb ${\eta^b}$ (long_name='Investment-specific technology shock')
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eg ${\eta^g}$ (long_name='Spending shock')
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eqs ${\eta^i}$ (long_name='Investment-specific technology shock')
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em ${\eta^m}$ (long_name='Monetary policy shock')
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epinf ${\eta^{p}}$ (long_name='Price markup shock')
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ew ${\eta^{w}}$ (long_name='Wage markup shock')
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;
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parameters curvw ${\varepsilon_w}$ (long_name='Curvature Kimball aggregator wages')
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cgy ${\rho_{ga}}$ (long_name='Feedback technology on exogenous spending')
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curvp ${\varepsilon_p}$ (long_name='Curvature Kimball aggregator prices')
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constelab ${\bar l}$ (long_name='steady state hours')
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constepinf ${\bar \pi}$ (long_name='steady state inflation rate')
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constebeta ${100(\beta^{-1}-1)}$ (long_name='time preference rate in percent')
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cmaw ${\mu_w}$ (long_name='coefficient on MA term wage markup')
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cmap ${\mu_p}$ (long_name='coefficient on MA term price markup')
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calfa ${\alpha}$ (long_name='capital share')
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czcap ${\psi}$ (long_name='capacity utilization cost')
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csadjcost ${\varphi}$ (long_name='investment adjustment cost')
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ctou ${\delta}$ (long_name='depreciation rate')
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csigma ${\sigma_c}$ (long_name='risk aversion')
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chabb ${\lambda}$ (long_name='external habit degree')
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ccs ${d_4}$ (long_name='Unused parameter')
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cinvs ${d_3}$ (long_name='Unused parameter')
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cfc ${\phi_p}$ (long_name='fixed cost share')
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cindw ${\iota_w}$ (long_name='Indexation to past wages')
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cprobw ${\xi_w}$ (long_name='Calvo parameter wages')
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cindp ${\iota_p}$ (long_name='Indexation to past prices')
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cprobp ${\xi_p}$ (long_name='Calvo parameter prices')
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csigl ${\sigma_l}$ (long_name='Frisch elasticity')
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clandaw ${\phi_w}$ (long_name='Gross markup wages')
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crdpi ${r_{\Delta \pi}}$ (long_name='Unused parameter')
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crpi ${r_{\pi}}$ (long_name='Taylor rule inflation feedback')
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crdy ${r_{\Delta y}}$ (long_name='Taylor rule output growth feedback')
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cry ${r_{y}}$ (long_name='Taylor rule output level feedback')
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crr ${\rho}$ (long_name='interest rate persistence')
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crhoa ${\rho_a}$ (long_name='persistence productivity shock')
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crhoas ${d_2}$ (long_name='Unused parameter')
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crhob ${\rho_b}$ (long_name='persistence risk premium shock')
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crhog ${\rho_g}$ (long_name='persistence spending shock')
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crhols ${d_1}$ (long_name='Unused parameter')
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crhoqs ${\rho_i}$ (long_name='persistence risk premium shock')
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crhoms ${\rho_r}$ (long_name='persistence monetary policy shock')
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crhopinf ${\rho_p}$ (long_name='persistence price markup shock')
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crhow ${\rho_w}$ (long_name='persistence wage markup shock')
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ctrend ${\bar \gamma}$ (long_name='net growth rate in percent')
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cg ${\frac{\bar g}{\bar y}}$ (long_name='steady state exogenous spending share')
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;
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// fixed parameters
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ctou=.025;
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clandaw=1.5;
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cg=0.18;
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curvp=10;
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curvw=10;
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// estimated parameters initialisation
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calfa=.24;
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cbeta=.9995;
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csigma=1.5;
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cfc=1.5;
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cgy=0.51;
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csadjcost= 6.0144;
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chabb= 0.6361;
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cprobw= 0.8087;
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csigl= 1.9423;
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cprobp= 0.6;
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cindw= 0.3243;
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cindp= 0.47;
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czcap= 0.2696;
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crpi= 1.488;
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crr= 0.8762;
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cry= 0.0593;
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crdy= 0.2347;
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crhoa= 0.9977;
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crhob= 0.5799;
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crhog= 0.9957;
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crhols= 0.9928;
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crhoqs= 0.7165;
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crhoas=1;
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crhoms=0;
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crhopinf=0;
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crhow=0;
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cmap = 0;
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cmaw = 0;
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constelab=0;
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%% Added by JP to provide full calibration of model before estimation
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constepinf=0.7;
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constebeta=0.7420;
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ctrend=0.3982;
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model(linear);
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//deal with parameter dependencies; taken from usmodel_stst.mod
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#cpie=1+constepinf/100; %gross inflation rate
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#cgamma=1+ctrend/100 ; %gross growth rate
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#cbeta=1/(1+constebeta/100); %discount factor
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#clandap=cfc; %fixed cost share/gross price markup
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#cbetabar=cbeta*cgamma^(-csigma); %growth-adjusted discount factor in Euler equation
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#cr=cpie/(cbeta*cgamma^(-csigma)); %steady state net real interest rate
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#crk=(cbeta^(-1))*(cgamma^csigma) - (1-ctou); %R^k_{*}: steady state rental rate
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#cw = (calfa^calfa*(1-calfa)^(1-calfa)/(clandap*crk^calfa))^(1/(1-calfa)); %steady state real wage
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//cw = (calfa^calfa*(1-calfa)^(1-calfa)/(clandap*((cbeta^(-1))*(cgamma^csigma) - (1-ctou))^calfa))^(1/(1-calfa));
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#cikbar=(1-(1-ctou)/cgamma); %k_1 in equation LOM capital, equation (8)
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#cik=(1-(1-ctou)/cgamma)*cgamma; %i_k: investment-capital ratio
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#clk=((1-calfa)/calfa)*(crk/cw); %labor to capital ratio
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#cky=cfc*(clk)^(calfa-1); %k_y: steady state output ratio
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#ciy=cik*cky; %consumption-investment ratio
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#ccy=1-cg-cik*cky; %consumption-output ratio
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#crkky=crk*cky; %z_y=R_{*}^k*k_y
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#cwhlc=(1/clandaw)*(1-calfa)/calfa*crk*cky/ccy; %W^{h}_{*}*L_{*}/C_{*} used in c_2 in equation (2)
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#cwly=1-crk*cky; %unused parameter
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#conster=(cr-1)*100; %steady state federal funds rate ($\bar r$)
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// flexible economy
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[name='FOC labor with mpl expressed as function of rk and w, flex price economy']
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0*(1-calfa)*a + 1*a = calfa*rkf+(1-calfa)*(wf) ;
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[name='FOC capacity utilization, flex price economy']
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zcapf = (1/(czcap/(1-czcap)))* rkf ;
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[name='Firm FOC capital, flex price economy']
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rkf = (wf)+labf-kf ;
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[name='Definition capital services, flex price economy']
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kf = kpf(-1)+zcapf ;
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[name='Investment Euler Equation, flex price economy']
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invef = (1/(1+cbetabar*cgamma))* ( invef(-1) + cbetabar*cgamma*invef(1)+(1/(cgamma^2*csadjcost))*pkf ) +qs ;
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[name='Arbitrage equation value of capital, flex price economy']
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pkf = -rrf-0*b+(1/((1-chabb/cgamma)/(csigma*(1+chabb/cgamma))))*b +(crk/(crk+(1-ctou)))*rkf(1) + ((1-ctou)/(crk+(1-ctou)))*pkf(1) ;
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[name='Consumption Euler Equation, flex price economy']
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cf = (chabb/cgamma)/(1+chabb/cgamma)*cf(-1) + (1/(1+chabb/cgamma))*cf(+1) +((csigma-1)*cwhlc/(csigma*(1+chabb/cgamma)))*(labf-labf(+1)) - (1-chabb/cgamma)/(csigma*(1+chabb/cgamma))*(rrf+0*b) + b ;
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[name='Aggregate Resource Constraint, flex price economy']
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yf = ccy*cf+ciy*invef+g + crkky*zcapf ;
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[name='Aggregate Production Function, flex price economy']
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yf = cfc*( calfa*kf+(1-calfa)*labf +a );
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[name='Wage equation, flex price economy']
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wf = csigl*labf +(1/(1-chabb/cgamma))*cf - (chabb/cgamma)/(1-chabb/cgamma)*cf(-1) ;
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[name='Law of motion for capital, flex price economy (see header notes)']
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kpf = (1-cikbar)*kpf(-1)+(cikbar)*invef + (cikbar)*(cgamma^2*csadjcost)*qs ;
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// sticky price - wage economy
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[name='FOC labor with mpl expressed as function of rk and w, SW Equation (9)']
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mc = calfa*rk+(1-calfa)*(w) - 1*a - 0*(1-calfa)*a ;
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[name='FOC capacity utilization, SW Equation (7)']
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zcap = (1/(czcap/(1-czcap)))* rk ;
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[name='Firm FOC capital, SW Equation (11)']
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rk = w+lab-k ;
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[name='Definition capital services, SW Equation (6)']
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k = kp(-1)+zcap ;
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[name='Investment Euler Equation, SW Equation (3)']
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inve = (1/(1+cbetabar*cgamma))* (inve(-1) + cbetabar*cgamma*inve(1)+(1/(cgamma^2*csadjcost))*pk ) +qs ;
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[name='Arbitrage equation value of capital, SW Equation (4)']
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pk = -r+pinf(1)-0*b
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+ (1/((1-chabb/cgamma)/(csigma*(1+chabb/cgamma))))*b
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+ (crk/(crk+(1-ctou)))*rk(1)
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+ ((1-ctou)/(crk+(1-ctou)))*pk(1) ;
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[name='Consumption Euler Equation, SW Equation (2)']
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c = (chabb/cgamma)/(1+chabb/cgamma)*c(-1)
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+ (1/(1+chabb/cgamma))*c(+1)
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+((csigma-1)*cwhlc/(csigma*(1+chabb/cgamma)))*(lab-lab(+1))
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- (1-chabb/cgamma)/(csigma*(1+chabb/cgamma))*(r-pinf(+1) + 0*b) +b ;
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[name='Aggregate Resource Constraint, SW Equation (1)']
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y = ccy*c+ciy*inve+g + 1*crkky*zcap ;
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[name='Aggregate Production Function, SW Equation (5)']
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y = cfc*( calfa*k+(1-calfa)*lab +a );
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[name='New Keynesian Phillips Curve, SW Equation (10)']
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pinf = (1/(1+cbetabar*cgamma*cindp)) * ( cbetabar*cgamma*pinf(1) +cindp*pinf(-1)
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+((1-cprobp)*(1-cbetabar*cgamma*cprobp)/cprobp)/((cfc-1)*curvp+1)*(mc) ) + spinf ;
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[name='Wage Phillips Curve, SW Equation (13), with (12) plugged for mu_w']
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w = (1/(1+cbetabar*cgamma))*w(-1)
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+(cbetabar*cgamma/(1+cbetabar*cgamma))*w(1)
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+(cindw/(1+cbetabar*cgamma))*pinf(-1)
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-(1+cbetabar*cgamma*cindw)/(1+cbetabar*cgamma)*pinf
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+(cbetabar*cgamma)/(1+cbetabar*cgamma)*pinf(1)
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+(1-cprobw)*(1-cbetabar*cgamma*cprobw)/((1+cbetabar*cgamma)*cprobw)*(1/((clandaw-1)*curvw+1))*
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(csigl*lab + (1/(1-chabb/cgamma))*c - ((chabb/cgamma)/(1-chabb/cgamma))*c(-1) -w)
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+ 1*sw ;
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[name='Taylor rule, SW Equation (14)']
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r = crpi*(1-crr)*pinf
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+cry*(1-crr)*(y-yf)
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+crdy*(y-yf-y(-1)+yf(-1))
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+crr*r(-1)
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+ms ;
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[name='Law of motion for productivity']
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a = crhoa*a(-1) + ea;
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[name='Law of motion for risk premium']
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b = crhob*b(-1) + eb;
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[name='Law of motion for spending process']
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g = crhog*(g(-1)) + eg + cgy*ea;
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[name='Law of motion for investment specific technology shock process']
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qs = crhoqs*qs(-1) + eqs;
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[name='Law of motion for monetary policy shock process']
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ms = crhoms*ms(-1) + em;
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[name='Law of motion for price markup shock process']
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spinf = crhopinf*spinf(-1) + epinfma - cmap*epinfma(-1);
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epinfma=epinf;
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[name='Law of motion for wage markup shock process']
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sw = crhow*sw(-1) + ewma - cmaw*ewma(-1) ;
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ewma=ew;
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[name='Law of motion for capital, SW Equation (8) (see header notes)']
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kp = (1-cikbar)*kp(-1)+cikbar*inve + cikbar*cgamma^2*csadjcost*qs ;
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// measurement equations
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[name='Observation equation output']
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dy=y-y(-1)+ctrend;
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[name='Observation equation consumption']
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dc=c-c(-1)+ctrend;
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[name='Observation equation investment']
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dinve=inve-inve(-1)+ctrend;
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[name='Observation equation real wage']
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dw=w-w(-1)+ctrend;
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[name='Observation equation inflation']
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pinfobs = 1*(pinf) + constepinf;
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[name='Observation equation interest rate']
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robs = 1*(r) + conster;
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[name='Observation equation hours worked']
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labobs = lab + constelab;
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end;
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steady_state_model;
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dy=ctrend;
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dc=ctrend;
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dinve=ctrend;
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dw=ctrend;
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pinfobs = constepinf;
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robs = (((1+constepinf/100)/((1/(1+constebeta/100))*(1+ctrend/100)^(-csigma)))-1)*100;
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labobs = constelab;
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end;
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shocks;
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var ea;
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stderr 0.4618;
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var eb;
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stderr 1.8513;
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var eg;
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stderr 0.6090;
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var eqs;
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|
stderr 0.6017;
|
|
var em;
|
|
stderr 0.2397;
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|
var epinf;
|
|
stderr 0.1455;
|
|
var ew;
|
|
stderr 0.2089;
|
|
end;
|
|
|
|
|
|
estimated_params;
|
|
// PARAM NAME, INITVAL, LB, UB, PRIOR_SHAPE, PRIOR_P1, PRIOR_P2, PRIOR_P3, PRIOR_P4, JSCALE
|
|
// PRIOR_SHAPE: BETA_PDF, GAMMA_PDF, NORMAL_PDF, INV_GAMMA_PDF
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|
stderr ea,0.4618,0.01,3,INV_GAMMA_PDF,0.1,2;
|
|
stderr eb,0.1818513,0.025,5,INV_GAMMA_PDF,0.1,2;
|
|
stderr eg,0.6090,0.01,3,INV_GAMMA_PDF,0.1,2;
|
|
stderr eqs,0.46017,0.01,3,INV_GAMMA_PDF,0.1,2;
|
|
stderr em,0.2397,0.01,3,INV_GAMMA_PDF,0.1,2;
|
|
stderr epinf,0.1455,0.01,3,INV_GAMMA_PDF,0.1,2;
|
|
stderr ew,0.2089,0.01,3,INV_GAMMA_PDF,0.1,2;
|
|
crhoa,.9676 ,.01,.9999,BETA_PDF,0.5,0.20;
|
|
crhob,.2703,.01,.9999,BETA_PDF,0.5,0.20;
|
|
crhog,.9930,.01,.9999,BETA_PDF,0.5,0.20;
|
|
crhoqs,.5724,.01,.9999,BETA_PDF,0.5,0.20;
|
|
crhoms,.3,.01,.9999,BETA_PDF,0.5,0.20;
|
|
crhopinf,.8692,.01,.9999,BETA_PDF,0.5,0.20;
|
|
crhow,.9546,.001,.9999,BETA_PDF,0.5,0.20;
|
|
cmap,.7652,0.01,.9999,BETA_PDF,0.5,0.2;
|
|
cmaw,.8936,0.01,.9999,BETA_PDF,0.5,0.2;
|
|
csadjcost,6.3325,2,15,NORMAL_PDF,4,1.5;
|
|
csigma,1.2312,0.25,3,NORMAL_PDF,1.50,0.375;
|
|
chabb,0.7205,0.001,0.99,BETA_PDF,0.7,0.1;
|
|
cprobw,0.7937,0.3,0.95,BETA_PDF,0.5,0.1;
|
|
csigl,2.8401,0.25,10,NORMAL_PDF,2,0.75;
|
|
cprobp,0.7813,0.5,0.95,BETA_PDF,0.5,0.10;
|
|
cindw,0.4425,0.01,0.99,BETA_PDF,0.5,0.15;
|
|
cindp,0.3291,0.01,0.99,BETA_PDF,0.5,0.15;
|
|
czcap,0.2648,0.01,1,BETA_PDF,0.5,0.15;
|
|
cfc,1.4672,1.0,3,NORMAL_PDF,1.25,0.125;
|
|
crpi,1.7985,1.0,3,NORMAL_PDF,1.5,0.25;
|
|
crr,0.8258,0.5,0.975,BETA_PDF,0.75,0.10;
|
|
cry,0.0893,0.001,0.5,NORMAL_PDF,0.125,0.05;
|
|
crdy,0.2239,0.001,0.5,NORMAL_PDF,0.125,0.05;
|
|
constepinf,0.7,0.1,2.0,GAMMA_PDF,0.625,0.1;//20;
|
|
constebeta,0.7420,0.01,2.0,GAMMA_PDF,0.25,0.1;//0.20;
|
|
constelab,1.2918,-10.0,10.0,NORMAL_PDF,0.0,2.0;
|
|
ctrend,0.3982,0.1,0.8,NORMAL_PDF,0.4,0.10;
|
|
cgy,0.05,0.01,2.0,NORMAL_PDF,0.5,0.25;
|
|
calfa,0.24,0.01,1.0,NORMAL_PDF,0.3,0.05;
|
|
end;
|
|
|
|
stoch_simul(order=1,irf=0,periods=0);
|
|
options_.qz_criterium = 1;
|
|
|
|
indx = [M_.nstatic+(1:M_.nspred)]';
|
|
indy = 1:M_.endo_nbr';
|
|
|
|
SS.A = oo_.dr.ghx(indx,:);
|
|
SS.B = oo_.dr.ghu(indx,:);
|
|
SS.C = oo_.dr.ghx(indy,:);
|
|
SS.D = oo_.dr.ghu(indy,:);
|
|
|
|
[CheckCO,minnx,minSS] = get_minimal_state_representation(SS,0);
|
|
|
|
Sigmax_full = lyapunov_symm(SS.A, SS.B*M_.Sigma_e*SS.B', options_.lyapunov_fixed_point_tol, options_.qz_criterium, options_.lyapunov_complex_threshold, 1, options_.debug);
|
|
Sigmay_full = SS.C*Sigmax_full*SS.C' + SS.D*M_.Sigma_e*SS.D';
|
|
|
|
Sigmax_min = lyapunov_symm(minSS.A, minSS.B*M_.Sigma_e*minSS.B', options_.lyapunov_fixed_point_tol, options_.qz_criterium, options_.lyapunov_complex_threshold, 1, options_.debug);
|
|
Sigmay_min = minSS.C*Sigmax_min*minSS.C' + minSS.D*M_.Sigma_e*minSS.D';
|
|
|
|
([Sigmay_full(:) - Sigmay_min(:)]')
|
|
sqrt(([diag(Sigmay_full), diag(Sigmay_min)]'))
|
|
dx = norm( Sigmay_full - Sigmay_min, Inf);
|
|
if dx > 2e-8
|
|
error(sprintf('something wrong with minimal state space computations, as numerical error is %d',dx))
|
|
else
|
|
fprintf('numerical error for moments computed from minimal state system is %d\n',dx)
|
|
end
|