//*************************************************************************************** //********************REPLICATION OF THE FRBNY DSGE MODEL******************************** //********************by BERTHEAU, FRANCESCHI, LE FUR, PANON***************************** //*************************************************************************************** //****************************************************** //******************DECLARATION OF THE VARIABLES******** //******************AND PARAMETERS********************** //****************************************************** var c // Consumption L // Labor supply (total available hours are normalized to one) // m // Mt(j): Money holdings xi // Xi_t: multiplicator w // Wt(j) is the wage paid to Lt(j) wtilde // Households that can optimize at time t choose a wage Wt(j)tilde to maximize (6) y // Final good pi // Inflation mc // Nominal marginal cost k // Effective capital qk // Tobin's q, price of installed capital i // Investments (amount) kbar // Installed capital (bought by entrepreneurs from capital producers at the end of current period) rk // Rental rate for unit of effective capital u // Capital utilisation rate r // Nominal interest rate (gross, so R>1) Rktilde // Nominal return to capital for entrepreneurs (gross, so Rktilde>1) n // Entrepreneurs' net worth z // Growth rate of productivity phi // Preference shifter lambda_f // Net mark-up that intermediate firms would like to charge over marginal costs mu // Marginal Efficiency of Investment shock sigma_omega // Spread-shock: an increase in volatility increases the perceived riskiness of borrowers-->increases the cost of capital g // Government spending Y_obs // Real Output growth (% annualised) L_obs // Hours worked (%) LS_obs // Labor Share (%) pi_core_obs // Inflation (% annualised) FFR_obs // Interest rate (% annualised) SP_obs // Spread (% annualised) ; varexo ez // Innovation of the unit-rooted productivity growth process ephi // Innovation of the labour supply shock elambda_f // Innovation of the intermediate goods producers mark-up shock emu // Innovation of the MEI shock esigma_omega // Innovation of the default cut-off shock (idiosyncratic random disturbance to entrepreneurs' capital productivity) eg // Innovation of the government spending (demand shock) eps_R ; // Innovation of the current period unanticipated shock (mon. policy) parameters alpha beta nu_l nu_m h lambda_w zeta_w chi pi_star gamma delta gamma_e zeta_spb lambda_f_ss rho_r psi_pi psi_y ddS dda zeta_p zeta_spsigma_omega zeta_nRktilde zeta_nr zeta_nn zeta_nqk zeta_nsigma_omega rho_z rho_lambda_f rho_mu rho_g rho_sigma rho_phi r_ss L_adj SP_star LS_star g_ss; beta = 0.9950; // Consumer's discount rate (from CMR) nu_l = 2; // Curvature on Disutility of Labor nu_m = 2; // Curvature on Utility of money h = .7; // Habit persistence zeta_w = .8; // Calvo wage rigidity lambda_w= .3; // Affects the elasticity of substitution between differentiated labor services.(mark-up wages) chi = .1; // Weight on utility of money pi_star = 2; // Inflation target. Useful for people stuck with their former wages (with proba zeta_w ) gamma =2.75/100; // exp(gamma) is growth rate of the economy (useful for people stuck with their former wages) delta = .025; // Depreciation rate gamma_e=.99; // Fraction of surviving entrepreneurs to the shock at each period zeta_spb=0.05; // Elasticity of spread to leverage lambda_f_ss=0.15; // Steady state value of the mark up shock rho_r=0.5; // Persistence of moneatry policy psi_pi=2; // Monetary policy stuff psi_y=0.2; // Monetary policy stuff ddS=4; // Second derivative of S dda=0.2; // Second derivative of a zeta_p=0.75; // Calvo parameter: share of firms that can't adjust their prices alpha= .33; // Elasticity of capital with respect to output rho_z=0.4; // Coefficient of the AR(1) process rho_phi=0.75; // Coefficient of the AR(1) process rho_lambda_f=0.75; // Coefficient of the AR(1) process rho_mu=0.75; // Coefficient of the AR(1) process rho_g=0.75; // Coefficient of the AR(1) process rho_sigma=0.75; // Coefficient of the AR(1) process r_ss=1.5; // Steady state (gross) nominal interest rate L_adj=253.5; // Captures the units of measured hours worked SP_star=2; // Spread in steady state LS_star=exp(-58); // Labour share in steady state g_ss=0.3; // Government spending in steady state //To be found on 7.46-7.47 p. 39 of the notes that we used - These values come directly from the Matlab code published zeta_spsigma_omega=0.0293; //elasticity of the spread to volatility of the spread shock zeta_nRktilde=1.5884; //elasticity of net worth to return on capital zeta_nr=0.5861; //elasticity of net worth to nominal interest rate zeta_nqk=0; //elasticity of net worth to the cost of capital zeta_nn=1.0023; //elasticity of net worth to net worth itself zeta_nsigma_omega=0.004; //elasticity of net worth to the volatility of sigma_omega (variance of the shock to entrepreneurs) //NB: parameters "chi" and "pi_star" are not used //*********************************************** //******************MODEL************************ //*********************************************** model(linear); //Shorthand for steady state values; some are derived directly from the published code, namely n_ss, L_ss, g_ss, #rk_ss=beta^(-1)*exp(gamma) - (1-delta); // Rental rate of capital #w_ss= (1/(1+lambda_f_ss)*(alpha^alpha)*(1-alpha)^(1-alpha)* // Optimal wage rk_ss^(-alpha))^(1/(1-alpha)); #L_ss=1; // Labour supply at SS #ratioIK_ss=1-((1-delta)/(exp(gamma))); // Investment to capital ratio #upsilon_ss=1; // #n_ss=11.9196; // Steady state entrepreneurs' net worth #k_ss=(alpha/(1-alpha))*(w_ss/rk_ss)*L_ss; // Capital stock #kbar_ss=exp(gamma)*k_ss; // Effective capital #i_ss=kbar_ss*ratioIK_ss; // Investments #y_ss=(k_ss^alpha)*(L_ss^(1-alpha)); // Output (with fixed cost it'd include PHI) #c_ss=(y_ss/g_ss)-i_ss; // Consumption #realised= Rktilde-pi; // Shorthand for realised return on capital #spread = Rktilde(+1) - r; // Shorthand for expected excess return on capital z=rho_z*z(-1)+ez; phi=rho_phi*phi(-1)+ephi; lambda_f=rho_lambda_f*lambda_f(-1)+elambda_f; mu=rho_mu*mu(-1)+emu; sigma_omega=rho_sigma*sigma_omega(-1)+esigma_omega; g=rho_g*g(-1)+eg; //Here we put the loglin'd equations from the published paper xi=r+xi(+1)-z(+1)-pi(+1); // Consumption euler equation (exp(gamma)-h*beta)*(exp(gamma)-h)*xi = -(exp(2*gamma)-beta*h^2)*c +h*exp(gamma)*c(-1) - h*exp(gamma)*z + beta*h*exp(gamma)*c(+1) + beta*h*exp(gamma)*z(+1); // Marginal utility of consumption kbar(-1)=-(1-(i_ss/k_ss))*z + (1-(i_ss/k_ss))*kbar(-2) + (i_ss/k_ss)*mu + (i_ss/k_ss)*i; // Capital stock k= u-z +kbar(-2); // Effective capital rk_ss*rk=dda*u; // Capital utilisation; dda is a parameter i= (1/(1+beta))*(i(-1) - z) + (beta/(1+beta))*(i(+1) + z(+1)) + (1/((1+beta)*ddS*exp(2*gamma)))*qk + (1/((1+beta)*ddS*exp(2*gamma)))*mu; // Optimal investment decision; ddS is a parameter realised=(rk_ss/(rk_ss+(1-delta)))*rk + ((1-delta)/(rk_ss+(1-delta)))*qk -qk(-1); // Realised real return to capital spread= zeta_spb*(qk + kbar(-1) - n) + zeta_spsigma_omega*sigma_omega; // Spread n=zeta_nRktilde*(Rktilde-pi)-zeta_nr*(r(-1)-pi)+zeta_nqk*(qk(-1)+kbar(-2))+ zeta_nn*n(-1) - gamma_e*(upsilon_ss/n_ss)*z - zeta_nsigma_omega*sigma_omega(-1); // Entrepreneurs' net worth w=w(-1)-pi+((1-zeta_w)/zeta_w)*wtilde; // Evolution of the aggregate nominal wage (1 + nu_l*((1 + lambda_w)/lambda_w))*wtilde + (1 + zeta_w*beta*nu_l*((1 + lambda_w)/lambda_w))*w = zeta_w*beta* (1 + nu_l*((1 + lambda_w)/lambda_w))*(wtilde(+1) + w(+1)) + phi + (1 - zeta_w*beta)*(nu_l*L - xi) + zeta_w*beta*(1 + nu_l* ((1 + lambda_w)/lambda_w))*(pi(+1) + z(+1)); // Optimal reset wage pi = beta*pi(+1) + (((1 - zeta_p*beta)*(1 - zeta_p))/zeta_p)*mc + (1/zeta_p)*(((1 - zeta_p*beta)*(1 - zeta_p)*lambda_f_ss)/(1 + lambda_f_ss))*lambda_f; // Phillips curve mc=(1-alpha)*w+alpha*rk; // Marginal cost (or labour share) y=alpha*k+(1-alpha)*L; // Production fuction k=w-rk+L; // Capital labour ratio y = g + (c_ss/(c_ss + i_ss))*c + (i_ss/(c_ss + i_ss))*i + ((rk_ss*k_ss)/(c_ss + i_ss))*u; // Resource constraint r=rho_r*r(-1)+(1-rho_r)*(psi_pi*(pi+pi(-1)+pi(-2)+pi(-3))+psi_y*(y+z+ // Policy rule +z(-1)+z(-2)-y(-4)+z(-3)))+ eps_R+eps_R(+4); //Measurement equations Y_obs=400*(y-y(-1)+z+gamma); L_obs=100*(L+log(L_adj)); LS_obs=100*(L+w-y+log(LS_star)); pi_core_obs=400*(pi+log(pi_star)); FFR_obs=400*(r+log(r_ss)); SP_obs=400*(spread +SP_star); end; varobs Y_obs L_obs LS_obs pi_core_obs FFR_obs SP_obs; observation_trends; Y_obs (z+gamma); L_obs(log(L_adj)); LS_obs (w-y+log(LS_star)); pi_core_obs (log(pi_star)); FFR_obs (log(r_ss)); SP_obs (SP_star); end; //*********************************************** //******************ESTIMATION******************* //*********************************************** // ****PRIORS**** // As in table 1 of the paper estimated_params; alpha, beta_pdf, .33, .02; zeta_p, beta_pdf, .75, .1; delta, beta_pdf, .025, .01; ddS, gamma_pdf, 4, 1.5; h, beta_pdf, .7, .05; dda, gamma_pdf, .2, .10; nu_l, gamma_pdf, 2, .75; nu_m, gamma_pdf, 2, .75; zeta_w, beta_pdf, .75, .1; r_ss, gamma_pdf, 1.5, 1; psi_pi, gamma_pdf, 2, 0.25; psi_y, gamma_pdf, 0.2, 0.1; rho_r, beta_pdf, .5, .2; pi_star, normal_pdf, 2, 0.25; SP_star, gamma_pdf, 2, 0.5; zeta_spb, beta_pdf, .05, .02; // Exogenous processes - level gamma, gamma_pdf, 2.75/100, .5; chi, gamma_pdf, 0.1, 0.1; g_ss, gamma_pdf, 0.3, 0.1; L_adj, normal_pdf, 253.5, 5; // Exogenous processes - autocorrelation rho_z, beta_pdf, 0.4, 0.25; rho_phi, beta_pdf, 0.75, 0.15; rho_lambda_f, beta_pdf, 0.75, 0.15; rho_mu, beta_pdf, 0.75, 0.15; rho_g, beta_pdf, 0.75, 0.15; rho_sigma, beta_pdf, 0.75, 0.15; // Exogenous processes - standard deviation stderr ez, inv_gamma_pdf, 0.3, 4; stderr ephi, inv_gamma_pdf, 3, 4.00; stderr elambda_f, inv_gamma_pdf, 0.2, 4.00; stderr emu, inv_gamma_pdf, 0.75, 4.00; stderr eg, inv_gamma_pdf, 0.5, 0.4; stderr esigma_omega, inv_gamma_pdf, 0.05, 4.00; end; check; shocks; //*********************************************** //******************SHOCKS*********************** //*********************************************** //Set of shocks, expected to be 7 var ez; stderr .3; var ephi; stderr 3; var elambda_f; stderr .2; var emu; stderr .75; var esigma_omega; stderr .05; var eg; stderr .5; var eps_R; stderr .2; //var end; stoch_simul(periods=1000, irf=20); estimated_params_init(use_calibration); end; estimation(datafile=datadelnegro2, mh_replic=20000, mh_jscale=0.038376, mode_compute=6);