%************************************************************************* % Title: Bayesian Estimation of a Small Open Economy Dynamic Stochastic % General Equilibrium (DSGE) Model % By: Nicola Babarcich % Date: April 2014 % Senior Thesis %************************************************************************ %************************************************************************* % PREAMBLE %************************************************************************* % VARIABLE DECLARATIONS var y c s psi z mc i pi pi_h pi_f y_star mc_star i_star pi_star a a_star x x_star e y_n y_n_star s_n obs_y obs_pi obs_i obs_s obs_z obs_y_star obs_pi_star obs_i_star; varexo epsilon_a % domestic productivity shock epsilon_i % domestic interest rate shock epsilon_s % exchange rate shock epsilon_pih % domestic inflation shock epsilon_pif % import inflation shock epsilon_astar % foreign productivity shock epsilon_istar % foreign interest rate shock epsilon_pistar; % foreign inflation shock % PARAMETER DECLARATIONS parameters % Domestic parameters beta % lifetime discount factor sigma % inverse elasticity of intertemporal substitution varphi % inverse elasticity of intertemporal labor supply b % degree of habit persistence rho % elasticity of substitution between domestic and imported goods alpha % share of foreign goods in consumption theta_h % domestic producer Calvo parameter theta_f % domestic importer Calvo parameter theta_star % foreign Calvo parameter rho_a % technology process coefficient % Domestic monetary policy parameters rho_i % interest rate persistence phi_pi % response to inflation phi_y % response to the output gap phi_e % response to the nominal exchange rate % All foreign parameters rho_a_star % foreign technology process coefficient rho_i_star % foreign interest rate persistence phi_pi_star % foreign response to inflation phi_y_star % foreign response to the output gap phi_e_star; % foreign response to the nominal exchange rate % INITIAL PARAMETER CALIBRATION % SEE: Alp and Elekdag (2011), Gali and Monacelli (2005) beta = 0.99; sigma = 1; varphi = 3; b = 0.9; rho = 1; alpha = 0.25; theta_h = 0.75; theta_f = 0.75; theta_star = 0.65; rho_a = 0.9; rho_i = 0.5; phi_pi = 1.5; phi_y = 0.5; phi_e = 0.2; rho_a_star = 0.9; rho_i_star = 0.5; phi_pi_star = 1.5; phi_y_star = 0.5; phi_e_star = 0.2; %************************************************************************* % MODEL %************************************************************************* model(linear); % Domestic inflation pi_h = beta*(1-theta_h)*pi_h(+1) + theta_h*pi_h(-1) + (((1-beta*theta_h)*(1-theta_h))/theta_h)*mc + epsilon_pih; % Imported inflation pi_f = beta*(1-theta_f)*pi_f(+1) + theta_f*pi_f(-1) + (((1-beta*theta_f)*(1-theta_f))/theta_f)*psi + epsilon_pif; % CPI inflation pi = (1-alpha)*pi_h + alpha*pi_f; % Domestic marginal cost mc = varphi*y + alpha*s - (1+varphi)*a + (sigma/(1-b))*(c-b*c(-1)); % Real exchange rate z = psi + (1-alpha)*s; % Terms of trade s - s(-1) = pi_f - pi_h + epsilon_s; % International risk sharing c = b*c + y_star - b*y_star + (1/sigma)*(1-b)*(psi + (1-alpha)*s); % Uncovered interest parity (i - pi(+1)) - (i_star - pi_star(+1)) = z(+1) - z + epsilon_s; % Goods market clearing y = (1-alpha)*c + alpha*y_star + (2-alpha)*alpha*rho*s + alpha*rho*psi; % Monetary policy rule i = rho_i*i(-1) + (1-rho_i)*(phi_pi*pi + phi_y*x + phi_e*(e-e(-1))) + epsilon_i; % Output gap x = y - y_n; % Technology process a = rho_a*a(-1) + epsilon_a; % Foreign Euler equation y_star = b*y_star(-1) + y_star(+1) - b*y_star - (1/sigma)*(1-b)*(i_star - pi_star(+1)); % Foreign inflation pi_star = beta*(1-theta_star)*pi_star(+1) + theta_star*pi_star(-1) + (((1-beta*theta_star)*(1-theta_star))/theta_star)*mc_star+epsilon_pistar; % Foreign marginal cost mc_star = varphi*y_star + alpha*s - (1+varphi)*a_star + (sigma/(1-b))*(y_star-b*y_star(-1)); % Foreign monetary policy rule i_star = rho_i_star*i_star(-1) + (1-rho_i_star)*(phi_pi_star*pi_star + phi_y_star*x_star + phi_e_star*(e(-1)-e)) + epsilon_istar; % Foreign output gap x_star = y_star - y_n_star; % Foreign technology process a_star = rho_a_star*a_star(-1) + epsilon_astar; % Nominal exchange rate e = z - z(-1) + pi - pi_star; % Measurement equations % Domestic output obs_y = y - y(-1); % Domestic inflation obs_pi = pi; % Domestic interest rate obs_i = i; % TOT obs_s = s - s(-1); % Real XR obs_z = z - z(-1); % Foreign output obs_y_star = y_star - y_star(-1); % Foreign inflation obs_pi_star = pi_star; % Foreign interest rate obs_i_star = i_star; % Output gap % Domestic definitions s_n = (((b*sigma)*(1+(varphi*alpha*rho)*(2-alpha)))/((sigma*(1+(varphi*alpha*rho)*(2-alpha)))+(varphi*(1-b)*(1-alpha)^2)))*s_n(-1) + ((sigma*(1+varphi))/((sigma*(1+(varphi*alpha*rho)*(2-alpha)))+(varphi*(1-b)*(1-alpha)^2)))*(a-b*a(-1)-(a_star-b*a_star(-1))); % Domestic output gap y_n = ((1+varphi)/varphi)*(a-a_star) - (s_n/varphi) + y_n_star; % Foreign output gap y_n_star = a_star*(((1-b)*(1-varphi))/(sigma+varphi*(1-b))) + ((b*sigma)/(sigma+varphi*(1-b)))*y_n_star(-1); end; %************************************************************************* % CHECK BLANCHARD-KAHN CONDITIONS % Conditions satisfied %************************************************************************* resid(1); check; %************************************************************************* % OBSERVED VARIABLES %************************************************************************* varobs obs_y obs_pi obs_i obs_s obs_z obs_y_star obs_pi_star obs_i_star; %************************************************************************* % INITIAL VALUES %************************************************************************* initval; y = 0; c = 0; s = 0; psi = 0; z = 0; mc = 0; i = 0; pi = 0; pi_h = 0; pi_f = 0; y_star = 0; mc_star = 0; i_star = 0; pi_star = 0; a = 0; a_star = 0; x = 0; x_star = 0; e = 0; y_n = 0; y_n_star = 0; obs_y = 0; obs_pi = 0; obs_i = 0; obs_s = 0; obs_z = 0; obs_y_star = 0; obs_pi_star = 0; obs_i_star = 0; end; %************************************************************************* % SHOCKS % Set shock stdev = 0.03 according to Alp and Elekdag (2011) %************************************************************************* % shocks; % IN TERMS OF STANDARD DEVIATION % var epsilon_a; stderr 2; % var epsilon_i; stderr 2; % var epsilon_s; stderr 2; % var epsilon_pih; stderr 2; % var epsilon_pif; stderr 2; % var epsilon_astar; stderr 2; % var epsilon_istar; stderr 2; % var epsilon_pistar; stderr 2; % IN TERMS OF VARIANCE % var epsilon_a = 0.03^2; % var epsilon_i = 0.03^2; % var epsilon_s = 0.03^2; % var epsilon_pih = 0.03^2; % var epsilon_pif = 0.03^2; % var epsilon_astar = 0.03^2; % var epsilon_istar = 0.03^2; % var epsilon_pistar = 0.03^2; %end; %************************************************************************* % ESTIMATED PARAMETERS %************************************************************************* estimated_params; sigma, normal_pdf, 2.0, 0.75; varphi, normal_pdf, 2.0, 0.5; b, beta_pdf, 0.7, 0.2; theta_h, beta_pdf, 0.5, 0.2; theta_f, beta_pdf, 0.5, 0.2; theta_star, normal_pdf, 0.65, 0.05; rho_a, beta_pdf, 0.8, 0.1; rho_i, beta_pdf, 0.6, 0.2; phi_pi, gamma_pdf, 1.45, 0.3; phi_y, gamma_pdf, 0.325, 0.2; phi_e, normal_pdf, 0.1, 0.05; rho_a_star, beta_pdf, 0.8, 0.1; rho_i_star, beta_pdf, 0.6, 0.2; phi_pi_star, gamma_pdf, 1.45, 0.3; phi_y_star, gamma_pdf, 0.325, 0.2; phi_e_star, normal_pdf, 0.1, 0.05; stderr epsilon_a, inv_gamma_pdf, 2, inf; stderr epsilon_i, inv_gamma_pdf, 2, inf; stderr epsilon_s, inv_gamma_pdf, 2, inf; stderr epsilon_pih, inv_gamma_pdf, 2, inf; stderr epsilon_pif, inv_gamma_pdf, 2, inf; stderr epsilon_astar, inv_gamma_pdf, 2, inf; stderr epsilon_istar, inv_gamma_pdf, 2, inf; stderr epsilon_pistar, inv_gamma_pdf, 2, inf; end; %************************************************************************* % COMPUTATION %************************************************************************* %stoch_simul(irf=40, periods=20000); %************************************************************************* % ESTIMATION %************************************************************************* estimation(datafile=ThesisData, order=1, mh_replic=10000, mh_nblocks=2, conf_sig=0.95, bayesian_irf, mh_jscale=0.5, mh_drop=0.4, mode_compute=6, nograph);