# RBC model: deterministic vs stochastic simulations

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## Description

This video is part of a series of videos on the baseline Real Business Cycle model and its implementation in Dynare. In this video I focus on simulations and discuss the difference between the deterministic and stochastic model framework of Dynare. I provide intuition how Dynare “solves” or “simulates” these different model frameworks and guidance on when to run either deterministic or stochastic simulations. Then I show how to simulate various scenarios in the baseline RBC model.

In the deterministic case (i.e. under perfect foresight), this videos covers

• unexpected or pre-announced temporary shocks
• unexpected or pre-announced permanent shocks
• return to equilibrium by using Dynare’s perfect_foresight_setup and perfect_foresight_solver (i.e. the old simul) commands and the shocks, initval, endval and histval blocks. I show what happens in MATLAB’s workspace and to Dynare’s output structure oo_.

In the stochastic case, this videos covers

• impulse-response-functions (irf)
• variance decompositions
• theoretical vs. simulated moments
• data simulation by using Dynare’s stoch_simul command and the shocks block. I show what happens in MATLAB’s workspace and to Dynare’s output structures oo_ and oo_.dr. Lastly, the difference between Dynare’s declaration and DR (decision-rule) ordering of variables is covered.

## Timestamps

Theory

• 01:06 - Deterministic vs. stochastic model framework
• 08:01 - When to use which framework?

Deterministic Simulation in Dynare

• 11:47 - Overview of Dynare commands for deterministic simulations
• 13:58 - Getting ready in Dynare
• 15:00 - Scenario 1: Unexpected temporary TFP shock
• 15:25 - What does perfect_foresight_setup do?
• 17:39 - What does perfect_foresight_solver do?
• 19:12 - What happens in MATLAB’s workspace?
• 19:54 - What happens in Dynare’s output structure oo_?
• 21:43 - Simulated_time_series is a dseries object
• 22:51 - Scenario 2: Sequence of temporary pre-announced shocks
• 24:56 - Why simul is a depreciated syntax; better use perfect_foresight_setup and perfect_foresight_solver!
• 26:20 - dsample command
• 27:14 - Scenario 3: Unexpected permanent shock
• 28:47 - Values of 0 can cause errors as log(0) is inf; double check your initval and endval blocks!
• 30:45 - Don’t forget to adjust steady-state computations to be dependent on value of exogenous variables (if they are different than 0)
• 32:27 - Scenario 4: Pre-announced permanent shock

Stochastic Simulation in Dynare

• 36:26 - Overview of Dynare commands for stochastic simulations
• 38:28 - Impulse-Response-Function (IRF) of TFP shock
• 39:39 - Adding a preference shock to the model
• 41:38 - Impulse-Response-Function (IRF) of preference shock
• 42:08 - What happens in MATLAB’s console?
• 42:35 - Theoretical moments with periods=0 option
• 43:06 - What happens in Dynare’s oo_ structure
• 43:51 - What happens in Dynare’s oo_.dr structure
• 44:53 - Difference between declaration and DR (decision rule) order
• 46:07 - Simulate data and simulated moments with periods option

Outro & References

• 47:01 - Outro
• 47:52 - References

Presentation

## Codes

function l = rbc_steady_state_helper(L0, w,C_L,ETAC,ETAL,PSI,GAMMA)
options = optimset('Display','off','TolX',1e-10,'TolFun',1e-10);
l = fsolve(@(l) w*C_L^(-ETAC) - PSI/GAMMA*(1-l)^(-ETAL)*l^ETAC , L0,options);
end


### rbc_nonlinear_common.inc

@#define LOGUTILITY = 0

var
y     ${Y}$        (long_name='output')
c     ${C}$        (long_name='consumption')
k     ${K}$        (long_name='capital')
l     ${L}$        (long_name='labor')
a     ${A}$        (long_name='productivity')
r     ${R}$        (long_name='interest Rate')
w     ${W}$        (long_name='wage')
iv    ${I}$        (long_name='investment')
mc    ${MC}$       (long_name='marginal Costs')
;

model_local_variable
uc    ${U_t^C}$
ucp   ${E_t U_{t+1}^C}$
ul    ${U_t^L}$
fk    ${f_t^K}$
fl    ${f_t^L}$
;

varexo
epsa  ${\varepsilon^A}$   (long_name='Productivity Shock')
;

parameters
BETA  ${\beta}$  (long_name='Discount Factor')
DELTA ${\delta}$ (long_name='Depreciation Rate')
GAMMA ${\gamma}$ (long_name='Consumption Utility Weight')
PSI   ${\psi}$   (long_name='Labor Disutility Weight')
@#if LOGUTILITY != 1
ETAC  ${\eta^C}$ (long_name='Risk Aversion')
ETAL  ${\eta^L}$ (long_name='Inverse Frisch Elasticity')
@#endif
ALPHA ${\alpha}$ (long_name='Output Elasticity of Capital')
RHOA  ${\rho^A}$ (long_name='Discount Factor')
;

% Parameter calibration
ALPHA = 0.35;
BETA  = 0.99;
DELTA = 0.025;
GAMMA = 1;
PSI   = 1.6;
RHOA  = 0.9;
@#if LOGUTILITY == 0
ETAC  = 2;
ETAL  = 1;
@#endif

model;
%marginal utility of consumption and labor
@#if LOGUTILITY == 1
#uc  = GAMMA*c^(-1);
#ucp  = GAMMA*c(+1)^(-1);
#ul = -PSI*(1-l)^(-1);
@#else
#uc  = GAMMA*c^(-ETAC);
#ucp  = GAMMA*c(+1)^(-ETAC);
#ul = -PSI*(1-l)^(-ETAL);
@#endif

%marginal products of production
#fk = ALPHA*y/k(-1);
#fl = (1-ALPHA)*y/l;

[name='intertemporal optimality (Euler)']
uc = BETA*ucp*(1-DELTA+r(+1));
[name='labor supply']
w = -ul/uc;
[name='capital accumulation']
k = (1-DELTA)*k(-1) + iv;
[name='market clearing']
y = c + iv;
[name='production function']
y = a*k(-1)^ALPHA*l^(1-ALPHA);
[name='marginal costs']
mc = 1;
[name='labor demand']
w = mc*fl;
[name='capital demand']
r = mc*fk;
[name='total factor productivity']
log(a) = RHOA*log(a(-1)) + epsa;
end;

% ------------------------ %
% ------------------------ %

a = exp(epsa/(1-RHOA));
mc = 1;
r = 1/BETA + DELTA -1;
K_L = (mc*ALPHA*a/r)^(1/(1-ALPHA));
w = mc*(1-ALPHA)*a*K_L^ALPHA;
IV_L = DELTA*K_L;
Y_L = a*(K_L)^ALPHA;
C_L = Y_L - IV_L;
@#if LOGUTILITY==1
l = GAMMA/PSI*C_L^(-1)*w/(1+GAMMA/PSI*C_L^(-1)*w);
@#else
L0 = 1/3;
@#endif
c  = C_L*l;
y  = Y_L*l;
iv = IV_L*l;
k  = K_L*l;

end;


### rbc_nonlinear_common1.inc

@#define LOGUTILITY = 0

var
y     ${Y}$        (long_name='output')
c     ${C}$        (long_name='consumption')
k     ${K}$        (long_name='capital')
l     ${L}$        (long_name='labor')
a     ${A}$        (long_name='productivity')
r     ${R}$        (long_name='interest Rate')
w     ${W}$        (long_name='wage')
iv    ${I}$        (long_name='investment')
mc    ${MC}$       (long_name='marginal Costs')
z
;

model_local_variable
uc    ${U_t^C}$
ucp   ${E_t U_{t+1}^C}$
ul    ${U_t^L}$
fk    ${f_t^K}$
fl    ${f_t^L}$
;

varexo
epsa  ${\varepsilon^A}$   (long_name='Productivity Shock')
epsz
;

parameters
BETA  ${\beta}$  (long_name='Discount Factor')
DELTA ${\delta}$ (long_name='Depreciation Rate')
GAMMA ${\gamma}$ (long_name='Consumption Utility Weight')
PSI   ${\psi}$   (long_name='Labor Disutility Weight')
@#if LOGUTILITY != 1
ETAC  ${\eta^C}$ (long_name='Risk Aversion')
ETAL  ${\eta^L}$ (long_name='Inverse Frisch Elasticity')
@#endif
ALPHA ${\alpha}$ (long_name='Output Elasticity of Capital')
RHOA  ${\rho^A}$ (long_name='Discount Factor')
RHOZ
;

% Parameter calibration
ALPHA = 0.35;
BETA  = 0.99;
DELTA = 0.025;
GAMMA = 1;
PSI   = 1.6;
RHOA  = 0.9;
@#if LOGUTILITY == 0
ETAC  = 2;
ETAL  = 1;
@#endif
RHOZ=0.5;

model;
%marginal utility of consumption and labor
@#if LOGUTILITY == 1
#uc  = z*GAMMA*c^(-1);
#ucp  = z(+1)*GAMMA*c(+1)^(-1);
#ul = -z*PSI*(1-l)^(-1);
@#else
#uc  = z*GAMMA*c^(-ETAC);
#ucp  = z(+1)*GAMMA*c(+1)^(-ETAC);
#ul = -z*PSI*(1-l)^(-ETAL);
@#endif

%marginal products of production
#fk = ALPHA*y/k(-1);
#fl = (1-ALPHA)*y/l;

[name='intertemporal optimality (Euler)']
uc = BETA*ucp*(1-DELTA+r(+1));
[name='labor supply']
w = -ul/uc;
[name='capital accumulation']
k = (1-DELTA)*k(-1) + iv;
[name='market clearing']
y = c + iv;
[name='production function']
y = a*k(-1)^ALPHA*l^(1-ALPHA);
[name='marginal costs']
mc = 1;
[name='labor demand']
w = mc*fl;
[name='capital demand']
r = mc*fk;
[name='total factor productivity']
log(a) = RHOA*log(a(-1)) + epsa;
log(z) = RHOZ*log(z(-1)) + epsz;
end;

% ------------------------ %
% ------------------------ %
z=1;
a = exp(epsa/(1-RHOA));
mc = 1;
r = 1/BETA + DELTA -1;
K_L = (mc*ALPHA*a/r)^(1/(1-ALPHA));
w = mc*(1-ALPHA)*a*K_L^ALPHA;
IV_L = DELTA*K_L;
Y_L = a*(K_L)^ALPHA;
C_L = Y_L - IV_L;
@#if LOGUTILITY==1
l = GAMMA/PSI*C_L^(-1)*w/(1+GAMMA/PSI*C_L^(-1)*w);
@#else
L0 = 1/3;
@#endif
c  = C_L*l;
y  = Y_L*l;
iv = IV_L*l;
k  = K_L*l;

end;


### rbc_nonlinear_det1.mod

@#include "rbc_nonlinear_common.inc"

% -------------------- %
% Unexpected TFP shock %
% -------------------- %

shocks;
var epsa; periods 1; values -0.1;
end;

% % make sure everything is set up correctly!
% perfect_foresight_setup(periods=4);
% oo_.exo_simul
% oo_.endo_simul

perfect_foresight_setup(periods=300);
perfect_foresight_solver;

rplot c iv y;
rplot l w;
rplot r;
rplot k;
rplot a;


### rbc_nonlinear_det2.mod

@#include "rbc_nonlinear_common.inc"

% ----------------------- %
% Pre-announced TFP shock %
% ----------------------- %
shocks;
var epsa;
periods    4,  5:8;
values  0.04, 0.01;
end;

% % make sure everything is set up correctly!
% perfect_foresight_setup(periods=8);
% oo_.exo_simul
% oo_.endo_simul

perfect_foresight_setup(periods=300);
perfect_foresight_solver;

dsample 100;
rplot c iv y;
rplot l w;
rplot r;
rplot k;
rplot a;


### rbc_nonlinear_det3.mod

@#include "rbc_nonlinear_common.inc"

% ------------------------------------------------ %
% Permanent shock: TFP increases permanently by 5% %
% ------------------------------------------------ %
initval;
epsa=0;
end;

endval;
epsa = (1-RHOA)*log(1.05);
end;

% make sure everything is set up correctly!
% perfect_foresight_setup(periods=8);
% oo_.exo_simul
% oo_.endo_simul

perfect_foresight_setup(periods=300);
perfect_foresight_solver;
dsample 100;
rplot c iv y;
rplot l w;
rplot r;
rplot k;
rplot a;


### rbc_nonlinear_det4.mod

@#include "rbc_nonlinear_common.inc"

% -------------------------------------------------------------- %
% Pre-announced permanent shock: TFP increases permanently by 5% %
% -------------------------------------------------------------- %
initval;
epsa=0;
end;

endval;
epsa = (1-RHOA)*log(1.05);
end;

shocks;
var epsa; periods 1:5; values 0;
end;

% make sure everything is set up correctly!
% perfect_foresight_setup(periods=8);
% oo_.exo_simul
% oo_.endo_simul

perfect_foresight_setup(periods=300);
perfect_foresight_solver;
dsample 100;
rplot c iv y;
rplot l w;
rplot r;
rplot k;
rplot a;


### rbc_nonlinear_det5.mod

@#include "rbc_nonlinear_common.inc"
% ---------------------- %
% ---------------------- %
histval;
k(0)=10;
a(0)=1;
end;

% make sure everything is set up correctly!
% perfect_foresight_setup(periods=4);
% oo_.exo_simul
% oo_.endo_simul

perfect_foresight_setup(periods=300);
perfect_foresight_solver;
dsample 100;
rplot c iv y;
rplot l w;
rplot r;
rplot k;
rplot a;


### rbc_nonlinear_stoch1.mod

@#include "rbc_nonlinear_common.inc"

% -------------------- %
% Unexpected TFP shock %
% -------------------- %

shocks;
var epsa = 0.04^2;
end;

stoch_simul(order=1,irf=30,periods=0) y c iv a;


### rbc_nonlinear_stoch2.mod

@#include "rbc_nonlinear_common1.inc"