Practicing Dynare
Francisco Barillas
New York University
Dept. of Economics
Riccardo Colacito
University of North Carolina
Dept. of Finance
Sagiri Kitao
University of Southern California
FBE Deptartment
Christian Matthes
New York University
Dept. of Economics
Thomas J. Sargent
New York University
Dept. of Economics
Yongseok Shin
University of Wisconsin
Dept. of Economics
July 28, 2007
Abstract
This paper teaches Dynare by applying it to approximate equilibria and estimate nine dynamic economic models.
We hope that it illustrates once again the efficacy of learning by doing.
1. Introduction
This paper describes nine examples that illustrate how Dynare can approximate the solutions of dynamic rational
expectations models, simulate them, and estimate them by maximum likelihood and Bayesian methods. Section
2 approximates and estimates a one-sector stochastic growth model. Section 3 approximates and estimates a twocountry
stochastic growth model. Section 4 follows chapter 11 of Ljungqvist and Sargent (2004) in studying the
effects of foreseen fiscal policy in a non-stochastic growth model. Section 5 estimates a rational expectations model
of hyperinflation originally formulated by Sargent (1977). Section 6 solves and estimates the permanent income model
of Hall (1988). This application is interesting, among other reasons, for the way that it illustrates how Dynare can
implement the `diffuse Kalman filter' needed in situations in which an initial endogenous state variable is unknown
and the model has a unit root. Section 7 solves and estimates the Ryoo and Rosen (2004) rational expectations model
of a market for engineers. Section 8 estimates a model of consumption growth proposed by Bansal and Yaron (2004).
Section 9 solves and estimates the Hansen, Sargent, and Tallarini (1999) model of robust permanent income and
pricing. Section 10 solves and estimates the Bansal and Yaron (2004) of asset prices and long-run risk in consumption
and dividends. Appendix A tells how the reader can obtain the file examples.zip that contain the *.mod and data
files that we used to generate these examples.
Some of our examples bear eloquent witness to the technological improvements brought by Dynare. For example,
the maximum likelihood estimates in Sargent (1977) and Hansen, Sargent, and Tallarini (1999) were time-consuming
and painful to obtain originally. Dynare has reduced the time and the pain.
We thank Michele Juillard for helpful comments on an earlier draft.
Authors' emails: Barillas: fvb201@nyu.edu, Colacito: riccardo.colacito@unc.edu, Kitao: sk910@nyu.edu, Matthes, cm1518@nyu.edu,
Sargent: ts43@nyu.edu, Shin: yshin@ssc.wisc.edu.
1
2. The Neoclassical growth model
2.1 The model
We study a widely used stochastic neoclassical growth model with leisure (see, for example, Cooley and Prescott
(1995)). A representative household's problem is
max
{ct,lt}
t=0
E0

t=1
t-1 c
t (1 - lt)1- 1-
1 - 
subject to the resource constraint
ct + it = ezt
k
t l1t
(1)
the law of motion for capital
kt+1 = it + (1 - )kt (2)
and the stochastic process for productivity
zt = zt-1 + st (3)
with t  N(0, 2
).
The system of equations characterizing an equilibrium is form by equations (1), (2) and (3), the Euler intertemporal
condition
c
t (1 - lt)1- 1-
ct
= Et
c
t+1(1 - lt+1)1- 1-
ct+1
(1 + ezt
k-1
t l
t - ) (4)
and an optimality condition for supply of labor
1 - 

ct
1 - lt
= (1 - )ezt
k
t lt
. (5)
Use (1) and (2) to get
kt+1 = ezt
k
t l1t
- ct + (1 - )kt. (6)
An equilibrium is characterized by a system of four equations (3), (4), (5) and (6).
2.2 Calibration and approximation
2.2.1 Calibration
To estimate the policy function, parameters that enter the model are calibrated as in Table 1. For more on the
calibrations of the related parameters, please see, for example, Cooley and Prescott (1995).
2.2.2 Approximation
The following instructions implement this model in Dynare.
periods 1000;
var c k lab z;
varexo e;
parameters bet the del alp tau rho s;
bet = 0.987;
the = 0.357;
del = 0.012;
alp = 0.4;
2
tau = 2;
rho = 0.95;
s = 0.007;
model;
(c^the*(1-lab)^(1-the))^(1-tau)/c=bet*((c(+1)^the*(1-lab(+1))^(1-the))^(1-tau)/c(+1))*
(1+alp*exp(z(-1))*k(-1)^(alp-1)*lab^(1-alp)-del);
c=the/(1-the)*(1-alp)*exp(z(-1))*k(-1)^alp*lab^(-alp)*(1-lab);
k=exp(z(-1))*k(-1)^alp*lab^(1-alp)-c+(1-del)*k(-1);
z=rho*z(-1)+s*e;
end;
initval;
k = 1;
c = 1;
lab = 0.3;
z = 0;
e = 0;
end;
shocks;
var e;
stderr 1;
end;
steady;
stoch_simul(dr_algo=0,periods=1000); datasaver('simudata',[]);
Parameter Calibration
 0.987
 0.357
 0.012
 0.4
 2
 0.95
s 0.007
 1
Table 1: Parameters' calibration
Dynare displays the results reported in Table 2 that provide the coefficients of the approximated policy functions
and transition functions.
3
k z c lab
Constant 54.615371 0 0.960775 0.543716
k (-1) 0.983153 0 0.011727 -0.001668
z (-1) 1.978406 0.950000 0.317480 0.228716
e -0.000935 0.007000 0.000454 -0.000162
k (-1),k (-1) -0.000038 0 -0.000030 0.000011
z (-1),k (-1) 0.011128 0 0.002996 0.000762
z (-1),z (-1) 1.253721 0 0.096224 -0.046394
e,e -0.000003 0 0.000001 0
k (-1),e -0.000002 0 0.000002 0.000001
z (-1),e -0.001076 0 0.000430 -0.000056
Table 2: Policy and transition functions.
0 10 20 30 40
0
0.5
1
1.5
2
2.5
3
x 10
-3 c
0 10 20 30 40
-0.05
0
0.05
0.1
0.15
0.2
k
0 10 20 30 40
-5
0
5
10
15
20
x 10
-4 lab
0 10 20 30 40
0
2
4
6
8
x 10
-3 z
Figure 1: Impulse response function to a technology shock
2.3 Estimation
The simulated data series for consumption are used to estimate some unknown parameters of the model. Suppose we
would like to estimate the preference parameters  and , and the stochastic process for productivity, summarized
by two parameters  and . The following code instructs Dynare to conduct the estimation.
periods 1000;
var c k lab z;
varexo e;
4
parameters bet del alp rho the tau s;
bet = 0.987;
the = 0.357;
del = 0.012;
alp = 0.4;
tau = 2;
rho = 0.95;
s = 0.007;
model;
(c^the*(1-lab)^(1-the))^(1-tau)/c=bet*((c(+1)^the*(1-lab(+1))^(1-the))^(1-tau)/c(+1))
*(1+alp*exp(z(-1))*k(-1)^(alp-1)*lab^(1-alp)-del);
c=the/(1-the)*(1-alp)*exp(z(-1))*k(-1)^alp*lab^(-alp)*(1-lab);
k=exp(z(-1))*k(-1)^alp*lab*(1-alp)-c+(1-del)*k(-1);
z=rho*z(-1)+s*e;
end;
initval;
k = 1;
c = 1;
lab = 0.3;
z = 0;
e = 0;
end;
shocks;
var e;
stderr 1;
end;
steady;
check;
estimated_params;
stderr e,inv_gamma_pdf, 0.95,inf;
rho,beta_pdf,0.93,0.02;
the,normal_pdf,0.3,0.05;
tau,normal_pdf,2.1,0.3;
end;
varobs c;
estimation(datafile=simudata,mh_replic=10000);
The priors used in estimation are as in Table 3. Table 4 is one of the outputs of Dynare which presents summary
statistics of posterior distribution.
5
0 2 4 6
0
0.5
1
1.5
2
0.85 0.9 0.95 1
0
5
10
15
20
25
0.1 0.2 0.3 0.4 0.5
0
2
4
6
8
1 1.5 2 2.5 3 3.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
 
 
Figure 2: Priors.
0 1 2 3
0
0.5
1
1.5
2
2.5
3
0.85 0.9 0.95 1
0
10
20
30
40
0.15 0.2 0.25 0.3 0.35 0.4
0
50
100
150
1 1.5 2 2.5 3 3.5
0
0.5
1
1.5
2
 
 
Figure 3: Posterior.
6
Parameter Distribution Mean Std.Dev.
 Beta 0.93 0.02
 Normal 0.3 0.05
 Normal 2.1 0.3
 Inv. Gamma 0.95 inf.
Table 3: Priors
prior mean post. mean conf. interval prior dist prior std
 0.930 0.9480 0.9319 0.9645 beta 0.0200
 0.300 0.3589 0.3540 0.3645 norm 0.0500
 2.100 2.0046 1.6532 2.3356 norm 0.3000
 0.950 1.0227 0.8321 1.2175 invg Inf
Table 4: Posterior.
7
3. International Business Cycle Model
We study a simplified version of the two-country production model in Kim and Kim (2003).
3.1 The model
Two countries are identical ex-ante and markets are complete. A solution to a Pareto planner's problem is characterized
by the following equations
C1,t = C2,t
C-
1,t = EtC-
1,t+1 A1,t+1K-1
1,t+1 + 1 - 
C-
2,t = EtC-
2,t+1 A2,t+1K-1
2,t+1 + 1 - 
A1,tK
1,t + A2,tK
2,t = C1,t + C2,t + K1,t+1 - (1 - )K1,t + K2,t+1 - (1 - )K2,t
ln A1,t+1 =  ln A1,t + 1,t+1
ln A2,t+1 =  ln A2,t + 2,t+1
The first three equalities are the first-order conditions with regard to the consumption (Ci,t), and the fourth one
is the world resource constraint. The last two describe the law of motion for the technological shock (Ai,t). Ki,t
denotes the capital stock of country i at time t.
3.2 Calibration and approximation
3.2.1 Calibration
The following parameter values are used.
Parameter Calibration
 0.98
 0.05
 0.4
 0.85
1
0.08
2
0.08
Table 5: Parameter Values
Dynare computes the policy function as a second-order approximation around the (log) steady-state. The .mod
Dynare file of the system is as follows.
periods 1000;
var c1 c2 k1 k2 a1 a2;
varexo e1 e2;
parameters gamma delta alpha beta rho;
gamma=2;
delta=.05;
alpha=.4;
beta=.98;
rho=.85;
model;
c1=c2;
exp(c1)^(-gamma) =...
8
beta*exp(c1(+1))^(-gamma)*(alpha*exp(a1(+1))*exp(k1)^(alpha-1)+1-delta);
exp(c2)^(-gamma) =...
beta*exp(c2(+1))^(-gamma)*(alpha*exp(a2(+1))*exp(k2)^(alpha-1)+1-delta);
exp(c1)+exp(c2)+exp(k1)-exp(k1(-1))*(1-delta)+exp(k2)-exp(k2(-1))*(1-delta)...
= exp(a1)*exp(k1(-1))^alpha+exp(a2)*exp(k2(-1))^alpha;
a1=rho*a1(-1)+e1; a2=rho*a2(-1)+e2; end;
initval;
k1=2.8;
k2=2.8;
c1=.8;
c2=.8;
a1=0;
a2=0;
e1=0;
e2=0;
end;
shocks;
var e1;
stderr .08;
var e2;
stderr .08;
end;
steady;
stoch_simul(dr_algo=0,periods=1000); datatomfile('simudata',[]);
The output is given as a matrix of coefficients for the policy function. From this matrix, for example, one can
construct the optimal consumption rule for Country 1:
^c1,t = 0.824 + 0.243(^k1,t + ^k2,t) + 0.140(^a1,t + ^a2,t) + 0.062(^k2
1,t-1 + ^k2
2,t-1) - 0.110^k1,t
^k2,t
- 0.015(^a1,t
^k1,t + ^a2,t
^k2,t) - 0.032(^a1,t
^k2,t + ^a2,t
^k1,t) + 0.044(^a2
1,t + ^a2
2,t) - 0.040^a1,t^a2,t,
where ^y denotes the deviation of ln Y from its log steady-state. Note that Dynare breaks ^ai,t into ^ai,t-1 and i,t, so
the output from Dynare looks unnecessarily messy.
Dynare also produces the associated impulse response functions of the system. Figure 4 illustrate the response
of the system to a one-standard-deviation shock to 1.
The last two lines of the code generates sample paths of the variables governed by the approximated policy
functions. The simulated process is saved as a .m file.
3.3 Estimation
Now assume that we do not know the true parameter values. We can estimate the unknown parameters from the
saved data, using the Bayesian econometrics function of Dynare.
We will estimate 4 parameters , , 1
, and 2
. Hence, we have to specify the prior distribution of each
parameter. In this exercise,  and  are given a normal prior, while the priors of 1
and 2
are inverse gamma
distributions. The Dynare code is as follows.
periods 1000;
var c1 c2 k1 k2 a1 a2;
varexo e1 e2;
parameters gamma delta alpha beta rho;
9
0 20 40
0
0.02
0.04
0.06
0.08
0.1
a1
0 20 40
0.004
0.006
0.008
0.01
0.012
0.014
0.016
c1
0 20 40
0.004
0.006
0.008
0.01
0.012
0.014
0.016
c2
0 20 40
0.01
0.02
0.03
0.04
0.05
0.06
k1
0 20 40
-0.06
-0.04
-0.02
0
0.02
k2
Figure 4: Impulse response function (1)
gamma=2;
delta=.05;
alpha=.4;
beta=.98;
rho=.85;
model;
c1=c2; exp(c1)^(-gamma) =...
beta*exp(c1(+1))^(-gamma)*(alpha*exp(a1(+1))*exp(k1)^(alpha-1)+1-delta);
exp(c2)^(-gamma) =...
beta*exp(c2(+1))^(-gamma)*(alpha*exp(a2(+1))*exp(k2)^(alpha-1)+1-delta);
exp(c1)+exp(c2)+exp(k1)-exp(k1(-1))*(1-delta)+exp(k2)-exp(k2(-1))*(1-delta)...
= exp(a1)*exp(k1(-1))^alpha+exp(a2)*exp(k2(-1))^alpha;
a1=rho*a1(-1)+e1; a2=rho*a2(-1)+e2; end;
initval;
k1=2.8;
k2=2.8;
c1=.8;
c2=.8;
a1=0;
a2=0;
e1=0;
e2=0;
end;
shocks;
var e1;
stderr .08;
var e2;
10
stderr .08;
end;
steady;
estimated_params;
rho, normal_pdf, .84,.05; %
alpha, normal_pdf, .38, .03;
stderr e1, inv_gamma_pdf, .078, inf;
stderr e2,inv_gamma_pdf,.082, inf;
end;
varobs c1 k2;
estimation(datafile=simudata,mh_replic=1000,mh_jscale=.5);
Given that there are two exogenous shock variables, we use two observables ln C1 and ln K2. The result of the
estimation is summarized in Figure 5, where the posterior distribution (in the darker line) of each parameter is
contrasted against the given prior distribution. The result can be compared to the true parameter values in Table 5.
0 0.1 0.2 0.3 0.4 0.5
0
20
40
60
80
100
120
140
SE_{e1}
0 0.1 0.2 0.3 0.4 0.5
0
50
100
150
200
SE_{e2}
0.6 0.7 0.8 0.9 1
0
10
20
30
40
50
rho
0.25 0.3 0.35 0.4 0.45 0.5
0
20
40
60
80
100
alpha
Figure 5: Prior v. Posterior Distribution
4. Fiscal policies in the growth model
This section describes a suite of Dynare programs that replicate the transition experiments performed in chapter 11
of Ljungqvist and Sargent (2004).
4.1 The model
A deterministic growth model has inelastic labor supply, an exogenous stream of government expenditures, and
several kinds of distorting taxes. All variables are as described in chapter 11 of Ljungqvist and Sargent (2004). A
representative agent maximizes

t=0
t c1-
t
1 - 
,   (0, 1)
11
Feasible allocations satisfy
gt + ct + kt+1  Ak
t + (1 - ) kt.
The household budget constraint is

t=0
{qt (1 + ct) + (1 - it) qt [kt+1 - (1 - ) kt]} 

t=0
{rt (1 - kt) kt + wt}
and the government budget constraint is

t=0
qtgt 

t=0
{ctqtct - itqt [kt+1 - (1 - ) kt] + rtktkt + wt}
The following conditions characterize an equilibrium:
ct = Ak
t + (1 - ) kt - kt+1 - gt (7)
qt = t
ct
/ (1 + ct) (8)
rt/qt = Ak-1
t (9)
wt/qt = Ak
t - ktAk-1
t (10)
Rt+1 =
(1 + ct)
(1 + ct+1)
(1 + it+1)
(1 + it)
(1 - ) +
(1 - kt+1)
(1 - it)
Ak-1
t+1 (11)
st/qt = (1 - kt) Ak-1
t + (1 - ) (12)
ct
= ct+1Rt+1
(13)
Because the other endogenous variables can be expressed as functions of kt and ct, in the programs below we use
only a subset of the equilibrium conditions to compute equilibrium paths.
4.2 Parameter values
We set the parameters and the baseline values of the exogenous variables at values set in chapter 11 of Ljungqvist
and Sargent (2004), as described in the following table are taken from RMT2.
Parameter/exogenous variable Values
 0.95
 0.2
A 1
 0.33
 2
g 0.2
c 0
i 0
k 0
4.3 Transition experiments
4.3.1 A permanent increase in g (Figure 11.3.1).
The Dynare code below reproduces Figure 11.3.1. which looks at the impact of a permanent increase of 0.2 in g at
t=10.
// This program replicates figure 11.3.1 from chapter 11 of RMT2 by
// Ljungqvist and Sargent.
// y_ records the simulated endogenous variables in alphabetical order
// ys0_ records the initial steady state
12
// ys_ records the terminal steady state
// We check that these line up at the end points
// Note: y_ has ys0_ in the first column, ys_ in the last column, which explains
// why it has 102 elements.
// The sample of size 100 is in between.
// Warning: All endogenous variables dated time t are treated as jump variables
// in Dynare. So k in the program corresponds to k_{t+1}
// and the same timing holds for the taxes.
//Declares the endogenous variables;
var c k;
//declares the exogenous variables
//investment tax credit, consumption tax, capital tax, government spending
varexo taui tauc tauk g;
parameters bet gam del alpha A;
bet=.95; // discount factor
gam=2; // CRRA parameter
del=.2; // depreciation rate
alpha=.33; // capital's share
A=1; // productivity
// Alignment convention:
// g tauc taui tauk are now columns of ex_. Because of a bad design decision
// the date of ex_(1,:) doesn't necessarily match the date in y_.
// Whether they match depends on the number of lag periods in endogenous versus
// exogenous variables.
// In this example they match because tauc(-1) and taui(-1) enter the model.
// These decisions and the timing conventions mean that y_(:,1) records the
// initial steady state, while y_(:,102) records the terminal steady state
// values. For j > 2, y_(:,j) records [c(j-1) .. k(j-1) .. G(j-1)] where k(j-1)
// means end of period capital in period j-1, which equals k(j) in chapter 11
// notation. Note that the jump in G occurs in y_(;,11), which confirms this
// timing. The jump occurs now in ex_(11,1)
model;
// equation 11.3.8.a
k=A*k(-1)^alpha+(1-del)*k(-1)-c-g;
// equation 11.3.8e + 11.3.8.g
c^(-gam)=
bet*(c(+1)^(-gam))*((1+tauc(-1))/(1+tauc))*((1-taui)*(1-del)/(1-taui(-1))+
((1-tauk)/(1-taui(-1)))*alpha*A*k(-1)^(alpha-1));
end;
initval; k=1.5; c=0.6; g = 0.2; tauc = 0; taui = 0; tauk = 0; end;
steady; // put this in if you want to start from the initial
// steady state, comment it out to start from the indicated values
13
endval; // The following values determine the new steady state
// after the shocks.
k=1.5; c=0.4; g =.4; tauc =0; taui =0; tauk =0; end;
steady; // We use steady again and the enval provided are initial
// guesses for Dynare to compute the ss.
// The following lines produce a g sequence with a once and for all jump in g shocks
// we use shocks to undo that for the first 9 periods and leave g at
// it's initial value of 0.2
var g; periods 1:9; values 0.2; end;
// now solve the model
simul(periods=100);
// Note: y_ records the simulated endogenous variables in alphabetical order
// ys0_ records the initial steady state
// ys_ records the terminal steady state
// check that these line up at the end points
y_(:,1) -ys0_(:) y_(:,102) - ys_(:)
// Compute the initial steady state for consumption to later do the plots.
co=ys0_(var_index('c')); ko = ys0_(var_index('k'));
// g is in ex_(:,1) since it is stored in alphabetical order
go = ex_(1,1)
// The following equation compute the other endogenous variables use in the
// plots below. Since they are function of capital and consumption, so we can
// compute them from the solved model above.
// These equations were taken from page 333 of RMT2
rbig0=1/bet;
rbig=y_(var_index('c'),2:101).^(-gam)./(bet*y_(var_index('c'),3:102).^(-gam));
rq0=alpha*A*ko^(alpha-1);
rq=alpha*A*y_(var_index('k'),1:100).^(alpha-1);
wq0=A*ko^alpha-ko*alpha*A*ko^(alpha-1);
wq=A*y_(var_index('k'),1:100).^alpha-y_(var_index('k'),1:100).*alpha*A.*...
y_(var_index('k'),1:100).^(alpha-1);
sq0=(1-ex_(1,4))*A*alpha*ko^(alpha-1)+(1-del);
sq=(1-ex_(1:100,4)')*A*alpha.*y_(var_index('k'),1:100).^(alpha-1)+(1-del);
//Now we plot the responses of the endogenous variables to the shock.
figure subplot(2,3,1) plot([ko*ones(100,1)
y_(var_index('k'),1:100)' ]) // note the timing: we lag capital to
correct for syntax title('k') subplot(2,3,2) plot([co*ones(100,1)
y_(var_index('c'),2:101)' ]) title('c') subplot(2,3,3)
plot([rbig0*ones(100,1) rbig' ]) title('R') subplot(2,3,4)
plot([wq0*ones(100,1) wq' ]) title('w/q') subplot(2,3,5)
plot([sq0*ones(100,1) sq' ]) title('s/q') subplot(2,3,6)
plot([rq0*ones(100,1) rq' ]) title('r/q')
14
0 50 100
1.4
1.6
1.8
2
2.2
k
0 50 100
0.4
0.45
0.5
0.55
0.6
0.65
c
0 50 100
1
1.01
1.02
1.03
1.04
1.05
1.06
1.07
R
0 50 100
0.76
0.78
0.8
0.82
0.84
0.86
w/q
0 50 100
1
1.01
1.02
1.03
1.04
1.05
1.06
1.07
s/q
0 50 100
0.2
0.21
0.22
0.23
0.24
0.25
0.26
r/q
Figure 6: A permanent increase of g (Figure 11.3.1).
The following figure shows the path of the endogenous variables along the transition to the new steady state.
Next we show the other transition experiments and only the parts of the code that need to be changed for each
experiment.
4.3.2 A permanent increase in c (Figure 11.3.2).
A permanent increase of c at t=10 of 20 per cent. (Figure 11.3.2)
initval;
k=1.5;
c=0.6;
g = 0.2;
tauc = 0;
taui = 0;
tauk = 0;
end;
steady;
endval;
k=1.5;
c=0.6;
g = 0.2;
tauc =0.2;
taui =0;
tauk =0;
end;
15
0 50 100
1.3
1.35
1.4
1.45
1.5
k
0 50 100
0.6
0.61
0.62
0.63
0.64
0.65
0.66
0.67
c
0 50 100
0.85
0.9
0.95
1
1.05
1.1
R
0 50 100
0.735
0.74
0.745
0.75
0.755
0.76
0.765
0.77
w/q
0 50 100
1.05
1.055
1.06
1.065
1.07
1.075
s/q
0 50 100
0.25
0.255
0.26
0.265
0.27
0.275
0.28
r/q
Figure 7: A permanent increase of c (Figure 11.3.2).
steady;
shocks;
var tauc;
periods 1:9;
values 0;
end;
4.3.3 A permanent increase in i (Figure 11.5.1).
A permanent increase of i at t=10 of 20 percent. (Figure 11.5.1)
initval;
k=1.5;
c=0.6;
g = 0.2;
tauc = 0;
taui = 0;
tauk = 0;
end;
steady;
endval;
k=1.5;
c=0.6;
16
0 50 100
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
k
0 50 100
0.58
0.6
0.62
0.64
0.66
c
0 50 100
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
R
0 50 100
0.76
0.78
0.8
0.82
0.84
0.86
w/q
0 50 100
1
1.01
1.02
1.03
1.04
1.05
1.06
1.07
s/q
0 50 100
0.2
0.21
0.22
0.23
0.24
0.25
0.26
r/q
Figure 8: A permanent increase of i (Figure 11.5.1).
g =0.2;
tauc =0;
taui =0.20;
tauk =0;
end;
steady;
shocks;
var taui;
periods 1:9;
values 0;
end;
4.3.4 A permanent increase in k (Figure 11.5.2).
A permanent increase of k at t=10 of 20 percent. (Figure 11.5.2)
initval;
k=1.5;
c=0.6;
g = 0.2;
tauc = 0;
taui = 0;
tauk = 0;
end;
steady;
17
0 50 100
1
1.1
1.2
1.3
1.4
1.5
k
0 50 100
0.6
0.62
0.64
0.66
0.68
c
0 50 100
1
1.02
1.04
1.06
1.08
R
0 50 100
0.68
0.7
0.72
0.74
0.76
0.78
w/q
0 50 100
1.02
1.03
1.04
1.05
1.06
1.07
1.08
1.09
s/q
0 50 100
0.25
0.26
0.27
0.28
0.29
0.3
0.31
0.32
r/q
Figure 9: A permanent increase of k (Figure 11.5.2).
endval;
k=1.5;
c=0.6;
g =0.2;
tauc =0;
taui =0;
tauk = 0.2;
end;
steady;
shocks;
var tauk;
periods 1:9;
values 0;
end;
4.3.5 One time increase in g (Figure 11.7.1).
A one time pulse of g at t=10 of 0.2. (Figure 11.7.1)
initval;
k=1.5;
c=0.6;
g = 0.2;
tauc = 0;
18
0 50 100
1.4
1.45
1.5
1.55
1.6
k
0 50 100
0.62
0.625
0.63
0.635
0.64
0.645
0.65
c
0 50 100
1.04
1.045
1.05
1.055
1.06
1.065
R
0 50 100
0.74
0.75
0.76
0.77
0.78
w/q
0 50 100
1.04
1.045
1.05
1.055
1.06
1.065
s/q
0 50 100
0.24
0.245
0.25
0.255
0.26
0.265
0.27
r/q
Figure 10: One time increase of g (Figure 11.7.1).
taui = 0;
tauk = 0;
end;
steady;
endval;
k=1.5;
c=0.6;
g = 0.2;
tauc =0;
taui =0;
tauk =0;
end;
steady;
shocks;
var g;
periods 10;
values 0.4;
end;
4.3.6 One time increase in i (Figure 11.7.2).
A one time pulse of i at t=10 of 20 percent. (Figure 11.7.2)
initval;
19
0 50 100
1.48
1.5
1.52
1.54
1.56
1.58
1.6
k
0 50 100
0.58
0.6
0.62
0.64
0.66
c
0 50 100
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
R
0 50 100
0.76
0.765
0.77
0.775
0.78
w/q
0 50 100
1.044
1.046
1.048
1.05
1.052
1.054
1.056
s/q
0 50 100
0.244
0.246
0.248
0.25
0.252
0.254
0.256
r/q
Figure 11: One time increase of i (Figure 11.7.2).
k=1.5;
c=0.6;
g = 0.2;
tauc = 0;
taui = 0;
tauk = 0;
end;
steady;
endval;
k=1.5;
c=0.6;
g =0.2;
tauc =0;
taui =0;
tauk =0;
end;
steady;
shocks;
var taui;
periods 10;
values 0.2;
end;
20
5. The demand for money during hyperinflations
Sargent (1977) specified and estimated a joint stochastic process for money growth and inflation that was designed
to make the adaptive expectations model of Cagan (1956) consistent with rational expectations. The model has the
form
mt - pt = t + ut (14)
t =
1 - 
1 - L
(pt - pt-1) =
1 - 
1 - L
xt (15)
t = mt - mt-1 (16)
with  < 0, m being the log of money supply,  the expectation of inflation, p the log of the price level and u an
erro term whose specification is described below. The hypothesis of rational expectations imposes
t = Etxt+1. (17)
Sargent (1977) reverse engineered a process for xt such that Cagan's adaptive expectations scheme is equal to rational
expectations. The following conditions ensure the rationality of Cagan's expectations scheme
ut = ut-1 + t (18)
t =
1 - 
1 - L
xt + t (19)
 and  are both serially uncorrelated random error terms with mean zero and variances 2
 and 2
 , respectively. For
estimating the model, it is convenient to represent it in the following form
xt
t
=
1 0
1 -  
xt-1
t-1
+
a1t
a2t
- I
a1t-1
a2t-1
(20)
The a terms are linked to the random terms above via
a1t
a2t
=
1 -1
1 + (1 - ) -(1 - )
( + (1 - ))-1 t
t
(21)
The code below estimates representation (20) by maximum likelihood:
// this program estimates the model in
// "The Demand for Money during Hyperinflations under Rational Expectations: I"
// by T. Sargent, IER 1977 using maximum likelihood
// variables are defined as follows:
// x=p_t-p_{t-1}, p being the log of the price level
// mu=m_t-m_{t-1}, m being the log of money supply
// note that in contrast to the paper eta and epsilon
// have variance 1 (they are multiplied by the standard deviations)
var x mu a1 a2; varexo epsilon eta; parameters alpha lambda sig_eta
sig_epsilon; lambda=.5921; alpha=-2.344; sig_eta=.001;
sig_epsilon=.001;
model;
x=x(-1)-lambda*a1(-1)+(1/(lambda+alpha*(1-lambda)))*sig_epsilon*epsilon
-(1/(lambda+alpha*(1-lambda)))*sig_eta*eta;
mu=(1-lambda)*x(-1)+lambda*mu(-1)-lambda*a2(-1)+(1+alpha*(1-lambda))/(lambda+alpha*(1-lambda))
*sig_epsilon*epsilon-(1-lambda)/(lambda+alpha*(1-lambda))*sig_eta*eta;
a1=(1/(lambda+alpha*(1-lambda)))*sig_epsilon*epsilon-(1/(lambda+alpha*(1-lambda)))*sig_eta*eta;
a2=(1+alpha*(1-lambda))/(lambda+alpha*(1-lambda))*sig_epsilon*epsilon
-(1-lambda)/(lambda+alpha*(1-lambda))*sig_eta*eta;
21
0.5 0.55 0.6 0.65 0.7 0.75
-40
-35
-30
-25
-20
-15
lambda
-6 -5.5 -5 -4.5 -4 -3.5
-38
-37
-36
-35
-34
-33
-32
alpha
0.2 0.25 0.3 0.35
-37.5
-37
-36.5
-36
-35.5
-35
sig_eta
0.05 0.06 0.07 0.08 0.09
-37.5
-37
-36.5
-36
-35.5
-35
sig_epsilon
Figure 12: Negative of the Log-Likelihood function for the hyperinflation model
end; steady; shocks;
var eta; stderr 1; var epsilon; stderr 1;
end;
estimated_params;
// ML estimation setup
// parameter name, initial value, boundaries_low, ..._up;
lambda, .5, 0.25, 0.75; alpha, -2, -8, -0.1; sig_eta, .0001, 0.0001,
0.3; sig_epsilon, .0001, 0.0001, 0.3; end;
varobs mu x; unit_root_vars x;
estimation(datafile=cagan_data,first_obs=1,nobs=34,mh_replic=0,mode_compute=2,mode_check);
The data for the estimation are from Cagan (1956) for Germany1
. The shape of the negative of the log-likelihood
function is plotted in figure 12.
Parameter Estimate s.d. t-stat
 0.6617 0.0535 12.3754
 -4.7654 3.0689 1.5528
 0.2447 0.2170 1.1275
 0.0727 0.0100 7.2946
Table 6: Results from Maximum likelihood
Next we estimate this model using Bayesian techniques. For this we replace the last part of the code above with
1The data are demeaned, allowing us to ignore constant terms in the model
22
0 0.5 1 1.5
0
2
4
6
8
10
lambda
-10 -8 -6 -4 -2 0
0
0.05
0.1
0.15
0.2
0.25
alpha
0 0.2 0.4 0.6 0.8 1
0
0.5
1
1.5
2
2.5
sig_eta
0.2 0.4 0.6 0.8
0
10
20
30
40
50
sig_epsilon
Figure 13: Posteriors and priors for the hyperinflation model
estimated_params;
// Bayesian setup
lambda, uniform_pdf, 0.68, .5;
alpha, uniform_pdf, -5, 2;
sig_eta,uniform_pdf, .5, 0.25;
sig_epsilon, uniform_pdf, .5, 0.25;
end;
varobs mu x; unit_root_vars x;
estimation(datafile=cagan_data,nobs=34,mh_replic=25000,mh_nblocks=1,mh_jscale=1,mode_compute=4);
Note that the priors we have chosen are reasonably loose, but given that there are only 34 observations we have
opted to use somewhat informative priors. The posteriors and priors are plotted in figure 13.
6. Solving and estimating Hall's permanent income model
This note describes a suite of Dynare programs to simulate and estimate the permanent income model of Hall(1988).
6.1 The model
A consumer faces the following decision problem:
max
{ct,kt}
t=0
-
1
2

t=0
t
(ct - bt)
2
23
subject to the following constraints
ct + kt = Rkt-1 + dt
dt = dt-1 + (1 - ) d + d
bt = bbt-1 + (1 - b) b + b
We assume that R = 1 and that there is no depreciation, which leads to the following law of motion for
investment:
it = kt - kt-1
From the Lagrangian for this problem one obtains the following first order conditions:
ct = ct+1
ct = bt - ct
where ct is the marginal utility of consumption or the Lagrange multiplier.
Note that this problem has a unit root and so there exist no steady state. However, we can supply ONE solution
corresponding to an initial k that we give to Dynare.
The following Dynare code simulates data from this model:
periods 5000;
var c k mu_c b d in;
varexo e_d e_b;
parameters R rho rho_b mu_b mu_d; R=1.05;
rho = 0;
mu_b=30;
mu_d=5;
rho_b = 0;
model(linear);
c+k = R*k(-1) + d;
mu_c = b - c;
mu_c=mu_c(+1);
d= rho*d(-1)+ mu_d*(1-rho) + e_d;
b=(1-rho_b)*mu_b+rho_b*b(-1)+e_b;
in = k - k(-1);
end;
//With a unit root, there exists no steady state. Use the following trick.
// Supply ONE solution corresponding to the initial k that you named.
initval;
d=mu_d;
k=100;
c = (R-1)*k +d;
mu_c=mu_b-c;
b=mu_b;
end;
24
1 2 3 4 5 6 7 8 9 10
0
0.5
1
b
1 2 3 4 5 6 7 8 9 10
-0.5
0
0.5
1
c
1 2 3 4 5 6 7 8 9 10
-1
-0.5
0
in
Figure 14: A shock to b
shocks;
var e_d;
stderr 1;
var e_b;
stderr 1;
end;
steady;
check;
stoch_simul(dr_algo=1, order=1, periods=500, irf=10);
save data_hall c in;
Next, we proceed to estimate a subset of the parameters of the model by ML using simulated data obtained from
the previous code. To do so we add the following few lines of code. Notice how we use the command unit root vars
to initialize the Kalman filter with a diffuse prior. We do this because k is not stationary ­ Hall's model with R = 1
has a solution for k that has a unit root and that is co-integrated with consumption.
estimated_params;
// ML estimation setup
// parameter name, initial value, boundaries_low, ..._up;
rho, -0.0159, -0.9, 0.9;
R, 1.0074, 0, 1.5;
end;
25
1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
d
1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
in
Figure 15: A shock to d
varobs c in;
// declare the unit root variables for diffuse filter
unit_root_vars k;
estimation(datafile=data_hall,first_obs=101,nobs=200,mh_replic=0,mode_compute=4,mode_check);
We proceed to estimate the same subset of the parameters using Bayesian techniques: This is implemented in
Dynare by adding the following few lines of code.
estimated_params;
rho, beta_pdf, .1, 0.2;
R, normal_pdf,1.02,0.05;
end;
varobs c in;
// declare the unit root variables for diffuse filter
unit_root_vars k;
estimation(datafile=data_hall,first_obs=101,nobs=200,mh_replic=1000,mh_nblocks=2,mh_jscale=2);
26
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
1000
2000
3000
4000
rho
0.9 0.95 1 1.05 1.1 1.15
0
200
400
600
800
R
Figure 16: priors and posteriors from the Bayesian estimation of Hall's model
7. Solving and estimating Ryoo and Rosen's (2004) model of the market
for engineers
This section describes a suite of Dynare programs to simulate and estimate a model of Aloysius Siow (1984) and
Jaewoo Ryoo and Sherwin Rosen (2004) of the labor market for engineers.
7.1 The model
Ryoo and Rosen studied a partial equilibrium model determining a stock of 'engineers' Nt; the number of new
entrants into engineering school, st; and the wage level wt of engineers.2
It takes k periods of schooling to become
an engineer. The model consists of the following components:
The present value of wages of an engineer:
Pt = (1 - ) Pt+1 + (1 - )
k
k
wt+k
A supply of new engineering students:
st = a0 + a1Pt + est
A time-to-build structure of the education process:
Nt = (1 - ) Nt-1 + st-k
A demand curve for engineers:
Nt = d0 - d1wt + edt
The following Dynare code simulates data from this model:
2Sherwin Rosen showed this version of the model to Sargent in Chicago in fall 1995.
27
periods 500;
var s N P W;
varexo e_s e_d;
parameters a0 a1 delta d0 d1 bet k;
a0=10;
a1=1;
d0=1000;
d1=1;
bet=.99;
delta=.02;
model(linear);
s=a0+a1*P+e_s; // flow supply of new entrants
N=(1-delta)*N(-1) + s(-4); // evolution of the stock
N=d0-d1*W+e_d; // stock demand equation
P=bet*(1-delta)*P(+1)+bet^4*(1-delta)^4*W(+4); // present value of
//wages
end;
initval;
s=0;
N=0;
P=0;
W=0;
end;
shocks;
var e_d;
stderr 1;
var e_s;
stderr 1;
end;
steady;
check;
stoch_simul(dr_algo=1, order=1, periods=500, irf=10);
save data_rosen s N P W;
Note that in this code we set k = 4. Dynare does not let you use a parameter as lag length.
Next, we estimate a subset of the parameters of the model by ML using simulated data obtained from the previous
code. To do so we add the following lines of code.
estimated_params;
a1, .5, -10, 10;
d1, .5, -20, 40;
stderr e_d, .5, .05, 5;
stderr e_s, .3, .05, 6; end;
// these are the ranges for the parameters
varobs W N;
28
2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
N
2 4 6 8 10
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
P
2 4 6 8 10
-0.4
-0.2
0
0.2
0.4
0.6
s
2 4 6 8 10
-0.5
-0.4
-0.3
-0.2
-0.1
0
W
Figure 17: A shock to est
estimation(datafile=data_rosen,first_obs=101,nobs=200,mh_replic=0,mode_compute=4,mode_check);
Figure plots the negative of the log-likelihood function for the parameters.
We proceed to estimate the same subset of the parameters using Bayesian techniques. We implemented this in
Dynare by adding the following lines of code.
estimated_params;
a1, gamma_pdf, .5, .5;
d1, gamma_pdf, 2, .5;
stderr e_d,inv_gamma_pdf,.5, 30;
stderr e_s,inv_gamma_pdf,.2, 30;
// The syntax for to input the priors is the following:
// variable name, prior distribution, parameters of distribution.
end;
varobs W N;
estimation(datafile=data_rosen,first_obs=101,nobs=200,mh_replic=10000,mh_nblocks=2,
mh_jscale=2,mode_compute=0,mode_file=rosen_estimateML2_mode);
29
1 1.2 1.4 1.6 1.8
400
405
410
415
SE_e_d
0.7 0.8 0.9 1 1.1 1.2
400
405
410
415
SE_e_s
0.9 1 1.1 1.2 1.3 1.4
403
403.5
404
404.5
405
405.5
406
a1
1 1.2 1.4 1.6 1.8
400
405
410
415
420
d1
Figure 18: Negative of the Log-Likelihood for Ryoo and Rosen's model
8. Estimation of Bansal and Yaron(2004) consumption growth process
Let consumption growth have the state space representation
xt = xt-1 + xx,t
ct = c + xt-1 + cc,t
where the shocks c,t and x,t are i.i.d. standard normal. We use quarterly growth rate of per capita U.S. consumption
over the period 1947.2-2003.3 to estimate the four unknown parameters. The following Dynare code does this by
maximum likelihood.
var x y;
varexo e_x e_u;
parameters rho sig_x sig_u mu_y;
rho = .98;
mu_y=.015;
sig_x=0.00025;
sig_u=.0078;
model(linear);
x=rho*x(-1) + sig_x*e_x;
y=mu_y + x(-1) +sig_u*e_u;
end;
initval;
30
0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
2
2.5
3
SE_e_d
0.2 0.4 0.6 0.8 1 1.2 1.4
0
1
2
3
4
5
6
7
SE_e_s
0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
2
a1
1 2 3
0
0.5
1
1.5
2
d1
Figure 19: priors and posteriors from the Bayesian estimation of Ryoo and Rosen's model
31
x=0;
y=mu_y;
end;
steady;
shocks;
var e_x;
stderr 1;
var e_u;
stderr 1;
end;
estimated_params;
// ML estimation setup
// parameter name, initial value, boundaries_low, ..._up;
rho, 0, -0.999, 0.999; // use this for unconstrained max
likelihood
// rho, .980, .975, .999 ; // use this for long run risk model
// sig_x, .0004,.0001,.05 ; // use this for the long run risk model
sig_x, .0005, .00000000001, .01; // use this for unconstrained
max likelihood sig_u, .007,.001, .1; mu_y, .014, .0001, .04;
end;
varobs y;
estimation(datafile=data_consRicardoypg,first_obs=1,nobs=227,mh_replic=0,mode_compute=4,mode_check);
// mode_compute=4 uses Chris Sim's csminwel algorithm to maximize the likelihood function.
// Using mode_compute=1 which uses the Matlab function fmincon or using mode_compute=2 which uses
// Lester Ingber's Adaptive Simulated Annealing algorithm produce almost identical results. However,
// model_compute=3 which uses fminunc gives the wrong answer.
Table 7 reports the estimated parameter from Dynare.
Parameter Estimate s.d. t-stat
rho 0.5623 0.1927 2.9179
sigx 0.0030 0.0011 2.6921
sigu 0.0040 0.0007 5.4519
muy 0.0056 0.0005 10.7613
Table 7: Results from Maximum likelihood
Figure 20 also plots the negative of the log-likelihood function along several dimensions.
Next, we estimate the same model using Bayesian techniques and two different sets of priors. The first set of
priors are tight around the calibrated values used in Bansal and Yaron (2004). To accomplish this, we can simply
change the following lines of code from our MLE program above.
estimated_params;
rho, beta_pdf, .98, .01;
mu_y, uniform_pdf, .005, .0025;
sig_u, inv_gamma_pdf, .003, inf;
sig_x, inv_gamma_pdf, .003, inf;
32
0.4 0.5 0.6 0.7 0.8
-873
-872.5
-872
-871.5
rho
2 2.5 3 3.5 4
x 10
-3
-873
-872.5
-872
-871.5
-871
sig_x
3 3.5 4 4.5 5
x 10
-3
-873
-872
-871
-870
-869
-868
sig_u
4 5 6 7
x 10
-3
-873
-872.5
-872
-871.5
-871
-870.5
mu_y
Figure 20: Negative of the Log-Likelihood function for the consumption growth model
33
0.92 0.94 0.96 0.98 1 1.02
0
10
20
30
40
rho
0 5 10
x 10
-3
0
50
100
150
200
mu_y
5 10 15
x 10
-3
0
200
400
600
800
1000
1200
1400
sig_u
0 5 10 15
x 10
-3
0
200
400
600
800
1000
1200
1400
sig_x
Figure 21: Prior and Posteriors for the Bansal and Yaron (2004) endownment process
// The syntax for to input the priors is the following:
// variable name, prior distribution, parameters of distribution.
end;
varobs y;
estimation(datafile=data_consRicardoypg,first_obs=1,nobs=227,mh_replic=5000,mh_nblocks=1,mh_jscale=1)
Figure 21 plots the prior and the posterior densities for the estimated parameters. The posteriors are plotted in
black and priors in gray. A vertical green line denotes a posterior mean.
Next, we set vague priors for  and for y.
rho, beta_pdf, .5, .2;
mu_y, uniform_pdf, .005, .0025;
sig_u, inv_gamma_pdf, .003, inf;
sig_x, inv_gamma_pdf, .003, inf;
Figure 22 plots the prior and the posterior densities for the estimated parameters. Notice that with a more diffuse
prior on , the posteriors for y are tighter.
9. Robust permanent income and pricing
Hansen, Sargent, and Tallarini (1999) study consumption, saving, and security market prices in a permanent income
model in which the agents mistrust their specification of the economic environment. The preferences of a
34
0 0.2 0.4 0.6 0.8 1
0
0.5
1
1.5
2
rho
2 4 6 8
x 10
-3
0
200
400
600
800
mu_y
0 5 10 15
x 10
-3
0
100
200
300
400
500
600
sig_u
0 5 10 15
x 10
-3
0
100
200
300
400
500
sig_x
Figure 22: Prior and Posteriors for the Bansal and Yaron (2004) endownment process
35
representative consumer are defined in terms of the following risk-sensitivity recursion:
Ut = ut + Rt(Ut+1) (22)
Rt(Ut+1) 
2

log Et[exp(
Ut+1
2
)] (23)
with ut being the per-period utility function. Note that  = 0 corresponds to Von Neumann-Morgenstern preferences.
The authors use a linear quadratic framework described below. Furthermore, they show that for a given (, ) pair
there exists another pair (~, 0) that gives the same equilibrium quantities. This observation simplifies solution and
estimation of the model significantly. For the  = 0 case and HST's linear-quadratic habit persistence model,
preferences become
U0 = E0

t=0
t
(-(st - b)2
) (24)
with b being a subsistence point and s defined as
st = (1 + )ct - ht-1 (25)
ht = hht-1 + (1 - h)ct (26)
where ct is current period consumption and ht-1 the habit stock. There is a linear production technology:
ct + it = kt-1 + dt (27)
kt = kkt-1 + it (28)
dt = d + ^dt + dt (29)
dt = (1 + 2) dt-1 - 12
dt-2 + cd  e1t (30)
^dt = (1 + 2) ^dt-1 - 12
^dt-2 + c^d  e2t (31)
e1t and e1t are standard normal iid shocks. Finally, we define the gross return on capital as
R  k +  (32)
and impose
R = 1. (33)
To solve the planning problem, we formulate a Lagrangian with multipliers 2t
st on (25), 2t
ht on (26) and 2t
ct
on (27) (after substituting for investment it). The first order necessary conditions are:
st = b - st (34)
ct = (1 + )st + (1 - h)ht (35)
ht = Et[hht+1 - st+1] (36)
ct = ct+1 (37)
plus (25)-(28). The following code estimates the model using maximum likelihood:
// Estimates the Hansen Sargent and Tallarini model by maximum likelihood.
var s c h k i d dhat dbar mus muc muh gamma R; varexo e_dhat e_dbar;
parameters lambda deltah deltak mud b bet phi1 phi2 cdbar alpha1
alpha2 cdhat;
bet=0.9971;
deltah=0.682;
lambda=2.443;
alpha1=0.813;
alpha2=0.189;
36
phi1=0.998;
phi2=0.704;
mud=13.710;
cdhat=0.155;
cdbar=0.108;
b=32;
deltak=0.975;
model(linear);
R=deltak+gamma;
R*bet=1;
s=(1+lambda)*c-lambda*h(-1);
h=deltah*h(-1)+(1-deltah)*c;
k=deltak*k(-1)+i;
c+i=gamma*k(-1)+d;
mus=b-s;
muc=(1+lambda)*mus+(1-deltah)*muh;
muh=bet*(deltah*muh(+1)-lambda*mus(+1));
muc=bet*R*muc(+1);
d=mud+dbar+dhat;
dbar=(phi1+phi2)*dbar(-1) - phi1*phi2*dbar(-2) +
cdbar*e_dbar;
dhat=(alpha1+alpha2)*dhat(-1) alpha1*alpha2*dhat(-2)+
cdhat*e_dhat;
end;
shocks;
var e_dhat;
stderr 1;
var e_dbar;
stderr 1;
end;
estimated_params;
bet, .91, .9, .99999;
deltah, 0.4, 0.1, 0.8;
lambda, 2, 0.1, 50;
alpha1, 0.8, 0.6, 0.99999;
alpha2, 0.2,0.01,0.5;
phi1, 0.8, 0.6, 0.99999;
phi2, 0.5, 0.3, 0.9;
mud, 10, 1,50;
cdhat, 0.1, 0.05, 0.2;
cdbar, 0.1, 0.05, 0.2;
end;
varobs c i;
estimation(datafile=dataHST,first_obs=1,nobs=500,mode_compute=4,mode_check);
To estimate the model using Bayesian techniques simply replace the last part of the code with
37
0.9 0.95 1
-1
0
1
2
x 10
8 bet
0.4 0.6 0.8
-900
-880
-860
-840
deltah
2 2.5 3 3.5
-890
-880
-870
-860
lambda
0.7 0.8 0.9 1
-900
-880
-860
-840
alpha1
0.1 0.15 0.2
-881.8
-881.7
-881.6
-881.5
alpha2
0.7 0.8 0.9 1
-5
0
5
10
x 10
7 phi1
0.4 0.6 0.8 1
-900
-800
-700
-600
phi2
10 15 20
-1000
-500
0
mud
0.1 0.15 0.2
-900
-880
-860
-840
cdhat
Figure 23: negative of the log-likelihood for the robust permanent income model
estimated_params;
bet,uniform_pdf, .9499999999, 0.0288675134306;
deltah,uniform_pdf, 0.45, 0.202072594216;
lambda,uniform_pdf,25.05,14.4048892163;
alpha1,uniform_pdf, 0.8, 0.115470053809;
alpha2,uniform_pdf, 0.25, 0.144337567297;
phi1,uniform_pdf,0.8,0.115470053809;
phi2,uniform_pdf, 0.5, 0.288675134595;
mud,uniform_pdf, 24.5, 14.1450815951;
cdhat,uniform_pdf,0.175,0.0721687836487;
cdbar,uniform_pdf, 0.175, 0.0721687836487;
end;
varobs c i;
estimation(datafile=dataHST,nobs=500,mode_check,mh_replic=10000,mh_nblocks=1,mh_jscale=0.2);
The sliced negative of the log-likelihood function is given by pictures (23) and (24). The maximization routine
used here (mode compute=4) is Chris Sims' csminwel. Note that this procedure uses a random number generator,
so different runs of this code can lead to different estimates (the routine might get stuck in a local extremum). Thus,
it is important to run the code several times.
38
0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12 0.125 0.13
-885
-880
-875
-870
-865
-860
-855
-850
cdbar
Figure 24: negative of the log-likelihood for the robust permanent income model
10. Asset pricing with risks for the long run
10.1 Introduction
Following Bansal and Yaron (2004), we specify dividend and consumption growth as containing a small but highly
persistent predictable component. These dynamics, when combined with Epstein and Zin (1989) preferences, help
explain financial phenomena including the equity premium puzzle. What allows this specification to be successful
is the ability of Epstein and Zin preferences to separate risk aversion from intertemporal elasticity of substitution,
possibly allowing both to be larger than one. Dynare enables us to compute policy functions for the response of
price-dividend ratios to an increase in the long-run predictable component of dividend and consumption growth.
Using the specification just described, this impact turns out to be positive. In economic terms, the intertemporal
substitution effect dominates the wealth effect. In response to higher expected growth rates, agents buy more assets
and this increases the wealth to consumption ratio. In the standard power utility (constant relative risk aversion)
model, the wealth effect dominates the substitution effect. Dynare shows that the policy functions for the latter
specification display the opposite sign for the feedback of the predictable component of growth on price-dividend
ratios. Hence the former specification helps explain high equity premia, because when the IES is larger than one,
agents fear that a reduction in economic growth prospects will lower asset prices.
10.2 Setup of the model
Specify preferences as follows:
Ut = (1 - )C
1-

t +  Et U1-
t+1
1


1-
(38)
where
 =
1 - 
1 - 1/
(39)
Here  reflects the degree of risk aversion, while  take into account intertemporal elasticity of substitution. When
 = 1, preferences collapse to the standard CRRA case. Hence this model can be regarded as a general one.
There are N assets in this economy, whose one period ahead return is denoted as Rj,t+1, j = 1, ..., N. The
share of investment in asset j is denoted as j,t. Hence there are N - 1 linearly independent elements in t =
39
1,t 2,t ... N,t

, since portfolio shares must satisfy:
N
j=1
j,t = 1, t (40)
The representative agent is endowed with an initial stock of the consumption asset A0 that can be either consumed
or invested in the N assets in the competitive market. Therefore, her wealth evolves according to:
At+1 = (At - Ct)
tRt (41)
Following Epstein and Zin (1991), first order conditions to price an asset that entitles to a claim on a stream of
consumption good imply:
vc,t = Et 
e[- 
 ct+1+(-1)rc,t+1](1 + vc,t+1)ect+1
(42)
where
ct = log(Ct) - log(Ct-1)
rc,t+1 = log Rc,t+1
Rc,t+1 =
(1 + vc,t+1)ect+1
vc,t
vc,t =
Pc,t
Ct
Similarly, first order condition to price an asset that entitles to a dividend Dt in each period imply:
vd,t = Et 
e[- 
 ct+1+(-1)rc,t+1](1 + vd,t+1)edt+1
(43)
where
dt = log(Dt - Dt-1)
vd,t =
Pd,t
Dt
We close the model, by specifying exogenous processes for consumption and dividend growth:
ct = c + xt-1 + c,t (44)
dt = d + dxxt-1 + d,t (45)
xt = xxt-1 + x,t (46)
where the shocks c,t, d,t and s,t are i.i.d. normal with mean zero and standard deviations c, d and x, respectively.
For stationarity |x| < 1. The economy is completely described by equations (42)-(46).
10.3 Calibration and simulation
We calibrate the parameters of the model in two different ways. Our guidelines in doing this are:
* use a reasonable risk aversion
* model the process of x as highly persistent
One important feature that, following the lines of Bansal and Yaron (2004), we would like to display is the different
way in which xt affects price-dividend ratios in the presence of Epstein and Zin utility functions. As we stressed
before the channel through which a highly persistent and predictable component of consumption and dividend
growth combined with Epstein-Zin preferences helps explaining the equity premium puzzle is the possibility of
having intertemporal elasticity of substitution and risk aversion greater than one at the same time. Hence we
want to experiment both a calibration where this condition is met and a standard constant relative risk aversion
preferences environment, where risk aversion greater than one implies IES smaller than unity. The latter specification
is a particular case in our model, as it only requires to put  = 1.
40
10.3.1 CRRA specification
We calibrate parameters as in Table 8. The numbers are the same as in Bansal and Yaron (2004) except for risk
IES, that is now the inverse of risk aversion.
Parameter Calibration
 0.133
 1
 7.5
 0.99
c 0.0015
c 0.0078
d 0.0015
dx 3
d 0.0351
x 0.979
x 0.0003432
Table 8: Parameters' calibration. The value of  is calibrated to satisfy equation (39).
The Dynare code reported below implements a first order approximation of the model.
periods 500;
var dc, dd, v_c, v_d, x; varexo e_c, e_x, e_d;
parameters DELTA THETA PSI MU_C MU_D RHO_X LAMBDA_DX;
DELTA=.99;
PSI=1.5;
THETA=(1-7.5)/(1-1/PSI);
MU_C=0.0015;
MU_D=0.0015;
RHO_X=.979;
LAMBDA_DX=3;
model;
v_c = DELTA^THETA *exp((-THETA/PSI)*dc(+1)+(THETA-1)*log((1+v_c(+1))*exp(dc(+1))/v_c))* (1+v_c(+1))*exp(dc(+1)); %
v_d = DELTA^THETA * exp((-THETA/PSI)*dc(+1) + (THETA-1)*log((1+v_c(+1))*exp(dc(+1))/v_c) ) * (1+v_d(+1))*exp(dd(+1)); %
dc = MU_C + x(-1) + e_c;
dd = MU_D + LAMBDA_DX*x(-1) + e_d; %
x = RHO_X * x(-1) + e_x;
end;
initval;
v_c=15;
v_d=15;
dc=MU_C;
dd=MU_D;
x=0;
e_c=0;
e_x=0;
e_d=0;
end;
shocks;
var e_c;
stderr .0078;
var e_x;
stderr .0078*.044;%
var e_d;
stderr .0078*4.5;
end;
steady(solve_algo=0); check;
stoch_simul(dr_algo=1, order=1, periods=1000, irf=30);
datasaver('simudata',[]);
Dynare computes policy and transition functions as reported in Table 9. Since these equations are expressed in
41
terms of xt-1, it is possible to express the system in terms of time t state variable using equation (46):
vc,t = 50.005 - 8086.753(xt - x)
vd,t = 50.005 - 5598.521(xt - x)
Equations (44)-(46) complete the description of the solution.
x c d vc vd
Constant 0 0.0015 0.0015 50.005844 50.005844
x (-1) 0.979 1 3 -7916.931499 -5480.952577
ec 0 1 0 0 0
ed 0 0 1 0 0
ex 1 0 0 -8086.753319 -5598.521529
Table 9: Policy and transition functions, CRRA calibration (Matlab output).
It is also interesting to look at the impulse response functions to a shock to xt. Figure 25 reports the response
of the variables of the system to a one standard deviation shock to x. According to the rationale discussed above,
both price-dividend and price-consumption ratios fall.
0 10 20 30
0
0.5
1
1.5
2
2.5
3
3.5
x 10
-4
0 10 20 30
0
0.2
0.4
0.6
0.8
1
1.2
x 10
-3
0 10 20 30
-2.8
-2.6
-2.4
-2.2
-2
-1.8
-1.6
-1.4
0 10 20 30
-2
-1.8
-1.6
-1.4
-1.2
-1
0 10 20 30
1.5
2
2.5
3
3.5
x 10
-4 x
c d v
c
v
d
Figure 25: Impulse response function to a shock to xt (CRRA preferences).
10.3.2 Epstein-Zin specification
This section focuses on the more general Epstein-Zin setup. In particular, we allow both risk aversion and IES to be
larger than one.  and, via equation (39),  are the only two parameters to differ between Table 8 and Table 10.
42
Parameter Calibration
 1.5
 -19.5
 7.5
 0.99
c 0.0015
c 0.0078
d 0.0015
dx 3
d 0.0351
x 0.979
x 0.0003432
Table 10: Parameters' calibration. The value of  is calibrated to satisfy equation (39).
By running the Dynare program with the change in the calibration of  and , the policy functions for vc and
vd change to:
vc,t = 104.209 + 1146.215(xt - x)
vd,t = 104.209 + 8023.510(xt - x)
Therefore, a shock to xt will now have a positive effect on price-dividend ratios. Separating risk aversion from
intertemporal substitution and modelling both as greater than unity produces the effect of changing the sign of the
impact of the forecastable part of consumption and dividend growth on price-dividend ratios. The IRF reported in
Figure 26 stress this finding.
0 10 20 30
0
0.5
1
1.5
2
2.5
3
3.5
x 10
-4
0 10 20 30
0
0.2
0.4
0.6
0.8
1
1.2
x 10
-3
0 10 20 30
0.2
0.25
0.3
0.35
0.4
0 10 20 30
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
0 10 20 30
1.5
2
2.5
3
3.5
x 10
-4 x
c d vc
v
d
Figure 26: Impulse response function to a shock to xt (CRRA preferences).
43
10.4 Bayesian estimation
We perform a Bayesian estimation of the general model, using simulated series. The model has three shocks c, x
and d and three observables vd, c and d. The priors that we use are reported in Table 11. A slight modification
to the program reported in the previous pages enables us to estimate the parameters of the model in Dynare. The
following code was been used:
var dc, dd, v_c, v_d, x;
varexo e_c, e_x, e_d;
parameters DELTA THETA PSI MU_C MU_D RHO_X LAMBDA_DX;
DELTA=.99;
PSI=1.5;
THETA=(1-7.5)/(1-1/PSI);
MU_C=0.0015;
MU_D=0.0015;
RHO_X=.979;
LAMBDA_DX=3;
model;
v_c = DELTA^THETA * exp((-THETA/PSI)*dc(+1) + (THETA-1)*log((1+v_c(+1))*exp(dc(+1))/v_c) ) * (1+v_c(+1))*exp(dc(+1)); %
v_d = DELTA^THETA * exp((-THETA/PSI)*dc(+1) + (THETA-1)*log((1+v_c(+1))*exp(dc(+1))/v_c) ) * (1+v_d(+1))*exp(dd(+1)); %
dc = MU_C + x(-1) + e_c; %
dd = MU_D + LAMBDA_DX*x(-1) + e_d; %
x = RHO_X * x(-1) + e_x;
end;
initval;
v_c=15;
v_d=15;
dc=MU_C;
dd=MU_D;
x=0;
e_c=0;
e_x=0;
e_d=0;
end;
shocks;
var e_d;
stderr .001;
var e_c;
stderr .001;
var e_x;
stderr .001;
end;
steady;
estimated_params;
DELTA, beta_pdf, 0.99,.001;
THETA,normal_pdf,-7.5, 0.005;
PSI,normal_pdf,1.5, 0.005;
MU_C,normal_pdf,0.001, 0.005;
MU_D,normal_pdf,0.001, 0.005;
RHO_X,normal_pdf,.98, 0.05;
LAMBDA_DX,normal_pdf,3, 0.05;%
stderr e_d,inv_gamma_pdf,.01, inf; %
stderr e_x,inv_gamma_pdf,.001, inf;
stderr e_c,inv_gamma_pdf,.0001, inf;
end;
varobs v_d dd dc;
estimation(datafile=simudata,mh_replic=5000);
44
In terms of Dynare's output Figure 27 displays priors' distributions, while Figure 28 reports a comparison between
prior's and posterior's distributions. Since the data generating process is exactly the model that we are estimating,
it is not very surprising that the distribution of the posterior has very little variance around the true parameters.
Parameter Prior Distribution Mean Std. Dev.
 Normal 1.5 .005
 Normal -19.5 .005
 Beta 0.99 .001
c Normal 0.0015 .005
c Inv. Gamma 0.0078 inf
d Normal 0.0015 .005
dx Normal 3 .05
d Inv. Gamma 0.0351 inf
x Normal 0.979 .05
x Inv. Gamma 0.0003432 inf
Table 11: Priors.
0 0.05 0.1
0
50
100
150
0 0.005 0.01
0
500
1000
1500
0 0.5 1
x 10
-3
0
5000
10000
15000
0.98 1
0
200
400
600
-7.55 -7.5 -7.45
0
20
40
60
80
1.5
0
20
40
60
80
-0.02 0 0.02
0
20
40
60
80
-0.02 0 0.02
0
20
40
60
80
0.5 1 1.5
0
2
4
6
8
2.8 3 3.2
0
2
4
6
8
d
x
c

  c
d
x
dx
Figure 27: Priors.
Dynare also produces a table of summary statistics of the estimation. As already seen from the Figures, the mean
of the posterior distributions is almost coincidental with the correct values and the estimated confidence interval is
very tight around these values. Table 12 reports these numbers.
Prior Mean Post. Mean Confidence Interval Prior dbn Post std dev
 0.99 0.9896 0.988 0.991 Beta 0.001
 -7.5 -7.501 -7.5102 - 7.4934 Normal 0.005
 1.5 1.4981 1.4898 1.5049 Normal 0.005
c 0.001 0.0012 0.0001 0.0021 Normal 0.005
d 0.001 0.0017 -0.0004 0.0044 Normal 0.005
x 0.98 0.9786 0.9753 0.9813 Normal 0.05
dx 3 2.9963 2.9205 3.0812 Normal 0.05
d 0.01 0.0368 0.0355 0.038 Inv. Gamma Inf
x 0.001 0.0003 0.0003 0.0003 Inv. Gamma Inf
c 0 0.0076 0.0074 0.0079 Inv. Gamma Inf
Table 12: Posterior.
45
0 0.05 0.1
-200
0
200
400
600
0 0.005 0.01
0
5
10
x 10
6
0 0.005 0.01
0
5000
10000
15000
0.98 1
0
200
400
600
-7.55 -7.5 -7.45
0
50
100
1.4 1.5 1.6
0
20
40
60
80
-0.02 0 0.02
0
200
400
600
-0.02 0 0.02
0
100
200
300
0.5 1 1.5
0
100
200
300
2.5 3 3.5
0
2
4
6
8

d

x
c

  
c

d
x
dx
Figure 28: Posterior.
A. List of Dynare Files
This appendix lists the dynare files used in this paper. All files and the data used are contained in the file examples.zip.
In case simulated data was used in the estimation one first needs to run the relevant dynare code to simulate the
data.
* GrowthApproximate.mod - solves the neoclassical growth model and simulates data
* GrowthEstimate.mod - estimates the neoclassical growth model
* TwocountryApprox.mod - solves the two country production economy and simulates data
* TwocountryEstim.mod - estimates the two country production economy
* Fig11XX.mod - solves a deterministic growth model and reproduces the graph XX in chapter 11 of Ljungqvist
and Sargent (2004)
* sargent77.mod - solves the model in Sargent (1977)
* sargent77ML.mod, sargent77Bayes.mod - estimate the model in Sargent (1977) using maximum likelihood and
Bayesian MCMC methods, respectively
* hall1.mod - solves the model in Hall (1988) and simulates data
* hall1estimateML.mod, hall1estimateBayes.mod - estimate the model in Hall (1988) using maximum likelihood
and Bayesian MCMC methods, respectively
* rosen.mod - solves the model in Ryoo and Rosen (2004) and simulates data
* rosenestimateML.mod, rosenestimateBayes.mod - estimate the model in Ryoo and Rosen (2004) using maximum
likelihood and Bayesian MCMC methods, respectively
46
ˇ BansalYaronML.mod, BansalYaronBayes.mod - estimate the consumption-growth process of Bansal and Yaron
(2004) using maximum likelihood and Bayesian MCMC methods, respectively
* HSTML.mod, HSTBayes.mod - estimate the model in Hansen, Sargent, and Tallarini (1999) using maximum
likelihood and Bayesian MCMC methods, respectively
* AssetPricingApproximation.mod - solves the model of Bansal and Yaron (2004) and simulates data
* AssetPricingEstimate.mod - estimates the model of Bansal and Yaron (2004)
47
References
Bansal, R., R. Gallant, and G. Tauchen (2007). Rational pessimism, rational exuberance and markets for macro
risks. Working Paper.
Bansal, R. and A. Yaron (2004, 08). Risks for the long run: A potential resolution of asset pricing puzzles. Journal
of Finance 59(4), 1481­1509.
Burnside, C. (1998, March). Solving asset pricing models with gaussian shocks. Journal of Economic Dynamics and
Control 22(3), 329­340.
Cagan, P. (1956). The monetary dynamics of hyperinflation. In M. Friedman (Ed.), Studies in the Quantity Theory
of Money. University of Chicago Press.
Cooley, T. F. and E. C. Prescott (1995). Economic growth and business cycles. In T. F. Cooley (Ed.), Frontiers of
Business Cycle Research. Princeton University Press.
Epstein, L. G. and S. E. Zin (1989, July). Substitution, risk aversion, and the temporal behavior of consumption
and asset returns: A theoretical framework. Econometrica 57(4), 937­69.
Epstein, L. G. and S. E. Zin (1991, April). Substitution, risk aversion, and the temporal behavior of consumption
and asset returns: An empirical analysis. Journal of Political Economy 99(2), 263­86.
Hall, R. E. (1988, April). Intertemporal substitution in consumption. Journal of Political Economy 96(2), 339­57.
Hansen, L. P., T. J. Sargent, and J. Tallarini, Thomas D (1999, October). Robust permanent income and pricing.
Review of Economic Studies 66(4), 873­907.
Kim, J. and S. H. Kim (2003, August). Spurious welfare reversals in international business cycle models. Journal of
International Economics 60(2), 471­500.
Ljungqvist, L. and T. J. Sargent (2004). Recursive Macroeconomic Theory, Second edition. Cambridge, Ma.: MIT
Press.
Ryoo, J. and S. Rosen (2004, February). The engineering labor market. Journal of Political Economy 112(S1),
S110­S140.
Sargent, T. J. (1977, February). The demand for money during hyperinflations under rational expectations: I.
International Economic Review 18(1), 59­82.
Schmitt-Grohe, S. and M. Uribe (2004). Solving asset pricing models with gaussian shocks. Journal of Economic
Dynamics and Control 28, 755­775.
Siow, A. (1984, May). Occupational choice under uncertainty. Econometrica 52(3), 631­45.
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