Seitani(2013)Toolkit for DSGE

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Matlab Toolkit for Simulating

Dynamic Stochastic General Equilibrium Models

Haruki Seitani*

August 2013

Abstract

Behavior of a dynamic stochastic general equilibrium model can be best

understood by working out an approximated solution. A set of Matlab codes is

designed for 1) log-linearizing equilibrium conditions around a steady-state, 2) solving

the system of expectational difference equations by applying eigenvalue decomposition,

and 3) simulating the model by calculating impulse-response functions and generating

artificial time series data. Taking a typical real business cycle model as an example, it is

shown how a calibrated model provides an insight into refinement of business cycle

theory.

*Economic and Social Research Institute, Cabinet Office, Government of Japan. Opinionsexpressed here are the authors alone and do not represent the views of the organization.


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1. Introduction

As Campbell (1994) advocates, behavior of a dynamic stochastic general

equilibrium (DSGE) model can be best understood by working out an approximatedsolution. In the DSGE analysis, it has become a common exercise to log-linearly

approximate equilibrium conditions derived from dynamic optimization of economic

agents around a deterministic steady state. Such approximated equilibrium conditions

typically take a form of a system of first-order expectational difference equations.

Solution to the approximated system governs development of economic states and

economic agents decisions functional on those states. Behavior of a DSGE model can

then be investigated by simulating the approximated model with artificially generated

data.

This Matlab toolkit linearizes equilibrium conditions around a steady state and

finds a set of policy functions and laws of motion of state variables as a solution to a

system of first-order expectational difference equations. Once policy functions are

obtained, one can simulate the model by giving it exogenous stochastic shocks to state

variables. The toolkit calculates response of the model economy to such a shock and

compares variation and correlation among aggregate economic variables.

A typical real business cycle (RBC) model is taken as an example of DSGE

models here not because it provides a better explanation of actual economic activities

but because it has been a work-force of development of almost all DSGE approach to

business-cycle analyses.

Many authors have already provided this sort of guidance and toolkits and the

present deliverable is entirely based on the contributions of the predecessors1. Thus this

paper does neither offer any original insights nor provide complete treatment of

alternative solution techniques. The purpose of it is to serve as a quick reference that

relates economic theory, guide for mathematical technique, its application, and literature

in a succinct manner.

2. Real Business Cycle Model

2.1. Setup of the Model

The model is based on a perfectly competitive economy populated by identical

infinitely-lived households and firms, so that their economic choices reduce to decisions

made by a single representative agent.

1

See Hartley, Hoover, and Salyer (1998), Uhlig (1999), Kato (2002). Dejong and Dave (2011)compare various alternative solution techniques.


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Households own and lease capital stock, tk , and sell labor (measured by hours

worked),tl , to firms respectively at the rental rate of capital and real wage rate. They

allocate their income into purchases of consumption goods, tc , and capital goods, ti ,

and their time into labor and leisure so that their lifetime expected utility is maximized.

Their preferences are represented by an identical utility function as follows:

00

0

0 1lnln1,t

tt

t

t

tt

t lcElcUE , 10 . (1)

Capital stock owned by households depreciates at a constant rate, .

Firms have an access to a constant-returns-to-scale production technology that is

subject to random technology shocks; specifically the production technology takes the

Cobb-Douglas type functional form and the technology shocks follow a first-order

autoregressive process (AR(1)) with a log-normally distributed disturbance.

1, ttttttt lkzlkfzy , (2)

11 exp ttt zz

, zt N ,0 . (3)

2.2. Deriving the Equilibrium Conditions

Since the economic structure above satisfies complete markets, price-taking

behavior, convexity of a production set, and convexity and local nonsatiation of

preferences, the fundamental theorems of welfare economics is applicable; a Pareto

optimal allocation coincides with a competitive equilibrium.2

The Pareto optimum is the solution to the following social-planners problem:

00 1lnlnMax

t

tt

tlcE ,

tttttt ylkzic 1subject to ,

ttt ikk 1and 1 .

This infinite horizon problem can be solved by exploiting its recursive structure; the

2 See Cooley and Prescott (1994).


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social planners problem is recast as the following dynamic programming:

11,, ,1lnlnMax, 1 tttttlkctt zkVElczkV ttt ,

1subject to1

1 tttttt klkzkc ,

where V denotes the optimized value function, or in this context the indirect utility

function of the household.

A solution to this problem has to satisfy the necessary conditions and aggregate

resource constraint below.

t

ttt

t c

lkz

l

1

1 . (4)

111 1

1

1

11

1

ttt

t

t

t

lkzc

Ec

. (5)

tttttt cklkzk

11

1 . (6)

The first equation that relates the current variables to each other is called

intratemporal optimality condition. It implies that the marginal rate of substitution

between labor and consumption has to equal the marginal product of labor. The second

equation that connects the current and one-period ahead variables is called

intertemporal optimality condition. The left-hand side of the equation represents the

marginal utility loss of investing in capital while the right-hand side represents the

expected marginal utility gain from it; these costs and benefits have to be equalized at

the optimum. These intra- and intertemporal optimality conditions, together with the

resource constraint and the dynamic low of motion for the technology shock, govern the

equilibrium behavior of the model economy.

3. Approximation and Solution

Since the solution to the social planners problem exists and is unique (Prescott

1986) but is NOT analytical, approximation helps us make the model operational. One

of the widely exercised methods is to take a log-linear approximation of the equilibrium


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conditions around the steady-state values of consumption, capital, labor, and

technology3.

3.1. Steady State of the Economy

A steady state equilibrium for the model economy is one in which the

technology shock is assumed to be constant: tzt ,1 , and the values of capital, labor,

and consumption are constant: tccllkk tt ,,, . The steady-state values of

capital, labor, and consumption are solution to the following equations obtained by

imposing these conditions on the necessary conditions for the optimality.

c

lk

l

1 . (7)

k

ylk

1111

. (8)

cyclkk 1 . (9)

3.2. Log-Linear Approximation of the Equilibrium Conditions

Recall that the first-order Taylor series expansion of a function xf around a

point x takes the following form.

xxxfxfxf .

Taking natural logarithmic of x and noting that xx lnexp , natural logarithmic of

xf can be approximated as follows.

xx

xf

xxfxfxf lnlnlnln

,

x

x

xf

xxf

xf

xflnln . (10)

Suppose that x deviates from x by only a small number, . Then,

3 Log-linearization can be justified as a good approximation as far as

the stochastic behavior of the model does not push the economy too far from the steady-state

behavior. It is also possible to approximate the model at higher order; for example, Schmitt-Grohand Uribe (2004) analyze a solution to second-order approximated conditions.


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x

xxx

x

x

x ~1lnlnln

.

Thus xxln can be interpreted as percentage deviation of x from x , x~ . Note

also that xfxxf is the elasticity of xf with respect to x . Thus equation

(10) represents percentage deviation of xf from xf due to the deviation of x .

This can easily be generalized to multivariate cases:

ii

n

i n

i

i

n

n

n

xx

xxfx

xxxf

xxfxxf ln

,,,,

,,,,ln

1 1

1

1

1

. (11)

Applying this technique to the equilibrium conditions from (4) to (6) and the process of

technology shocks, (3), delivers the following log-linearly approximated counterparts.

1) Intratemporal Optimality Condition

ttttt clkzl ~

~~~~ . (12)

2) Intertemporal Optimality Condition

.~1~1~~

~

1

1~

1

1

~

1

~~

1111

111

11

111

11

111

11

1

tttttttt

tttt

ttttt

lEkEzEk

ycE

lElk

lk

kElk

lk

zElk

lkcEc

(13)

The last expression uses the steady-state equilibrium condition:

k

ylk 11 , and


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1111 lk .

3) Aggregate Resource Constraint

.~~1~~

~

1

~

1

1

~

1

~

1

~

11

1

1

1

1

1

1

tttt

tt

tttt

ck

clkz

k

y

ccklk

cl

cklk

lk

kcklk

lkz

cklk

lkkE

(14)

The final line of the equation results from the steady-state relationships:

ylk 1 , and

kcklk 11 .

4) Technology Shock Process

11~~

ttt zz .

Taking expectations of both sides of the last equation,

ttt zzE~~

1 . (15)

Let

tttt zklc ~~~~tY . Then the equilibrium conditions can be written as a system

of first-order expectational difference equations:

1tt YBYA tE , (16)

where


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000

1

0001

111

ky

ky

ky

kcA ,

1000

0100

111

0000

k

y

k

y

k

y

B .

Premultiplying both sides of equation (16) by 1A yields:

1t1

t YBAY tE . (17)

3.3. Solving the Approximated Model

The trouble with solving the system of equations is that terms related to other

variables appear one another in each individual equation. It would be much easier to

solve if only we were able to transform these equations so that they are decoupled

each other.

This indeed possible by applying eigenvalue decomposition to BA 1 .

1t1

1t1

t YQQYBAY

tt EE , (18)

where is a diagonal matrix whose diagonal entries are eigenvalue of BA 1 , and

Q is a matrix whose columns are the associated eigenvectors4. Let tt

1 YYQ . Then,

1 tt YY tE

1

1

1

1

4

3

2

1

~

~

~

~

000

000

000

000

~

~

~

~

tt

tt

tt

tt

t

t

t

t

zE

kE

lE

cE

z

k

l

c

Thus the transformation successfully decouples the equations.

4 This technique of changing variables is discussed in Chapter 23 of Simon and Blume (1994).


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Notice that the system of equations generally involves both control (or jump)

and state (or predetermined) variables. In this case, there are two control variables,tc

and tl , and two state variables, tk and tz . Suppose that in a general case tY

consists of n control variables, tX , and m state variables, tS :

t

t

tS

XY .

Let 1ijQ and i be partitioned matrices of

1Q and whose numbers of rows and

columns are respectively conformable to those of X and S .

1t

1t

1

22

1

21

1

12

1

11

S

X

t

t

1

22

1

21

1

12

1

11

S

X

QQ

QQ

O

O

S

X

QQ

QQtE .

1t

1t

S

X

t

t

S

X

O

O

S

X

tE (19)

Notice that equation (19) implies:

t

t

1

S

1

X

t

t

S

X

O

O

S

X

T

T

T

Tt

E . (20)

Existence and uniqueness of stable rational expectations solutions require that all

eigenvalues contained in S have to be greater than unity so that

O 1S

T

Tlim ,

while those contained in X have to be smaller than unity (Blanchard and Kahn

1980)..


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1) Solution to Control Variables

It directly follows from equation (19) that:

t1XTt XX T

tE

.

Provided that the Blanchard-Kahn condition is satisfied, since all eigenvalues in X is

greater than one, it has to be true in order for TtX

tE not to explode that:

OSQXQOX t1

12t

1

11t

.

This leads to the solution to control variables:

tXt112111t SPSQQX 1 . (21)

2) Solution to State Variables

It also follows from equation (19) that:

1t1221t1211S1t1221t121

t

1

St

SQXQSQXQ

SS

tt

t

EE

E 1

.

Substituting (21) into above equation,

tX1211221S1tX121122 SPQQSPQQ tE

Then the solution to state variables is given by:

tStX1211221SX1211221t SPSPQQPQQS

1

tE . (22)

3.4.

Matlab Codes


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This subsection explains some key lines of Matlab codes written in the toolkit.

The complete lines are provided in Appendix. The codes draw a direct line with those in

Kato (2002).

3.4.1. Linearization Part

In Matlab, linearization of equilibrium conditions is implemented by taking

Jacobian of the system of equations from (3) to (6) and evaluating it at the steady-state

values:

%---- Differentiation ----%

J = jacobian ( eqcon,xx ) ;

%---- Evaluation at the Steady State ----%

cc = CS ;

ca = CS ;

lc = LS ;

la = LS ;

kc = KS ;

ka = KS ;

zc = Z ;

za = Z ;

coef = eval ( J ) ;

3.4.2. Solution Part

The Matlab function named bksolve returns mn ,,, SX PP by

implementing the eigenvalue decomposition:

[ Q, L ] = eig (inv(A)*B) ;

Then the eigenvalue matrix L is vectorized and sorted in ascending order; columns of

the matrix Q, eigenvectors associated with eigenvalues, are also rearranged in line with

the same order:

L = diag ( L ) ;

[ L, order ] = sort ( diag ( L ), ascending ) ;


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Q = Q ( : , order) ;

The following lines count the number of control and state variables in the system, n andm.

n = 0;

for i = 1:length ( L )

if abs ( L ( i ) ) < 1.0000

n = n + 1;

end

end

m = length ( L ) - n;

Then the inverse matrix of Q is partitioned into four blocks while L is divided into

stable and unstable roots.

%--- Partition Matrices ---%

QI11 = QI(1:n, 1:n);

QI21 = QI(n+1:m+n, 1:n);

QI12 = QI(1:n, n+1:m+n);

QI22 = QI(n+1:m+n, n+1:m+n);

%--- Extracting Stable Roots ---%

L1 = L ( 1:n, : ) ;

L2 = L ( n+1:m+n, : ) ;

LS = diag ( L2 ) ;

The rest of the code calculates XP and SP according to formulae (21) and (22) (see

Appendix).

4. Simulation with a Calibrated Model

Once one obtains the approximated solution to a model, she can exercise a

quantitative analysis by assigning numerical values to parameters. The strategy for

finding numerical parameter values is called calibration.


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By imposing restriction on the mapping between theory and data, calibration

enables us to measure the distance between an artificial model economy and its actual

counterpart in that it identify in which dimensions the former succeeds or fails inmatching with the latter. Results of the exercise serve as guidance about refinements of

the theoretical model or improvements in the measurements of the actual economy

(Cooley 1997).5

4.1. Parametarization

Cooley (1997) provides some rules for good calibration. According to them,

it is important to set parameter values so that the behavior of the model economy

matches features of the measured data such that one cannot abstract from. In the

context of the RBC model here, parameter values should be chosen so that the Kaldors

stylized facts of economic growth are mimicked, for the underlying structure of the

model is the neoclassical growth framework. Also, some information about individual

choices over time such as total hours worked could be useful for setting parameter

values related to preference specifications.

The parameter values employed here are:

007.0,95.0,025.0,36.0,0.2,99.0 z .

These parameter values imply the following steady-state values, which are broadly

consistent with the post-war U.S. experience:

11.1,9.10,30.0,83.0 yklc .

4.2. Impulse-Response Function

Then one can investigate propagation and amplification mechanisms of the model

economy by looking at impulse-response functions. Figure 1 displays the behavior of

output, consumption, investment, hours worked, and labor productivity in response to a

unitary shock to technology. As can be seen in the figure, consumption shows the

smallest response among them. This is due to the optimizing behavior of households

who have risk-averse preferences and thus want to smooth their consumption. The other

side of the same coin is the response of investment, which is the largest among all

5 This is an advantage of calibration exercise over classical statistical inference. Since the

distribution theory for a standard statistical test of a model is derived under the null hypothesis, it isunlikely to provide much information about how to proceed if the test rejects the model.


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variables. Turning eyes to behavior of labor, one can see that hours worked show a

positive response to the technology shock but it is not greater than that of output. This is

an outcome of interaction between income- and substitution effects. While the formereffect induces the household to reduce labor supply, the latter gives incentive for

working more. Since the latter dominates the former, the net effect is positive.

While these results are intuitively comprehensible, it has been pointed out that the

impulse-response functions of the RBC model have properties inconsistent with their

empirical counterparts in some dimensions. For example, the calibrated model

predicts positive comovement among output, hours worked, and labor productivity in

response to a positive technology shock. However, Gal (1999) points out that an

empirical impulse-response function estimated by a vector autoregression (VAR) shows

a negative response of hours-worked. In addition, while Cogley and Nason (1995)

report that a trend reverting component of U.S. output has a hump-shaped

moving-average representation, the theoretical impulse-response of output does not

show any such a property; it peaks at the first period in which the shock hits the

economy.

Figure 1

0 5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Horizon

%DeviationfromS

teadyState

Impulse Response to a Unitary Technology Shock

Output

Consumption

Investment

Hours-worked

Labor Productivity


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4.3. Simulated Second Moments

Another common simulation exercise is second moment comparison between

artificial data and their actual counterparts. Figure 2 shows an artificial time paths forconsumption, output, and investment, which are detrended by the Hodrick-Prescott filter.

It confirms that investment fluctuates more volatilely than output while consumption is

smoother than output, as is also true in actual data.

But at the same time, a careful look at precise statistics sheds lights on some

defects about the model. Table 1 reports the standard deviations of simulated series of

output, consumption, investment, hours worked, and labor productivity, and correlations

among them, comparing with actual data. It should be noted that 1) the calibrated model

predict considerably less volatile output than actually observed, 2) the predicted hours

worked is also less volatile than actual, and 3) it generates a strong comovement

between hours-worked and labor productivity, which has not been observed in the U.S.

experience (Hansen and Wright 1992) .

These failures mainly results from the fact that the model economy relies solely

on a single technology shock to generate fluctuation; not only the driving force of

business cycle is underestimated but also the behavior of the labor market is too much

simplified. That is, wage (=productivity) and hours worked in the model fluctuate in

response to shifts in labor demand only, which is triggered by technology shocks. A

strong positive correlation between them represents a stable upward sloping labor

supply curve traced out by a labor demand curve shifting over time. This calibration

exercise suggests that the model should take account of other impulse factors which

have multifaceted influence on a labor market. Hansen and Wright (1992) introduce a

shock to home production technology and stochastic government expenditure as a

demand shock. They both affect the households choice on participation in labor market

so that labor supply curve shifts. Hansen and Wright demonstrate that the correlation

between hours and productivity is mitigated under the environment in which both labor

supply and demand are shifting incessantly.


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Figure 2

0 50 100 150-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Time Period

%DeviationfromS

teadyState

Simulated Time Series Data

Output

Consumption

Investment

Table 1 Second Moments of Simulated and Actual US Data

Standard Error

Output

y

Consumption

c / y

Investment

i / y

Hours

l / y

Productivity

w / y

Simulated 0.0135 0.31 3.12 0.51 0.50

Actual 0.0192 0.45 2.78 0.78-0.96 0.45-0.57

Correlation with Output

Output Consumption Investment Hours Productivity

Simulated 1.00 0.90 0.99 0.98 0.98Actual 1.00 0.71 0.73 0.82-0.90 0.31-0.63

Standard Error and Correlation between Hours-worked and Labor Productivity

Hours

l / w

Correlation

Simulated 1.02 0.92

Actual 1.37-2.15 -0.14-0.07* Simulated moments are the sample means of statistics computed for 300 times simulations with 150 periods.* Actual data are reported in Hansen and Wright (1992). Sample period is 1947:1-1991:3. Statistics related to hours

and productivity take different figures depending on which of two sources of data (household survey orestablishment survey) is used.


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5. Summary

A DSGE framework provides a foundation for a logically coherent analysis on

how stochastic shocks are amplified and propagated through optimizing behavior ofeconomic agents. Since equilibrium conditions in a DSGE model generally take

non-linear forms, approximation would be helpful for making them easy to handle.

A common exercise is to log-linearly approximate intra- and intertemporal

optimal conditions around a steady state. A resulting approximated version of the model

takes a form of a system of first-order expectational difference equations, which can be

solved relying on eigenvalue decomposition.

This paper illustrated how the technique works by taking a typical RBC model for

an instance. Calibration of the model reveals that the RBC theory succeeds in

explaining some features of an actual economy but fails in other important dimensions

such as behavior of the labor market. The results of the quantitative exercise points to a

direction in which the theory should proceed in order to reach better understanding of

business cycle phenomena.


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Reference

Blanchard, Oliver J., and Charles M. Kahn. The Solution of Linear Difference Modelsunder Rational Expectations.Econometrica, Vol. 48, No. 5, 1980, 1305-1311.

Campbell, John Y. Inspecting the Mechanism: An Analytical Approach to the

Stochastic Growth Model. Journal of Monetary Economics, Vol. 33, Issue 3,

1994, 463-506.

Cogley, Timothy, and James M. Nason. Output Dynamics in Real-Business-Cycle

Models.American Economic Review, Vol. 85, No. 3, 1995, 492-511.

Cooley, Tomas F. Calibrated Models. Oxford Review of Economic Policy, Vol. 13,

Issue 3, 1997, 55-69.

_________., and Edward C. Prescott. Economic Growth and Business Cycles. Tomas

F. Cooley ed., Frontiers of Business Cycle Research, Princeton University Press,

Princeton, New Jersey, 1994, 1-38.

Dejong, David N., and Chetan Dave. Structural Macroeconometrics: second edition,

Princeton University Press, Princeton, New Jersey, 2011.

Gal, Jordi. Technology, Employment, and the Business Cycle: Do Technology Shocks

Explain Aggregate Fluctuations? American Economic Review, Vol. 89, No. 1,

1999, 249-271.

Hansen, Gary D., and Randall Wright. The Labor Market in Real Business Cycle

Theory. Federal Reserve Bank of Minneapolis Quarterly Review, Vol. 16, No. 2,

1992, 2-12.

Hartley, James E., Kevin D. Hoover, and Kevin D. Salyer. A Users Guide to Solving

Real Business Cycle Models. Real Business Cycles, A Reader, Routledge,

London and New York, 1998, 43-54.

Kato, Ryo. A User Guide for Matlab Code for An RBC Model Solution and

Simulation. mimeo, 2002.

Schmitt-Groh, Stephanie and Martin Uribe. Solving Dynamic General Equilibrium

Models Using a Second-Order Approximation to the Policy Function.Journal of

Economic Dynamics and Control, Vol. 28, Issue 4, 2004, 755-775.

Prescott, Edward C. Theory Ahead of Business Cycle Measurement.

Carnegie-Rochester Conference Series on Public Policy, Vol. 25, autumn, 1986,

11-44.

Simon, Carl P., and Lawrence Blume. Mathematics for Economists. W. W. Norton &

Company, London and New York, 1994.

Uhlig, Harrold. A Toolkit for Analysing Non-linear Dynamic Stochastic Models


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Easily. R. Marimon and A. Scott eds. Computational Methods for the Study of

Dynamic Economies, Oxford and New York, Oxford University Press, 1999,

30-61.


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Appendix: Matlab Code for Simulating a DSGE Model

A-1: Main Program modelsim.m

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Basic Real Business Cycle Model %%%% Matlab Code Written by Haruki Seitani %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

clear all%==================================================%% Initialization: Model Settings %%==================================================%%--- Deep Parmeter Values ---%alpha = 0.36 ; % Capital Income Sharebeta = 0.99; % Subjective Discount rate

delta = 0.025; % Capital Depreciation RatePsi = 2.0 ; % Marginal Disutility of LaborZ = 1.0 ; % Mean of Aggregate Technologysigma_z = 0.007; % Standard Deviation of Technologyrho = 0.95 ; % Autocorrelation of Technology Shocks

%--- Variales ---%syms cc lc kc zc ca la ka za;xx = [ca la ka za cc lc kc zc];% Variables with "a" / "c" denote forward/current variables, respectively% 1) Control Variables% c : Consumpton

% l : Hours-worked% 2) State Variables% k : Capital% z : Technology

varname = char('Consumption', 'Hours-worked', 'Capital', 'Technology');

%--- Simulation Settings ---%% 1) Impulse-Response Analysish = 24; % Horizons of Impulse-Response Functionhit = 2; % Period in which an innovation occursimp = varname(4,:); % Variable hit by an innovationSI = [0, 1]'; % Innovation Vector

% 2) 2nd Moments Comparisonmcreps = 300; % # of Monte-Carlo Simulationssize = 150; % Size of simulated time seriesdisc = 300; % First "disc" observations will be discarded.vars = 5; % # of variables among which 2nd moments are analyzed

% 5 variables: y, c, i, l, y/llambda = 1600; % Smoothing parameter of HP filter

disp('%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%')disp(' Toolkit for a Standard RBC Model ')

disp('%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%')


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disp('Welcome!')disp('This is a toolkit for solving and simulating an approximated DSGE model.')disp(' ')disp('First of all, the steady state of the model economy will be calculated.' )disp('Hit any key when ready......')pause;

%==================================================%% Calculation of the Steady State %%==================================================%disp('=========================================')disp(' Steady State ')disp('=========================================')disp(' K/Y CS KS LS YS IS')

KY = 1/((1-beta*(1-delta))/(alpha*beta*Z));CY = 1-delta*KY;LS = 1/(1+Psi/((1-alpha)/CY));KS = LS*(alpha*beta*Z/(1-(beta*(1-delta))))^(1/(1-alpha));YS = Z*(KS^alpha)*(LS^(1-alpha));IS = delta*KS;CS = YS - IS;

s = [CS, LS, KS, Z];

disp([KY, CS, KS, LS, YS, IS]);

disp('Then the equilibrium conditions are linearized around the steady state,')disp('and the solution to the approximated system will be found.')disp('Hit any key when ready......')pause;

%===================================================%% Equilibrium Conditions %%===================================================%%---- Intratemporal Optimality Condition ----%intra = Psi/(1-lc) - (1-alpha)*zc*((kc/lc)^(alpha))*(1/cc);

%---- Intertemporal Optimality Condition ----%inter = beta*(alpha*za*(la/ka)^(1-alpha) + (1-delta))*(1/ca) - (1/cc);

%---- Resource Constraint ----%resource = zc*(kc^alpha)*(lc^(1-alpha)) + (1-delta)*kc - cc - ka;

%---- Dynamics of Technology Shock ----%tech = za - zc^rho;

%---- Optimal Conditions ----%eqcon = [intra; inter ; resource; tech ];


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%===================================================%% Linearization and Solving the Model %%===================================================%%---- Differentiation ----%J = jacobian(eqcon,xx);

%---- Evaluation at the Steady State ----%cc = CS;ca = CS;kc = KS;ka = KS;lc = LS;la = LS;zc = Z;za = Z;

coef = eval(J);SS = [ s ; s ; s ; s];

%---- Solving the Model ----%[PX, PS, n, m] = bksolve(SS,s,coef);

%===================================================%% Impulse-Response Function %%===================================================%disp('Next, responses to a technology shock are to be displayed.')disp('Hit any key when ready......')

disp(' ')pause;

SC = SI; % SI will initially be used as the current state vector% SC will be revised recursively.

S = zeros(h, m); % Simulated state variables% which will be revised as the simulation proceeds.

for i = hit:h % Shock occurs in s-th period; matrix S is revised from 6throw.SA = PS*SC ; % New state vectorS(i,:) = SA'; % Revise i-th row of state matrixSC = S(i,:)' ; % Use revised i-th row of state matrix

% as state vector in next period.end

S(hit,:) = SI; % Simulated endogenous & exogenous state variablesX = (PX*S')'; % Simulated jump variables

Y = [X, S];

ct = Y(:, 1);lt = Y(:, 2);kt = Y(:, 3);

zt = Y(:, 4);


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yt = Y(:, 4) + alpha*Y(:, 3) + (1-alpha)*Y(:, 2); % Outputit = (YS/IS)*yt - (CS/IS)*Y(:, 1); % Investmentwt = yt - Y(:,2); % Marginal Product of Labor

%---- Graphics ----%t = 1:length(yt);set(gca, 'Fontsize', 12);plot(t, yt,'k-', t, ct,'m--+', t, it,'r-.o', t, lt,'b:s', t, wt,'c:*',...

'Linewidth', 1.5, 'Markersize', 4.5');axis([0,h,0,Inf]);xlabel('Horizon');ylabel('% Deviation from Steady State');title('Impulse Response to a Unitary Technology Shock', 'Fontsize', 14);legend('Output', 'Consumption', 'Investment', 'Hours-worked', 'Labor

Productivity');

%===================================================%% 2nd Moments of Simulated Time Series %%===================================================%disp('Finally, the model will be simulated to show 2nd moments of artificial time

series.')disp('Hit any key when ready......')pause;

sstd = zeros(mcreps,vars);sstdy = zeros(mcreps,vars);scorr = zeros(mcreps,vars);

scorrpl = zeros(mcreps, 2);

for i = 1:mcreps%--- Generating Shock Variables ---%

eps = zeros(ssize+disc, m);shock = sigma_z*randn(ssize+disc,1);eps(:, 2) = shock;

%--- Generating State Variables ---%SI = [0 0]';Sc = SI;S = zeros(ssize+disc, m);for j = 1:ssize+disc

Sa = PS*Sc+(eps(j,:))' ;S(j,:) = Sa';Sc = S(j,:)' ;

endS(1,:) = SI;

%--- Generating Control Variables ---%X = (PX*S')';

Y = [X, S];kt = Y(disc+1:disc+ssize, 3);zt = Y(disc+1:disc+ssize, 4);

ct = Y(disc+1:disc+ssize, 1);


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lt = Y(disc+1:disc+ssize, 2);yt = zt + alpha*kt + (1-alpha)*lt;it = (YS/IS)*yt - (CS/IS)*ct;wt = yt - lt;

simy = [yt, ct, it, lt, wt];

%--- Hodrick-Prescott Filter ---%simy = hpfilter(simy, lambda)*simy;

%--- Second Moments of Simulated Series ---%sstd(i,:) = std(simy);sstdy(i,:) = sstd(i,:)/sstd(i,1);correlations = corrcoef(simy);scorr(i,:) = correlations(1,:);scorrpl(i,:) = [sstd(i,4)/sstd(i,5), correlations(4,5)];

end

%--- Monte-Carlo Average of Simulated Moments ---%disp('=========================================')disp(' Second Moments of Simulated Series ')disp('=========================================')disp(' ')disp('--- S.D. of y, c, i, l, y/l ---')MCSTD = mean(sstd);disp(MCSTD);

disp('--- S.D. Relative to y ---')

MCSTDY = mean(sstdy);disp(MCSTDY);

disp('--- Correlation with y ---')MCCORR = mean(scorr);disp(MCCORR);

disp('--- S.D.(l vs y/l) & Corr. b/w l and y/l ---')MCCORRPL = mean(scorrpl);disp(MCCORRPL);

%--- Graphics ---%t = 1:length(yt);

set(gca, 'Fontsize', 12);plot(t, yt,'k-', t, ct,'m:+', t, it,'r-.o',...

'Linewidth', 1.0, 'Markersize', 3.0');xlabel('Time Period');ylabel('% Deviation from Steady State');title('Simulated Time Series Data', 'Fontsize', 14);legend('Output', 'Consumption', 'Investment');


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A-2: Difference Equations Solver bksolve.m

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Solving First-Order Difference Equations %%%% by Blanchard and Kahn(1980)'s Method %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [PX, PS, n, m] = bksolve(SS,s,coef)%==============================================%% Coefficient Matrices %%==============================================%%---- Coefficients Associated with Current Variables ----%A = [ coef(:, 1+size(coef,2)/2:size(coef,2))].*SS ;

%---- Coefficients Associated with Forward Variables ----%

B = [ -coef(:, 1:size(coef,2)/2)].*SS ;

% NOTE% A*Y(t) = B*Y(t+1)% Y(t) = inv(A)*B*Y(t+1)

%==============================================%% Eigenvalue Decomposition %%==============================================%

%--- Eigenvalue Decomposition ---%[Q,L] = eig(inv(A)*B); % columns of Q : eigenvectors

% diagonal matrix L : eigenvalues% Q*L*inv(Q) = inv(A)*B

%--- Reordering Eigenvalues and Eigenvectors ---%disp('=========================================')disp(' Eigenvelues ')disp('=========================================')

L = diag(L); % vectorize eigenvaluesdisp(L);[L, order] = sort(abs(L), 'ascend');

% sort eigenvalues in ascending orderQ = Q(:, order); % reorder eigenvectors associated with their eigenvalues

QI = inv(Q); % Q-inverse

%--- # of Eigenvalues Inside/Outside a Unit Circle ---%% 1) # of Control Variables X(t)n = 0;for i = 1:length(L)

if abs(L(i)) < 1.00000n = n + 1;

endend

% 2) # of State Variables S(t)

m = length(L)-n;


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%--- Partition Matrices ---%QI11 = QI(1:n, 1:n);QI21 = QI(n+1:m+n, 1:n);QI12 = QI(1:n, n+1:m+n);QI22 = QI(n+1:m+n, n+1:m+n);

%--- Extracting Stable Roots ---%L1 = L(1:n, :);L2 = L(n+1:m+n, :);VL = diag(L(n+1:n+m,:));

%==============================================%% Solution to the System %%==============================================%

%--- Solution to Control Variables ---%disp('==========================================')disp(' Coefficient Matrix for Control Variables ')disp('==========================================')

PX = -inv(QI11)*QI12;disp(PX);% QI11*X(t) + QI12*S(t) = 0% X(t) = -inv(QI11)*QI12*S(t)

%--- Solution to State Variables ---%disp('==========================================')

disp(' Coefficient Matrix for State Variables ')disp('==========================================')

PS = inv(QI22+QI21*PX)*inv(VL)*(QI22+QI21*PX);PS = real(PS);disp(PS);


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A-3: Hodrick-Prescott Filter hpfilter.m

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% Hodrick-Prescott Filter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [hp_y]=hpfilter(y,lambda);

%--- Input Arguments ---% lambda: smoothing parameter% y: original series%% NOTE% lambda = 100 for yearly data% lambda = 1600 for quarterly data% lambda = 14400 for monthly data% are commonly used.%%--- Output ---% hp_y: filtering matrix%% NOTE% hp_y*y = y_hpfiltered

if size(y,1)

Seitani(2013)Toolkit for DSGE

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