讲解Programming留学生、Matlab程序语言调试、辅导Matlab、讲解Simulate Dynamic Models
讲解Jav
International Macroeconomics
Contents
1 Programming I: Introduction to Matlab 1
2 Programming II: Use of Dynare to Simulate Dynamic Models 8
2.1 Baseline Theoretical Example . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Steady Operationalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Back to Dynamic Equations in Example Model . . . . . . . . . . . . . . . . 12
2.4 Summary of Parameters and Steady State Values . . . . . . . . . . . . . . . 13
2.5 Dynare Timing Convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.6 The Dynare Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6.1 Declaring Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6.2 Declaring Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6.3 Declaring the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6.4 Initial Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6.5 Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6.6 Stochastic Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6.7 Why Do All This? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6.8 Some Additional Things to Note . . . . . . . . . . . . . . . . . . . . . 18
2.7 Matlab Code for General Control and Obtaining Results . . . . . . . . . . . 20
1 Programming I: Introduction to Matlab
This section is designed as a “do it yourself” part so that you can familiarize yourself
with the use of Matlab.
Open Matlab, and you will get something like in the figure below (exact appearance
varies by Matlab version).
1? Note that the directory to which Matlab is pointing is on top in the middle, in the case
of the example above C:, and you can always change the directory to operate Matlab
from any directory that you wish. How to change the directory may differ depending
on the version of Matlab. In the case in the figure above there is a dropdown menu
just right C: (see the downward pointing arrow on the right of the directory path).
In the top left part of the Matlab interface shown above there is the “New Script”
button. Pressing this, or also going through File → New → Script you can create a
new .m file, which is what is used to run Matlab code.
In the command window (in the case of the figure above it’s right in the middle) you
can see the command prompt after which Matlab code is introduced.
A few icons to the right of the New Script button is the New Variable butt, which, of
course, is used to create a new variable.
We’ll go over all of the preceding Matlab notes in greater detail following below.
For now, note from the figure above that there are 5 panels: (1.) Current Folder
(top left); (2.) Details (bottom left); (3.) Command Window (in the middle); (4.)
Workspace (top right); (5.) Command History (bottom right).
– You can move around these panels and organize them in whatever way you like,
but these are the most useful windows to keep handy.
2– If any of these windows is not visible you can activate them by using the Layout
button, which in the case of the figure above is to the right of the New Variable
button.
Matlab interprets all variables (or parameters) as matrices.
In the command window type (all codes I indicate should be written in Matlab verbatim)
X = 1 and press enter (note that the variable X, which you just created, now
appears in the Workspace window). Matlab interprets this as a 1 × 1—that is, 1 row
and 1 column).
Note: Matlab understands capitalization. For example, type in the command prompt
x = 1 and you’ll see that now both X and x exist as different variables.
Now, let’s create a row vector of 1 × 5 (i.e., a vector with 1 row and 5 columns). Write
in the command prompt Y = [1 2 3 4 5] and press enter.
To obtain instead a column vector of 5 × 1 (i.e., a vector with 5 rows and 1 column)
type in the command prompt, for instance, Z = [6; 7; 8; 9; 10] (in this case the
semi colons serve to indicate to Matlab that we are moving on to a new row in the
vector).
You can transpose a vector or a matrix. Type Z = Z’ in the command prompt and
press enter (now Z is a vector with 1 row and 5 columns; note that the name Z now
refers to the new form of this variable and no longer to its original column vector
format).
Now, let’s create a 2×5 matrix. Write in the command prompt W = [1 2 3 4 5; 6 7
8 9 10] and press enter. Given what we’ve done so far we could have also generated
W by writing in the command prompt W = [Y; Z] (try it).
If you want to create some variable like we did before but we don’t want the result
to appear in the command window, simply put in a semi colon after the relevant
command. For instance, type W2 = [1; 2]; in the command prompt, press enter,
and then type W2 in the command prompt and press enter to “see” this vector in the
command window (so, in the preceding step we in fact created the vector W2 but its
statement was suppressed from the command window at the time of its creation).
Of course, as long as the dimensions of a matrix allow it, you can perform standard
algebraic operations using Matlab.
Let’s start all over. In the command window type clear all and press enter; note
that all variables that were in the workspace panel have been cleared. Yet, there’s still
clutter in the command window. To get rid of this clutter type clc and press enter.
Now, type in the command prompt X = [1; 2; 3]; and press enter; and do the same
for Y = [4; 5; 6]; and Z = [7 8 9];
Finally, type in W = 2; and press enter.
3 What we’ve done just now is create two 3 × 1 matrices, one 1 × 3 matrix, and a scalar
matrix.
Let’s perform some addition. Write in the command prompt XplusY = X + Y and
press enter.
You can only add matrices that have the same number of rows and columns.
For instance, write in the command prompt XplusZ = X + Z and press enter (see what
happens).
The exception to the preceding addition rule occurs in the case of scalar matrices. For
example, write in the command prompt XplusW = X + W and see what happens.
– These two matrices have different dimensions, but since W is a scalar matrix Matlab
adds W to each element of X.
Subject to matrix algebra rules, you can perform any algebraic operation that you
want, including:
– Addition ( + );
– Subtraction ( -- );
– Multiplication ( * );
– Division ( / );
– and exponentiation ( ^ ).
In Matlab there is a difference between multiplication (or division or exponentiation)
“per se” and:
– Multiplication (or division or exponentiation) “element by element.”
– We now proceed to explore this difference.
Write in the command prompt X = [X Y] and press enter; note that we have created
a new matrix X that is 3 × 2 and replaces the old version of X but using first that old
version to create its new version X.
Let’s move on to other fun stuff.
Write in the command prompt Z(1,4) = 10 and press enter; what we’ve just done is
add an element to Z by telling Matlab to put in the first row, fourth column of Z the
number 10.
Now, write in the command prompt Z(2,1:3) = Z(1,2:4) and press enter; you’ll see
that what we’ve just done is create a second row for Z telling Matlab to fill the second
row, columns 1 through 3 of Z (the colon in Matlab language roughly means “through”
and in this case “from 1 to 3”) with the elements of row 1, columns 2 through 4 of Z.
4 Note also that any element that one does not explicitly assign content to will automatically
be filled with a 0 (in our example the element in the second row, fourth column
of Z).
Write in the command prompt XtimesZ = X*Z and press enter.
The preceding step is “multiplication per se.” X is 3 × 2 matrix and Z is a 2 × 4 so the
result of multiplying them is a 3 × 4 matrix.
For matrix multiplication to be possible, the matrix on the left hand side of the multiplication
has to have the same number of columns as the matrix on the right hand
side of the multiplication.
For example, type in the command prompt XtimesY = X*Y and press enter (see what
happens).
Just like with addition, the exception to this multiplication rule is when we’re dealing
with multiplication and one of the matrices involved is a scalar matrix. To see this
issue write in the command prompt XtimesW = X*W (note that each element of the
matrix X has been multiplied by the scalar W and we obtain as a result a 3 × 2 matrix).
Up until now we’ve been creating matrices and calculating stuff using the method
of naming said stuff prior to creating but if we just want to see the result of some
operation simply type, for instance, X*Z and press enter.
Now, write X*X and press enter; you’ll see that you obtain an error message since the
dimensions of the matrix do not allow multiplication per se.
Instead, type in the command prompt X.*X and press enter. What’s happened now is
that the operation does take place given the .* notation, which indicates that multiplication
must be carried out “element by element” from the point of view that each
element of a matrix is multiplied by the same corresponding element in the other
matrix.
– Of course, for element by element multiplication to be feasible the matrices that
are being multiplied have to have the same number of rows and columns.
All of the preceding regarding the difference between “per se” and “element by element”
also applies to division and exponentiation.
We’ll wrap up with a few additional things that are useful to know.
If you want to know the size of a matrix write in the command prompt, for instance,
size(Z), which will yield as a result the number of rows and columns (in that order)
of the matrix Z.
Continuing with our examples, if you want to see the elements in the first row of the
matrix Z type in the command window Z(1,:) and press enter.
5– In this case the colon indicates that all elements of the indicated column are to
be selected.
Similarly, if you want to see all the elements of the second column of Z type in the
command prompt Z(:,2) and press enter.
Of course, if you wanted to see all elements in the first row, columns 2 through 4 of Z
you write in the command prompt Z(1,2:4) and press enter.
– Etc.
Finally, if you want to create a matrix of zeros with n rows and m columns, where n
and m are integers, you write in the command prompt zeros(n,m) and press enter.
Similarly, if you want to create a matrix of “ones” with n rows and m columns you
write in the command prompt ones(n,m) and press enter.
Instead of writing all of the previous in the command window it’s much better to create
an “.m file” to run things for us. Let’s go ahead and do that. Go to File → New →
Script and you’ll see a new window pop up like in the figure below.
This new window/file will be saved automatically with the extension .m and is known
as an “.m file.”
Now, go to “Save” and create a folder wherever you like that’s called, for example,
“Intro to Matlab” and save in this folder your .m file giving it the name, for instance,
“Practice”
6 Now, change the Matlab directory to point to the new folder that you just created.
Note that in the .m file there is a green arrow around the top middle part (see the
figure above). By clicking this arrow the .m file will “run.”
As an example, copy and paste the following (which is basically all that we’ve done so
far) in your .m file, save it, and then press the noted green arrow so that the file runs.
clear all
clc
%Create a square matrix X
X = [1 2 3; 4 5 6; 7 8 9];
%Find out what size X has
size(X)
%Create a matrix that’s equal to the third row of X
XColThree = X(:,3);
%Create a matrix that’s equal to the first row, second column of X
XRowsOneAndTwo = X(1:2,:);
%Obtain X squared
X^2
%Obtain the element-by-element square of X
X.^2
%Create a column matrix Y
Y = [10; 11; 12];
%Multiply X and Y
X*Y
%Create a 1x2 matrix of zeros
zeros1 = zeros(1,2);
%create a 2x2 matrix of ones
ones1 = ones(2,2);
In the code above some variables have been suppressed from the command window
using the semi colon notations and other have not, on purpose, just so you can see the
different combos that can be obtained.
Note that everything preceded by % is interpreted by Matlab as a comment and ignored
when running the code.
Our .m file from above creates a bunch of variables in the Workspace panel.
Find the “Save Workspace” button, which is by the “New Variable” button. This
button serves to save the variables that are in the Workspace (alternatively you can
go to File → Save Workspace as...).
Save your workspace in your Intro to Matlab folder giving it the name, for instance,
results.mat (note that the extension .mat is used automatically to save these types of
files).
7 Now, close your .m file and write and enter clear all and then clc
To load the .mat file that you created where you have your saved variables you can
now type load results
As you’ll see, all the variables that we had created are once more in the workspace.
Also, if you want to reopen your .m simply go to File → Open and select the file from
the folder you saved it in.
Similarly, instead of using the load command to load your .mat file you can go to File
→ Open and open your file by locating it where you saved it.
2 Programming II: Use of Dynare to Simulate Dynamic
Models
Dynare is not a program in itself, but, rather, a collection of Matlab Codes.
To start working with Dynare, open Matlab and check what version of Dynare you
have installed, for example, 4.5.7, and type the following in the Matlab command
promptaddpath c:\dynare\ 4.5.7\matlab and press enter.
To use Dynare you have to create .mod files (these files can be obtained by opening a
script in Matlab and then saving it using the extension .mod instead of .m).
Suppose that our Dynare file is called RBC.mod
Make sure that this file is saved where the Matlab directory is pointing to.
To run the Dynare file, simply type in the Matlab command prompt
dynare RBC
and press enter.
Note: for more examples and manuals you can go to the Dynare website: www.dynare.org.
I also posted a few lecture notes from other professor to the class website.
2.1 Baseline Theoretical Example
We’ll use as an example a typical closed economy RBC model developed from the point
of view of a centralized planning problem.
The planner’s problem is:(1)
where: C is consumption; H is work hours; K is the capital stock; Et
is the expectation
operator; β ∈ (0, 1) is the subjective parametric discount factor; γ > 0 is a parameter
that governs the disutility of labor; ε is the Frisch elasticity of labor supply; Y is
output; I is investment; G ≥ 0 is the parametric value of government consumption;
δ > 0 is the depreciation rate of capital; A is technology; α ∈ (0, 1) is a parameter;
ρ ∈ (0, 1) is the productivity process persistence parameter; and ξ ～ N(0, σ2). Of course, the constraints relevant to the optimization problem can be combined into
a single constraint by getting rid of investment and output in the first constraint:
Ct + G + Kt+1(1 δ)Kt = AtH. (2)
9 Combining the first and second first order conditions:(3)
Combining the first and third first order conditions:(4)
Jointly, equations (1), (2), (3), and (4) are a system of 4 equations in the 4 key variables:
A, C, H, and K. In steady state, these equations imply that:
– NOTE: Steady state is a situation in which any variable xt has the same
value across periods, i.e., xt = xt+k = x for any integer value of k.
– Since A simply becomes a parameter, in fact the system reduces to one of 3
equations in the 3 key unknowns: C, H, and K. Indeed, note given steady state:
ln At = ρ ln At?1 + ξt
SS
→ ln A = ρ ln A + 0,
i.e., in stead state the shocks to the productivity process are nill, meaning that
in steady state
(1 ρ) ln A = 0 → ln A = 0 → A = 1.
10 Note also that Y as a variable is not essential to solve the system because it is itself
a function of the system’s key variables. So, we can treat Y as a “definition:” Similarly, investment doesn’t appear in any useful place in steady state, so we can treat
it as a definition as well. Using the capital equation of motion: I ≡ δK.
Also, since this is a competitive model, if we were interested in backing out the rental
rate and the wage we could define. Rt ≡ (1 α) AtHα
Typically in these models all variables are normalized by a country’s population, and,
as we’ve assumed implicitly throughout, the price of consumption, and therefore output
and investment, is normalized to 1.
2.2 Steady Operationalization
In order to operationalize the model in steady state system of equations (5) through
(7) we need to assign values to the parameters: β; α; γ; δ; G; and ε. That is, we need
to calibrate the model.
The choice of certain parameters depends on the timing frequency that we assume for
the model’s operationalization. We’ll focus on a quarterly calibration.
The choice of certain parameters is obvious. For example, the Frisch elasticity is usually
set at ε = 4 so that results are consistent with empirical evidence.
Also, cross country data suggest that in general α = 2/3, i.e., the share of labor in
total income is two thirds.
Assuming that β =1
, where r is the real interest rate and historically equal to about
5% in the U.S., then given that the calibration is at quarterly frequency we obtain a
value of r such that: The quarterly depreciation rate in the U.S. is about δ = 0.025, so we assume this value.
Finally, the calibration of the parameters γ and G is not necessarily obvious.
– Recall that the parameter γ affects the disutility of labor, so we can use this
parameter to achieve some target regarding hours worked.
11 In the U.S., since 1960 average work hours per population are approximately
1700 per year. The number of hours in a year is approximately 8760. So,
we can normalize total work hours in a year to 1 and impose that in steady
state H is equal to the average fraction of work hours to total available hours.
That is, one of our calibration target will can be to choose γ such that H =
1700/8760.
Also, the parameter G reflecting government consumption will affect directly
the level of private consumption C. In the U.S. since 1960 the average ratio
of private consumption to output C/Y is approximately 0.63, so our other
calibration target is C/Y =0.63. G will be chosen to hit this target.
– With a simple enough model like this one, you can solve for the implied
values of γ and G necessary to hit these targets by hand. The resulting
values are γ = 8.2156 and G = 0.0847. In a more complex model you would
use Matlab’s algorithm for solving systems of nonlinear equations to solve for the
values of γ and G corresponding to the calibration targets (we will not be doing
this in this class).
– As an example, and to make my life easier, I solve for these values using Matlab’s
“fsolve” algorithm. I wrote a function called “RBCSTEADY.m”, which is called
by “FINDSTEADY.m”. Both can be found in Week 7 folder at class website.
You don’t have to use them, but they are useful and you can always adapt these
codes to more complicated problems.
2.3 Back to Dynamic Equations in Example Model
Recall the key equations from the dynamic version of the model that we used as an
example in the previous section:
and the definition:
It ≡ Kt+1 + (1 δ)Kt.
NOTE: I modified the third equation only for the purposes of demonstration and so
that it actually makes sense to have investment in here, otherwise investment can be
completely done without as far as solving the model is concerned.
122.4 Summary of Parameters and Steady State Values
We’ll use the parameters chosen before. In addition, as is standard in the literature
we’ll assume that at quarterly frequency ρ = 0.95 and σ = 0.01. So, we have:
Parameters
ε α β δ γ ρ σ G
4 2/3 0.988 0.025 8.2156 0.95 0.01 0.0847
We’ll also need for reference the steady state values from the calibrated version of the
model. As can be shown using pen and paper:
Steady State Values
C H K Y
0.3662 0.1941 5.2167 0.5813
and remember that in steady state A = 1.
2.5 Dynare Timing Convention
Dynare is set up with an internal timing convention that requires that we rewrite the
model slightly. In particular, Dynare requires that any predetermined variable (that
is, a state variable, such as the capital stock) be lagged one period. So, for instance,
if in the model a state variable appears with the subscript t then we need to rewrite
this with the subscript t ? 1, and if a state variable appears with the subscript t + 1
then we need to rewrite this with the subscript t. All told, this timing convention is
how we tell Dynare that a state variable is such.
In our model, the only state variable is the capital stock.
So, as far as coding in Dynare goes, our key equations need to be written as such:
as well as the definition:
It ≡ Kt(1 δ)Kt1.
With these “proper” equations in hand we’re ready to start writing our Dynare code
(which, recall, takes place using a .mod file).
13 Note: Just like in Matlab, if an expression starts with % Dynare will ignore it when
solving the model.
2.6 The Dynare Code
2.6.1 Declaring Variables
The first thing we have to do is declare the model variables.
Dynare automatically groups variables as follows: static variables (that is, variable
that only enter the system with a subscript t); purely predetermined variables (that
is, variables that only enter the system with subscripts t and t ? 1); variables that are
both predetermined and forward looking (that is, variables that enter the system with
subscripts t 1, t, and t + 1); and variables that are both static and forward looking
(that is, variables that only enter the system with subscripts t and t + 1).
To decare the endogenous variables we write var followed by these variables.
In our example:
var C H K Z Y I; (The semi colon is necessary!!!).
Then, we declare the exogenous variables, which in our case only amounts to the error
term in the AR(1) process ξ (xi) starting with the expression varexo.
In our example:
varexo xi; (The semi colon is necessary!!!).
2.6.2 Declaring Parameters
Immediately after declaring the variables and we declare the parameters starting with
the expression parameters and followed below by the parameter values. (In all cases
the semicolon is necessary!!!).
In our example:
parameters e a B d g rho sigma G;
e = 4;
a = 2/3;
B = 0.988;
d = 0.025;
g = 8.2156;
rho = 0.95;
sigma = 0.01;
G = 0.1025;
142.6.3 Declaring the Model
Having declared the variables and parameters the next step is declaring the model.
To do so, we write model followed by the model equations and finalizing with the
expression end
– The model that we’re solving is nonlinear, so Dynare will solve it using a linear
approximation whose order we can choose.
To write down the model, if a variable x appears with the subscript t you just write
x. If a variable x appears with the subscript t ? 1 you write down x (?1). And if a
variable x appears with the subscript t + 1 you write x (+1).
For our example we declare the model as follows. (The semi colons are necessary!!!).
model;
I = K - ( 1 - d )*K(-1);
Y = A*( H^a )*( K(-1)^( 1 - a ) );
log( A ) = rho*log( A(-1) ) + xi;
C + I + G = Y;
H = ( ( a/g )*( Y/C ) )^( e/( 1 + e ) );
1/C = B*( 1/C(+1) )*( ( 1 - a )*Y(+1)/K + ( 1 - d ) );
end;
One thing to note is that the IRFs that Dynare generates are in terms of level deviations
from steady state. In other words, the IRF for a variable x, x
IRF , is graphed as
x
IRF = xt x
SS, where SS denotes steady state.
However, often times what one is really interested in seeing, and what is usually presented
in research, is IRFs in terms of percent deviations from steady state. This
problem can be solved doing one of these two:
1. Declare the model as above, then after dynare runs, go and find dynare’s stored
values for each variable, then take the logarithm of those values and steady state
values, and plot the difference.
2. Or use the following trick and write every variable as if they are logs. This
way, you will not have to do any extra calculation. Resulting impulse response
functions will be in terms of percentage deviations. Thus, I will write the model
as follows:
152.6.4 Initial Values
Dynare will (try to) solve for the model’s steady state and use it to initialize the model
simulation. However, Dynare’s algorithm for finding steady states is very sensitive to
initial values, which is why we first find the steady state using pen and paper, or in
the case of a complex model using Matlab.
To declare the initial values we start with the command initval, then we write down
the initial values, and finally we write end as follows. (The semi colons are necessary!!!).
initval;
C = 0.3662;
H = 0.1941;
K = 5.2167;
A = 1;
Y = 0.5813;
end;
Because I declared the variables as logs, initial values should also be declared as logs:
2.6.5 Shocks
The next step is to specify the variance of the productivity shocks. This part of the
code starts with the command shocks, followed by the variance specification, and
ending with the statement end. (The semi colons are necessary!!!).
16shocks;
var xi = sigma^2;
end;
In the next step we write steady so that Dynare calculates the steady state of the
model’s endogenous variables as follows. (The semi colon is necessary!!!).
steady;
2.6.6 Stochastic Simulation
In order to go ahead with the model’s stochastic simulation we write down the command
stoch simul, which solves the model, generates IRFs, etc.
Dynare automatically uses a second order approximation to simulate the model.
There are several options that can be used after this last command. However, if you
only write down stoch simul you will obtain the following:
1. Steady state values of the endogenous variables;
2. A summary of model variables by type;
3. A covariance matrix of the shocks (in our example we would just obtain a scalar);
4. The functions, as they’re known, of policy and transition;
5. First and second theoretical moments;
6. A matrix of theoretical correlations;
7. Theoretical autocovariances up to order 5;
8. IRFs in response to a 1 standard deviation in the productivity process (Dynare
uses a Choleski decomposition).
? In our example we’ll do the following. (The semi colon is necessary!!!).
stoch simul(order=1,irf=40,periods=240); order = 1 makes Dynare solve the model
using a first order approximation. (Unless a model is exceedingly complicated it is not necessary
to use a second order approximation, which, recall, is what Dynare does automatically);
irf=40 makes Dynare generate and graph impulse responses for 80 periods (in our case, quarters);
periods=240 makes Dynare generate simulated time series data for 240 periods (in our
case, quarters). If you want Dynare to minimize the amount of information that will appear
in Matlab’s command window following the simulation routine, use also noprint option.
172.6.7 Why Do All This?
Static models have closed form solutions, such as a typical micro optimization problem
in which you can solve explicitly for each variable.
Dynamic models do not have closed form solutions, unless they are imposed to be in
steady state; so, there is no way to solve the model explicitly for each variable other
than in steady state, which is ultimately why programs like Dynare come in. The true
solution to a dynamic model involves sequences of data implied by the model, i.e., time
series data.
And, to see if a dynamic model works we want to see if the data generated by the
model are consistent with their empirical counterpart.
Using empirical data we can obtain key statistical moments and, using a VAR, we can
also obtain IRFs.
In turn, using Dynare we can obtain the theoretical counterpart of these data and
therefore compare the performance of the model with the reality that it’s trying to
explain in order to gauge how good of a model we’ve come up with.
Even more importantly, if we are happy with how our dynamic model works,
then we can use the model as a laboratory for policy analysis, for instance,
understanding what happens when there is a change in productivity or, say,
government consumption.
2.6.8 Some Additional Things to Note
Given the Dynare timing convention, the IRFs that Dynare generates for predetermined
(i.e., state) variables show the evolution of these variables one period ahead.
So, in our example what we’ll see is the IRF for capital jumping at the time of the
productivity shock, while in fact since capital is a state variable it can’t move at the
moment of the shock. In the following section we’ll see how to correct for it.
In spite of the impact of Dynare’s timing convention for the IRFs that are generated,
the simulated data generated by Dynare is in the correct temporal order.
Once we run the Dynare code, Dynare saves in Matlab’s workspace the simulated time
series for variable under their name, for instance in the case of the artificial time series
for consumption you’ll see simply C, and the IRFs of each variable are stored as the
variable name followed by an underscore and the name of the shock that the variable
is responding to, so in our case the IRF for consumption will be C xi
Also, if you look at the set of variables generated by Dynare, you’ll be able to find the
steady state values under oo → dr -→ ys and the variables there appear in the same
vector column order as they do in the Matlab command window following Dynare’s
run.
18 Some items we get by running the dynare code:
1. Moments of the simulated variables:
2. Correlations between simulated variables:
3. Autocorrelations of simulated variables:
194. And of course Impulse Response Functions (IRFs):
2.7 Matlab Code for General Control and Obtaining Results
First I’m going to show how to use a Matlab script file to write a Dynare code and
then save it as a .mod file.
Second, we’ll run the Dynare code by writing, in our example, dynare RBCDynare in
Matlab’s command window (that’s the name I’ll give our Dynare file).
Then, instead of just writing in Matlab dynare RBCDynare to run the model we’ll create
an .m file (CONTROL.m to be more specific) that will serve the purpose of controlling
a bunch of additional things we’ll want to do, including running the Dynare file and
then generating other stuff of interest with the Dynare simulated data. In particular,
what we’ll do with the general control code is the following:
1. Make the Dynare code run;
2. Generate and graph the correct IRF for capital with it having the same length as
the other IRFs, that is, 40 periods;
3. Extract the steady states generated by Dynare;
204. Obtain the cyclical component of the variables Y , C, I, and H using an HP filter
with smoothing parameter 1600 (The Dynare code that we developed generates
data in levels and quarterly frequency; the HP filter code will be provided).
5. Generate graphs of the cyclical components of the variables using the Matlab
command subplot.
6. Generate a graph of the natural log of output along with its trend component.
7. Use the cyclical component of variables to create a table with:
(a) Their standard deviation;
(b) Their correlation with output;
(c) The one lag autocorrelation of each variable.
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