欢迎使用Markdown编辑器

经管之家：Do the best economic and management education！
Tht best econometrics and Computational economy
for example,DSGE Statial econometrics

再次测试使用Markdown 制作论坛文章但是还没掌握好LATEX，其中彩色代码部分已经书写进去，就差LATEX的公式书写正确。这是第二次使用这个新功能，明年在人大经济论坛公开免费讲课DSGE将会使用这个Mardown编辑器生成脚好看的文件。如果可以可以考虑使用英文，在google 或者 baidu 能搜到到我的文章。

我以后会考虑原创DSGE 动态随机一般均衡模型建模吧，包括写自己的程序。今天上传的是李向阳老师的代码。其中插入的LATEX 公式又不是李向阳老师的程序，是Dynare 研究团队里的论文的程序生成的 .tex 文件。原本在我的电脑里的 Texworks 是可以生成公式的。但是插入这个页面是无法显示正确的公式，大概明年二月可以完全掌握使用方法。

如何插入一段漂亮的代码片

选择一款你喜欢的代码片高亮样式，下面展示同样高亮的 代码片.

def get_abs(x):
    if not isinstance(x, (int, float)):
        raise TypeError('bad operand type')
    if x >= 0:
        return x
    else:
        return -x

Rstudio code

    library(rugarch)
    spec=ugarchspec() #spection garch model 

MATLAB code

%% Simple SPBC Model // Read Me First
%
% This tutorial is a collection of example files for a simple sticky-price
% business cycle model (SPBC). The model file and m-files included in the
% tutorial describe a typical workflow in building and operating a
% small-scale DSGE models for practical policy analysis and forecasting.
%

%% How to Best Run This Tutorial?
%
% Each m-file in this tutorial is split into what is called "code sections"
% in Matlab. A code cell is a shorter block of code performing a specific
% task, separated from other code cells by a double percent sign, `%%`
% (usually with a title and brief introduction added). By default, the
% cells are visually separated from each other by a horizontal rule in the
% Matlab editor.
%
% Instead of running each m-file from the command window, or executing this
% `read_me_first` as a whole, do the following. Open one tutorial m-file in
% the Matlab editor. Arrange the editor window and the command window next
% to each other so that you can see both of them at the same time. Then run
% the m-file cell by cell. This will help you watch closely what exactly
% is going on.
%
% To execute one particular cell, place the cursor in that cell (the
% respective block of code will get highlighted), and select "Run Current
% Section" from a contextual menu (upon a right click on the mouse), or
% pressing a keyboard shortcut (which differ on different systems and
% Matlab versions). To learn more on code sections, search Matlab
% documentation for "code section".

clc;

%% Simple Sticky Price Business Cycle Model File
%
% This is a model file for a simple sticky-price business model. The model
% file describes variables, parameters and equations. Note that the model
% file does not specifies the tasks that will be performed with the model.
% The tasks will be set up in separate m-files, using standard Matlab
% functions and IRIS functions.
%
% You cannot run the model file itself. Instead, load the file into Matlab
% using the function |model| function. See the m-file |read_model| for more
% details.
%
% The IRIS model files can be syntax-highlighted in the Matlab editor; this
% makes the files easier to read. To this end, associate the model file
% extension(s) (which can be anything) with the editor. Open the menu File
% - Preferences, and click on the Editor/Debuger - Language tab. Use the
% Add button in the File extensions section to add new extensions, e.g.
% 'model'. Then restart the editor, and that's it.

edit simple_SPBC.model;

%% Read and Solve Model
%
% Create a model ob ject by loading the model file `Simple_SPBC.model`,
% assign parameters to the model ob ject, find its steady state, and compute
% the first-order accurate solution. The model ob ject is then saved to a
% mat file, and ready for further experiments.

% edit read_model.m;
read_model;

%% Get Information About Model ob ject
%
% Use the function `get` (and a few others) to access various pieces of
% information about the model and its properties, such as variable names,
% parameter values, equations, lag structure, or the model eigenvalues. Two
% related topics are furthermore covered in separate files:
% assigning/changing parameters and steady-state values, and accessing
% model solution matrices.

% edit get_info_about_model.m;
get_info_about_model;

%% Assign and Change Parameters and Steady States
%
% Assign or change the values of parameters and/or steady states of
% variables in a model ob ject using a number of different ways. Under
% different circumstances, different methods of assigning parameters may be
% more convenient (but they, of course, all do the same job).

% edit change_parameters_and_sstates.m;
change_parameters_and_sstates;

%% Model Solution Matrices
%
% Describe and retrieve the state-space form of a solved model. IRIS uses a
% state-space form with two modifications. First, the state-space system is
% transformed so that the transition matrix is upper triangular
% (quasi-triangular). Second, the effect of future anticipated shocks can
% be directly computed upon request, and added to the system stored in the
% model ob ject.

% edit know_all_about_solution.m;
know_all_about_solution;

%% Find and Describe Balanced Growth Path
%
% The SPBC.model is a BGP model: It does not have a stationary long run.
% Instead, it has two unit roots, introduced through the productivity
% process, and the general nominal price level. To deal with BGP models,
% there is absolutely no need to stationarise them. They can be worked with
% directly in their non-stationary forms.
       
% edit play_with_bgp.m;
play_with_bgp;

%% Simulate Simple Shock Responses
%
% Simulate a simple shock both as deviations from control and in full
% levels, and report the simulation results.

% edit simulate_simple_shock.m;
simulate_simple_shock;

%% More Complex Simulation Experiments
%
% Simulate the differences between anticipated and unanticipated future
% shocks, run experiments with temporarily exogenised variables, and show
% how easy it is to examine simulations with mutliple different
% parameterisations.

% edit simulate_complex_shocks.m;
simulate_complex_shocks;

%% Simulate Permanent Change in Inflation Target
%
% Simulate a permanent change in the inflation target, calculate the
% sacrifice ratio, and run a simple parameter sensitivity exercise using
% model ob jects with multiple parameterizations.

% edit simulate_disinflation.m;
simulate_disinflation;

%% Monte-Carlo Stochatic Simulations
%
% Draw random time series from the model distribution, and compare their
% sample properties against the unconditional model-implied models. Keep in
% mind that this is a purely simulation exercise, and no observed data
% whatsoever are involved.

% edit resample_from_model.m;
resample_from_model;

%% Import CSV Data Files and Prepare Data
%
% Load basic data from CSV data files into databases where each series is
% represented by a tseries (time series) ob ject. Prepare the data to be
% used later with the model: seasonally adjust, convert to quaterly
% periodicity, and create model-consistent variable names.

% edit read_data.m;
read_data;

%% Simulate Fisher Info Matrix and Test Parameter Identification
%
% Calculate estimates of the Fisher information matrix. The Fisher matrix
% is a property of the model itself, and is independent of any data. It
% represents the maximum amount of information one can hope for to find in
% the data in case the data are really generated by the model DGP.
%
% Compare two approaches: a time-domain approach, and a frequency-domain
% approach. Use the singular value decomposition to learn more about which
% parameters (or combinations of them) are identified the best or the
% worst.

% edit fisher_information_matrix.m;
fisher_information_matrix;

%% Run Bayesian Parameter Estimation
%
% Use bayesian methods to estimate some of the parameters. First, set up
% our priors about the individual parameters, and locate the posterior
% mode. Then, run a posterior simulator (adaptive random-walk Metropolis)
% to obtain the whole distributions of the parameters.

% edit estimate_params.m;
estimate_params;

%% Posterior Simulator with 'saveEvery=' Option
%
% In this file, we show two features of convenience when running larger
% posterior simulations. First, the posterior simulator can be run with the
% option `'saveEvery='` to split the simulated posterior chain into smaller
% bits and saving them each in a separate data file. This is a way to get
% around possible out-of-memory problems when simulating larger models
% and/or longer chains. Second, a large posterior simulation can executed
% incrementally in smaller chunks, with the final state of one simulation
% being used as the initial state for the next one.

% edit more_on_poster_simulator.m;
more_on_poster_simulator;

%% Kalman Filtering and Historical Simulations
%
% Run the Kalman filter on the historical data to back out unobservable
% variables (such as the productivity process) and shocks, and perform a
% number of analytical exercises that help understand the inner workings of
% the model.

% edit filter_hist_data.m;
filter_hist_data;

%% More on Kalman Filter
%
% Run more advanced Kalman filter exercises. Split the data sample into two
% sub-samples, and pass information from one to the other. Run the filter
% with time-varying std deviations of some shocks. Evaluate the likelihood
% function and the contributions of individual time periods to the overall
% likelihood.

% edit more_on_kalman_filter.m;
more_on_kalman_filter;

%% Forecasts with Judgmental Adjustments
%
% Use the Kalman filtered data as the starting point for forecasts, both
% unconditional and conditional, i.e. with various types of judgmental
% adjustments.

% edit forecasts_with_judgment.m;
forecasts_with_judgment;

%% Compare Second Moment Properties in Model and Data
%
% Compute and compare several second-moment properties of the estimated
% model and the data. Describe the data using an estimated VAR; this also
% allows to evaluate sampling uncertainty of the empirical estimates using
% bootstrap methods.

% edit compare_model_and_data.m;
compare_model_and_data;

Dynare code from Xiangyang LI

%Anticipated   shock
%adapted by Xiangyang Li@SCC, 2017-3-4
close all; //close all existing figures
var 
K,  //capital stock
Y, //output
N, // labour
C, //consumption
A, // level technology
a, // log technology
STerm//simplifying variables
;

%usually, exogenous variables has steady state zero
varexo epsA;

parameters beta, theta, tau, alpha, psi, delta, rho, Astar;

beta    =   0.99; //discount factor
theta   =   0.35; //weight  of consumption in utility
tau     =   2; //risk aversion
alpha   =   0.35; //share of capital in production
psi     =  -0.1; //elasticity of substitution of captial and labor
delta   =   0.025; //depreciation rate
rho     =   0.85; //autocorrelation of productivity, i.e. technology persisitence
Astar =   1;// steady state of level of productivity or technology;

model(block,bytecode,cutoff=0);

  //1: Production Technology
  Y = A*(alpha*(K(-1)^psi)+(1-alpha)*(N^psi))^(1/psi);
  
  // 2: Level Technology shock
  A = Astar*exp(a);

  // 3: Log-level Technology
  a = rho*a(-1) + epsA;

  // 4. Resource Constraint
  K = Y-C + (1-delta)*K(-1);

  // 5: Labor supply equation
  ((1-theta)/theta)*(C/(1-N)) - (1-alpha)*(Y/N)^(1-psi);

  // 6: Euler Equation
  (((C^theta)*((1-N)^(1-theta)))^(1-tau))/C  = STerm(1);

  // 7: Simplifying variable
  STerm = beta*((((C^theta)*((1-N)^(1-theta)))^(1-tau))/C)*(alpha*((Y/K(-1))^(1-psi))+(1-delta));

end;

steady_state_model;
a = epsA/(1-rho);
A = Astar*exp(a);
Y_K=((1/beta-1+delta)/alpha)^(1/(1-psi));
C_K=Y_K-delta;
N_K=(((Y_K/A)^psi-alpha)/(1-alpha))^(1/psi);
Y_N=Y_K/N_K;
C_N=C_K/N_K;

% Compute steady state of the endogenous variables.
N=1/(1+C_N/((1-alpha)*theta/(1-theta)*Y_N^(1-psi)));
C=C_N*N;
K=N/N_K;
Y=Y_K*K;
STerm=beta*((((C^theta)*((1-N)^(1-theta)))^(1-tau))/C)
             *(alpha*((Y/K)^(1-psi))+1-delta);
end;

%the economy starts from the inital steady states
%immediately followed by a steady command and this is equvilent to
%initval block with endogenous variables are equal to steady states;
initval;
epsA = 0;
end;

steady;

%in period 5, technology increases by 8% permanently
%and this is anticipated;
%followed immediately by a steady command and this is equvilent to
%endval block with endogenous variables are equal to steady states;
endval;
epsA = (1-rho)*log(1.08);
end;
steady;

%a shocks block is used to maintain technology remains at
%its initial level during periods 1-4;
shocks;
var epsA;
periods 1:4;
values 0;
end;

simul(periods=98);

%built-in Dynare function
%Plots the simulated trajectory of one or several variables.
rplot C ;
rplot K;

// Latex file produce
//r.write();
//r.compile();

DSGE iris toolbox code

%% Assign and Change Parameters and Steady States
%
% Assign or change the values of parameters and/or steady states of
% variables in a model ob ject using a number of different ways. Under
% different circumstances, different methods of assigning parameters may be
% more convenient (while being all equivalent).

%% Clear Workspace
%
% Clear workspace, close all graphics figures, clear command window, and
% check the IRIS version.

clear
close all
clc
irisrequired 20180131
%#ok<*NOPTS>
%#ok<*NASGU>


%% Read Model File and Assign Parameters to Model ob ject
%
% The easiest way to assign or change parameters is simply by using the
% dot-reference, i.e. the name of the model ob ject dot the name of the
% parameter.

m = model('simple_SPBC.model');

m.alpha = 1.03^(1/4);
m.beta = 0.985^(1/4);
m.gamma = 0.60;
m.delta = 0.03;
m.pi = 1.025^(1/4);
m.eta = 6;
m.k = 10;
m.psi = 0.5;

m.chi = 0.80;
m.xiw = 60;
m.xip = 80;
m.rhoa = 0.90;

m.rhor = 0.8;
m.kappap = 2.5;
m.kappan = 0.1;

m.Short_ = 0;
m.Wage_ = 0;

m.std_Mp = 0;
m.std_Mw = 0;
m.std_Ea = 0.1/100;

%% Assign Parameter Database When Reading Model File
%
% Create first a database with the desired parameter values 
% (or use an existing one, for example), and assign the database when
% reading the model file, i.e. when calling the function |model( )|,
% by using the option |Assign=|.

P = struct( );

P.alpha = 1.03^(1/4); 
P.beta = 0.985^(1/4);
P.gamma = 0.60;
P.delta = 0.03;
P.pi = 1.025^(1/4);
P.eta = 6;
P.k = 10;
P.psi = 0.5;

P.chi = 0.80;
P.xiw = 60;
P.xip = 80;
P.rhoa = 0.90;

P.rhor = 0.8;
P.kappap = 2.5;
P.kappan = 0.1;

P.Short_ = 0;
P.Wage_ = 0;

P.std_Mp = 0;
P.std_Mw = 0;
P.std_Ea = 0.1/100;

m = model('simple_SPBC.model', ...
    'assign=', P); 

%% Assign Parameter Database After Reading Model File
%
% Here, use again a parameter database, but assign the database after
% reading the model file, in a separate call to the function |assign( )|.

P = struct( );

P.alpha = 1.03^(1/4);
P.beta = 0.985^(1/4);
P.gamma = 0.60;
P.delta = 0.03;
P.pi = 1.025^(1/4);
P.eta = 6;
P.k = 10;
P.psi = 0.5;

P.chi = 0.80;
P.xiw = 60;
P.xip = 80;
P.rhoa = 0.90;

P.rhor = 0.8;
P.kappap = 2.5;
P.kappan = 0.1;

P.Short_ = 0;
P.Wage_ = 0;

P.std_Mp = 0;
P.std_Mw = 0;
P.std_Ea = 0.1/100;

m = model('simple_SPBC.model');

m = assign(m, P); 

%% Change Parameters in Model ob ject
%
% There are several ways how to change some of the parameters. All the
% following three blocks of code do exactly the same.
%
% * Refer directly to the model ob ject using a model-dot-name notation.

m.chi = 0.9;
m.xip = 100;

%%%
%
% * Use the |assign( )| function and specify name-value pairs.

m = assign(m, 'chi=', 0.9, 'xip=', 100); 

%%%
%
% * Create a database with somenew values, and run |assign( )|.

P = struct( );
P.chi = 0.9;
P.xip = 100;
m = assign(m, P);

%%%
%
% Reset the parameters to their original values.

m.chi = 0.8;
m.xip = 80;

%% Fast Way to Repeatedly Change Parameters
%
% If you need to iterate over a number of different parameterisations, use
% the fast version of the function |assign( )|. First, initialise the fast
% |assign( )| by specifying the list of parameters (and nothing else).
% Then, use |assign( )| repeatedly to pass different sets of values (in the
% same order) to the model ob ject. Compare the time needed to assign 1,000
% different pairs of values for two parameters.

load MAT/read_model.mat m;

chis = linspace(0.5, 0.95, 1000);
xips = linspace(60, 200, 1000);

assign(m, {'chi', 'xip'}); 

tic
for i = 1 : 1000
   m = assign(m, [chis(i), xips(i)]); 
end
toc

tic
for i = 1 : 1000
   m.chi = chis(i);
   m.xip = xips(i);
end
toc

%% Assign or Change Steady State Manually
%
% If you wish to manually change some of the steady-state values (or, for
% instance, assign all of them because they have been computed outside the
% model), treat the steady-state values the same way as parameters.
%

m = sstate(m, 'Growth=', true, 'Solver=', {'IRIS', 'Display=', 'Off'});
chksstate(m)
disp('Steady-state database')
sstate_database = get(m, 'sstate')

%%%
%
% Change both the levels and growth rates of |Y| and |C| using the
% model-dot-name notation.

m.Y = 2 + 1.01i;
m.Pk = 10 + 1.05i;

%%%
%
% Change the steady states for |Y| and |C| using the function |assign| with
% name-pair values.

m = assign(m, 'Y', 2+1.01i, 'Pk', 10+1.05i);

%%%
%
% Do the same as above but separately for the levels and growth rates.

m = assign(m, '-level', 'Y', 2, 'Pk', 10);
m = assign(m, '-growth', 'Y', 1.01, 'Pk', 1.05);

%%%
%
% Change the steady states by creating a database with the new values, and
% passing the database in |assign|.

P = struct();
P.Y = 2 + 1.01i;
P.Pk = 10 + 1.05i;
m = assign(m, P);

%%%
%
% Note that the newly assigned steady states are, of course, not consistent
% with the model.

disp('Check steady state -- it does not hold');
[flag, listOfEquations] = chksstate(m, 'Error=', false);
flag
listOfEquations

%%%
%
% Reset the steady state to the original values.

m = assign(m, sstate_database);
disp('Check steady state; it holds');
chksstate(m)

如何创建一个注脚

一个具有注脚的文本。¹

tulipsliu的注释。²

KaTeX数学公式

您可以使用渲染LaTeX数学表达式 KaTeX:

Gamma公式展示 $\Gamma(n) = (n-1)!\quad\forall n\in\mathbb N$ 是通过欧拉积分

$\Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt\,.$

My Katex [A DSGE Euantations]

Equation 1

$0=\frac{\frac{{{\epsilon_w}}\, \left(1+{{\tau_c}}\right)\, {{N}_{t}}^{{{\varphi}}}}{{{C}_{t}}^{\left(-{{\sigma}}\right)}}}{{{\frac{W}{P}}_{t}}}+\left(1-{{\epsilon_w}}\right)\, {{\tau^{s}_w}}\, \left(1-{{\tau_n}_{t}}\right)-\frac{\frac{{{\Pi}_{t}}\, {{\phi_w}}\, \left(\frac{{{\Pi^w}_{t}}}{(\bar{{\Pi}})}-1\right)}{(\bar{{\Pi}})}\, {{Y}_{t}}}{{{N}_{t}}\, {{\frac{W}{P}}_{t}}}+\frac{\frac{{{\beta}}\, {{Z}_{t+1}}\, {{C}_{t+1}}^{\left(-{{\sigma}}\right)}}{{{C}_{t}}^{\left(-{{\sigma}}\right)}\, {{Z}_{t}}}}{{{N}_{t}}\, {{\frac{W}{P}}_{t}}}\, \frac{{{\Pi^w}_{t+1}}\, {{\phi_w}}\, \left(\frac{{{\Pi^w}_{t+1}}}{(\bar{{\Pi}})}-1\right)}{(\bar{{\Pi}})}\, {{Y}_{t+1}}$

Eqation 2

${{N^d}_{t}}={{N}_{t}}$

Equation 3

${{\Pi^w}_{t}}={{\Pi}_{t}}\, \frac{{{\frac{W}{P}}_{t}}}{{{\frac{W}{P}}_{t-1}}}$

Equation 4

${{Q}_{t}}=\frac{{{\beta}}\, \left(\frac{{{C}_{t+1}}}{{{C}_{t}}}\right)^{\left(-{{\sigma}}\right)}\, \frac{{{Z}_{t+1}}}{{{Z}_{t}}}}{{{\Pi}_{t+1}}}$

Equation 5

${{R^n}_{t}}=\frac{1}{{{Q}_{t}}}$

Equation 6

${{Y}_{t}}={{A}_{t}}\, \left(\frac{{{N^d}_{t}}}{{{S}_{t}}}\right)^{1-{{\alpha}}}$

Equation 7

${{R^n}_{t}}={{\Pi}_{t+1}}\, {{R^{r}}_{t}}$

Equation 8

${{R^n}_{t}}=\frac{1}{{{\beta}}}\, \left({{\Pi}_{t}}^{\frac{{{\lambda_{w}}}}{{{\lambda_{w}}}+{{\lambda_{p}}}}}\, {{\Pi^w}_{t}}^{\frac{{{\lambda_{p}}}}{{{\lambda_{w}}}+{{\lambda_{p}}}}}\right)^{{{\phi_{\pi}}}}$

Equation 9

${{C}_{t}}={{Y}_{t}}\, \left(1-\frac{{{\phi_w}}}{2}\, \left(\frac{{{\Pi^w}_{t}}}{(\bar{{\Pi}})}-1\right)^{2}\right)$

Equation 10

$log\left({{A}_{t}}\right)={{\rho_a}}\, log\left({{A}_{t-1}}\right)+{{\varepsilon_a}_{t}}$

Equation 11

$log\left({{Z}_{t}}\right)={{\rho_{z}}}\, log\left({{Z}_{t-1}}\right)-{{\varepsilon_z}_{t}}$

Equation 12

${{\nu}_{t}}={{\rho_{\nu}}}\, {{\nu}_{t-1}}+{{\varepsilon_\nu}_{t}}$

Equation 13

${{mc}_{t}}=\frac{{{\frac{W}{P}}_{t}}\, {{\tau^{s}_p}}}{{{S}_{t}}\, \frac{{{Y}_{t}}\, \left(1-{{\alpha}}\right)}{{{N^d}_{t}}}}$

Equation 14

$1={{\theta_p}}\, {{\Pi}_{t}}^{{{\epsilon_p}}-1}+\left(1-{{\theta_p}}\right)\, {{\Pi^*}_{t}}^{1-{{\epsilon_p}}}$

Equation 15

${{S}_{t}}=\left(1-{{\theta_p}}\right)\, {{\Pi^*}_{t}}^{\frac{\left(-{{\epsilon_p}}\right)}{1-{{\alpha}}}}+{{\theta_p}}\, {{\Pi}_{t}}^{\frac{{{\epsilon_p}}}{1-{{\alpha}}}}\, {{S}_{t-1}}$

Equation 16
${{\Pi^*}_{t}}^{1+{{\epsilon_p}}\, \frac{{{\alpha}}}{1-{{\alpha}}}}=\frac{{{\epsilon_p}}\, \frac{{{x_1}_{t}}}{{{x_2}_{t}}}}{{{\epsilon_p}}-1}$

Equation 17
${{x_1}_{t}}={{mc}_{t}}\, {{Y}_{t}}\, {{C}_{t}}^{\left(-{{\sigma}}\right)}\, {{Z}_{t}}+{{\beta}}\, {{\theta_p}}\, {{\Pi}_{t+1}}^{{{\epsilon_p}}+\frac{{{\alpha}}\, {{\epsilon_p}}}{1-{{\alpha}}}}\, {{x_1}_{t+1}}$

Equation 18

${{x_2}_{t}}={{Y}_{t}}\, {{C}_{t}}^{\left(-{{\sigma}}\right)}\, {{Z}_{t}}+{{\beta}}\, {{\theta_p}}\, {{\Pi}_{t+1}}^{{{\epsilon_p}}-1}\, {{x_2}_{t+1}}$

Equation 19

${{\log(Y)}_{t}}=log\left({{Y}_{t}}\right)$

Equation 20

${{\log(W/P)}_{t}}=log\left({{\frac{W}{P}}_{t}}\right)$

Equation 21

${{\log(N)}_{t}}=log\left({{N}_{t}}\right)$

Equation 22

${{\pi^{ann}}_{t}}=4\, log\left({{\Pi}_{t}}\right)$

Equation 23

$4
{{\pi^{w,ann}}{t}}=4, log\left({{\Pi^w}{t}}\right)
$$

Equation 24

}
${{i^{ann}}_{t}}=4\, log\left({{R^n}_{t}}\right)$

Equation 25

${{r^{r,ann}}_{t}}=4\, log\left({{R^{r}}_{t}}\right)$

Equation 26

${{M/P}_{t}}=\frac{{{Y}_{t}}}{{{R^n}_{t}}^{{{\eta}}}}$

Equation 27

${{\log(A)}_{t}}=log\left({{A}_{t}}\right)$

Equation 28
${{\log(\Pi)}_{t}}=log\left({{\Pi}_{t}}\right)$

Equation 29

${{\log(\Pi_w)}_{t}}=log\left({{\Pi^w}_{t}}\right)$

Equation 30

${{\log(Z)}_{t}}=log\left({{Z}_{t}}\right)$

Equation 31

${{\tau_n}_{t}}=\left(1-{{\rho_{\tau^n}}}\right)\, {{\bar {\tau^n}}}+{{\rho_{\tau^n}}}\, {{\tau_n}_{t-1}}+{{\varepsilon_{\tau^n}}_{t}}$

Equation 32

${{\mathbb{W}}_{t}}={{Z}_{t}}\, \left(log\left({{C}_{t}}\right)-\frac{{{N}_{t}}^{1+{{\varphi}}}}{1+{{\varphi}}}\right)$

% Equation 33

${{X^{W}}_{t}}=1$

Equation 34

${{\mathbb{V}}_{t}}={{\mathbb{W}}_{t}}+{{\beta}}\, {{\mathbb{V}}_{t+1}}$

Equation 35

${{\mathbb{V^{nat}_{gap}}}_{t}}={{Z}_{t}}\, \left(log\left(\left(1-{{\lambda}}\right)\, {{Y^{nat}}_{t}}\right)-\frac{{{N^{nat}}_{t}}^{1+{{\varphi}}}}{1+{{\varphi}}}\right)+{{\beta}}\, {{\mathbb{V^{nat}_{gap}}}_{t+1}}$
Equation 36

${{\mathbb{V^{gap}}}_{t}}={{\mathbb{V}}_{t}}-{{\mathbb{V^{nat}_{gap}}}_{t}}$
Equation 37
${{MPN}_{t}}=\frac{\left(\frac{{{N^d}_{t}}}{{{S}_{t}}}\right)^{1-{{\alpha}}}\, {{A}_{t}}\, \left(1-{{\alpha}}\right)}{{{N^d}_{t}}}$

Equation 38
${{MRS}_{t}}=\frac{{{N}_{t}}^{{{\varphi}}}\, \frac{1+{{\tau_c}}}{1+{{\tau_n}_{t}}}}{{{C}_{t}}^{\left(-{{\sigma}}\right)}}$
Equation 39
$\frac{\frac{{{\epsilon_w}}\, \frac{\frac{1+{{\tau_c}}}{1+{{\tau_n}_{t}}}\, {{N^{nat}}_{t}}^{{{\varphi}}}}{\left({{A}_{t}}\, {{N^{nat}}_{t}}^{1-{{\alpha}}}\right)^{\left(-{{\sigma}}\right)}}}{{{\epsilon_w}}-1}}{{{\tau^{s}_w}}}=\frac{\frac{\left({{\epsilon_p}}-1\right)\, {{A}_{t}}\, \left(1-{{\alpha}}\right)\, {{N^{nat}}_{t}}^{\left(-{{\alpha}}\right)}}{{{\epsilon_p}}}}{{{\tau^{s}_p}}}$
Equation 40
${{Y^{nat}}_{t}}={{A}_{t}}\, {{N^{nat}}_{t}}^{1-{{\alpha}}}$

Equation 41
${{\log{Y^{gap}}}_{t}}=log\left({{Y}_{t}}\right)-log\left({{Y^{nat}}_{t}}\right)$

• 参考资料
  [1]:Tulipsliu,2019-12-4《# 关于R语言下tGarch-copula-CoVaR 系统风险程序与教学》7523839-1-1.html
  [2]:Tulipsliu,2019-09-20,《# MATLAB 最新的意大利人TommasoBelluzzo-SystemicRisk-2b4ebd0程序分享》