bamMCMC

Purpose

Main procedure for conducting MCMC-Gibbs style sampling of models.

Format

bamOut0 = bamMCMC(myData, MCCtl0)
Parameters:

  • myData (struct) –

    An instance of the dgpControl structure. For an instance of the dgpcontrol structure named myData, the members are:

    model

    String, specifies model type. Must be one of the following GAUSS supported model specifications:

    • ”Linear” - Univariate or multivariate linear model.

    • ”Auto” - Univariate linear model with autoregressive error terms.

    • ”HB” - Hierarchical bayes estimation.

    • ”Probit” - Binary choice probit model.

    • ”Logit” - Binary choice logit model.

    numObs

    Scalar, number of observations to be generated per subject.

    numSubjects, NTot

    Scalar, total number of subjects and total number of observations, respectively.

    numMix

    Scalar, number of mixing components.

    trueBeta

    Matrix, numX x numY, true beta coefficients for linear dependency.

    trueSigma

    Matrix, numX x numY, true standard deviation of error terms.

    trueLambda

    Matrix, numX x numX, upper triangular matrix for HB model standard deviation of coefficients in stage one equation.

    trueTheta

    Matrix, numZ x numX, HB linear dependency coefficients in stage one equation.

    Member

    Description

    trueInterceptX

    Scalar, intercept value for all x equations.

    trueTrendX

    Scalar, coefficient on linear trend in x equation.

    trueInterceptZ

    Scalar, intercept value for all z equations.

    trueTrendZ

    Scalar, coefficient on linear trend in z equation.

  • MCCtl0 (struct) –

    An instance of the bayesControl structure. For an instance of the bayesControl structure named MCCtl0 the members are:

    MCCtl0.model

    String, specifies model type. Must be one of the following GAUSS supported model specifications:

    • ”Linear” - Univariate or multivariate linear model.

    • ”Auto” - Univariate linear model with autoregressive error terms.

    • ”HB” - Hierarchical bayes estimation.

    • ”Probit” - Binary choice probit model.

    • ”Logit” - Binary choice logit model.

    MCCtl0.savedIter

    Scalar, number of sampler iterations to be saved.

    MCCtl0.saveSkip

    Scalar, number of sampler iterations to skip between saving iterations.

    MCCtl0.burnIter

    Scalar, number of sampler burn-in iterations.

    MCCtl0.MLE

    Scalar, indicator parameter set to one to use MLE to find initial values.

    MCCtl0.printOut

    Scalar, indicator parameter set to one to print output details to screen.

    MCCtl0.graphOut

    Scalar, indicator parameter set to one to produce post sampler graphs of posterior.

    MCCtl0.Intercept

    Scalar, parameter indicating deterministic trends to include in the model:

    • 0 for deterministic trends

    • 1 for intercept only

    • 2 for trend and intercept

    • 3 for trend only

    MCCtl0.numX

    Scalar, number of dependent variables, when relevant.

    MCCtl0.numZ

    Scalar, number of first stage Z dependent variables, when relevant.

    MCCtl0.numLags

    Scalar, number of autoregressive lags, when relevant.

    MCCtl0.numMix

    Scalar, number of mixtures components, when relevant.

    MCCtl0.numSubjects

    Scalar, number of subjects, when relevant.

Returns:

bamOut0 (struct) –

An instance of the bayesOut structure. For an instance of the bayesOut structure named bamOut0, the members are:

bamOut0.estBeta

Matrix, stored draws of the posterior distribution of beta.

bamOut0.estSigma

Matrix, stored draws of the posterior distribution of sigma.

bamOut0.estRho

Matrix, stored draws of the posterior distribution of rho.

bamOut0.estLambda

Matrix, stored draws of the posterior distribution of lambda.

bamOut0.estTheta

Matrix, stored draws of the posterior distribution of theta.

bamOut0.estPhi

Matrix, stored draws of the posterior distribution of phi.

bamOut0.estZProb

Matrix, stored draws of the posterior distribution of individual level class membership probabilities.

bamOut0.betaMean

Matrix, means of the posterior distributions of beta coefficients.

bamOut0.betaStd

Matrix, standard deviation of the posterior distributions of beta coefficients.

bamOut0.sigmaMean

Matrix, means of the posterior distributions of sigma.

bamOut0.sigmaStd

Matrix, standard deviation of the posterior distributions of sigma.

bamOut0.rhoMean

Matrix, means of the posterior distributions of rho coefficients.

bamOut0.rhoStd

Matrix, standard deviation of the posterior distributions of rho coefficients.

bamOut0.phiMean

Matrix, means of the posterior distributions of phi coefficients.

bamOut0.phiStd

Matrix, standard deviation of the posterior distributions of phi coefficients.

bamOut0.lambdaMean

Matrix, means of the posterior distributions of lambda coefficients.

bamOut0.lambdaStd

Matrix, standard deviation of the posterior distributions of lambda coefficients.

bamOut0.thetaMean

Matrix, means of the posterior distributions of theta coefficients.

bamOut0.thetaStd

Matrix, standard deviation of the posterior distributions of theta coefficients.

bamOut0.yhat

Matrix, predicted y-values using the posterior coefficients distributions.

bamOut0.resid

Matrix, residuals using predicted y-values.

bamOut0.cy

Matrix, correlation between y and y-hat.

bamOut0.cy2

Matrix, r-squared.

bamOut0.estLL

Matrix, estimation log-likelihood.

Examples

Linear model with AR error terms

Step One: Data Generation

The first step when using bamMCMC() is to identify the data for the model. For this model, we will use the dataGen() procedure to generate our data. This data must then be placed in the dgpOut structure.

new;
cls;
library bet;

// Data generation control structure
struct dgpControl GC0;

// Specify autoregressive model
GC0.Model = "AUTO";

// Turn on plotting of generated data
GC0.PlotData = 1;

// Specify number of observations
GC0.NumObs = 100;

// Specify model parameters
GC0.TrueBeta = -3.1|2|-1.0|3;
GC0.TrueRho = 0.3|-0.5;

// Error variance
GC0.TrueSigmaE = 2;

// X Error Variance
GC0.TrueSigmaX = 2;
GC0.TrueTrendX = 0;
GC0.TrueInterceptX = 0;

// Call data generation function
struct dgpOut myData;
myData = DataGen(GC0);

Step Two: Initialize Parameters for MCMC

Next, we set up the specifications for the MCMC simulation.

// Declare instance of the bayesControl structure
struct bayesControl MCCtl0;

// Specify AR(2) model
MCCtl0.model = "AUTO";
MCCtl0.numLags = 2;

// Number of saved iterations
MCCtl0.SavedIter = 4000;

// Save skipped iterations
MCCtl0.SaveSkip = 1;

// Number of burn-in iterations
MCCtl0.BurnIter = 1000;

// No intercept
MCCtl0.InterceptX = 0;

// Turn of MLE for start values
MCCtl0.MLE = 0;

// Control printing and graphs
MCCtl0.printGraph = 1;
MCCtl0.printOut = 1;

Step Three: Perform MCMC

The final step is to call bamMCMC() to perform MCMC simulation.

// Step Three: MCMC
struct bayesOut BAMSt0;
bamOut0 = bamMCMC(myData, MCCtl0);

Linear model with loaded data

In this example, the loadd() procedure is used to load the model data.

Step One: Load data

new;
library bet;

// Load data from gbs_auto.gdat file
data = loadd(__FILE_DIR $+ "gbs_ato.gdat");

// Call data generation function
struct dgpOut myData;
mydata.ydata = data[., "Y"];
myData.xdata = ones(rows(data), 1)~data[., "X"];

Step Two: Initialize Parameters for MCMC

// Declare instance of the bayesControl structure
struct bayesControl MCCtl0;

// Specify AR(2) model
MCCtl0.model = "AUTO";
MCCtl0.numLags = 1;

// Number of saved iterations
MCCtl0.SavedIter = 4000;

// Save skipped iterations
MCCtl0.SaveSkip = 1;

// Number of burn-in iterations
MCCtl0.BurnIter = 1000;

// No intercept
MCCtl0.InterceptX = 0;

// Turn off MLE for start values
MCCtl0.MLE = 0;

// Control printing and graphs
MCCtl0.printGraph = 1;
MCCtl0.printOut = 1;

Step Three: Perform MCMC

// Define storage structure
struct bayesOut BAMSt0;
BAMSt0 = bamMCMC(myData, MCCtl0);

Remarks

The bamMCMC() procedure is the main procedure for model estimation in BET. It will can be used to estimate a number of different models. The type of model to be estimated is specified in the bayesControl structure.

Estimation with the BET library using bamMCMC() requires three steps:

  1. Data loading or generation

    The BET library allows you to input data using standard GAUSS data loading tools, such as loadd(). However, it also provides a complete suite of data generation tools that allow users to specify true data parameters and build hypothetical data sets for analysis. Whether user defined or GAUSS generated, the dgpOut structure is used to input data into the bamMCMC() procedure.

  2. Initialize the MCMC

    The next step is to setup the parameters of the MCMC simulation using the bayesControl structure. This includes the:

    • Model

    • Number of saved iterations

    • Number of iterations to skip

    • Number of burn-in iterations

    • Total number of iterations

    • Inclusion of intercept

    • Plotting behavior

  3. Perform bayesian analysis

    The final step is to call the bamMCMC() procedure using dgpOut data structure along with the bayesControl structure. In this step, GAUSS performs Markov Chain Monte Carlo numerical simulation, combined with assumed statistical structures and priors, to numerically compute parameter posterior distributions.

    In addition to producing graphs of all MCMC iterations for all parameters and posterior distributions for all parameters, this procedure has one return structure the bayesOut structure. The bayesOut structure includes:

    • Draws for all parameters at each iteration

    • Posterior Mean for all parameters

    • Posterior standard deviation for all parameters

    • Predicted values

    • Residuals

    • Correlation matrix between \(Y`\) and \(\hat{Y}\)

    • PDF values and corresponding PDF grid for all posterior distributions

    • Log-likelihood value (when applicable)

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