Posterior distribution of lambda

 

The GAUSS Bayesian Estimation Tools package provides a suite of tools for estimation and analysis of a number of pre-packaged models. The internal GAUSS Bayesian models provide quickly accessible, full-stage modeling including data generation, estimation, and post-estimation analysis. Modeling flexibility is provided through control structures for setting modeling parameters, such as burn-in periods, total iterations and others.

 

Data generation tools for building hypothetical data sets:

Univariate and multivariate linear models

Autoregressive error terms (AR)

Hierarchical Bayes (HB)

Probit and logit data

 

Supported models for Markov Chain Monte Carlo (MCMC) Estimation:

Univariate and multivariate linear models

Autoregressive error terms (AR)

Hierarchical Bayes (HB)

Probit model

Dynamic two-factor model

Structural vector autoregressive (SVAR)

 

Flexible, user defined MCMC estimation parameters including:

Number of saved iterations

Skipped iterations

Burn-in iterations

Total number of iterations

Inclusion of intercept

Optional graph and results output

Elective maximum likelihood estimation (MLE) initialization

 

Thorough computations including:

Draws for all parameters at each iteration

Posterior mean of parameters

Posterior standard deviation of parameters

Predicted variable values and residuals

Correlation matrix between observed and predicted data

PDF values and corresponding PDF graphs

Log-likelihood values (when applicable)

 

 

Sample output report for probit model

 

Model Type: Probit regression model
*************************************************************
Possible underlying (unobserved) choice generation:
Agent selects one alternative:
Y[ij] = X[j]*beta_i + epsilon[ij]
epsilon[ij]~N(0,Sigma)
*************************************************************
Y[ij] is mvar vector
Y[ij] is utility from subject i, choice set j, alternative k
where	i = 1, ..., numSubjects
			j = 1, ..., numChoices
			k = 1, ..., numAlternatives - 1
*************************************************************
X[j] is numAlternative x rankX for choice j
*************************************************************
Pick alternative k if:
Y[ijk] > max( Y[ijl] )
for all k < mvar+1 and l not equal to k
Select base alternative if max(Y)<0
*************************************************************
Observed model:
*************************************************************
Choice vector C[ij] is a numAlternative vector of 0/1
beta_i = Theta'Z[i] + delta[i]
delta[i]~N(0,Lambda)
*************************************************************



Summary stats of independent data

*****************************************
Summary stats for X variables
*****************************************

        Variable        Mean              STD              MIN              MAX 
           X1          0.33333          0.47538             0                1 
           X2          0.33333          0.47538             0                1 
           X3          0.33333          0.47538             0                1 
           X4          0.28648          0.20641        -0.083584          0.71157 
           X5          0.083333         0.59065            -1                1 

*****************************************
Summary stats for Z variables
*****************************************

        Variable        Mean            STD             MIN             MAX 
           Y1         -0.10328         1.1582         -6.1714          3.7266 
           Y2         -0.23821         1.1428         -6.1295          3.2853 
           Y3         -0.28473         1.2776         -5.4752           4.58 

*****************************************
Summary stats for dependent variables
*****************************************

        Variable        Mean             STD             MIN              MAX 
           Y1         -0.10328          1.1582         -6.1714          3.7266 
           Y2         -0.23821          1.1428         -6.1295          3.2853 
           Y3         -0.28473          1.2776         -5.4752            4.58 

***********************************
MCMC Analysis Setup
***********************************
Total number of iterations:     1100.0 
Total number of saved iterations:     1000.0 
Number of iterations in transition period:     100.00 
Number of iterations between saved iterations:     0.0000 
Number of obs:    60.000 
Number of independent variables:    5.0000 
(excluding deterministic terms)
Number of dependent variables:    3.0000 


********************************
MCMC Analysis Results
********************************

***********************************
Error Standard Deviation
***********************************
Variance-Covariance Means(Sigma)

        Equation        Y1               Y2               Y3 
           Y1         0.20831         0.078641         -0.12772 
           Y2         0.078641        0.26217          -0.078051 
           Y3        -0.12772        -0.078051                1 

***********************************
Error Standard Deviation
***********************************
Variance-Covariance Means (Lambda)

 Equation        Beta1        Beta2           Beta3         Beta4            Beta5 
  Beta1       0.038024       0.0084823      0.0050414     -0.010463       -0.0044786 
  Beta2       0.0084823      0.038058       0.0061952     -0.0098521       0.0017846 
  Beta3       0.0050414      0.0061952      0.080755      -0.0086755       0.016158 
  Beta4      -0.010463      -0.0098521     -0.0086755      0.10271        -0.010493 
  Beta5      -0.0044786      0.0017846      0.016158      -0.010493        0.046216 

***********************************
Theta for Z Equation     1.0000 
***********************************

        Variable         PostMean          PostSTD 
          Theta1          0.53176          0.43012 
          Theta2          0.43195          0.35411 
          Theta3        -0.011848       0.00015526 
          Theta4          -2.0511          -1.9772 
          Theta5           1.0605           1.1038 

***********************************
Theta for Z Equation     2.0000 
***********************************

        Variable         PostMean          PostSTD 
          Theta1          0.90016          0.79037 
          Theta2          0.37388          0.19278 
          Theta3         -0.32424         -0.37066 
          Theta4          0.69154          0.85307 
          Theta5         -0.26623         -0.19126 

***********************************
Theta for Z Equation     3.0000 
***********************************

        Variable         PostMean          PostSTD 
          Theta1         -0.24998          -0.2454 
          Theta2         -0.22883         -0.19728 
          Theta3        -0.043585         0.026509 
          Theta4         -0.29718         -0.30046 
          Theta5          0.52032          0.50741 

 

Platform: Windows, Mac, and Linux

 

Requirements: GAUSS/GAUSS Engine/GAUSS Light v13.1 or higher