# GAUSS Bayesian Estimation Tools

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.

## GAUSS Bayesian internal models include

• Univariate and multivariate linear models
• Linear models with auto-correlated error terms
• HB Interaction and HB mixture models
• Probit models
• Logit models
• Dynamic two-factor model
• SVAR models with sign restrictions

Users may load data into GAUSS for estimation and analysis using standard intrinsic GAUSS procedures. However, in addition, the Bayesian Analysis Module includes a data generation feature that allows users to specify true data parameters to build hypothetical data sets for analysis.

## Individual modeling

Users can meet individual modeling needs by specifying key controls for the estimation algorithm including:

• Number of saved iterations
• Number of iterations to skip
• Number of burn-in iterations
• Total number of iterations
• Inclusion of an intercept

## Easy to interpret stored results

The Bayesian application module stores all results in a single output structure. In addition the Bayesian module graphs draws of all parameters and the posterior distributions for all parameters.

• 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 Yhat
• PDF values and corresponding PDF grid for all posterior distributions
• Log-likelihood value (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

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