Linear Estimation Methods

Linear models include:
AR Autoregressive errors
ARCH Autoregressive conditional heteroscedastic errors
OLS Ordinary least squares
PLS Partial least squares
POISSON Poisson regression
QR Quantal response (logit, probit, ordered)
STEPWISE Stepwise regression
SURE Seemingly unrelated regression estimation
VAR Vector autoregressive
2SLS Two stage least squares 
3SLS Three stage least squares 
Descriptive statistics, elasticities, automatic treatment of missing values, weighted analysis and White or Newey-West robust standard errors are standard. Lag specification such as y(-1) is supported, as are PDL structures. Diagnostics for single equations include Godfrey's test for residual serial correlation, Ramsey's RESET test for functional form, Jarque-Bera's test for normality of residuals, Breusch-Pagan test for heteroscedasticity, and Chow's test for stability.

Non-linear Estimation Methods

Non-linear models include:

FIML Full information maximum likelihood
GMM Generalized method of moments
ML Maximum likelihood
NLS Non linear least squares

Step-algorithms include:

BFGS Broyden-Fletcher- Goldfarb-Shanno algorithm
BHHH Berndt-Hall-Hall- Hausman algorithm
DFP Davidon-Fletche-Powell algorithm
DW Dennis-Wolkowicz algorithm
GA Genetic algorithm
GAUSS Linearized in variables algorithm
GN Gauss Newton algorithm
GO Global optimization
NM Nelder-Meade search algorithm
NM Newton-Raphson algorithm
SA Simulated annealing

Step-size methods include:

LS Line search
QP Quadratic programming
Trust region

Both the White and Newey-West robust estimators are supported, as well as the Murphy-Topel two step estimation. The defaults used during non-linear estimation can be altered heuristically during execution, or through a script file. Gradients, Hessian, and Jacobian are estimated numerically as the default; however they can be written as a procedure by the user. Coefficient restrictions can be imposed with PARAM, and investigated using ANALYZ, which can be used following either linear or non-linear estimation.  Descriptive statistics, automatic treatment of missing values, and weighted analysis are standard. 

Non-linear Processes

The maximum likelihood (ML) procedure permits the estimation of any specified likelihood - Gaussx includes examples for a number of non-linear processes:

AGARCH Asymmetric GARCH process
ANN Artificial neural network
ARCH Autoregressive conditional heteroscedastic process
ARFIMA Autoregressive fractional integrated moving average process
ARIMA Autoregressive integrated moving average process
ARMA Autoregressive moving average process
DBDC Double-bounded dichotomous choice process 
EGARCH Exponential GARCH process
FIGARCH Fractionally integrated GARCH process 
FMNP Feasible multinomial probit 
FPF Frontier production function process
GARCH GARCH process 
IGARCH Integrated GARCH process 
KALMAN Kalman filter 
LOGIT Binomial logit process
MGARCH Multivariate GARCH process 
MNL Multinomial logit 
MNP Multinomial probit
MSM Markov switching models
MVN Multivariate normal process 
NEGBIN Negative binomial process
NPE Non parametric estimate
ORDLGT Ordered logit process
ORDPRBT Ordered probit process
PGARCH Power GARCH process
POISSON Poisson process
PROBIT Binomial multivariate probit process
SV Stochastic volatility process
TGARCH Truncated GARCH process
TOBIT Tobit process
VARMA Vector autoregressive moving average process
WHITTLE Local Whittle process

Constrained Optimization

Constrained optimization is supported under FIML, GMM, ML, NLS and RSM.  The parameter constraints can be linear or non-linear. The estimation is undertaken using sequential quadratic programming. The constrained confidence region for any specified confidence level for each parameter is calculated. 

Automatic Differentiation

A choice of numeric, analytic, or symbolic derivatives is available for FIML, ML and NLS. The default method of deriving gradients and Hessians is numeric, using finite differencing. Analytic derivatives are specified by the user, while symbolic derivatives are calculated using the automatic differentiation capability of Maple 9. Symbolic gradients and Hessians can be saved as procedures and reused.  Analytic differentiation work only for Gaussx for Windows, and requires Maple 9 or higher.

Time Series Analysis

A complete range of time series analysis is available under Gaussx, including  ARMA, ARIMA and ARFIMA for single equations, and VAR and VARMA for multiple equations.   ARIMA includes full identification, estimation and forecasting with graphical presentation. Systems of transfer functions can be specified, with a separate moving average structure for each equation.  Markov switching models (MSM) include AR components and non-linear state equations.  Spectral analysis is also supported

LDV Models

Linear LDV models include binomial probit, multinomial logit, and ordered logit and probit; in each case the marginal effects and elasticities, and their variances, evaluated at the mean, are available. Diagnostics include Pearson Residual test, Deviance test, Bera, Jarque and Lee normality test (probit),  Hosmer-Lemeshow test, as well as concordance/discordance measures.  For both probit and logit, Mills ratio is available allowing correction for selection bias. Heckman's two step procedure (HECKIT) incorporates Greene's covariance correction. Non linear multinomial logit and probit (MNL and MNP) are available using ML; for the latter, high dimensional integration is carried out either exactly using the GAUSS CDFMVN function, or through simulation using the smooth recursive simulator.  Double-bounded (DBDC) models are also supported. For models with large number of alternatives, feasible multinomial probit (FMNP), which does not require parameterization of the covariance matrix, is available for both ranked and non-ranked data.  

       Panel Models

PANEL estimates the coefficients of a linear regression model for panel data.. Both fixed effects and random effects (error components model) for balanced and unbalanced models are supported. The Hausman test for testing the orthogonality of the random effects and the regressors, and the Bresuch Pagan test for random effects are implemented.

GARCH Models

A variety of Arch and Garch models are supported; these include linear ARCH, single equation non-linear ARCH, AGARCH, EGARCH, FIGARCH, GARCH, IGARCH, PGARCH and TGARCH.  Residuals can be distributed normal, Student-t, or GED.  Garch in the mean, leverage options, and MA residuals are all supported. Stability and positive variance is secured using the constrained optimization facilities.
Multivariate GARCH (MGARCH) estimated over a system of equations, with the option of weakly exogenous variables, is also supported, under both the VEC and BEKK formulation. MGARCH-M is also available.

Duration/Survival Models

Survival models (life data regression) are estimated using ML for both uncensored or censored data. Supported parametric models include: 
BETA Beta process
COX Cox proportional hazards process 
EXPON Exponential process
GAMMA Gamma process
GOMPERTZ Gompertz process
GUMBEL Gumbel (largest extreme value) process
INVGAUSS Inverse Gaussian process
LOGISTIC Logistic process
LOGLOG Loglogistice process
LOGNORM Log normal process 
NORMAL Normal process
PARETO Pareto process
PEARSON Pearson process
SEV Smallest extreme value process
WEIBULL Weibull process

Survival measures, based on the last survival model estimation, include  survival rate,  inverse survival rate,  hazard rate, cumulative hazard rate and cumulative failure rate. For each measure, the rate, the standard error, and the lower and upper confidence band are reported for each observation. Residuals include ordinary, standardized, Cox-Snell, deviance, martingale, Schoenfeld, scaled Schoenfeld and score. The indices can be linear or non-linear.

The Cox proportional hazards model supports the same survival and residual measures; ties are treated using the Breslow, Efron, discrete and exact methods.

Non-parametric processes  (support the same survival measures, using either the Kaplan-Meier or Nelson-Aalan algorithms.

Exponential Smoothing 

Methods include single, double, Holt-Winters, and seasonally additive or multiplicative Holt-Winters. Smoothing parameters can be user specified or optimally estimated by Gaussx.


Denosing of signals and time series is accomplished using wavelet shrinkage methods.  Thresholds include universal, minimax and SURE.

Non-parametric Analysis

Non-parametric and semiparametric analysis under Gaussx permits the estimation of the window width and the weights in the semiparametric index using cross validation under maximum likelihood. For the single index case, the FFT is used to speed calculation. Conditional response coefficients are determined for the density, conditional mean, discrete and smeared case.

Neural Networks

The hidden and output weights in a feed forward network with a single hidden layer are estimated using non-linear optimization, rather than back propagation. Transfer functions include Arctan, Gaussian, Halfsine, Linear, Sigmoid, Step, and Tanh. Output processing includes levels, density, and maximum.

Kalman Filter

Analysis with the Kalman Filter allows for the estimation of state vectors, with smoothing,  time varying transition matrices (ie. each element is a function), and the estimation of the elements of the Kalman matrices using ML.  Stochastic Volatility models (SV) are estimated using quasi ML based on a Kalman Filter model.

Robust Estimation

Robust estimation (ROBUST) of linear models when the distribution of the residual is unknown is undertaken using Quantile Regression (interior point algorithm), as well as using reiterated weighted least squares for Least Absolute Deviation, Huber's t Function, Ramsay's E Function, Andrew's Wave Function, and Tukey's Biweight.  The parameter covariance matrix is estimated using bootstrapping.


Static and dynamic forecast values and residuals are available for all estimations. Systems of non-linear equations can be solved statically or dynamically. An impulse response function is available for VAR models.  OLS  forecasts also include prediction error, bounds, studentized residuals,  Cook's D, HAT, DFFITS and DFBETASML forecasts include log likelihood,  GARCH forecasts include conditional variance, QR forecasts include probabilities and category, and ARFIMA forecasts include both naive and best linear predictor.  After an estimation, FORCST can evaluate the predicted value and standard errors for variables that are non-linear functions of estimated parameters.


Monte-Carlo simulation (MCS) can be carried out over a block of code, using both bootstrap and jackknife methods.  Output for the selected variables is shown dynamically on the screen, and final output includes cumulants and quantiles.  Latin Hypercube Sampling (LHS) is provided as an alternative to MCS, and allows for nearly orthogonal and correlated sampling.h


For Bayesian analysis, Markev Chain Monte Carlo (MCMC) is carried out over user supplied distributions and priors. Diagnostics include Geweke numerical standard error, relative numerical efficiency and a Chissq test for stability.  Examples include:
AR(k) with hetersoscedastic residuals
Binomial probit
Heteroscedastic binomial probit
Multinomial probit
OLS, residuals distributed normal
OLS, residuals distributed t
OLS, heteroscedastic residuals 


STATLIB consists of a set of  procs for evaluating density functions, which can be used from GAUSS or Gaussx; these provide the PDF, the CDF, the inverse CDF and random sampling from over 60 distributions.

  Continuous Distributions

beta beta4  boxcox  burr
cauchy chisq  chisq_scaled erf 
expon f_scaled  fatiguelife 
fisk  foldednormal frechet  gamma 
ged  gengamma  genlogistic  genpareto 
gumbel halfnormal invgamma  invgauss 
johnson_sb  johnson_sl  johnson_su laplace 
levy loggamma  logistic  loglog 
lognorm  maxwell  ncchisq  ncf 
nct  normal  pareto  pearson 
pert  power  rayleigh  reciprocal 
sev  skewnormal  students_t  t_scaled 
triangular  uniform  vonmises  weibull 

  Discrete Distributions

bernoulli binomial geometric hypergeom
logarithmic negbin poisson rectangular


Sampling from a truncated multivariate normal, multivariate t, and Wishart distribution,  sampling from a specified cdf, correlated sampling using COPULA and multivariate  random sampling (MVRND), as well as sampling with and without replacement are also available.  


Bitwise  AND, EQV, OR XOR, NOT, and SHFT are supported, as are conversions from decimal to base, and base to decimal. Quasi random variables (Sobol) are also supported.

Financial, Engineering and Economic Tools

Tools include:
AMORT Amortization schedule
FRONTIER Markowitz efficient frontier
FV Future value
GINI Gini coefficients
LP Linear programming
MCALC Mortgage calculation
ME Maximum entropy
PV Present value
RSM Response surface methodology and multi response optimization
SOLVE Solve a system of equations
SPECTRAL Power spectrum estimation
WELFARE Consumer surplus (CV, EV, MS and deadweight loss)

Econometric Tests

The TEST command includes both parametric and nonparametric tests:
AD Anderson-Darling normality test, with censoring
ANOVA Analysis of Variance
ARCH Engle's ARCH test.
BARTLETT Bartlett's Test for equality of variances:
BKW Belsley-Kuh-Walsh collinearity test
BP Breusch Pagan homoscedasticity test 
CHISQ Chi squared test.
CHOW Chow stability test.
DF Dickey-Fuller unit root test.
EG Engle-Granger cointegration test
F F test
FTEST Linear restriction test
GRANGER Granger causality test
HANSEN Hansen test of overidentifying restrictions.
HAUSMAN Hausman specification test
JB Jarque-Bera normality test.
JTEST Davidson and MacKinnon's J-Test for non-nested estimations.
KPSS KPSS stationarity test.
LBQ Ljung-Box Q test. 
LM Lagrange Multiplier test
LRT Likelihood ratio test
NW Newey West D test
PIT Probability integral transformation test
PPC Probability plot correlation test
RECURS CUSUM and CUSUM-squared tests for stability
SF Shapiro-Francia normality test, with censoring
SW Shapiro-Wilks normality test, with censoring
THEIL Thiel's decomposition of two vectors
TTEST T test
WALD Wald test
WELCH Welch's test for equality of means
BF  Brown-Forsythe test of scale 
CONOVER  Conover test for treatment
FRIEDMAN  Friedman test for treatment
KS Kolmogorov-Smirnov test
KURTOSIS  Kurtosis test
KW  Kruskal-Wallis test of location
LEVENE  Levene test of scale
MOOD  Mood's test of location
MW  Mann Whitney U test of location
OBRIEN  O'Briens test of scale
RUNS  Runs test of randomness
SIGN  Sign test for treatment 
SKEWNESS Skewness test
WALSH  Walsh test for treatment
WILCOXON  Wilcoxon test for treatment


Data Handling and Conversion

Memory allocation and all file control is handled automatically. Data size for non-AR estimation is limited by disk capacity only. External data can be imported as delineated ASCII, packed ASCII,  binary, Lotus, Excel, GAUSS data files, GAUSS format files, and Gaussx save files. Data can be exported as ASCII, binary, Gaussx or GAUSS data files. Under Windows, import/export is available for Lotus, Excel, Quattro, Dbase, Symphony, Paradox, Foxpro, Clipper and both delineated and packed ASCII. Variables in a Gaussx dataset can be user selected with the KEEP or DROP commands.

Data Creation and Transformation

Data transformation (GENR, FEVAL) permits the use of all GAUSS operations and all the GAUSS functions, such as FFT, all GAUSS distributions, random number generators, etc. Thus all the power of GAUSS is available in Gaussx.  However sample selection (SMPL) makes coding far simpler, and data input/output is transparent. Stochastic data can be created using 15 distributions with DGP, and data can be filtered with 7 types of FILTER. Other tools include vector convolution and deconvolution, difference and inverse difference, and vector interpolation.

Descriptive Statistics

A number of descriptive measures are provided:
ANOVA N-way analysis of variance for fixed, random or mixed models.  Nested and interaction effects, variance components.
CATALOG Descriptive comment for each variable
CLUSTER Cluster groups and dendrogram
CORDIM Correlation dimension
COVA Means, standard deviations, minimum and maximum, sum, covariance and correlation matrices, autocorrelogram, partial autocorrelogram.
CROSSTAB Cross tabulation of data
DIVISIA Divisia indices
FREQ Frequency distributions
LYAPUNOV Lyapunov exponent
PRIN Principal components
SAMA Seasonal adjustment (including Census X12), 
SVD Singular value decomposition
TABULATE Tabulates data across two class variables 

Cluster Analysis

Cluster analysis creates an hierarchical cluster tree of the data, and optionally graphs the tree - a dendrogram. Five distance metrics and four linkage methods are available.


Graphical output (PLOT, GRAPH, COVA) is available using either Publication Quality Graphics or, if installed, GAUSSPlot.   Line and scatter graphs/plots are supported.  Gaussx  has full support for all GAUSS PQG routines, while for GAUSSPlot, interactive customization of a graph can be saved and used in subsequent sessions. 

Programming Features

All GAUSS commands, logical goto, DO loops, and GAUSS procs can be used within a  Gaussx file. In addition, Gaussx provides a number of programming commands; these include macro definitions for formulae, LOOP control for multisectored data, GROUP control (like BY in SAS) and recursive LIST names.  A timer control is available to simulate realtime analysis. 

Symbolic Algebra

Symbolic algebra can be used for symbolic differentiation and integration, exact linear algebra, and equation solving. Gaussx uses Maple and/or Mathematica to permit GAUSS to undertake symbolic manipulation. Simply include the Maple/Mathematica statements within the command file, select the statements, and click the Maple or Mathematica button. This works only for Gaussx for Windows, and requires Maple, rev 4 or higher or Mathematica, rev 3 or higher.


A number of mathematical tools to augment GAUSS
ACF Autocorrelation function
ACV Autocovariance function
COMBS All k combinations of a vector
DECONV Deconvolution
INTERP Interpolation
INVERT Find the inverse values of a function.
ISCHAR Tests for a character vector
PERMS All permutations of a vector
POLYDIV Polynomial division
POLYINV Polynomial inversion


GAUSS statements can be included within the command file. Gaussx variables can be made global (FETCH), and global variables can be stored in the Gaussx  workspace (STORE). Thus maximum flexibility is achieved by being able to mix GAUSS and Gaussx commands. User written procs can be included within Gaussx formula definitions. In addition, most GAUSS application modules can be run directly from a Gaussx file.

Extending GAUSSX

The complete source code, written in GAUSS, is included. Thus even if you don't want all the features of Gaussx, you can extract a particular procedure and use it in your own GAUSS programs -- procedures such as inverse cumulative normal density function, Gibbs sampling, smooth recursive simulator (GHK), multivariate normal rectangle probabilities for any dimension, random sampling from a multivariate truncated distribution, maximum entropy estimation, quasi random sequences, bitwise arithmetic and more. And because of its modular design, you can also add your own procedures to Gaussx, or modify any Gaussx procedure to fit your requirements.

Project Management

Windows Project Control Screen
Project management is provided for Gaussx for Windows, with up to 100 separate applications, each associated with different file names and paths. Gaussx is network compatible - thus on a network, each client has its own project and configuration file. Project management can also be used to manage pure Gauss applications.

Help Facilities

During execution of a command file, pop-up help is available to explain the current screen, using Alt-H. Under  Windows, context sensitive help  (F1) is available to provide the complete syntax of each Gaussx command.

System Requirements

Gaussx can be run on a single machine or on a network, under either Windows, Unix or Mac.

GAUSSX for WINDOWS runs under Windows 2000, XP, Vista, Win7 and Win8.  Gaussx  for Windows requires  GAUSS for Windows 6.0 or higher, and about 7 MB of hard drive.  GAUSSX  supports both 32 bit and 64 bit versions of  GAUSS.

GAUSSX for UNIX and MAC runs in  Terminal mode. Networking is built in, so that individuals will each have their own configuration file. The econometric specifications for the Unix version is identical to the Windows version.  Gaussx for Unix has been designed to be machine independent by writing the entire package in GAUSS. Thus, if your Unix machine runs GAUSS,  it will run Gaussx.  Gaussx requires Gauss for Unix 4 or higher, and about 1MB of hard drive.


Pricing and Ordering

The package includes: source code, menu driven installation, tutorial, 50 sample command files with index, compiled HTML help for syntax, and a complete 450 page manual in PDF format (with reference section and index).  A hard copy version of the manual is an optional extra.  Single-user, network and student versions are available. Academic prices start at about $225, with a 30 day, no-question refund policy, and free technical support by phone and the internet. For technical information, contact Econotron Software, and for ordering information, contact Aptech Systems.