## 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` 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.

## Denoising

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.

## Forecasting

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.

## Simulation

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

## Bayesian

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 Poisson Tobit SURE

## Distributions

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 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 step

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

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:
 Parametric 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 Non-Parametric 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.

## Graphics

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.

## Tools

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

## Mixing GAUSS and GAUSSX

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.