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 DFBETAS.  ML 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

  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.