GAUSS Mathematical and Statistical System

1. GAUSSMathematical and Statistical System • GAUSS 12 Windows Interface • Pages • Command, Source, Data, Debug, Help • Windows • Command Input, Program Output, … • Menus • File, Edit, Tool, … • Toolbars • Run, Debug, Edit, … • Operation • Interactive Mode: Using Command Page • Batch Mode: Using Source Page

2. GAUSS Basics • Basic Operations on Matrices + -  ^   .*   ./   %   !   *   / .<   .<=   .==   .>=   .>   ./=     <   <=   ==   >=   >   /= .not  .and   .or   .xor   not  and or   xor ~   |   .*.   *~ • Special Operators []   {}   :   .   '(transpose) • Useful Algrbra and Matrix Operations exp ln log  abs  sqrt  pi sin cos inv invpd(inverse) det(determinant) • Example • Least Squares: b=y/x

3. GAUSS ProgrammingUseful GAUSS Functions • System Functions: use, load, output • Data Generating Functions:ones, zeros, eye, seqa, seqm, rndu, rndn • Data Conversion Functions:reshape, selif, delif, vec, vech, xpnd, submat, diag, diagrv • Basic Matrix Functions: • Matrix Description:rows, cols, maxc, minc, meanc, median, stdc • Matrix Operations:sumc,cumsumc,prodc,cumprodc,sortc,sorthc,sortind • Matrix Computation:det,inv,invpd,solpd,vcx,corrx,cond,rank,eig,eigh • Probability and Statistical Functions:pdfn, cdfn, cdftc, cdffc, cdfchic, dstat, ols • Calculus Functions:gradp, hessp, intsimp, linsolve, eqsolve, sqpsolve

4. GAUSS ProgrammingControlling Execution Flow • If Statementif; then; else; elseif; endif; • For Loopfor i (start,stop,step); ... endfor; • Do Loopdo while ... endo;do until ... endo;

5. GAUSS ProgrammingWrite Your Own Functions • Single Line Functionfn fn_name(args) = code_for_function; • Procedureproc [[(nrets)=]] proc_name(arg_list);local list of local variables;... statements in the body of procedure;retp(ret_list);endp;

6. GAUSS ProgrammingExample 1 • Do you know the accuracy of your computer's numerical calculation? This example addresses this important problem. Suppose e is a known small positive number, and the 5x4 matrix X is defined as follows: Verify that the eigenvalues of X'X are 4+e2, e2, e2, and e2. How small of the value of e your computer will allow so that X'X can be inverted? (examle.1)

7. GAUSS ProgrammingExample 2 • Write a single-line GAUSS function to convert a quarterly time series into the annual series by taking the average of every four data points. How you extend the single-line version of time series conversion function to a multi-line procedure so that it can handle the conversion of more than one time series? (example.2)

8. GPE2 for GAUSS • GPE2 is a package of econometric procedures written in GAUSS. There are four main functions, driven by a set of global control variables: • ResetSet up global control (input and output) variables. • EstimateEstimation of a linear or generalized linear model. • ForecastForecasting based on a linear or generalized linear model. • OptimizeEstimation of a nonlinear model. • GpehelpOnline help of using GPE2.

9. GPE2 for GAUSSSelected Global Control Variables • Input Control Variables_names, _begin, _end, _rstat, _rtest, _rplot, _rlist,_const, _restr, _vcov, _hacv, _weight, _ivar, _dlags, _pdl, _eq, _id, _ar, _ma, _arma, _garch, _acf, _acf2, _nlopt, _method, _iter, _tol, _step, _conv, _fbrgin, _fend, _fstat, _fplot, _splag, _spw, _spwd • Output Control Variables__y, __x, __e, __b, __vb, __v, __rss, __r2, __f, __vf, __t, __a, __va

10. GPE2 for GAUSSA Typical Program Using GPE2 • /* • ** Comments on program title, purposes, and the usage of the program • */ • use gpe2; @ using GPE package (version 2) @ • // this must be the first executable statement • /* • ** Writing output to file or sending it to printer: • ** specify file name for output • */ • // Loading data: read data series from data files. • /* • ** Generating or transforming data series: • ** create and generate variables with data scaling or transformation • ** (e.g. y and x are generated here and will be used below) • */ • call reset; @ initialize global variables @ • /* • ** Set input control variables for model estimation • ** (e.g. _names for variable names, see Appendix A) • */ • call estimate(y,x); @ do model estimation @ • // variables y, x are generated earlier • /* • ** Retrieve output control variables for model evaluation and analysis • */ • /* • ** Set more input control variables if needed, for model prediction • ** (e.g. _b for estimated parameters) • */ • call forecast(y,x); @ do model prediction @ • end; @ important: don’t forget this @

11. GPE2 for GAUSSExamples • More than 70 examples covering linear and nonlinear least squares, instrumental variables, system of simultaneous linear equations, time series analysis, panel data, limited dependent variables, maximum likelihood, generalized methods of moments, and … • The latest extensions include spatial lag model estimation, hypothesis testing, and robust inference.

12. Software Demonstration • Installation • GAUSS Light 12.0 • GPE2 for GAUSS 12.0 • Example: Zellner and Revankar [1970]U.S. Transportation Equipment Industry • Cobb-Douglas Production Functionln(Q) = a + b ln(L) + g ln(K) + e • Generalized Cobb-Douglas Production Functionln(Q) + q Q = a + b ln(L) + g ln(K) + e

13. ExampleZellner and Revankar [1970] • Cobb-Douglas Production Function • OLS Estimator • Hypothesis Testing • Constant Returns to Scale? • Homoscedasticity? • Generalized Production Function • Output Effects? • Instrumental Variables • GAUSS/GPE2 Program and Data

14. References • K.-P. Lin, Computational Econometrics: GAUSS Programming for Econometricians and Financial Analysts, ETEXT Publishing, Los Angeles, 2001. • C.-F. Chung, Learning Econometrics with GAUSS, Institute of Economics, Academia Sinica, 2000. • A. Zellner and N. Revankar, "Generalized Production Functions," Review of Economic Studies, 1970, 241-250.