1. Spatial Econometric Analysis Using GAUSS2
Kuan-Pin Lin Portland State University

2. GAUSS Mathematical and Statistical System
Windows Interface
Windows
Command, Error, Log, …
Menu
File, Edit, Run, …, Help
Operation
Interactive Mode
Command (Input / Output)
Batch Mode
Writing Program
Online Help

3. 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

4. GAUSS Programming Useful 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

5. GAUSS Programming Controlling Execution Flow
If Statement if; then; else; elseif; endif;
For Loop for i (start,stop,step); ... endfor;
Do Loop do while ... endo; do until ... endo;

6. GAUSS Programming Write Your Own Functions
Single Line Function fn fn_name(args) = code_for_function;
Procedure proc [[(nrets)=]] proc_name(arg_list); local list of local variables; ... statements in the body of procedure; retp(ret_list); endp;

7. GAUSS Programming Example 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? (example.1)
one=ones(1,4);
e=1.0;
do until e < 1.0e-20;
x=one|(e.*eye(4));
print "e = " e;
r=eigrs(x'x); print r';
invx=invpd(x'x); print invx;
e=e/10;
endo;

8. GAUSS Programming Example 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)
fn qtoa1(x) = meanc(reshape(x,rows(x)/4,4)');
proc qtoa(x);
local r,c,y,i;
r = rows(x);
c = cols(x);
y = qtoa1(x[.,1]);
i = 2;
do until i > c;
y = y~qtoa1(x[.,i]);
i = i+1;
endo;
retp(y);
endp;

9. 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:
Reset Set up global control (input and output) variables.
Estimate Estimation of a linear or generalized linear model.
Forecast Forecasting based on a linear or generalized linear model.
Optimize Estimation of a nonlinear model.
Gpehelp Online help of using GPE2.

10. GPE2 for GAUSS Selected 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

11. GPE2 for GAUSS A Typical Program Using GPE2/*
** Comments on program title, purposes, and the usage of the program */
use using GPE package (version
// 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 initialize global
** Set input control variables for model estimation
** (e.g. _names for variable names, see Appendix A)
call do model
// 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 do model
important: don’t forget

12. GPE2 for GAUSS Examples
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.

13. Software Demonstration
Installation
GAUSS Light 10.0
GPE2 for GAUSS 10.0
Example: Zellner and Revankar [1970] U.S. Transportation Equipment Industry
Cobb-Douglas Production Function ln(Q) = a + b ln(L) + g ln(K) + e
Generalized Cobb-Douglas Production Function ln(Q) + q Q = a + b ln(L) + g ln(K) + e

14. Example Zellner 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

15. 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,