Demonstration of numerical issues in LS calculations.

Probably the worst algorithm to perform a least squares estimation is to evaluate

b = (X'X)^-1X'y

This is illustrated in http://amsterdamoptimization.com/models/regression/longley2.gms. The solution compares as follows:

 parameter  reference                 LS solver     b=inv(X'X)X'y
    B0     -3482258.63459582         -3.48226E+6       65317.000
    B1      15.0618722713733         15.06187227          0
    B2     -0.358191792925910E-01    -0.03581918          0
    B3     -2.02022980381683         -2.02022980          0
    B4     -1.03322686717359         -1.03322687          0
    B5     -0.511041056535807E-01    -0.05110411          0
    B6      1829.15146461355         1.829151E+3          0

A numerically sound method is documented in http://amsterdamoptimization.com/pdf/regression.pdf. The above model can then be coded as: http://amsterdamoptimization.com/models/regression/longley.gms. The results are listed in the column "LS Solver".