Accuracy: Nonlinear Least Squares Computation
Nonlinear regression is more difficult than the linear computations described above. B.D. McCullough (Journal of Applied Econometrics, 1999) surveyed numerous programs for their ability to solve nonlinear least squares problems. LIMDEP was able to solve nearly all the benchmark problems using only the program default settings, and all of the rest with only minor additional effort. The example below is one of the most difficult of the set. The (correct) solution from the more difficult starting values is routine.
The NIST data and solution file
Data: 1 Response (y)
1 Predictor (x)
16 Observations
Higher Level of Difficulty
Model: Exponential Class
y = b1 * exp[b2/(x+b3)] + e
Starting values Certified Values
Start 1 Start 2 Parameter Standard Deviation
b1 = 2 0.02 5.6096364710E-03 1.5687892471E-04
b2 = 400000 4000 6.1813463463E+03 2.3309021107E+01
b3 = 25000 250 3.4522363462E+02 7.8486103508E-01
Residual Sum of Squares: 8.7945855171E+01
Residual Standard Deviation: 2.6009740065E+00
Degrees of Freedom: 13
LIMDEP solution
READ ; Nobs=16 ; Nvar=2 ; Names=Ym10,Xm10$
3.478000E+04 5.000000E+01
2.861000E+04 5.500000E+01
2.365000E+04 6.000000E+01
1.963000E+04 6.500000E+01
1.637000E+04 7.000000E+01
1.372000E+04 7.500000E+01
1.154000E+04 8.000000E+01
9.744000E+03 8.500000E+01
8.261000E+03 9.000000E+01
7.030000E+03 9.500000E+01
6.005000E+03 1.000000E+02
5.147000E+03 1.050000E+02
4.427000E+03 1.100000E+02
3.820000E+03 1.150000E+02
3.307000E+03 1.200000E+02
2.872000E+03 1.250000E+02
NLSQ ; Lhs=Ym10
; Fcn=B1*EXP(B2/(Xm10+B3))
; Labels=B1,B2,B3 ; Dfc
; Start=2,400000,25000 ; Maxit=10000$
+-------------------------------------------------------------------+
|User Defined Optimization |
|Nonlinear least squares regression Weighting variable = none |
|Number of iterations completed =2050 |
|Dep. var. = YM10 Mean= 12432.06250 , S.D.= 9722.364270 |
|Model size: Observations = 16, Parameters = 3, Deg.Fr.= 13|
|Residuals: Sum of squares= 87.94585517 , Std.Dev.= 2.60097|
|Fit: R-squared= 1.000000, Adjusted R-squared = 1.00000|
| (Note: Not using OLS. R-squared is not bounded in [0,1]|
|Model test: F[ 2, 13] =********, Prob value = .00000|
+-------------------------------------------------------------------+
+---------+--------------+----------------+--------+---------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
+---------+--------------+----------------+--------+---------+
B1 .5609636411E-02 .15687857E-03 35.758 .0000
B2 6181.346355 23.308962 265.192 .0000
B3 345.2236349 .78485908 439.854 .0000