Accuracy: Linear Regression Computation
LIMDEP's linear regression computations are extremely accurate. The 'Filippelli problem' in the NIST benchmark problems is the most difficult of the set. Most programs are not able to do the computation at all. The assessment of another widely used package was as follows:
'Filippelli test: XXXXX found the variables so collinear that it dropped two of them - that is, it set two coefficients and standard errors to zero. The resulting estimates still fit the data well. Most other statistical software packages have done the same thing and most authors have interpreted this result as acceptable for this test.'
We don't find this acceptable. First, the problem is solvable. See LIMDEP's solution below using only the program defaults - just the basic regression instruction. Second, LIMDEP would not, on its own, drop variables from any regression and leave behind some arbitrarily chosen set that provides a 'good fit.' If the regression can't be computed within the (very high) tolerance of the program, we just tell you so. For this problem, LIMDEP does issue a warning, however. What you do next is up to you, not the program.
The NIST data and solution file
Dataset Name: Filippelli (filippelli.dat)
Procedure: Linear Least Squares Regression
Reference: Filippelli, A., NIST.
Data: 1 Response Variable (y)
1 Predictor Variable (x)
82 Observations
Higher Level of Difficulty
Model: Polynomial Class
11 Parameters (B0,B1,...,B10)
y = B0+B1*x+B2*(x**2) +...+B9*(x**9)+B10*(x**10)+e
Certified Regression Statistics
Standard Deviation
Parameter Estimate of Estimate
B0 -1467.48961422980 298.084530995537
B1 -2772.17959193342 559.779865474950
B2 -2316.37108160893 466.477572127796
B3 -1127.97394098372 227.204274477751
B4 -354.478233703349 71.6478660875927
B5 -75.1242017393757 15.2897178747400
B6 -10.8753180355343 2.23691159816033
B7 -1.06221498588947 0.221624321934227
B8 -0.670191154593408E-01 0.142363763154724E-01
B9 -0.246781078275479E-02 0.535617408889821E-03
B10 -0.402962525080404E-04 0.896632837373868E-05
Standard Deviation 0.334801051324544E-02
R-Squared 0.996727416185620
F Statistic 2162.43954511489
Certified Analysis of Variance Table
Source of Degrees of Sums of Mean
Variation Freedom Squares Squares
Regression 10 0.242391619837339 0.242391619837339E-01
Residual 71 0.795851382172941E-03 0.112091743968020E-04
LIMDEP solution
WARNING: Badly conditioned X. Condition value = .1818039D+13
+-----------------------------------------------------------------------+
| Ordinary least squares regression Weighting variable = none |
| Dep. var. = Y Mean= .8495756098 , S.D.= .5479337971E-01 |
| Model size: Observations = 82, Parameters = 11, Deg.Fr.= 71 |
| Residuals: Sum of squares= .7958513674E-03, Std.Dev.= .00335 |
| Fit: R-squared= .996727, Adjusted R-squared = .99627 |
| Model test: F[ 10, 71] = 2162.44, Prob value = .00000 |
+-----------------------------------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Constant -1467.489645 298.08453 -4.923 .0000
X1 -2772.179651 559.77986 -4.952 .0000 -6.1502375
X2 -2316.371131 466.47757 -4.966 .0000 40.058748
X3 -1127.973965 227.20427 -4.965 .0000 -273.41944
X4 -354.4782414 71.647866 -4.948 .0000 1938.1080
X5 -75.12420340 15.289718 -4.913 .0000 -14165.550
X6 -10.87531828 2.2369116 -4.862 .0000 106165.36
X7 -1.062215010 .22162432 -4.793 .0000 -812357.31
X8 -.6701911701E-01 .14236376E-01 -4.708 .0000 6324653.3
X9 -.2467810841E-02 .53561741E-03 -4.607 .0000 -49962743.
X10 -.4029625348E-04 .89663283E-05 -4.494 .0000 .39956133E+0