Accuracy: Linear Regression
LIMDEP and NLOGIT'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 and NLOGIT's solution below using only the program defaults - just the basic regression instruction. Second, LIMDEP and NLOGIT would not, on their 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 and NLOGIT do issue a warning, however. What you do next is up to you, not the program.
The NIST data and solution file Dataset Name: Filippelli (NIST-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 and NLOGIT solution WARNING: Badly conditioned X. Condition value = .2999482D+10 ----------------------------------------------------------------------------- Ordinary least squares regression ............ LHS=Y Mean = .84958 Standard deviation = .05479 No. of observations = 82 Degrees of freedom Regression Sum of Squares = .242392 10 Residual Sum of Squares = .795851E-03 71 Total Sum of Squares = .243187 81 Standard error of e = .00335 Fit R-squared = .99673 R-bar squared = .99627 Model test F[ 10, 71] = 2162.43959 Prob F > F* = .00000 Diagnostic Log likelihood = 356.90255 Akaike I.C. =-11.27452 Restricted (b=0) = 122.29336 Bayes I.C. =-10.95167 Chi squared [ 10] = 469.21839 Prob C2 > C2* = .00000 --------+-------------------------------------------------------------------- | Standard Prob. 95% Confidence Y| Coefficient Error t |t|>T* Interval --------+-------------------------------------------------------------------- Constant| -1467.49*** 298.0845 -4.92 .0000 -2051.72 -883.25 X1| -2772.18*** 559.7799 -4.95 .0000 -3869.33 -1675.03 X2| -2316.37*** 466.4776 -4.97 .0000 -3230.65 -1402.09 X3| -1127.97*** 227.2043 -4.96 .0000 -1573.29 -682.66 X4| -354.478*** 71.64787 -4.95 .0000 -494.905 -214.051 X5| -75.1242*** 15.28972 -4.91 .0000 -105.0915 -45.1569 X6| -10.8753*** 2.23691 -4.86 .0000 -15.2596 -6.4911 X7| -1.06222*** .22162 -4.79 .0000 -1.49659 -.62784 X8| -.06702*** .01424 -4.71 .0000 -.09492 -.03912 X9| -.00247*** .00054 -4.61 .0000 -.00352 -.00142 X10|-.40296D-04*** .8966D-05 -4.49 .0000 -.57870D-04 -.22723D-04 --------+-------------------------------------------------------------------- Note: nnnnn.D-xx or D+xx => multiply by 10 to -xx or +xx. Note: ***, **, * ==> Significance at 1%, 5%, 10% level. -----------------------------------------------------------------------------