|_* |_* Berndt, Chapter 3, Exercise 4 |_* |_* Estimation of cobb-douglas cost function for U.S. electric utility industry |_* Berndt, equation 3.16 |_* |_FILE 11 NERLOV UNIT 11 IS NOW ASSIGNED TO: NERLOV |_SAMPLE 1 145 |_READ (11) ORDER COSTS KWH PL PF PK / SKIPLINES=1 6 VARIABLES AND 145 OBSERVATIONS STARTING AT OBS 1 |_* |_* Part (a) |_* Perform data transformations |_* |_GENR LNCOSTS=LOG(COSTS) |_GENR LNKWH=LOG(KWH) |_GENR LNPL=LOG(PL) |_GENR LNPF=LOG(PF) |_GENR LNPK=LOG(PK) |_PRINT LNKWH LNKWH 0.6931472 1.098612 1.386294 1.386294 1.609438 2.197225 2.397895 2.564949 2.564949 3.091042 3.218876 3.218876 3.555348 3.663562 3.761200 4.143135 4.219508 4.394449 4.430817 4.290459 4.595120 4.615121 4.779123 4.787492 4.804021 4.867534 4.927254 5.003946 5.278115 5.283204 5.342334 5.365976 5.393628 5.455321 5.459586 5.533389 5.631212 5.669881 5.669881 5.686975 5.700444 5.780744 5.808142 5.823046 5.866468 5.866468 6.030685 6.040255 6.122493 6.182085 6.246107 6.309918 6.333280 6.338594 6.383507 6.508769 6.545350 6.577861 6.609349 6.678342 6.684612 6.694562 6.698268 6.751101 6.756932 6.812345 6.816736 6.828712 6.891626 6.898715 6.907755 7.001246 7.011214 7.019297 7.022868 7.036148 7.052721 7.061334 7.064759 7.102499 7.153834 7.163172 7.162397 7.193686 7.224753 7.258412 7.295735 7.311218 7.342779 7.407924 7.419381 7.485492 7.512618 7.513709 7.516433 7.488294 7.559038 7.565275 7.614805 7.629004 7.642044 7.707962 7.742402 7.758333 7.763446 7.769379 7.804251 7.806696 7.826842 7.835975 7.853993 7.865955 7.962067 8.004032 8.071531 8.097426 8.105308 8.159947 8.171317 8.241176 8.253488 8.297544 8.346879 8.367532 8.410498 8.468843 8.571113 8.572249 8.642592 8.644883 8.668884 8.699515 8.719154 8.721928 8.880864 8.972844 9.038246 9.064389 9.081029 9.157361 9.205931 9.348100 9.375516 9.572132 9.724301 |_* Note that these are not quite sorted in order -- look at observations 18-20, 82-83, and 93-96. These are probably due to transcription errors from the original data (which may also explain why our results do not completely match Nerlove's), since the rest of the series is sorted in order of output, from smallest to largest. |_* Part (b) |_* Estimate the cost function subject to restrictions |_* |_OLS LNCOSTS LNKWH LNPL LNPF LNPK / LOGLOG RESTRICT NOANOVA RSTAT COEF=BETA & | RESID=ERR GF LM REQUIRED MEMORY IS PAR= 23 CURRENT PAR= 500 OLS ESTIMATION 145 OBSERVATIONS DEPENDENT VARIABLE = LNCOSTS ...NOTE..SAMPLE RANGE SET TO: 1, 145 |_RESTRICT LNPL+LNPF+LNPK=1 |_END F TEST ON RESTRICTIONS= 0.57366 WITH 1 AND 140 DF P-VALUE= 0.45008 R-SQUARE = 0.9257 R-SQUARE ADJUSTED = 0.9241 VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.15348 STANDARD ERROR OF THE ESTIMATE-SIGMA = 0.39176 SUM OF SQUARED ERRORS-SSE= 21.640 MEAN OF DEPENDENT VARIABLE = 1.7247 LOG OF THE LIKELIHOOD FUNCTION(IF DEPVAR LOG) = -317.915 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 141 DF P-VALUE CORR. COEFFICIENT AT MEANS LNKWH 0.72069 0.1744E-01 41.33 0.000 0.961 0.9696 0.7207 LNPL 0.59291 0.2046 2.898 0.004 0.237 0.0514 0.5929 LNPF 0.41447 0.9895E-01 4.189 0.000 0.333 0.1046 0.4145 LNPK -0.73811E-02 0.1907 -0.3870E-01 0.969-0.003 -0.0005 -0.0074 CONSTANT -4.6908 0.8849 -5.301 0.000-0.408 0.0000 -4.6908 DURBIN-WATSON = 1.0154 VON NEUMANN RATIO = 1.0224 RHO = 0.49192 RESIDUAL SUM = 0.57954E-13 RESIDUAL VARIANCE = 0.15348 SUM OF ABSOLUTE ERRORS= 39.097 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.9257 R-SQUARE BETWEEN ANTILOGS OBSERVED AND PREDICTED = 0.9049 RUNS TEST: 45 RUNS, 70 POS, 0 ZERO, 75 NEG NORMAL STATISTIC = -4.7414 COEFFICIENT OF SKEWNESS = 1.2648 WITH STANDARD DEVIATION OF 0.2014 COEFFICIENT OF EXCESS KURTOSIS = 4.8225 WITH STANDARD DEVIATION OF 0.4001 GOODNESS OF FIT TEST FOR NORMALITY OF RESIDUALS - 12 GROUPS OBSERVED 1.0 0.0 4.0 15.0 20.0 35.0 39.0 19.0 4.0 3.0 2.0 3.0 EXPECTED 0.9 2.4 6.4 13.3 21.7 27.8 27.8 21.7 13.3 6.4 2.4 0.9 CHI-SQUARE = 23.7172 WITH 6 DEGREES OF FREEDOM, P-VALUE= 0.001 JARQUE-BERA ASYMPTOTIC LM NORMALITY TEST CHI-SQUARE = 166.6326 WITH 2 DEGREES OF FREEDOM, P-VALUE= 0.000 Because the RESTRICT option is used here, we obtain an estimate for Beta3 (the coefficient for LNPF). Nerlove imposes the restriction by transforming the variables, so he does not obtain an estimate for this parameter. As well, statistics such as the R-square are transformed (and so are invalid) in Nerlove's study, because they are based on a transformed dependent variable. There appears to be considerable evidence of misspecification here. The Durbin-Watson statistic is significant, despite the fact that this is not a time series! This suggests a systematic pattern in the residuals (which is also noted in part (e) below). As well, there is stong evidence that the residuals are not normally distributed. This is both a warning of possible misspecification, and a caution that hypothesis tests are not exact. |_* |_* Part (c-d) |_* Test hypotheses on returns to scale and price terms |_* |_CONFID LNKWH USING 95% AND 90% CONFIDENCE INTERVALS CONFIDENCE INTERVALS BASED ON T-DISTRIBUTION WITH 141 D.F. - T CRITICAL VALUES = 1.960 AND 1.645 NAME LOWER 2.5% LOWER 5% COEFFICIENT UPPER 5% UPPER 2.5% STD. ERROR LNKWH 0.6865 0.6920 0.72069 0.7494 0.7549 0.017 The confidence interval does not include the value of unity, which we can reject the hypothesis that the ocefficient of LNKWH has this value. |_TEST LNKWH=1 TEST VALUE = -0.27931 STD. ERROR OF TEST VALUE 0.17436E-01 T STATISTIC = -16.019558 WITH 141 D.F. P-VALUE= 0.00000 F STATISTIC = 256.62623 WITH 1 AND 141 D.F. P-VALUE= 0.00000 WALD CHI-SQUARE STATISTIC = 256.62623 WITH 1 D.F. P-VALUE= 0.00000 UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.00390 A test of the null hypothesis that the coefficient of LNKWH is unity is strongly rejected; p-value is 0 to 5 decimal places. Constant returns to scale is rejected. |_* Compute asymptotic significance test for alpha2 = beta2/betay |_TEST LNPK/LNKWH TEST VALUE = -0.10242E-01 STD. ERROR OF TEST VALUE 0.26467 T STATISTIC = -0.38695548E-01 WITH 141 D.F. P-VALUE= 0.96919 F STATISTIC = 0.14973454E-02 WITH 1 AND 141 D.F. P-VALUE= 0.96919 WALD CHI-SQUARE STATISTIC = 0.14973454E-02 WITH 1 D.F. P-VALUE= 0.96913 UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 1.00000 The parameter alpha2 = beta2/betay. A TEST command on this ratio gives the estimated value (-0.010), and tests the null pypothesis that the true value is zero. This hypothesis is accepted; p-value is 0.97. |_* Compute returns to scale parameter 1/betay |_GEN1 1/BETA(1) 1.3875639 The coefficients of the regression equation (the betas) are stored in the vector named BETA by the COEF=BETA option in the OLS command. The first element in this vector [BETA(1)] is the coefficient on the first variable, which is LNKWH. |_* |_* Part (e) |_* Plot residuals |_PLOT ERR LNKWH REQUIRED MEMORY IS PAR= 15 CURRENT PAR= 500 FOR MAXIMUM EFFICIENCY USE AT LEAST PAR= 17 145 OBSERVATIONS *=ERR M=MULTIPLE POINT 2.0000 | 1.7895 | * 1.5789 | * 1.3684 | 1.1579 | 0.94737 | * * 0.73684 | * * 0.52632 | * * * 0.31579 | * * * * * * *M 0.10526 | * * * *M ***MMMMMM -0.10526 | * M *MMMMMMMMMM -0.31579 | ****MM*MMMMM* -0.52632 | M **MM*MM -0.73684 | M* MM * * -0.94737 | -1.1579 | * -1.3684 | -1.5789 | -1.7895 | -2.0000 | ________________________________________ 0.000 3.000 6.000 9.000 12.000 LNKWH This pattern is consistent with a U-shaped cost curve. This is inconsistent with a constant coefficient for LNKWH -- a possibility that is explored in Exercise 5. |_STAT ERR LNKWH /PCOR NAME N MEAN ST. DEV VARIANCE MINIMUM MAXIMUM ERR 145 0.39214E-15 0.38766 0.15028 -1.0120 1.8192 LNKWH 145 6.5567 1.9128 3.6588 0.69315 9.7243 CORRELATION MATRIX OF VARIABLES - 145 OBSERVATIONS ERR 1.0000 LNKWH 0.21156E-16 1.0000 ERR LNKWH The OLS residuals have been stored in the data series named ERR by the RESID=ERR option in the OLS command. By design, the OLS residuals are orthoganal to all the independent variables, including LNKWH; otherwise, the SSE would not be minimized. Therefore, the small correleation coefficient has no significance. |_STOP TYPE COMMAND Updated October 8, 2008