|_* |_* Estimation by Weighted Least Squares (WLS) / Generalized Least Squares (GLS) |_* |_READ (DATA3-10) PROFITS SALES UNIT 88 IS NOW ASSIGNED TO: DATA3-10 ...SAMPLE RANGE IS NOW SET TO: 1 27 |_* |_* STEP 1: Estimate using OLS, saving residuals |_OLS PROFITS SALES /NOANOVA RESID=E REQUIRED MEMORY IS PAR= 2 CURRENT PAR= 500 OLS ESTIMATION 27 OBSERVATIONS DEPENDENT VARIABLE= PROFITS ...NOTE..SAMPLE RANGE SET TO: 1, 27 R-SQUARE = 0.4074 R-SQUARE ADJUSTED = 0.3837 VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.13874E+06 STANDARD ERROR OF THE ESTIMATE-SIGMA = 372.48 SUM OF SQUARED ERRORS-SSE= 0.34685E+07 MEAN OF DEPENDENT VARIABLE = 472.85 LOG OF THE LIKELIHOOD FUNCTION = -197.117 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 25 DF P-VALUE CORR. COEFFICIENT AT MEANS SALES 18.434 4.446 4.146 0.000 0.638 0.6383 0.8233 CONSTANT 83.575 118.1 0.7075 0.486 0.140 0.0000 0.1767 |_* |_* STEP 2: Check for the presence of heteroskedasticity |_DIAG /HET REQUIRED MEMORY IS PAR= 4 CURRENT PAR= 500 DEPENDENT VARIABLE = PROFITS 27 OBSERVATIONS REGRESSION COEFFICIENTS 18.4337564214 83.5752937937 HETEROSKEDASTICITY TESTS CHI-SQUARE D.F. P-VALUE TEST STATISTIC E**2 ON YHAT: 4.882 1 0.02713 E**2 ON YHAT**2: 4.214 1 0.04010 E**2 ON LOG(YHAT**2): 4.642 1 0.03120 E**2 ON X (B-P-G) TEST: BASED ON R2: 4.882 1 0.02713 BASED ON SSR: 11.191 1 0.00082 E**2 ON LAG(E**2) ARCH TEST: 0.017 1 0.89764 LOG(E**2) ON X (HARVEY) TEST: 12.539 1 0.00040 ABS(E) ON X (GLEJSER) TEST: 11.025 1 0.00090 |_* |_* STEP 3: Estimate an appropriate auxiliary regression |_* For example, the Breusch-Pagan-Godfrey specification E**2 on X |_* Save the calculated values of E**2 |_GENR E2 = E**2 |_?OLS E2 SALES / PREDICT=E2HAT |_* |_* STEP 4: Calculate appropriate weights for the WLS regression. |_* These weights should be proportional to the reciprocal of the |_* estimated variance of the disturbance for that observation. |_* (See SHAZAM manual, p. 95). |_* In the present case, this variance is estimated by E2HAT. |_GENR WT = 1 / E2HAT |_* |_* STEP 5: Check that all the weights are positive. |_* Skip observations with negative weights. |_STAT WT NAME N MEAN ST. DEV VARIANCE MINIMUM MAXIMUM WT 27 0.65142E-04 0.32133E-03 0.10325E-06 -0.37430E-03 0.15532E-02 |_SKIPIF (E2HAT .LE. 0.0001) OBSERVATION 9 WILL BE SKIPPED OBSERVATION 20 WILL BE SKIPPED OBSERVATION 25 WILL BE SKIPPED |_* |_* STEP 6: Estimate by Weighted Least Squares (WLS) |_OLS PROFITS SALES /NOANOVA WEIGHT=WT REQUIRED MEMORY IS PAR= 3 CURRENT PAR= 500 OLS ESTIMATION 24 OBSERVATIONS DEPENDENT VARIABLE= PROFITS ...NOTE..SAMPLE RANGE SET TO: 1, 27 SUM OF LOG(SQRT(ABS(WEIGHT))) = -24.461 R-SQUARE = 0.5327 R-SQUARE ADJUSTED = 0.5114 VARIANCE OF THE ESTIMATE-SIGMA**2 = 8584.7 STANDARD ERROR OF THE ESTIMATE-SIGMA = 92.654 SUM OF SQUARED ERRORS-SSE= 0.18886E+06 MEAN OF DEPENDENT VARIABLE = 168.46 LOG OF THE LIKELIHOOD FUNCTION = -166.165 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 22 DF P-VALUE CORR. COEFFICIENT AT MEANS SALES 18.362 3.667 5.008 0.000 0.730 0.7299 0.5272 CONSTANT 79.643 25.93 3.072 0.006 0.548 0.0000 0.4728 |_* |_* Compare with OLS using heteroskedasticity-consistent standard errors |_OLS PROFITS SALES /NOANOVA HETCOV REQUIRED MEMORY IS PAR= 3 CURRENT PAR= 500 OLS ESTIMATION 24 OBSERVATIONS DEPENDENT VARIABLE= PROFITS ...NOTE..SAMPLE RANGE SET TO: 1, 27 USING HETEROSKEDASTICITY-CONSISTENT COVARIANCE MATRIX R-SQUARE = 0.3758 R-SQUARE ADJUSTED = 0.3474 VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.15721E+06 STANDARD ERROR OF THE ESTIMATE-SIGMA = 396.49 SUM OF SQUARED ERRORS-SSE= 0.34586E+07 MEAN OF DEPENDENT VARIABLE = 510.87 LOG OF THE LIKELIHOOD FUNCTION = -176.594 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 22 DF P-VALUE CORR. COEFFICIENT AT MEANS SALES 18.938 5.936 3.190 0.004 0.562 0.6130 0.8706 CONSTANT 66.125 103.3 0.6403 0.529 0.135 0.0000 0.1294 |_* |_DELETE SKIP$ VARIABLE SKIP$ IS DELETED 27 WORDS RELEASED |_* |_* We can redo Steps 3-6 using alternative auxiliary regressions. |_* For example, let us use the Glejser specification ABS(E) on X. |_GENR ABSE = ABS(E) |_?OLS ABSE SALES / PREDICT=ABSEHAT |_* Since ABSEHAT is a predictor of the square root of the variance, |_* the appropriate weight is now the reciprocal of ABSEHAT**2. |_STAT ABSEHAT NAME N MEAN ST. DEV VARIANCE MINIMUM MAXIMUM ABSEHAT 27 229.50 140.69 19795. 58.925 546.32 |_SKIPIF (ABSEHAT .LE. 0.0001) |_GENR WT = 1 / ABSEHAT**2 |_OLS PROFITS SALES /NOANOVA WEIGHT=WT REQUIRED MEMORY IS PAR= 4 CURRENT PAR= 500 OLS ESTIMATION 27 OBSERVATIONS DEPENDENT VARIABLE= PROFITS ...NOTE..SAMPLE RANGE SET TO: 1, 27 SUM OF LOG(SQRT(ABS(WEIGHT))) = -10.419 R-SQUARE = 0.4461 R-SQUARE ADJUSTED = 0.4240 VARIANCE OF THE ESTIMATE-SIGMA**2 = 26348. STANDARD ERROR OF THE ESTIMATE-SIGMA = 162.32 SUM OF SQUARED ERRORS-SSE= 0.65870E+06 MEAN OF DEPENDENT VARIABLE = 209.81 LOG OF THE LIKELIHOOD FUNCTION = -185.110 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 25 DF P-VALUE CORR. COEFFICIENT AT MEANS SALES 17.074 3.805 4.487 0.000 0.668 0.6679 0.5585 CONSTANT 92.633 40.72 2.275 0.032 0.414 0.0000 0.4415 |_OLS PROFITS SALES /NOANOVA HETCOV REQUIRED MEMORY IS PAR= 4 CURRENT PAR= 500 OLS ESTIMATION 27 OBSERVATIONS DEPENDENT VARIABLE= PROFITS ...NOTE..SAMPLE RANGE SET TO: 1, 27 USING HETEROSKEDASTICITY-CONSISTENT COVARIANCE MATRIX R-SQUARE = 0.4074 R-SQUARE ADJUSTED = 0.3837 VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.13874E+06 STANDARD ERROR OF THE ESTIMATE-SIGMA = 372.48 SUM OF SQUARED ERRORS-SSE= 0.34685E+07 MEAN OF DEPENDENT VARIABLE = 472.85 LOG OF THE LIKELIHOOD FUNCTION = -197.117 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 25 DF P-VALUE CORR. COEFFICIENT AT MEANS SALES 18.434 5.317 3.467 0.002 0.570 0.6383 0.8233 CONSTANT 83.575 77.57 1.077 0.292 0.211 0.0000 0.1767 |_DELETE SKIP$ VARIABLE SKIP$ IS DELETED 27 WORDS RELEASED |_* |_* Repeat using the Harvey specification of LOG(E**2) on X. |_GENR LOGE2 = LOG(E**2) |_?OLS LOGE2 SALES /PREDICT=LOGE2HAT |_* LOGE2HAT is a predictor of the log of the variance, so the |_* appropriate weight is the reciprocal of EXP(LOGE2HAT). |_* Note that negative weights do not have to be checked in this case. |_GENR WT = 1 / EXP(LOGE2HAT) |_OLS PROFITS SALES /NOANOVA WEIGHT=WT REQUIRED MEMORY IS PAR= 4 CURRENT PAR= 500 OLS ESTIMATION 27 OBSERVATIONS DEPENDENT VARIABLE= PROFITS ...NOTE..SAMPLE RANGE SET TO: 1, 27 SUM OF LOG(SQRT(ABS(WEIGHT))) = -10.289 R-SQUARE = 0.3954 R-SQUARE ADJUSTED = 0.3712 VARIANCE OF THE ESTIMATE-SIGMA**2 = 30140. STANDARD ERROR OF THE ESTIMATE-SIGMA = 173.61 SUM OF SQUARED ERRORS-SSE= 0.75349E+06 MEAN OF DEPENDENT VARIABLE = 210.14 LOG OF THE LIKELIHOOD FUNCTION = -186.795 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 25 DF P-VALUE CORR. COEFFICIENT AT MEANS SALES 17.379 4.298 4.043 0.000 0.629 0.6288 0.6632 CONSTANT 70.786 48.00 1.475 0.153 0.283 0.0000 0.3368 |_OLS PROFITS SALES /NOANOVA HETCOV REQUIRED MEMORY IS PAR= 4 CURRENT PAR= 500 OLS ESTIMATION 27 OBSERVATIONS DEPENDENT VARIABLE= PROFITS ...NOTE..SAMPLE RANGE SET TO: 1, 27 USING HETEROSKEDASTICITY-CONSISTENT COVARIANCE MATRIX R-SQUARE = 0.4074 R-SQUARE ADJUSTED = 0.3837 VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.13874E+06 STANDARD ERROR OF THE ESTIMATE-SIGMA = 372.48 SUM OF SQUARED ERRORS-SSE= 0.34685E+07 MEAN OF DEPENDENT VARIABLE = 472.85 LOG OF THE LIKELIHOOD FUNCTION = -197.117 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 25 DF P-VALUE CORR. COEFFICIENT AT MEANS SALES 18.434 5.317 3.467 0.002 0.570 0.6383 0.8233 CONSTANT 83.575 77.57 1.077 0.292 0.211 0.0000 0.1767 |_STOP TYPE COMMAND