Welcome to SHAZAM - Version 9.0 - DEC 2003 SYSTEM=WIN-XP PAR= 2000 CURRENT WORKING DIRECTORY IS: C:\ECONOM~1\Berndt\CHAP8.DAT |_* Exercise 8.27 |_* |_FILE PATH C:\ESL\USER\ |_READ (DATA7-11) price age aircon baths bedrms cond corner culd dish fence firepl floors garage irreg lajolla lndry patio pool r | ooms sprink sqft view yard UNIT 88 IS NOW ASSIGNED TO: C:\ESL\USER\DATA7-11 ...SAMPLE RANGE IS NOW SET TO: 1 59 |_* Check the data have been read properly |_STAT /ALL NAME N MEAN ST. DEV VARIANCE MINIMUM MAXIMUM PRICE 59 228.63 112.44 12642. 110.00 590.00 AGE 59 16.186 11.068 122.50 1.0000 60.000 AIRCON 59 0.33898E-01 0.18252 0.33314E-01 0.0000 1.0000 BATHS 59 2.2797 0.72380 0.52389 1.0000 5.0000 BEDRMS 59 3.4576 0.79485 0.63179 2.0000 5.0000 COND 59 4.7627 0.75061 0.56341 2.0000 6.0000 CORNER 59 0.10169 0.30484 0.92928E-01 0.0000 1.0000 CULD 59 0.67797E-01 0.25355 0.64290E-01 0.0000 1.0000 DISH 59 0.89831 0.30484 0.92928E-01 0.0000 1.0000 FENCE 59 0.94915 0.22157 0.49094E-01 0.0000 1.0000 FIREPL 59 1.1186 0.52800 0.27878 0.0000 2.0000 FLOORS 59 1.2542 0.43917 0.19287 1.0000 2.0000 GARAGE 59 1.9322 0.61207 0.37463 0.0000 3.0000 IRREG 59 0.32203 0.47127 0.22209 0.0000 1.0000 LAJOLLA 59 0.49153 0.50422 0.25424 0.0000 1.0000 LNDRY 59 0.94915 0.22157 0.49094E-01 0.0000 1.0000 PATIO 59 1.0000 0.37139 0.13793 0.0000 2.0000 POOL 59 0.22034 0.41803 0.17475 0.0000 1.0000 ROOMS 59 2.9661 1.0822 1.1712 1.0000 5.0000 SPRINK 59 0.71186 0.45678 0.20865 0.0000 1.0000 SQFT 59 1991.5 617.19 0.38092E+06 950.00 3775.0 VIEW 59 0.52542 0.50364 0.25365 0.0000 1.0000 YARD 59 7472.7 5379.9 0.28944E+08 1530.0 36304. |_* Estimate initial model (Exercise 8.9) using OLS |_OLS price sqft yard pool /RESID=u REQUIRED MEMORY IS PAR= 15 CURRENT PAR= 2000 OLS ESTIMATION 59 OBSERVATIONS DEPENDENT VARIABLE= PRICE ...NOTE..SAMPLE RANGE SET TO: 1, 59 R-SQUARE = 0.4333 R-SQUARE ADJUSTED = 0.4024 VARIANCE OF THE ESTIMATE-SIGMA**2 = 7554.5 STANDARD ERROR OF THE ESTIMATE-SIGMA = 86.917 SUM OF SQUARED ERRORS-SSE= 0.41550E+06 MEAN OF DEPENDENT VARIABLE = 228.63 LOG OF THE LIKELIHOOD FUNCTION = -345.078 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 55 DF P-VALUE CORR. COEFFICIENT AT MEANS SQFT 0.90203E-01 0.2019E-01 4.467 0.000 0.516 0.4951 0.7857 YARD 0.53010E-02 0.2323E-02 2.282 0.026 0.294 0.2536 0.1733 POOL 36.007 27.38 1.315 0.194 0.175 0.1339 0.0347 CONSTANT 1.4411 38.99 0.3696E-01 0.971 0.005 0.0000 0.0063 |_DIAGNOS /HET REQUIRED MEMORY IS PAR= 25 CURRENT PAR= 2000 DEPENDENT VARIABLE = PRICE 59 OBSERVATIONS REGRESSION COEFFICIENTS 0.902029888687E-01 0.530102616510E-02 36.0067805465 1.44109465061 HETEROSKEDASTICITY TESTS CHI-SQUARE D.F. P-VALUE TEST STATISTIC E**2 ON YHAT: 0.044 1 0.83303 E**2 ON YHAT**2: 0.028 1 0.86797 E**2 ON LOG(YHAT**2): 0.047 1 0.82756 E**2 ON LAG(E**2) ARCH TEST: 0.103 1 0.74773 LOG(E**2) ON X (HARVEY) TEST: 5.336 3 0.14877 ABS(E) ON X (GLEJSER) TEST: 6.807 3 0.07830 E**2 ON X TEST: KOENKER(R2): 3.636 3 0.30353 B-P-G (SSR) : 14.694 3 0.00210 ...MATRIX INVERSION FAILED IN ROW 7 ...RESULTS MAY BE UNRELIABLE E**2 ON X X**2 (WHITE) TEST: KOENKER(R2): ********** 6 ********* B-P-G (SSR) : ********** 6 ********* ...MATRIX INVERSION FAILED IN ROW 7 ...RESULTS MAY BE UNRELIABLE E**2 ON X X**2 XX (WHITE) TEST: KOENKER(R2): ********** 9 ********* B-P-G (SSR) : ********** 9 ********* The White test computation fails because POOL is a dummy variable, and so is identical to its own squared value. |_* Compute White test manually |_GENR u2=u**2 |_GENR sqft2=sqft**2 |_GENR yard2=yard**2 |_GENR sqftyard=sqft*yard |_GENR sqftpool=sqft*pool |_GENR yardpool=yard*pool |_?OLS u2 sqft yard pool sqft2 yard2 sqftyard sqftpool yardpool |_GEN1 white=$N*$R2 ..NOTE..CURRENT VALUE OF $N = 59.000 ..NOTE..CURRENT VALUE OF $R2 = 0.25840 |_DISTRIB white /TYPE=CHI DF=8 CHI-SQUARE PARAMETERS- DF= 8.0000 MEAN= 8.0000 VARIANCE= 16.000 MODE= 6.0000 DATA PDF CDF 1-CDF WHITE ROW 1 15.246 0.18056E-01 0.94546 0.54541E-01 We cannot reject the null of homoskedasticity at 5%. |_* Estimate by FGLS, where var(u) ~ sqft |_GENR wt=1/sqft |_OLS price sqft yard pool /Weight=wt REQUIRED MEMORY IS PAR= 19 CURRENT PAR= 2000 OLS ESTIMATION 59 OBSERVATIONS DEPENDENT VARIABLE= PRICE ...NOTE..SAMPLE RANGE SET TO: 1, 59 SUM OF LOG(SQRT(ABS(WEIGHT))) = -1.4781 R-SQUARE = 0.3126 R-SQUARE ADJUSTED = 0.2751 VARIANCE OF THE ESTIMATE-SIGMA**2 = 8197.9 STANDARD ERROR OF THE ESTIMATE-SIGMA = 90.542 SUM OF SQUARED ERRORS-SSE= 0.45088E+06 MEAN OF DEPENDENT VARIABLE = 211.77 LOG OF THE LIKELIHOOD FUNCTION = -348.968 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 55 DF P-VALUE CORR. COEFFICIENT AT MEANS SQFT 0.74086E-01 0.2252E-01 3.290 0.002 0.405 0.4075 0.6315 YARD 0.45081E-02 0.2736E-02 1.648 0.105 0.217 0.2053 0.1450 POOL 39.412 28.72 1.372 0.176 0.182 0.1545 0.0407 CONSTANT 38.713 38.96 0.9936 0.325 0.133 0.0000 0.1828 |_* |_* Test for missing variables (see LM test in Section 6.14 and example 6.7) |_OLS u age aircon baths bedrms cond corner culd dish fence firepl floors garage irreg lajolla lndry patio pool rooms sprink sqft | view yard REQUIRED MEMORY IS PAR= 30 CURRENT PAR= 2000 OLS ESTIMATION 59 OBSERVATIONS DEPENDENT VARIABLE= U ...NOTE..SAMPLE RANGE SET TO: 1, 59 R-SQUARE = 0.6277 R-SQUARE ADJUSTED = 0.4002 VARIANCE OF THE ESTIMATE-SIGMA**2 = 4296.8 STANDARD ERROR OF THE ESTIMATE-SIGMA = 65.550 SUM OF SQUARED ERRORS-SSE= 0.15469E+06 MEAN OF DEPENDENT VARIABLE = 0.78280E-14 LOG OF THE LIKELIHOOD FUNCTION = -315.930 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 36 DF P-VALUE CORR. COEFFICIENT AT MEANS AGE -0.27415 1.234 -0.2221 0.825-0.037 -0.0358*********** AIRCON -41.603 58.27 -0.7139 0.480-0.118 -0.0897*********** BATHS 32.010 23.52 1.361 0.182 0.221 0.2737*********** BEDRMS -9.9956 18.90 -0.5287 0.600-0.088 -0.0939*********** COND 0.98789E-01 14.91 0.6626E-02 0.995 0.001 0.0009*********** CORNER 1.1261 30.86 0.3649E-01 0.971 0.006 0.0041*********** CULD 39.964 41.44 0.9644 0.341 0.159 0.1197*********** DISH -20.161 37.01 -0.5448 0.589-0.090 -0.0726*********** FENCE 67.272 68.01 0.9892 0.329 0.163 0.1761*********** FIREPL 38.285 21.34 1.794 0.081 0.286 0.2388*********** FLOORS -15.423 30.45 -0.5065 0.616-0.084 -0.0800*********** GARAGE -5.1004 19.51 -0.2615 0.795-0.044 -0.0369*********** IRREG 39.975 22.40 1.785 0.083 0.285 0.2226*********** LAJOLLA 126.15 25.01 5.043 0.000 0.643 0.7515*********** LNDRY 36.675 49.78 0.7367 0.466 0.122 0.0960*********** PATIO 6.5235 29.00 0.2249 0.823 0.037 0.0286*********** POOL 5.5653 25.27 0.2202 0.827 0.037 0.0275*********** ROOMS -2.9634 14.84 -0.1996 0.843-0.033 -0.0379*********** SPRINK 23.909 24.25 0.9861 0.331 0.162 0.1290*********** SQFT -0.57101E-01 0.4023E-01 -1.419 0.164-0.230 -0.4164*********** VIEW 17.937 19.96 0.8986 0.375 0.148 0.1067*********** YARD 0.29241E-03 0.3189E-02 0.9168E-01 0.927 0.015 0.0186*********** CONSTANT -118.79 118.5 -1.002 0.323-0.165 0.0000*********** |_GEN1 lm=$N*$R2 ..NOTE..CURRENT VALUE OF $N = 59.000 ..NOTE..CURRENT VALUE OF $R2 = 0.62771 |_DISTRIB lm /TYPE=CHI DF=19 CHI-SQUARE PARAMETERS- DF= 19.000 MEAN= 19.000 VARIANCE= 38.000 MODE= 17.000 DATA PDF CDF 1-CDF LM ROW 1 37.035 0.22635E-02 0.99214 0.78559E-02 We can clearly reject the null hypotheses that none of the additional variables belong in the model. But an inspection of the t-statistics reveals only one variable that is clearly significant: lajolla. |_* Add significant variables and reestimate |_OLS price sqft yard pool lajolla /RESID=u REQUIRED MEMORY IS PAR= 19 CURRENT PAR= 2000 OLS ESTIMATION 59 OBSERVATIONS DEPENDENT VARIABLE= PRICE ...NOTE..SAMPLE RANGE SET TO: 1, 59 R-SQUARE = 0.7072 R-SQUARE ADJUSTED = 0.6855 VARIANCE OF THE ESTIMATE-SIGMA**2 = 3975.6 STANDARD ERROR OF THE ESTIMATE-SIGMA = 63.052 SUM OF SQUARED ERRORS-SSE= 0.21468E+06 MEAN OF DEPENDENT VARIABLE = 228.63 LOG OF THE LIKELIHOOD FUNCTION = -325.599 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 54 DF P-VALUE CORR. COEFFICIENT AT MEANS SQFT 0.73740E-01 0.1483E-01 4.972 0.000 0.560 0.4048 0.6423 YARD 0.23244E-02 0.1736E-02 1.339 0.186 0.179 0.1112 0.0760 POOL 43.621 19.89 2.193 0.033 0.286 0.1622 0.0420 LAJOLLA 124.73 17.55 7.107 0.000 0.695 0.5593 0.2681 CONSTANT -6.5141 28.30 -0.2301 0.819-0.031 0.0000 -0.0285 |_DIAGNOS /HET REQUIRED MEMORY IS PAR= 32 CURRENT PAR= 2000 DEPENDENT VARIABLE = PRICE 59 OBSERVATIONS REGRESSION COEFFICIENTS 0.737404373535E-01 0.232441098049E-02 43.6209704664 124.726999651 -6.51409685364 HETEROSKEDASTICITY TESTS CHI-SQUARE D.F. P-VALUE TEST STATISTIC E**2 ON YHAT: 2.532 1 0.11153 E**2 ON YHAT**2: 2.030 1 0.15423 E**2 ON LOG(YHAT**2): 2.866 1 0.09045 E**2 ON LAG(E**2) ARCH TEST: 0.113 1 0.73714 LOG(E**2) ON X (HARVEY) TEST: 16.858 4 0.00206 ABS(E) ON X (GLEJSER) TEST: 25.340 4 0.00004 E**2 ON X TEST: KOENKER(R2): 9.225 4 0.05570 B-P-G (SSR) : 48.635 4 0.00000 ...MATRIX INVERSION FAILED IN ROW 8 ...RESULTS MAY BE UNRELIABLE E**2 ON X X**2 (WHITE) TEST: KOENKER(R2): ********** 8 ********* B-P-G (SSR) : ********** 8 ********* ...MATRIX INVERSION FAILED IN ROW 8 ...RESULTS MAY BE UNRELIABLE E**2 ON X X**2 XX (WHITE) TEST: KOENKER(R2): ********** 14 ********* B-P-G (SSR) : ********** 14 ********* |_* Compute White test manually |_GENR u2=u**2 |_GENR sqftlaj=sqft*lajolla |_GENR poollaj=lajolla*pool |_GENR YARDLAJ=LAJOLLA*YARD |_?OLS u2 sqft yard pool sqft2 yard2 sqftyard sqftpool yardpool sqftlaj poollaj yardlaj |_GEN1 white=$N*$R2 ..NOTE..CURRENT VALUE OF $N = 59.000 ..NOTE..CURRENT VALUE OF $R2 = 0.43175 |_DISTRIB white /TYPE=CHI DF=11 CHI-SQUARE PARAMETERS- DF= 11.000 MEAN= 11.000 VARIANCE= 22.000 MODE= 9.0000 DATA PDF CDF 1-CDF WHITE ROW 1 25.473 0.26387E-02 0.99223 0.77673E-02 The White test and the Harvey, Glesjer, and B-P-G test statistics are all significant. |_GENR wt=1/sqft |_OLS price sqft yard pool lajolla /Weight=wt REQUIRED MEMORY IS PAR= 21 CURRENT PAR= 2000 OLS ESTIMATION 59 OBSERVATIONS DEPENDENT VARIABLE= PRICE ...NOTE..SAMPLE RANGE SET TO: 1, 59 SUM OF LOG(SQRT(ABS(WEIGHT))) = -1.4781 R-SQUARE = 0.6489 R-SQUARE ADJUSTED = 0.6229 VARIANCE OF THE ESTIMATE-SIGMA**2 = 4264.8 STANDARD ERROR OF THE ESTIMATE-SIGMA = 65.306 SUM OF SQUARED ERRORS-SSE= 0.23030E+06 MEAN OF DEPENDENT VARIABLE = 211.77 LOG OF THE LIKELIHOOD FUNCTION = -329.149 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 54 DF P-VALUE CORR. COEFFICIENT AT MEANS SQFT 0.72515E-01 0.1624E-01 4.464 0.000 0.519 0.3989 0.6181 YARD 0.14263E-02 0.2019E-02 0.7064 0.483 0.096 0.0650 0.0459 POOL 53.342 20.80 2.564 0.013 0.329 0.2091 0.0550 LAJOLLA 126.43 17.58 7.192 0.000 0.699 0.5987 0.2825 CONSTANT -0.33878 28.62 -0.1184E-01 0.991-0.002 0.0000 -0.0016 |_* Retest for missing variables |_?OLS u age aircon baths bedrms cond corner culd dish fence firepl floors garage irreg lajolla lndry patio pool rooms sprink sqf | t view yard |_GEN1 lm=$N*$R2 ..NOTE..CURRENT VALUE OF $N = 59.000 ..NOTE..CURRENT VALUE OF $R2 = 0.27946 |_DISTRIB lm /TYPE=CHI DF=18 CHI-SQUARE PARAMETERS- DF= 18.000 MEAN= 18.000 VARIANCE= 36.000 MODE= 16.000 DATA PDF CDF 1-CDF LM ROW 1 16.488 0.69539E-01 0.44149 0.55851 We can accept the null that none of the remaining variables belong in the model. ..INPUT FILE COMPLETED..TYPE A NEW COMMAND OR TYPE: STOP Updated February 14, 2005.