Meta-Analysis of Airport Noise and Home Values: Problems and Prospects
Results from 20 studies in the US and Canada from 1967 to present, analyzing the impact of airport noise on home values. Factors such as noise depreciation index, structural and locational variables, and regression coefficients are considered in the analysis.
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Meta-Analysis and Meta- Regression Airport Noise and Home Values J.P. Nelson (2004). Meta-Analysis of Airport Noise and Hedonic Property Values: Problems and Prospects, Journal of Transport Economics and Policy, Vol. 38, Part 1, pp. 1-28.
Data Description Results from 20 Studies (containing 33 separate estimates), relating home prices to airport noise. All studies in US and Canada, from 1967 to present Regressions control for other factors including: structural variables (e.g. size), locational variables, local taxes, government services, and environmental quality. Primary Variable: Noise Depreciation Index (NDI) and its Regression coefficient (effect of increasing airport noise by 1 decibel on house cost). Positive coefficient implies that as noise increases, home value decreases. The units are percent depreciation.
Study Specific Variables / Models For each study (with several exceptions), there are: Noise Depreciation Index (NDI) and its estimated standard error Mean Real Property Value (Year 2000, US $1000s) An indicator of whether accessibility (to airport) adjustment was made (1 if No Adjustment, 0 if Adjustment was made) Sample Size (log scale) Indicator of whether the response (price) scale was linear (1 if Linear, 0 if Log) Indicator of whether airport was in Canada (1 if Canada, 0 if US) Models Considered Fixed and Random Effects Meta-Analyses with no covariates Meta-Regressions with predictors: Ordinary Least Squares with robust standard errors and Weighted Least squares
Data Study City 1 Baltimore 2 Los Angeles 3 NYC 4 NYC 5 Dallas 6 Dallas 7 San Francisco 8 San Jose 9 Minneapolis 10 DC 11 Reno 12 Winnipeg 13 St. Louis 14 Rochester 15 Rochester 16 Edmonton 17 Toronto 18 Toronto 19 Reno 20 DC 21 Buffalo 22 Cleveland 23 New Orleans 24 St. Louis 25 San Diego 26 San Francisco 27 Multiple 28 Atlanta 29 Atlanta 30 Boston 31 Montreal 32 Vancouver 33 Vancouver NDI_PCT NDI_SE 1.07 1.26 1.2 0.67 0.99 0.8 0.5 0.7 0.58 1.49 0.28 1.3 0.56 0.86 0.68 0.51 0.87 0.95 0.37 1.06 0.52 0.29 0.4 0.51 0.74 0.58 0.55 0.64 0.67 0.81 0.65 0.65 0.9 MPV2000 NoAccAdj ln(SmpSz)Linear 170.703 449.359 523.9 275.942 136.25 119.9 150.42 114.45 132.27 163.871 137.603 70.104 81.832 99.893 114.014 108.73 89.982 108.063 178.2 149.63 112.575 113.894 119.763 89.44 175.713 161.789 129.236 103.354 81.178 Canada 0.823 0.788 0 3.401197 0 3.178054 0 3.401197 0 3.401197 0 8.357963 0 7.146772 0 4.406719 0 4.584967 0 5.402677 0 3.332205 0 7.375256 1 7.399398 1 8.787678 1 5.986452 1 6.897705 1 5.863631 0 6.232448 0 6.415097 1 8.373785 0 3.951244 0 4.836282 0 5.220356 0 4.962845 0 4.727388 0 4.828314 0 5.030438 0 6.739337 1 5.513429 0 4.564348 1 5.598422 1 6.056784 0 6.46925 0 6.810142 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 Note: Due to missing data, analyses will be based on only 31 or 29 airports. 0.33 0.267 0.25 0.422 0.366 0.753 0.183 0.342 0.24 0.319 0.279 0.224 0.212 0.187 0.111 0.714 0.2 0.128 0.195 0.267 0.233 0.184 0.2 0.2 0.3 0.238 0.325 0.164 0.323 118.985 124.076
Meta-Analysis with No Covariates Fixed Effects Model Assumes that each airport has the same true NDI, and that all variation is due to sampling error Random Effects Model Allows true NDIs to vary among airports along some assumed Normal Distribution. Test for Homogeneity (Fixed Effects) can be conducted after estimating the mean (Hedges and Olkin, 1985, pp.122-123). s d "Data": ,..., NDI Estimates for each airport, with standard errors: d d 1 k i
Estimates and Tests k wd i i d 1 1 = = = 2 = Fixed Effects Estimator: 1 k i d w s F d F i k 2 s w w i i i = = 1 1 i i s d 95% Confidence Interval for True Effect: ; 1 d t k F F 2 = = Test for Homogeneity: : ... Where true effect for airport H i 0 1 k i ( ) k 2 ( ) = 2 Test Statistic: Reject , 1 Q w d d H k F 0 i i = 1 i k * w d ( ) 1 Q k i 1 i = = = * 2 R = Random Effects Estimator: max 0, 1 k i d w R d + 2 2 R k s i * w 2 w i k i i = 1 i = w 1 i i k = w 1 i i = 1 i
Estimates and Tests - Results s d 347.5701 598.8022 1 = = = = Study Weight 1 1.476387 1.579735 0.353841 1.469219 1.572064 2 1.610451 2.029168 0.743704 1.601926 2.018426 3 0 4 0 5 9.182736 9.090909 1.54029 8.912281 8.823158 6 14.02741 11.22193 0.6762 13.40595 10.72476 7 16 8 0.103535 15.19648 7.598239 8 5.615328 3.930729 0.080266 5.513022 3.859115 9 7.465138 4.32978 1.46E-06 7.285406 4.225535 10 1.76364 2.627824 1.459051 1.753421 2.612597 11 29.86055 8.360954 2.695379 27.17855 7.609995 12 8.549639 11.11453 4.42669 8.314714 10.80913 13 17.36111 9.722222 0.007255 16.41909 14 9.826947 8.451175 0.768001 9.517852 8.185353 15 12.8467 8.735756 0.127333 12.32351 8.379986 16 19.92985 10.16422 0.098894 18.69833 9.536147 17 22.24991 19.35742 1.865514 20.72594 18.03157 18 28.59676 27.16692 3.905543 26.12759 24.82121 19 81.16224 30.03003 3.594347 63.99706 23.67891 20 1.961569 2.079263 0.451113 1.948935 2.065871 21 25 13 0.091332 23.09217 12.00793 22 61.03516 17.7002 5.148725 50.79051 14.72925 23 26.29849 10.5194 0.856263 24.19566 9.678266 24 14.02741 7.153979 0.069606 13.40595 6.837037 25 18.41994 13.63075 0.468947 26 29.53686 17.13138 5.78E-06 26.91014 15.60788 27 25 13.75 0.023168 23.09217 12.70069 28 25 16 0.088678 23.09217 14.77899 29 11.11111 7.444444 0.089118 10.71757 7.180773 30 17.65412 14.29984 0.930315 16.68092 13.51155 31 9.467456 6.153846 0.045806 9.180231 32 37.18025 24.16716 0.179888 33 9.585063 8.626556 0.978799 9.290769 8.361692 Wt*NDI Q W* (W*)*NDI 0.5804 0.0409 d F F 598.8022 0 0 0 0 0 0 d_F 0.5804 s{d_F} 0.0409 Lower 0.4970 Upper 0.6639 Q X2(.05,30) 43.7730 P-value 0.3737 9.19469 31.8676 31.6876 (31 1) k = = = 2 R 2 i 0.003305 20160.58 w 20160.58 598.9022 = 598.8022 1 i 17.363 12.84862 s d 319.4793 541.3124 1 = = = = 0.5902 0.0430 d R R 541.3124 5.96715 33.1118 21.52267 d_R 0.5902 s{d_R} 0.0430 Lower 0.5024 Upper 0.6780 Sum 598.8022 347.5701 31.86761 541.3124 319.4793
Meta-Regressions Regressions to determine which (if any) factors are associated with NDI Three Models Fit: Ordinary Least Squares with robust standard errors (White s heteroscedastic-consistent standard errors) Weighted Least Squares with weights equal to the inverse variance of the NDI: wi= 1/s2{di} Weighted Least Squares with weights equal to the inverse standard error of the NDI: wi= 1/s{di} Model 1 based on k = 31 airports (2 have no Mean property values) Models 2 and 3 based on k = 29 airports (2 have no weights)
Specification Tests Conducted on Models - I Ramsey s RESET Test Used to test whether the model is correctly specified and does not involve any nonlinearities among the regressors. Step 1: Fit the Original Regression with all Predictors Step 2: Fit Regression with same predictors and squared (and possibly higher order) fitted values from first model. Conduct F-test or t-test on polynomial fitted value(s) : Model is correctly specifed (no excess interactions or polynomial terms needed) H 0 2 n ^ ^ ^ ^ ^ ( ) = + + + = = + Model 1: ... 1 Y X X SSE Y Y df n p 1 1 i i 0 1 p 1 1 1 i ip i = 1 i 2 2 q n ^ ^ ^ ^ ^ ^ ^ ^ ^ ( ) = + + + + + + = = + Model 2: ... ... Y X X Y Y SSE Y Y df n p q 2 1 1 2 i i i i 0 1 1 1 p q 1 2 2 i ip i = 1 i SSE df SSE df SSE SSE 1 2 1 q SSE p 2 1 = = = 1 2 Test Statistic: -value Pr F P F F ( ) p q + obs obs 1, q N SSE df 2 2 + ( ) n q 2
Specification Tests Conducted on Models - II White s Test for Heteroscedasticity Step 1: Fit the Original Regression with all Predictors Step 2: Fit Regression relating squared residuals from step 1 to the same predictors and squared values for all numeric predictors (other version includes interactions for general specification test) Compare nR2 with Chi-Square(df = # Predictors in Step 2) : Errors have constant variance (Homoscedastic) : Error variance is related to levels of predictors (Heteroscedastic) H H 0 A ^ ^ ^ ^ ^ = + + + = Model 1: ... Y X X e Y Y i i 0 1 p 1 i ip i i ^ ^ ^ ^ ^ ^ = + + + + + + 2 i 2 1 i 2 iq 2 2 Model 2: ... ... Test Statistic: e X X X X nR 0 1 11 p qq 1 i ip 2 p q + 2 2 -Value: Pr P nR Note: This assumes the first predictors are quantitative, remaining q are dummy variables p q
Specification Tests Conducted on Models - III Jarque-Bera Test ( = # of Observations) : Errors are normally distributed 1 i i i e Y Y e n n H 0 n ^ = = which will not be 0 under Weighted Least Squares e i = 1 i ( ) ( ) n n 1 n 1 n 3 4 e e e e i i = = = = 1 1 i i S K 3/2 2 ( ) ( ) n n 1 n 1 n 2 2 e e e e i i = = 1 1 i i 1 4 n ( ) 2 = + 2 3 JB S K 6 approx 2 2 For Large Samples, under normality: ~ JB
Ordinary Least Squares with Robust Standard Errors ( ) = + + + + = = 2 i ... 1,..., ~ 0, , 0 Y X X i n N COV i j 0 1 1 i i p ip i i i j Matrix Form: = = Y X ~ 1 1 X X X X + 2 1 Y Y 0 0 0 11 1 0 p 1 1 2 2 0 21 2 1 p 2 2 = = = = = Y X V V 1 X ( X 2 n Y 0 0 1 n np p n n ) , N Y X V Ordinary Least Squares Estimator: ( ( ^ ^ ^ ( ) ) ) ( ) 1 1 1 1 = = = = E V X'X X'Y X'X X'X X'X X'VX X'X ^ ^ ^ = = Y X e Y Y ^ White's Heteroscedastic-Consistent Estimator of : V 2 1 0 0 e 0 0 e ( 2 2 ^ ^ ) ( ) 1 1 = = V X'X X'S X X'X S 0 0 2 n 0 0 e
Model 1 OLS with Robust Standard Errors - I X'X X'Y 31 4705.119 1002521.277 875.112 23938.3417 54.878862 2008.946 619.94 8 172.8444 9 6 22.9 NDI_PCT Y-hat 1.07 0.951627 0.118373 1.26 1.134955 0.125045 1.2 1.16973 0.67 1.016613 -0.34661 0.99 0.68034 0.8 0.731334 0.068666 0.5 0.702232 -0.20223 0.7 0.67103 0.58 0.64079 -0.06079 1.49 0.764735 0.725265 0.28 0.544589 -0.26459 1.3 0.714737 0.585263 0.56 0.428329 0.131671 0.86 0.766925 0.093075 0.68 0.729682 -0.04968 0.51 0.816051 -0.30605 0.87 0.796452 0.073548 0.95 0.798404 0.151596 0.37 0.508714 -0.13871 1.06 0.724718 0.335282 0.52 0.657196 0.29 0.638639 -0.34864 0.4 0.655251 -0.25525 0.51 0.648403 0.74 0.696586 0.043414 0.58 0.677793 -0.09779 0.55 0.571497 0.64 0.606768 0.033232 0.67 0.651524 0.018476 0.65 0.998794 -0.34879 0.65 0.805561 -0.15556 e 4705.119 875.112 23938.34 2008.946 8 54.87886 1037.79 47.82732 38.43661 3 47.82732 3 38.43661 619.94 3827.847 8 3 3 5.57 0.03027 172.84441 123.0956 9 6 9 1 1 6 8.18 4.93 0.30966 INV(X'X) 1.0063712 -0.001648 0.0969673 -0.136707 0.0575894 -0.000571429 -0.055222 -0.004426 -0.018469 8.90125E-05 -0.043773 b1_OLS 0.831615 0.000618 -0.01058 -0.05044 0.186153 0.223628 0.02897 -0.00164751 0.0969673 -0.136707 0.057589 -0.01847 6.17173E-06 0.0001405 0.000148 -0.000571 0.000140504 0.2420189 -0.028023 -0.055222 -0.04377 0.00014764 -0.028023 0.022246 -0.004426 -0.00631 8.9E-05 0.22071 0.020633 -0.00631 0.020633 0.234809 X'S0X 1.9013654 264.26707 0.6066714 10.009632 0.3840985 0.6104469 264.2670727 0.6066714 10.00963 0.384099 0.610447 43058.45797 54.781244 1264.664 72.30259 54.64545 54.78124434 0.6066714 4.209069 0.132789 0.557857 1264.663947 4.209069 58.83819 2.149966 4.158308 72.30259062 0.1327887 2.149966 0.384099 0.121657 54.64545305 0.557857 4.158308 0.121657 0.610447 -0.1372 -0.1384 Robust V(b1) 0.0756089 -5.04034E-05 0.0101028 -0.011169 -0.007485 -0.00171 -5.04E-05 9.6628E-08 3.26E-06 0.0101028 3.25981E-06 0.0117445 -0.002238 -0.004483 0.007423 -0.011169 6.19844E-06 -0.002238 0.001772 0.001137 -0.00037 -0.007485 -1.20455E-05 -0.004483 0.001137 0.011225 0.000843 -0.00171 2.53499E-06 0.0074229 -0.000374 0.000843 B1_OLS 0.831615 0.306195 0.000618 0.000346 -0.01058 0.120678 -0.05044 0.046869 0.186153 0.117979 0.223628 0.135423 Robust_SE -0.0215 6.2E-06 -1.2E-05 2.53E-06 0.01479
Model 1 OLS with Robust Standard Errors - I Y'Y CM SSTO SSE1 SSR1 R^2 F* P SSE2 1.82363 1.023042 0.321888 RESET P 19.6666 16.91645161 2.7501484 1.901365 0.848783 0.308632 2.232035 0.08259 ^ ^ ^ ^ = + + + = = 31 6 = Model 1: ... 1.901365 25 Y X X SSE df 1 i 0 1 5 1 5 1 1 i i 2 ^ ^ ^ ^ ^ ^ = + + + + = = 31 7 = Model 2: ... 1.82363 24 Y X X Y SSE df 2 1 i i 0 1 5 1 5 2 2 i i SSE df SSE df = 1.901365 1.82 25 24 1.82363 363 1 2 = = = = 1 2 : 0 : 0 : 1.023042 Pr 1.023042 .321888 H H TS F 1,24 F 0 A obs SSE df 2 24 2 ^ ^ ^ ^ ^ ^ = + + + + + = 2 2 2 3 Model 3: ... 0.3327723 # Predictors = 7 ( 3 31 0.3327723 nR = = Y X X X X R 3 i 0 1 5 11 33 1 5 i i 1 3 s i i ) = = 2 2 7 : Homoscedastic Errors : Heteroscedastic Error : 10.31594 Pr 10.31594 .1714 H H TS W 0 0 obs SUMMARY OUTPUT e e^2 e^3 e^4 Regression Statistics Multiple R R Square Adjusted R Square 0.1297029 Standard Error Observations sum sum/n 4.56302E-14 1.901365 0.448932 0.482637 1.47194E-15 0.061334 0.014482 0.015569 0.5768642 0.3327723 nR^2 10.31594 df P-value 7 0.171365 0.1030444 White s Test 31 S K JB p-value 0.95337405 4.13857601 6.370556373 0.041366737 ANOVA df SS MS 0.0174 1.638713 0.174628 F Significance F Regression Residual Total 7 0.121801 23 0.244217 0.010618 30 0.366018 Coefficients Standard Error t Stat 1.048289 0.399649 2.623027 0.015204 0.221553 -2.23E-05 0.001145 -0.019462 0.0343538 0.051821 0.662937 0.513961 -0.072845 -0.326956 0.116265 -2.812161 -0.02973 0.048756 -0.609777 0.547987 -0.130588 0.105426 0.05381 1.959232 0.062316 -0.005888 -7.99E-07 1.8E-06 -0.444742 0.660663 -4.52E-06 0.0252295 0.009448 2.670349 0.013667 0.005685 P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% 1.875024985 0.221553 1.875025 0.98464 -0.00239 0.002345357 0.141552908 -0.072845 0.141553 0.00989 -0.567468 -0.086443432 -0.567468 -0.086443 0.071128473 -0.130588 0.071128 0.216740113 -0.005888 2.9176E-06 -4.52E-06 0.044774263 0.005685 0.044774 Intercept MPV20000 NoAccAdj ln(sampsz) linear Canada MPV^2 SS^2 -0.00239 0.002345 Jarque-Bera Test 0.21674 2.92E-06
Weighted Least Squares Models 2 and 3 Clearly Model 1 provides a poor fit (non-significant F- Statistic (p=.0826), R2=.3086) Models 2 and 3 Use Weighted Least Squares with weights equal to the Variances and the Standard Errors, respectively, of the NDI estimates from each study 0 w 0 0 w 1 0 2 = = = * * W Y WY X WX 0 0 w n ( ) ^ 1 ( ) 1 = = * * * * ' ' ' ' X X X Y X WWX X WWY W ^ ^ ^ = = * * * * * Y X e Y Y W * e Apply the Specification Tests to
Weighted Least Squares Model 2 wi = 1/s2{di} Weight 1.476387 1.579735 1.476387 252.0238 1.610451 2.029168 1.610451 723.6707 9.182736 9.090909 9.182736 1251.148 14.02741 11.22193 14.02741 1681.886 16 8 5.615328 3.930729 5.615328 642.6742 7.465138 4.32978 7.465138 987.4138 1.76364 2.627824 29.86055 8.360954 29.86055 4108.901 8.549639 11.11453 8.549639 599.3639 8.549639 63.26218 17.36111 9.722222 17.36111 1420.694 17.36111 152.5639 9.826947 8.451175 9.826947 981.6433 9.826947 58.82855 9.826947 12.8467 8.735756 12.8467 1464.704 19.92985 10.16422 19.92985 2166.972 19.92985 116.8613 22.24991 19.35742 22.24991 2002.091 28.59676 27.16692 28.59676 3090.251 81.16224 30.03003 81.16224 14463.11 81.16224 679.6351 1.961569 2.079263 1.961569 293.5096 25 13 25 2814.375 61.03516 17.7002 61.03516 6951.538 26.29849 10.5194 26.29849 3149.586 14.02741 7.153979 14.02741 1254.612 18.41994 13.63075 18.41994 3236.623 29.53686 17.13138 29.53686 4778.739 25 13.75 25 3230.9 25 16 25 2583.85 11.11111 7.444444 11.11111 901.9778 9.467456 6.153846 9.467456 1126.485 9.467456 57.34233 9.467456 9.467456 10.41716 -4.26331 37.18025 24.16716 37.18025 4613.177 0 240.5283 WY WLS_X Yhat_WLSe_WLS 0 1.161478 0.418256 0 1.233911 0.795258 0 6.403745 2.687164 0 10.11923 1.102695 0 7.004271 0.995729 0 2.457433 1.473296 0 3.141412 1.188367 0 0.805284 1.822541 0 11.45371 -3.09276 0 8.549639 6.391931 0 0 6.627859 3.094363 0 7.511386 0.939789 12.8467 0 0 19.92985 15.40238 -5.23816 0 22.24991 16.64376 2.713658 0 28.59676 21.24831 5.918609 0 0 30.91846 -0.88843 0 0 0 0 10.82779 2.172212 0 0 25.99098 -8.29079 0 0 0 0 6.132636 1.021343 0 0 7.877649 5.753105 0 0 12.55716 4.574217 0 0 9.904166 3.845834 0 0 11.02214 4.977864 0 0 4.899553 2.544891 0 5.021485 1.476387 0 5.118101 1.610451 0 76.74897 9.182736 0 100.2507 14.02741 0 70.50751 0 25.74609 0 40.33173 0 5.876811 0 220.2292 16 2406.72 0 0 0 0 0 1.76364 289.0095 4.7226 12.8467 88.61275 9.58533 -0.84957 0 138.6714 0 183.451 0 7.750637 0 120.907 0 318.6252 0 130.5153 0 66.31301 0 88.93724 0 148.5834 0 168.4834 25 137.8357 0 50.71498 0.8755 1.203763 11.3114 -0.79201 0 37.18025 27.53588 -3.36872
Weighted Least Squares Model 2 wi = 1/s2{di} X*'X* 19757.038 2751519.9 8335.2524 131596.23 637.10346 3255.1319 X*'Y* 2751519.887 8335.2524 131596.2 637.1035 3255.132 403665907.3 1350559.6 18941976 75747.11 363413.3 1350559.565 8335.2524 66384.55 351.2393 559.9278 18941975.6 66384.553 917542.1 4386.052 20687.22 75747.10628 351.23932 4386.052 637.1035 89.63272 363413.3061 559.92785 20687.22 89.63272 3255.132 9316.5 1253143 3557.226 61158.77 500.0303 2461.986 INV(X*'X*) 0.0024467 -6.63E-06 0.0007453 -7.94178E-07 0.0005534 -0.000132 -5.79E-05 9.06E-05 -0.000263 -1.45121E-06 -0.000132 -0.000234 2.94087E-06 -5.79E-05 -3.28E-05 0.001701 7.76E-05 -0.000154 3.05509E-06 9.058E-05 -5.85E-05 b1_WLS 0.533219 0.189334468 -8.9E-05 0.019578 0.090044761 -0.01864 0.034210126 0.331991 0.157868989 0.338948 0.083420045 SE -6.6269E-06 0.0007453 -0.000263 -0.000234 -0.00015 1.15101E-07 -7.94E-07 -1.45E-06 2.94E-06 3.06E-06 0.0012986 7.99E-05 -3.28E-05 -5.9E-05 7.76E-05 0.000475 Y*'Y* 5123.9964 CM Y*'P*Y* SSR_WLS SSE_WLS SSTO_WLSR^2_WLS F_WLS 393.792 336.9767 730.7687 0.538874 5.375574 0.002038343 P_WLS 4393.227757 4787.0197 Note that CM (Correction for the Mean) is different in WLS than OLS ( ) 2 iY n ( ) n Y 2 ( ) 1 = = = ' Y X X X ' where X Y X X OLS: ' is the first column of CM 1 1 1 1 1 OLS ( ) 1 = = * * 1 * 1 * 1 * 1 * * 1 * Y X X X X Y X X WX WLS: ' ' ' where is the first column of CM WLS
Model 2 Specification Tests RESET Test: Based on the following models: 2 * * 29 ^ ^ ^ ^ ^ = + + + = = = 29 6 = * i * 1 i * 5 i * Model 1: ... 336.9767 23 Y X X X SSE Y Y df 1 1 i i 0 1 5 0 1 1 i = 1 i 2 2 * * * 29 ^ ^ ^ ^ ^ ^ ^ = + + + + = = = 29 7 = * i * 1 i * 5 i * Model 2: ... 302.7271 22 Y X X X Y SSE Y Y df 2 1 2 i i i 0 1 5 0 2 2 i = 1 i = = 2.4890 Pr 2.4890 0.1289 F 1,22 F White's Test based on regression: 2 * ^ e ^ ^ ^ ^ ^ = + + + + + 2 1 i 2 3 i ... # of predictors = 7 X X X X i 0 1 2 11 33 1 5 i i ( ) = = = 2 2 7 29 0.2189 6.3479 Pr 6.347 9 .4998 nR Jarque-Bera Test: ( ) ( ) ( ) 2 3 4 1 29 1 29 1 29 29 29 29 = = = * i * * i * * i * 10.7414 29.9407 413.7327 e e e e e e = = = 1 S 1 JB 1 i i i = = = = 2 2 0.8505 3.5859 3.9110 Pr 3.9110 .1415 K
Model 3 WLS wi = 1/s{d_i} This is a more traditional weighting scheme than Model2 The fit however, for this analysis is not as good: R2 = 0.4131 Fobs = 3.2380, P = .0234 While for Model 2: R2 = 0.5389 Fobs = 5.3756, P = .0020
