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

DataStudy City NDI_PCT NDI_SE MPV2000 NoAccAdj ln(SmpSz) Linear Canada

1 Baltimore 1.07 0.823 170.703 0 3.401197 1 02 Los Angeles 1.26 0.788 449.359 0 3.178054 1 03 NYC 1.2 523.9 0 3.401197 1 04 NYC 0.67 275.942 0 3.401197 1 05 Dallas 0.99 0.33 136.25 0 8.357963 1 06 Dallas 0.8 0.267 119.9 0 7.146772 1 07 San Francisco 0.5 0.25 150.42 0 4.406719 0 08 San Jose 0.7 0.422 114.45 0 4.584967 0 09 Minneapolis 0.58 0.366 132.27 0 5.402677 0 0

10 DC 1.49 0.753 163.871 0 3.332205 0 011 Reno 0.28 0.183 137.603 0 7.375256 0 012 Winnipeg 1.3 0.342 70.104 1 7.399398 0 113 St. Louis 0.56 0.24 81.832 1 8.787678 0 014 Rochester 0.86 0.319 99.893 1 5.986452 1 015 Rochester 0.68 0.279 114.014 1 6.897705 1 016 Edmonton 0.51 0.224 108.73 1 5.863631 0 117 Toronto 0.87 0.212 89.982 0 6.232448 0 118 Toronto 0.95 0.187 108.063 0 6.415097 0 119 Reno 0.37 0.111 178.2 1 8.373785 0 020 DC 1.06 0.714 149.63 0 3.951244 0 021 Buffalo 0.52 0.2 112.575 0 4.836282 0 022 Cleveland 0.29 0.128 113.894 0 5.220356 0 023 New Orleans 0.4 0.195 119.763 0 4.962845 0 024 St. Louis 0.51 0.267 89.44 0 4.727388 0 025 San Diego 0.74 0.233 175.713 0 4.828314 0 026 San Francisco 0.58 0.184 161.789 0 5.030438 0 027 Multiple 0.55 0.2 129.236 0 6.739337 0 028 Atlanta 0.64 0.2 103.354 1 5.513429 0 029 Atlanta 0.67 0.3 81.178 0 4.564348 0 030 Boston 0.81 0.238 1 5.598422 1 031 Montreal 0.65 0.325 118.985 1 6.056784 1 132 Vancouver 0.65 0.164 124.076 0 6.46925 0 133 Vancouver 0.9 0.323 0 6.810142 0 1

Note: Due to missing data, analyses will be based on only 31 or 29 airports.

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).

1"Data": ,..., NDI Estimates for each airport, with standard errors: k id d s d

Estimates and Tests

212

1 1

0 1

1 1Fixed Effects Estimator:

95% Confidence Interval for True Effect: ; 12

Test for Homogeneity: : ... Where true effect for airport

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Test Statistic: Reject , 1

11Random Effects Estimator: max 0,

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Estimates and Tests - Results

d_F s{d_F} Lower Upper0.5804 0.0409 0.4970 0.6639

Study Weight Wt*NDI Q W* (W*)*NDI1 1.476387 1.579735 0.353841 1.469219 1.5720642 1.610451 2.029168 0.743704 1.601926 2.0184263 0 0 0 04 0 0 0 05 9.182736 9.090909 1.54029 8.912281 8.8231586 14.02741 11.22193 0.6762 13.40595 10.724767 16 8 0.103535 15.19648 7.5982398 5.615328 3.930729 0.080266 5.513022 3.8591159 7.465138 4.32978 1.46E-06 7.285406 4.225535

10 1.76364 2.627824 1.459051 1.753421 2.61259711 29.86055 8.360954 2.695379 27.17855 7.60999512 8.549639 11.11453 4.42669 8.314714 10.8091313 17.36111 9.722222 0.007255 16.41909 9.1946914 9.826947 8.451175 0.768001 9.517852 8.18535315 12.8467 8.735756 0.127333 12.32351 8.37998616 19.92985 10.16422 0.098894 18.69833 9.53614717 22.24991 19.35742 1.865514 20.72594 18.0315718 28.59676 27.16692 3.905543 26.12759 24.8212119 81.16224 30.03003 3.594347 63.99706 23.6789120 1.961569 2.079263 0.451113 1.948935 2.06587121 25 13 0.091332 23.09217 12.0079322 61.03516 17.7002 5.148725 50.79051 14.7292523 26.29849 10.5194 0.856263 24.19566 9.67826624 14.02741 7.153979 0.069606 13.40595 6.83703725 18.41994 13.63075 0.468947 17.363 12.8486226 29.53686 17.13138 5.78E-06 26.91014 15.6078827 25 13.75 0.023168 23.09217 12.7006928 25 16 0.088678 23.09217 14.7789929 11.11111 7.444444 0.089118 10.71757 7.18077330 17.65412 14.29984 0.930315 16.68092 13.5115531 9.467456 6.153846 0.045806 9.180231 5.9671532 37.18025 24.16716 0.179888 33.1118 21.5226733 9.585063 8.626556 0.978799 9.290769 8.361692

Sum 598.8022 347.5701 31.86761 541.3124 319.4793

347.5701 10.5804 0.0409

598.8022 598.8022F Fd s d

Q X2(.05,30) P-value31.8676 43.7730 0.3737

2 2

1

31.6876 (31 1)0.003305 20160.58

20160.58598.8022

598.9022

k

R ii

w

d_R s{d_R} Lower Upper0.5902 0.0430 0.5024 0.6780

319.4793 10.5902 0.0430

541.3124 541.3124R Rd s d

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)

0

2^ ^ ^ ^ ^

1 10 1 1 1 11

^ ^ ^ ^ ^ ^

2 10 1 11

: Model is correctly specifed (no excess interactions or polynomial terms needed)

Model 1: ... 1

Model 2: ...

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Y X X SSE Y Y df n p

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1Test Statistic: -value Pr

q n

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obs obsq N p q

Y SSE Y Y df n p q

SSE SSE SSE SSEdf df q

F P F FSSE SSEdf n p q

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)

0

^ ^ ^ ^ ^

0 1 1

^ ^ ^ ^2

0 1 1

: Errors have constant variance (Homoscedastic)

: Error variance is related to levels of predictors (Heteroscedastic)

Model 1: ...

Model 2: ...

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... Test Statistic:

-Value: Pr

Note: This assumes the first predictors are quantitative, remaining are dummy variables

qqi iq

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X X nR

P nR

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Specification Tests Conducted on Models - III

0

^

1

3 4

1 13/2

2

1

Jarque-Bera Test ( = # of Observations)

: Errors are normally distributed

1 which will not be 0 under Weighted Least Squares

1 1

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Ordinary Least Squares with Robust Standard Errors

20 1 1

211 1 01 1 1

221 2 12 2 2

1

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Matrix Form:

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Ordinary Least Squares Estimator:

White's Heteroscedastic-Consistent Estimator of :

0

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Model 1 – OLS with Robust Standard Errors - IX'X X'Y

31 4705.119 8 172.8444 9 6 22.94705.119 1002521.277 875.112 23938.34 2008.946 619.94 3827.847

8 875.112 8 54.87886 3 3 5.57172.84441 23938.3417 54.878862 1037.79 47.82732 38.43661 123.0956

9 2008.946 3 47.82732 9 1 8.186 619.94 3 38.43661 1 6 4.93

INV(X'X) b1_OLS1.0063712 -0.00164751 0.0969673 -0.136707 0.057589 -0.01847 0.831615-0.001648 6.17173E-06 0.0001405 0.000148 -0.000571 8.9E-05 0.0006180.0969673 0.000140504 0.2420189 -0.028023 -0.055222 -0.04377 -0.01058-0.136707 0.00014764 -0.028023 0.022246 -0.004426 -0.00631 -0.050440.0575894 -0.000571429 -0.055222 -0.004426 0.22071 0.020633 0.186153-0.018469 8.90125E-05 -0.043773 -0.00631 0.020633 0.234809 0.223628

X'S0X1.9013654 264.2670727 0.6066714 10.00963 0.384099 0.610447264.26707 43058.45797 54.781244 1264.664 72.30259 54.645450.6066714 54.78124434 0.6066714 4.209069 0.132789 0.55785710.009632 1264.663947 4.209069 58.83819 2.149966 4.1583080.3840985 72.30259062 0.1327887 2.149966 0.384099 0.1216570.6104469 54.64545305 0.557857 4.158308 0.121657 0.610447

Robust V(b1) B1_OLS Robust_SE0.0756089 -5.04034E-05 0.0101028 -0.011169 -0.007485 -0.00171 0.831615 0.306195-5.04E-05 9.6628E-08 3.26E-06 6.2E-06 -1.2E-05 2.53E-06 0.000618 0.000346

0.0101028 3.25981E-06 0.0117445 -0.002238 -0.004483 0.007423 -0.01058 0.120678-0.011169 6.19844E-06 -0.002238 0.001772 0.001137 -0.00037 -0.05044 0.046869-0.007485 -1.20455E-05 -0.004483 0.001137 0.011225 0.000843 0.186153 0.117979

-0.00171 2.53499E-06 0.0074229 -0.000374 0.000843 0.01479 0.223628 0.135423

NDI_PCT Y-hat e1.07 0.951627 0.1183731.26 1.134955 0.125045

1.2 1.16973 0.030270.67 1.016613 -0.346610.99 0.68034 0.30966

0.8 0.731334 0.0686660.5 0.702232 -0.202230.7 0.67103 0.02897

0.58 0.64079 -0.060791.49 0.764735 0.7252650.28 0.544589 -0.26459

1.3 0.714737 0.5852630.56 0.428329 0.1316710.86 0.766925 0.0930750.68 0.729682 -0.049680.51 0.816051 -0.306050.87 0.796452 0.0735480.95 0.798404 0.1515960.37 0.508714 -0.138711.06 0.724718 0.3352820.52 0.657196 -0.13720.29 0.638639 -0.34864

0.4 0.655251 -0.255250.51 0.648403 -0.13840.74 0.696586 0.0434140.58 0.677793 -0.097790.55 0.571497 -0.02150.64 0.606768 0.0332320.67 0.651524 0.0184760.65 0.998794 -0.348790.65 0.805561 -0.15556

Model 1 – OLS with Robust Standard Errors - IY'Y CM SSTO SSE1 SSR1 R^2 F* P SSE2 RESET P

19.6666 16.91645161 2.7501484 1.901365 0.848783 0.308632 2.232035 0.08259 1.82363 1.023042 0.321888

^ ^ ^ ^

1 0 1 51 5 1 1

2^ ^ ^ ^ ^ ^

2 10 1 51 5 2 2

1 2

1 20

2

2

Model 1: ... 1.901365 31 6 25

Model 2: ... 1.82363 31 7 24

1.901365 1.82

: 0 : 0 :

i i i

i ii i

A obs

Y X X SSE df

Y X X Y SSE df

SSE SSE

df dfH H TS F

SSE

df

1 3

1,24

^ ^ ^ ^ ^ ^2 2 2

3 0 1 5 11 331 5 3

0 0

36325 24

1.023042 Pr 1.023042 .3218881.82363

24

Model 3: ... 0.3327723 # Predictors = 7

: Homoscedastic Errors : Heteroscedastic Error

i ii i i

F

Y X X X X R

H H

2 23 7s : 31 0.3327723 10.31594 Pr 10.31594 .1714obsTS W nR

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.5768642 nR^2 df P-valueR Square 0.3327723 10.31594 7 0.171365Adjusted R Square0.1297029Standard Error 0.1030444Observations 31

ANOVAdf SS MS F Significance F

Regression 7 0.121801 0.0174 1.638713 0.174628Residual 23 0.244217 0.010618Total 30 0.366018

CoefficientsStandard Error t Stat P-value Lower 95% Upper 95% Lower 95.0%Upper 95.0%Intercept 1.048289 0.399649 2.623027 0.015204 0.221553 1.875024985 0.221553 1.875025MPV20000 -2.23E-05 0.001145 -0.019462 0.98464 -0.00239 0.002345357 -0.00239 0.002345NoAccAdj 0.0343538 0.051821 0.662937 0.513961 -0.072845 0.141552908 -0.072845 0.141553ln(sampsz) -0.326956 0.116265 -2.812161 0.00989 -0.567468 -0.086443432 -0.567468 -0.086443linear -0.02973 0.048756 -0.609777 0.547987 -0.130588 0.071128473 -0.130588 0.071128Canada 0.105426 0.05381 1.959232 0.062316 -0.005888 0.216740113 -0.005888 0.21674MPV^2 -7.99E-07 1.8E-06 -0.444742 0.660663 -4.52E-06 2.9176E-06 -4.52E-06 2.92E-06SS^2 0.0252295 0.009448 2.670349 0.013667 0.005685 0.044774263 0.005685 0.044774

e e^2 e^3 e^4sum 4.56302E-14 1.901365 0.448932 0.482637sum/n 1.47194E-15 0.061334 0.014482 0.015569

S 0.95337405K 4.13857601JB 6.370556373p-value 0.041366737

Jarque-Bera Test

White’sTest

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

1

2

1 1

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0 0

0 0

0 0

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Apply the Specification Tests to

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Weighted Least Squares – Model 2 – wi = 1/s2{di}Weight WY WLS_X Yhat_WLS e_WLS1.476387 1.579735 1.476387 252.0238 0 5.021485 1.476387 0 1.161478 0.4182561.610451 2.029168 1.610451 723.6707 0 5.118101 1.610451 0 1.233911 0.7952589.182736 9.090909 9.182736 1251.148 0 76.74897 9.182736 0 6.403745 2.68716414.02741 11.22193 14.02741 1681.886 0 100.2507 14.02741 0 10.11923 1.102695

16 8 16 2406.72 0 70.50751 0 0 7.004271 0.9957295.615328 3.930729 5.615328 642.6742 0 25.74609 0 0 2.457433 1.4732967.465138 4.32978 7.465138 987.4138 0 40.33173 0 0 3.141412 1.188367

1.76364 2.627824 1.76364 289.0095 0 5.876811 0 0 0.805284 1.82254129.86055 8.360954 29.86055 4108.901 0 220.2292 0 0 11.45371 -3.092768.549639 11.11453 8.549639 599.3639 8.549639 63.26218 0 8.549639 6.391931 4.722617.36111 9.722222 17.36111 1420.694 17.36111 152.5639 0 0 6.627859 3.0943639.826947 8.451175 9.826947 981.6433 9.826947 58.82855 9.826947 0 7.511386 0.939789

12.8467 8.735756 12.8467 1464.704 12.8467 88.61275 12.8467 0 9.58533 -0.8495719.92985 10.16422 19.92985 2166.972 19.92985 116.8613 0 19.92985 15.40238 -5.2381622.24991 19.35742 22.24991 2002.091 0 138.6714 0 22.24991 16.64376 2.71365828.59676 27.16692 28.59676 3090.251 0 183.451 0 28.59676 21.24831 5.91860981.16224 30.03003 81.16224 14463.11 81.16224 679.6351 0 0 30.91846 -0.888431.961569 2.079263 1.961569 293.5096 0 7.750637 0 0 0.8755 1.203763

25 13 25 2814.375 0 120.907 0 0 10.82779 2.17221261.03516 17.7002 61.03516 6951.538 0 318.6252 0 0 25.99098 -8.2907926.29849 10.5194 26.29849 3149.586 0 130.5153 0 0 11.3114 -0.7920114.02741 7.153979 14.02741 1254.612 0 66.31301 0 0 6.132636 1.02134318.41994 13.63075 18.41994 3236.623 0 88.93724 0 0 7.877649 5.75310529.53686 17.13138 29.53686 4778.739 0 148.5834 0 0 12.55716 4.574217

25 13.75 25 3230.9 0 168.4834 0 0 9.904166 3.84583425 16 25 2583.85 25 137.8357 0 0 11.02214 4.977864

11.11111 7.444444 11.11111 901.9778 0 50.71498 0 0 4.899553 2.5448919.467456 6.153846 9.467456 1126.485 9.467456 57.34233 9.467456 9.467456 10.41716 -4.2633137.18025 24.16716 37.18025 4613.177 0 240.5283 0 37.18025 27.53588 -3.36872

Weighted Least Squares – Model 2 – wi = 1/s2{di}X*'X* X*'Y*19757.038 2751519.887 8335.2524 131596.2 637.1035 3255.132 9316.52751519.9 403665907.3 1350559.6 18941976 75747.11 363413.3 12531438335.2524 1350559.565 8335.2524 66384.55 351.2393 559.9278 3557.226131596.23 18941975.6 66384.553 917542.1 4386.052 20687.22 61158.77637.10346 75747.10628 351.23932 4386.052 637.1035 89.63272 500.03033255.1319 363413.3061 559.92785 20687.22 89.63272 3255.132 2461.986

INV(X*'X*) b1_WLS SE0.0024467 -6.6269E-06 0.0007453 -0.000263 -0.000234 -0.00015 0.533219 0.189334468-6.63E-06 1.15101E-07 -7.94E-07 -1.45E-06 2.94E-06 3.06E-06 -8.9E-05 0.0012986

0.0007453 -7.94178E-07 0.0005534 -0.000132 -5.79E-05 9.06E-05 0.019578 0.090044761-0.000263 -1.45121E-06 -0.000132 7.99E-05 -3.28E-05 -5.9E-05 -0.01864 0.034210126-0.000234 2.94087E-06 -5.79E-05 -3.28E-05 0.001701 7.76E-05 0.331991 0.157868989-0.000154 3.05509E-06 9.058E-05 -5.85E-05 7.76E-05 0.000475 0.338948 0.083420045

Y*'Y* CM Y*'P*Y* SSR_WLS SSE_WLS SSTO_WLSR^2_WLS F_WLS P_WLS5123.9964 4393.227757 4787.0197 393.792 336.9767 730.7687 0.538874 5.375574 0.002038343

22 1

OLS

1

WLS

Note that CM (Correction for the Mean) is different in WLS than OLS

OLS: ' ' ' where is the first column of

WLS: ' ' ' where is

iYCM n Y

n

CM

1 1 1 1 1

* * * * * * *1 1 1 1 1

Y X X X X Y X X

Y X X X X Y X the first column of *X WX

Model 2 – Specification Tests

2* *29^ ^ ^ ^ ^* * * *

1 10 1 50 1 5 1 11

2* *^ ^ ^ ^ ^ ^* * * *

2 10 1 50 1 5 2

RESET Test: Based on the following models:

Model 1: ... 336.9767 29 6 23

Model 2: ...

i ii i i ii

i ii i i i

Y X X X SSE Y Y df

Y X X X Y SSE Y

2*29 ^

2 21

1,22

2*^ ^ ^ ^ ^ ^2 2

0 1 2 11 331 5 1 3

2 27

302.7271 29 7 22

2.4890 Pr 2.4890 0.1289

White's Test based on regression:

... # of predictors = 7

29 0.2189 6.3479 Pr 6.347

i

i

i i i i i

Y df

F F

e X X X X

nR

2 3 429 29 29* * * * * *

1 1 1

22

9 .4998

Jarque-Bera Test:

1 1 110.7414 29.9407 413.7327

29 29 29

0.8505 3.5859 3.9110 Pr 3.9110 .1415

i i ii i i

e e e e e e

S K JB

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