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Meta-Analysis and MetaRegression
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
NDI_PCT NDI_SE MPV2000 NoAccAdj ln(SmpSz) Linear
1 Baltimore
1.07
0.823 170.703
0 3.401197
2 Los Angeles
1.26
0.788 449.359
0 3.178054
3 NYC
1.2
523.9
0 3.401197
4 NYC
0.67
275.942
0 3.401197
5 Dallas
0.99
0.33
136.25
0 8.357963
6 Dallas
0.8
0.267
119.9
0 7.146772
7 San Francisco
0.5
0.25
150.42
0 4.406719
8 San Jose
0.7
0.422
114.45
0 4.584967
9 Minneapolis
0.58
0.366
132.27
0 5.402677
10 DC
1.49
0.753 163.871
0 3.332205
11 Reno
0.28
0.183 137.603
0 7.375256
12 Winnipeg
1.3
0.342
70.104
1 7.399398
13 St. Louis
0.56
0.24
81.832
1 8.787678
14 Rochester
0.86
0.319
99.893
1 5.986452
15 Rochester
0.68
0.279 114.014
1 6.897705
16 Edmonton
0.51
0.224
108.73
1 5.863631
17 Toronto
0.87
0.212
89.982
0 6.232448
18 Toronto
0.95
0.187 108.063
0 6.415097
19 Reno
0.37
0.111
178.2
1 8.373785
20 DC
1.06
0.714
149.63
0 3.951244
21 Buffalo
0.52
0.2 112.575
0 4.836282
22 Cleveland
0.29
0.128 113.894
0 5.220356
23 New Orleans
0.4
0.195 119.763
0 4.962845
24 St. Louis
0.51
0.267
89.44
0 4.727388
25 San Diego
0.74
0.233 175.713
0 4.828314
26 San Francisco
0.58
0.184 161.789
0 5.030438
27 Multiple
0.55
0.2 129.236
0 6.739337
28 Atlanta
0.64
0.2 103.354
1 5.513429
29 Atlanta
0.67
0.3
81.178
0 4.564348
30 Boston
0.81
0.238
1 5.598422
31 Montreal
0.65
0.325 118.985
1 6.056784
32 Vancouver
0.65
0.164 124.076
0 6.46925
33 Vancouver
0.9
0.323
0 6.810142
Canada
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.
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).
"Data": d1 ,..., d k  NDI Estimates for each airport, with standard errors: s di 
Estimates and Tests
k
Fixed Effects Estimator: d F 
wd
i
i 1
k
i
wi 
w
 
1
s d i 
s2 d F 
2
i
i 1
1
k
w
i 1
i


95% Confidence Interval for True Effect: d F  t  ; k  1 s d F
2

 
Test for Homogeneity: H 0 : 1  ...   k
k

Test Statistic: Q   wi di  d F
i 1

Where  i  true effect for airport i
Reject H 0   2  , k  1
2
k
w d
*
Random Effects Estimator: d R 
i 1
k
i
w
*
i 1
i
i
w*i 
1
s 2 di    2R









Q   k  1 
2
 R  max  0,

 k 2 

  wi  
 k
i 1

  wi   k


 i 1
  wi  

 i 1  

Estimates and Tests - Results
Study
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
Sum
Weight Wt*NDI Q
W*
(W*)*NDI
1.476387 1.579735 0.353841 1.469219 1.572064
1.610451 2.029168 0.743704 1.601926 2.018426
0
0
0
0
0
0
0
0
9.182736 9.090909 1.54029 8.912281 8.823158
14.02741 11.22193
0.6762 13.40595 10.72476
16
8 0.103535 15.19648 7.598239
5.615328 3.930729 0.080266 5.513022 3.859115
7.465138 4.32978 1.46E-06 7.285406 4.225535
1.76364 2.627824 1.459051 1.753421 2.612597
29.86055 8.360954 2.695379 27.17855 7.609995
8.549639 11.11453 4.42669 8.314714 10.80913
17.36111 9.722222 0.007255 16.41909 9.19469
9.826947 8.451175 0.768001 9.517852 8.185353
12.8467 8.735756 0.127333 12.32351 8.379986
19.92985 10.16422 0.098894 18.69833 9.536147
22.24991 19.35742 1.865514 20.72594 18.03157
28.59676 27.16692 3.905543 26.12759 24.82121
81.16224 30.03003 3.594347 63.99706 23.67891
1.961569 2.079263 0.451113 1.948935 2.065871
25
13 0.091332 23.09217 12.00793
61.03516 17.7002 5.148725 50.79051 14.72925
26.29849 10.5194 0.856263 24.19566 9.678266
14.02741 7.153979 0.069606 13.40595 6.837037
18.41994 13.63075 0.468947
17.363 12.84862
29.53686 17.13138 5.78E-06 26.91014 15.60788
25
13.75 0.023168 23.09217 12.70069
25
16 0.088678 23.09217 14.77899
11.11111 7.444444 0.089118 10.71757 7.180773
17.65412 14.29984 0.930315 16.68092 13.51155
9.467456 6.153846 0.045806 9.180231 5.96715
37.18025 24.16716 0.179888 33.1118 21.52267
9.585063 8.626556 0.978799 9.290769 8.361692
598.8022 347.5701 31.86761 541.3124 319.4793
dF 
347.5701
 0.5804
598.8022
d_F
0.5804
Q
31.8676
s{d_F}
0.0409
 
s dF 
1
 0.0409
598.8022
Lower
0.4970
Upper
0.6639
X2(.05,30)
43.7730
P-value
0.3737
31.6876  (31  1)
 
 0.003305
 20160.58 
598.8022  

 598.9022 
2
R
dR 
319.4793
 0.5902
541.3124
d_R
0.5902
s{d_R}
0.0430
 
s dR 
Lower
0.5024
k
w
i 1
2
i
 20160.58
1
 0.0430
541.3124
Upper
0.6780
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)
H 0 : Model is correctly specifed (no excess interactions or polynomial terms needed)
^
^
^
^
Model 1: Y i1   0   1 X i1  ...   p X ip
^


 SSE1    Yi  Y i1 

i 1 
n
2
2
^
^ 
^ 
Model 2: Y i 2   0   1 X i1  ...   p X ip   1  Y i1   ...   q 1  Y i1 
 
 
^
^
Test Statistic: Fobs
^
^
^
 SSE1  SSE2   SSE1  SSE2 
 df  df  

q 1

1
2





 SSE2 
 SSE2 
 df 


 2 
 n   p  q 
q
df1  n   p  1
^


 SSE2    Yi  Y i 2 

i 1 
n

P-value  Pr Fq 1, N  p  q   Fobs

2
df 2  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)
H 0 : Errors have constant variance (Homoscedastic)
H A : Error variance is related to levels of predictors (Heteroscedastic)
^
^
^
^
^
^
Model 1: Y i   0   1 X i1  ...   p X ip
^
2
i
^
^
 ei  Yi  Y i
^
^
Model 2: e   0   1 X i1  ...   p X ip   11 X  ...   qq X iq2
2
i1
 Test Statistic: nR22
P -Value: Pr   p2  q  nR22 
Note: This assumes the first q predictors are quantitative, remaining p  q are dummy variables
Specification Tests Conducted on Models - III
Jarque-Bera Test (n = # of Observations)
H 0 : Errors are normally distributed
1 n
e   ei which will not be 0 under Weighted Least Squares
n i 1
^
ei  Yi  Y i
S

1 n
ei  e

n i 1

3
1

e

e
n  i

 i 1

n


2
3/2
K

1 n
ei  e

n i 1

4
1

e

e
n  i

 i 1

n


2
n 2 1
2
JB   S   K  3 
6
4

approx
For Large Samples, under normality: JB ~  22
2
Ordinary Least Squares with Robust Standard Errors
Yi   0  1 X i1  ...   p X ip   i
i  1,..., n  i ~ N  0,  i2 
COV  i ,  j   0  i  j
Matrix Form:
 Y1 
Y 
Y   2
 
 
Yn 
Y  Xβ  ε
X1 p 
1 X 11
1 X

X
21
2
p

X




X np 
1 X n1
Y ~ N  Xβ, V 
 0 
 
1
β 
 
 
  p 
 1 
 
ε   2
 
 
 n 
 12 0

2
0

2
V  V ε  


0
 0
Ordinary Least Squares Estimator:
^
β   X'X  X'Y
1
^
^
Y  Xβ

^

^
E β   X'X  X'Xβ  β
V β   X'X  X'VX  X'X 
1
^
e  YY

^
White's Heteroscedastic-Consistent Estimator of V β :
^

^
V β   X'X  X'S 0 X  X'X 
1
1
e12

0
S0  


 0
0
e22
0
0

0


en2 
1
1
0

0


 n2 
Model 1 – OLS with Robust Standard Errors - I
X'X
NDI_PCT Y-hat
e
1.07 0.951627 0.118373
1.26 1.134955 0.125045
1.2 1.16973 0.03027
0.67 1.016613 -0.34661
0.99 0.68034 0.30966
0.8 0.731334 0.068666
0.5 0.702232 -0.20223
0.7 0.67103 0.02897
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 -0.1372
0.29 0.638639 -0.34864
0.4 0.655251 -0.25525
0.51 0.648403 -0.1384
0.74 0.696586 0.043414
0.58 0.677793 -0.09779
0.55 0.571497 -0.0215
0.64 0.606768 0.033232
0.67 0.651524 0.018476
0.65 0.998794 -0.34879
0.65 0.805561 -0.15556
31
4705.119
8
172.84441
9
6
X'Y
4705.119
8 172.8444
9
6
1002521.277 875.112 23938.34 2008.946
619.94
875.112
8 54.87886
3
3
23938.3417 54.878862 1037.79 47.82732 38.43661
2008.946
3 47.82732
9
1
619.94
3 38.43661
1
6
INV(X'X)
1.0063712 -0.00164751 0.0969673
-0.001648 6.17173E-06 0.0001405
0.0969673 0.000140504 0.2420189
-0.136707
0.00014764 -0.028023
0.0575894 -0.000571429 -0.055222
-0.018469 8.90125E-05 -0.043773
X'S0X
1.9013654
264.26707
0.6066714
10.009632
0.3840985
0.6104469
-0.136707
0.000148
-0.028023
0.022246
-0.004426
-0.00631
0.057589
-0.000571
-0.055222
-0.004426
0.22071
0.020633
-0.01847
8.9E-05
-0.04377
-0.00631
0.020633
0.234809
264.2670727
43058.45797
54.78124434
1264.663947
72.30259062
54.64545305
0.6066714
54.781244
0.6066714
4.209069
0.1327887
0.557857
10.00963
1264.664
4.209069
58.83819
2.149966
4.158308
0.384099
72.30259
0.132789
2.149966
0.384099
0.121657
0.610447
54.64545
0.557857
4.158308
0.121657
0.610447
Robust V(b1)
0.0756089 -5.04034E-05
-5.04E-05
9.6628E-08
0.0101028 3.25981E-06
-0.011169 6.19844E-06
-0.007485 -1.20455E-05
-0.00171 2.53499E-06
0.0101028
3.26E-06
0.0117445
-0.002238
-0.004483
0.0074229
-0.011169
6.2E-06
-0.002238
0.001772
0.001137
-0.000374
-0.007485
-1.2E-05
-0.004483
0.001137
0.011225
0.000843
-0.00171
2.53E-06
0.007423
-0.00037
0.000843
0.01479
22.9
3827.847
5.57
123.0956
8.18
4.93
b1_OLS
0.831615
0.000618
-0.01058
-0.05044
0.186153
0.223628
B1_OLS Robust_SE
0.831615 0.306195
0.000618 0.000346
-0.01058 0.120678
-0.05044 0.046869
0.186153 0.117979
0.223628 0.135423
Model 1 – OLS with Robust Standard Errors - I
Y'Y
19.6666
^
^
CM
SSTO
SSE1
SSR1
R^2
F*
P
16.91645161 2.7501484 1.901365 0.848783 0.308632 2.232035 0.08259
^
^
Model 1: Y i1   0   1 X i1  ...   5 X i 5
SSE1  1.901365 df1  31  6  25
^ 
Model 2: Y i 2   0   1 X i1  ...   5 X i 5    Y i1 
 
^
^
^
^
H 0 :   0 H A :   0 TS : Fobs
^
^
^
^
2
SSE2  1.82363 df 2  31  7  24
 SSE1  SSE2  1.901365  1.82363 
 df  df  

25  24
1
2


 1.023042
 SSE2 
1.82363 
 df 
 24 
 2 
^
^
^
Model 3: Y i 3   0   1 X i1  ...   5 X i 5   11 X i21   33 X i23
H 0 : Homoscedastic Errors
SSE2
RESET
P
1.82363 1.023042 0.321888
Pr  F1,24  1.023042  .321888
R32  0.3327723 # Predictors = 7
H 0 : Heteroscedastic Errors TS : Wobs  nR32  31 0.3327723  10.31594 Pr   72  10.31594  .1714
SUMMARY OUTPUT
White’s
Test
Regression Statistics
Multiple R
0.5768642
R Square
0.3327723
Adjusted R Square0.1297029
Standard Error 0.1030444
Observations
31
nR^2
df
10.31594
sum
sum/n
P-value
7 0.171365
ANOVA
df
Regression
Residual
Total
Intercept
MPV20000
NoAccAdj
ln(sampsz)
linear
Canada
MPV^2
SS^2
SS
MS
F Significance F
7 0.121801
0.0174 1.638713 0.174628
23 0.244217 0.010618
30 0.366018
Coefficients
Standard Error t Stat
1.048289 0.399649 2.623027
-2.23E-05 0.001145 -0.019462
0.0343538 0.051821 0.662937
-0.326956 0.116265 -2.812161
-0.02973 0.048756 -0.609777
0.105426 0.05381 1.959232
-7.99E-07 1.8E-06 -0.444742
0.0252295 0.009448 2.670349
P-value Lower 95%
0.015204 0.221553
0.98464 -0.00239
0.513961 -0.072845
0.00989 -0.567468
0.547987 -0.130588
0.062316 -0.005888
0.660663 -4.52E-06
0.013667 0.005685
Upper 95% Lower 95.0%
Upper 95.0%
1.875024985 0.221553 1.875025
0.002345357 -0.00239 0.002345
0.141552908 -0.072845 0.141553
-0.086443432 -0.567468 -0.086443
0.071128473 -0.130588 0.071128
0.216740113 -0.005888 0.21674
2.9176E-06 -4.52E-06 2.92E-06
0.044774263 0.005685 0.044774
S
K
JB
p-value
e
e^2
e^3
e^4
4.56302E-14 1.901365 0.448932 0.482637
1.47194E-15 0.061334 0.014482 0.015569
0.95337405
4.13857601
6.370556373
0.041366737
Jarque-Bera Test
Weighted Least Squares – Models 2 and 3
• Clearly Model 1 provides a poor fit (non-significant FStatistic (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
 w1
0
W


0
w2
0
βW   X ' X
^
*
^
^
Y  X βW
*
*
0
0 


wn 
0

* 1
Y*  WY
X*  WX
X* ' Y*   X ' WWX  X ' WWY
1
^
e  Y  Y*
*
*
Apply the Specification Tests to e*
Weighted Least Squares – Model 2 – wi = 1/s2{di}
Weight WY
WLS_X
1.476387 1.579735 1.476387
1.610451 2.029168 1.610451
9.182736 9.090909 9.182736
14.02741 11.22193 14.02741
16
8
16
5.615328 3.930729 5.615328
7.465138 4.32978 7.465138
1.76364 2.627824 1.76364
29.86055 8.360954 29.86055
8.549639 11.11453 8.549639
17.36111 9.722222 17.36111
9.826947 8.451175 9.826947
12.8467 8.735756 12.8467
19.92985 10.16422 19.92985
22.24991 19.35742 22.24991
28.59676 27.16692 28.59676
81.16224 30.03003 81.16224
1.961569 2.079263 1.961569
25
13
25
61.03516 17.7002 61.03516
26.29849 10.5194 26.29849
14.02741 7.153979 14.02741
18.41994 13.63075 18.41994
29.53686 17.13138 29.53686
25
13.75
25
25
16
25
11.11111 7.444444 11.11111
9.467456 6.153846 9.467456
37.18025 24.16716 37.18025
252.0238
723.6707
1251.148
1681.886
2406.72
642.6742
987.4138
289.0095
4108.901
599.3639
1420.694
981.6433
1464.704
2166.972
2002.091
3090.251
14463.11
293.5096
2814.375
6951.538
3149.586
1254.612
3236.623
4778.739
3230.9
2583.85
901.9778
1126.485
4613.177
0
0
0
0
0
0
0
0
0
8.549639
17.36111
9.826947
12.8467
19.92985
0
0
81.16224
0
0
0
0
0
0
0
0
25
0
9.467456
0
5.021485
5.118101
76.74897
100.2507
70.50751
25.74609
40.33173
5.876811
220.2292
63.26218
152.5639
58.82855
88.61275
116.8613
138.6714
183.451
679.6351
7.750637
120.907
318.6252
130.5153
66.31301
88.93724
148.5834
168.4834
137.8357
50.71498
57.34233
240.5283
1.476387
1.610451
9.182736
14.02741
0
0
0
0
0
0
0
9.826947
12.8467
0
0
0
0
0
0
0
0
0
0
0
0
0
0
9.467456
0
Yhat_WLS e_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
8.549639 6.391931
4.7226
0 6.627859 3.094363
0 7.511386 0.939789
0 9.58533 -0.84957
19.92985 15.40238 -5.23816
22.24991 16.64376 2.713658
28.59676 21.24831 5.918609
0 30.91846 -0.88843
0
0.8755 1.203763
0 10.82779 2.172212
0 25.99098 -8.29079
0 11.3114 -0.79201
0 6.132636 1.021343
0 7.877649 5.753105
0 12.55716 4.574217
0 9.904166 3.845834
0 11.02214 4.977864
0 4.899553 2.544891
9.467456 10.41716 -4.26331
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
2751519.887
403665907.3
1350559.565
18941975.6
75747.10628
363413.3061
8335.2524
1350559.6
8335.2524
66384.553
351.23932
559.92785
131596.2
18941976
66384.55
917542.1
4386.052
20687.22
637.1035
75747.11
351.2393
4386.052
637.1035
89.63272
INV(X*'X*)
0.0024467 -6.6269E-06 0.0007453 -0.000263 -0.000234
-6.63E-06 1.15101E-07 -7.94E-07 -1.45E-06 2.94E-06
0.0007453 -7.94178E-07 0.0005534 -0.000132 -5.79E-05
-0.000263 -1.45121E-06 -0.000132 7.99E-05 -3.28E-05
-0.000234 2.94087E-06 -5.79E-05 -3.28E-05 0.001701
-0.000154 3.05509E-06 9.058E-05 -5.85E-05 7.76E-05
3255.132
363413.3
559.9278
20687.22
89.63272
3255.132
X*'Y*
9316.5
1253143
3557.226
61158.77
500.0303
2461.986
-0.00015
3.06E-06
9.06E-05
-5.9E-05
7.76E-05
0.000475
b1_WLS SE
0.533219 0.189334468
-8.9E-05
0.0012986
0.019578 0.090044761
-0.01864 0.034210126
0.331991 0.157868989
0.338948 0.083420045
Y*'Y*
CM
Y*'P*Y* SSR_WLS SSE_WLS SSTO_WLSR^2_WLS F_WLS
P_WLS
5123.9964 4393.227757 4787.0197 393.792 336.9767 730.7687 0.538874 5.375574 0.002038343
Note that CM (Correction for the Mean) is different in WLS than OLS
 
OLS: CM OLS  n Y
2
Y 

i
n
2
 Y ' X1  X1 ' X 1  X 1 ' Y
1
WLS: CM WLS  Y* ' X*1  X*1 ' X*1  X*1 ' Y*
1
where X1 is the first column of X
where X*1 is the first column of X*  WX
Model 2 – Specification Tests
RESET Test: Based on the following models:
^*
^
^
^
Model 1: Y i1   0 X   1 X  ...   5 X
*
i0
*
i1
 ^* 
Model 2: Y i 2   0 X   1 X  ...   5 X    Y i1 
 
^*
^
*
i0
^
^
*
i1
2
 * ^* 
 SSE1    Yi  Y i1   336.9767 df1  29  6  23
i 1 

29
*
i5
*
i5
^
2
2
 * ^* 
 SSE2    Yi  Y i 2   302.7271 df 2  29  7  22
i 1 

29
F  2.4890 Pr  F1,22  2.4890  0.1289
White's Test based on regression:
2
^
^
^
^
^
 ^* 
2
2
 ei    0   1 X i1  ...   2 X i 5   11 X i1   33 X i 3
 
# of predictors = 7
nR 2  29  0.2189   6.3479 Pr   72  6.3479  .4998
Jarque-Bera Test:

1 29 * *
 ei  e
29 i 1

2
 10.7414

1 29 * *
 ei  e
29 i 1

3
 29.9407

1 29 * *
 ei  e
29 i 1

4
 S  0.8505 K  3.5859 JB  3.9110 Pr   22  3.9110  .1415
 413.7327
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