Testing for Assumptions (listed in 6.2) of the Disturbance
of the Population Regression
Assumption
a. Linearity
Plot
Statistic
(look for key patterns)
(summarize key
patterns)
Residual vs. Predicted
NA
Residual vs. Predicted
Breusch - Pagan
Histogram
Skewness
Normal plot (Q-Q or P-P)
Kurtosis
E( e | Xs) = 0
b. Constant Variance
Var(e | Xs) = 
c. Normality
Jarque – Bera
d. Independence
(See Appendix I)
(See Appendix II)
Line plot of the residual
Durbin - Watson
1
Appendix I: Testing for Normality By Using a Q-Q Plot
A natural question in applying a normal distribution is: how can we test whether the
the data actually come from a normal distribution? A simple method is to construct a
histogram, and compare the shape with the normal distribution that has the same mean
and the standard deviation as the sample mean and the sample standard deviation of the
data, respectively. Fortunately, several convenient statistical packages are available for
drawing both the histogram with the normal curve superimposed. As an example, the
figure below shows a histogram with a normal curve for recent 61 observations of the
monthly stock rate of return of Exxon.
20
10
Std. Dev = .05
Mean = .014
0
N = 61.00
-.125
-.075
-.100
-.025
-.050
.025
.000
.075
.050
.125
.100
.175
.150
The histogram uses only 61 observations, whereas the normal curve superimposed
depicts the histogram using infinitely many observations. Therefore, sampling errors
should show up as gaps between the two curves. The graph shows that the distribution of
the rate of return of Exxon appears not markedly different from a normal distribution at
least in the middle part.
The procedure described above is easy to understand, but is not effective for revealing
a subtle but systematic departure of the histogram from normality. A better graphical
check of normality is a normal probability plot. The plot can be easily developed using
Excel and we describe the process in below.
The first step is to sort the data from the lowest to the highest. Let n be the number of
observations. Then, the lowest observation, denoted as x(1) is the (1/n) th quantile of the
data. A quantile times 100 is the percentile, so x(1) is also the (1/n) x 100 th percentile of
the data. With this convention, however, the largest observation becomes the 100
percentile of the data, which presents a problem as the 100 percentile of a normal
2
distribution is infinity, the value that can never be assumed in observation. A suggested
choice is to define the i-th largest observation, x(i) as the (i/(n+1))th quantile, or the
(i/(n+1)) x 100 percentile of the data. In the Excel worksheet on the next page both
choices are computed for comparison. The next step is to determine for each observation
the corresponding quantile of the normal distribution that has the same mean and the
standard deviation as the data. The following Excel function is a convenient way to
determine the normal (i/(n+1)) th quantile, denoted as x’(i).
x’(i) = NORMINV(i/(n+1), sample mean, sample standard deviation).
i/(n+1)
x’(i)
This value is the expected quantile if the data come from a normal distribution. x(i)
should be close to x’(i) if the normality of the distribution is true. The quantile of the
normal distribution -- with the mean and the standard deviation equaling the sample
mean and the sample standard deviation, respectively -- are computed in the column
with the heading “Expected.”
A normal probability plot is a scatterplot of the data vs. the expected quantiles. the
plot is shown below. If the data indeed come from a normal distribution, then the
scatterplot should deviate in a random fashion from the reference line. Note that the 45
degree line serves as a convenient reference line for detecting a systematic departure
from normality.
3
0.2000
0.1500
0.1000
0.0500
0.0000
-0.1 -0.0 0.00 0.05 0.10
000 -0.0500
500
00
00
00
0.15
00
0.20
00
-0.1000
-0.1500
We show below two normal probability plots, one for a sample of 60 independent
observations from a uniform distribution over the interval (0, 1) and the other for a
sample of 60 independent observations from an exponential distribution with mean 0.5.
Probabilities at both tails of a uniform distribution is significantly larger than a normal
distribution. On the other hand, an exponential distribution is skewed. These departures
from the normality are quite common and the two plots show how each departure can be
detected by using the normal probability plots.
Normal Probability Plot of Data From a Uniform Distribution
The plot on the right is a normal
probability plot of observations from a
uniform distribution. The plot has an
elongated S shape.
1.0
0.9
0.8
U
n
i
f
o
r
m
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.5
ex punif
4
1.0
Normal Probability Plot of Data From an Exponential Distribution
The plot on the right is a normal probability
plot of observations from an exponential
distribution. The plot is convex.
2
E
x
p
o1
n
0
0
1
expexpo
5
2
Appendix II: Testing for Normality By Using a Jarque-Bera Statistic
A normal probability plot test can be inconclusive when the plot pattern is not clear.
In such case it is useful to compute a few numbers that measure non-normality. The
asymmetry of the distribution is measured by the skewness which is the third central
moment of the distribution:
x   3



3 E
  
The sample skewness is evaluated as follows:
ˆ3 

1 n xi  x 3

 
n i 1  ˆ 
where:
n
1
2
ˆ   xi  x 
n i 1
The skewness  3 is 0 for a symmetric population, as can been seen from the formula.
ˆ 3 is significantly different from 0, then one can infer
Therefore, the sample skewness 
that the population distribution is unlikely to be symmetric and hence not normal.
Another number that can be used to check the normality of the distribution is the the
fourth central moment of the distribution, called the kurtosis  4 :
x   4


4  E 
  
The sample kurtosis is computed as follows:
4
x  x 
ˆ 4  1  
 i



ˆ 
n i 1 
n
The kurtosis measures the amount of the tail probabilities of the distribution and equals 3
ˆ 4 is significantly
for a normal population distribution. Therefore, if the sample kurtosis 
6
different from 3, then one can infer that the population distribution is unlikelty to be not
normal.
ˆ 4 as follows:
ˆ 3 and 
The Jarque-Bera Statistic combines the two measures 
n  2 1
2
ˆ 3   
ˆ 4  3 

JB 

6 
4
For a large number of observations, JB higher than 6 suggests that the population
distribution is unlikely to be normal.
The worksheet below illustrates the computation of these statistics for the rate of
return of the Weyerhaeuser stock for recent 60 months.
7
Computing the Skewness, the Kurtosis and the Jarque-Bera Statistc for Recent 60
Monthly Return of the Weyerhaeuser Stock
Month
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
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
AVERAGE
STDEVP
Weyerhaeuser
0.27020
0.09449
0.08873
-0.02731
-0.10706
0.02551
0.02139
0.08088
-0.05442
-0.27098
-0.06645
0.10320
-0.00968
0.13487
-0.10145
-0.02065
-0.02000
0.11735
-0.08037
-0.02010
-0.00513
0.01753
0.02051
0.01005
0.08657
-0.04167
0.02899
0.10986
0.01709
-0.07143
0.16018
-0.01181
-0.05976
-0.06610
0.00459
0.00913
-0.10679
0.00000
0.06154
-0.04155
0.11735
-0.06849
-0.04216
-0.15026
-0.02439
-0.07875
0.07586
0.12179
0.06514
0.00543
0.04324
0.11606
0.14085
-0.11934
0.05794
-0.01339
z-score
2.8469
0.9295
0.8667
-0.3996
-1.2699
0.1768
0.1318
0.7810
-0.6955
-3.0586
-0.8267
1.0246
-0.2073
1.3701
-1.2086
-0.3269
-0.3198
1.1789
-0.9787
-0.3209
-0.1575
0.0898
0.1223
0.0081
0.8431
-0.5563
0.2147
1.0973
0.0850
-0.8810
1.6464
-0.2305
-0.7537
-0.8229
-0.0515
-0.0019
-1.2669
-0.1016
0.5699
-0.5550
1.1789
-0.8490
-0.5617
-1.7413
-0.3677
-0.9609
0.7262
1.2275
0.6092
-0.0423
0.3703
1.1649
1.4354
-1.4039
0.5307
-0.2477
0.00452
0.00631
-0.14932
0.17021
0.00931
0.09164
-0.0522
-0.0328
-1.7310
1.7558
0.00000
JB=
(z-score)
23.0741
0.8031
0.6509
-0.0638
-2.0478
0.0055
0.0023
0.4764
-0.3364
-28.6127
-0.5649
1.0756
-0.0089
2.5722
-1.7656
-0.0349
-0.0327
1.6386
-0.9373
-0.0331
-0.0039
0.0007
0.0018
0.0000
0.5992
-0.1721
0.0099
1.3211
0.0006
-0.6839
4.4625
-0.0122
-0.4282
-0.5572
-0.0001
0.0000
-2.0333
-0.0010
0.1851
-0.1709
1.6386
-0.6120
-0.1772
-5.2795
-0.0497
-0.8873
0.3830
1.8495
0.2261
-0.0001
0.0508
1.5806
2.9572
-2.7669
0.1494
-0.0152
3
(z-score)
65.6902
0.7464
0.5641
0.0255
2.6004
0.0010
0.0003
0.3721
0.2339
87.5141
0.4670
1.1021
0.0018
3.5242
2.1339
0.0114
0.0105
1.9318
0.9173
0.0106
0.0006
0.0001
0.0002
0.0000
0.5052
0.0957
0.0021
1.4496
0.0001
0.6025
7.3469
0.0028
0.3227
0.4586
0.0000
0.0000
2.5760
0.0001
0.1055
0.0949
1.9318
0.5196
0.0995
9.1929
0.0183
0.8526
0.2782
2.2702
0.1378
0.0000
0.0188
1.8412
4.2447
3.8843
0.0793
0.0038
-0.0001
0.0000
-5.1869
5.4131
-0.03913
1.43903
0.0000
0.0000
8.9787
9.5046
3.75464
4
0.3000 0
0.2000 0
0.1000 0
0.0000 0
-0.2000
-0.1000
0.0000
-0.1000 0
-0.2000 0
-0.3000 0
8
0.1000
0.2000
0.3000