Anon

Building and Testing a Multiple Linear Regression Model 93

From the second implication above we can see that violation of the
normality assumption makes hypothesis testing suspect. More specifically,
if the assumption is violated, the t-tests explained in Chapter 3 will not be
applicable.
Typically the following three methodologies are used to test for normal-
ity of the error terms (1) chi-square statistic, (2) Jarque-Bera test statistic, and
(3) analysis of standardized residuals.

Chi-Square Statistic The chi-square statistic is defined as

(^) χ^2
2
1

=

()− ⋅

= ⋅

∑

nnp

np

ii

i i

k

(4.17)

where some interval along the real numbers is divided into k segments of
possibly equal size. The pi indicate which percentage of all n values of the
sample should fall into the ith segment if the data were truly normally dis-
tributed. Hence, the theoretical number of values that should be inside of
segment i is n · pi. The ni are the values of the sample that actually fall into
that segment i. The test statistic given by equation (4.17) is approximately
chi-square distributed with k – 1 degrees of freedom. As such, it can be
compared to the critical values of the chi-square distribution at arbitrary
α-levels. If the critical values are surpassed or, equivalently, the p-value of
the statistic is less than α, then the normal distribution has to be rejected for
the residuals.

Jarque-Bera Test Statistic The Jarque-Bera test statistic is not simple to cal-
culate manually, but most computer software packages have it installed.
Formally, it is

(^) JB=+n S ()K−













6

3

4

2

2

(4.18)

with

(^) S n
xx
n
xx
i
n
i
n

=

−

()−









=

=

∑

∑

1

1

3

1

2

1

3

2

()

(4.19)