T
his appendix covers some basic mathematics that are used in econometric analy
sis. We summarize various properties of the summation operator, study properties
of linear and certain nonlinear equations, and review proportions and percents.
We also present some special functions that often arise in applied econometrics, includ
ing quadratic functions and the natural logarithm. The first four sections require only
basic algebra skills. Section A.5 contains a brief review of differential calculus; while a
knowledge of calculus is not necessary to understand most of the text, it is used in some
endofchapter appendices and in several of the more advanced chapters in Part III.
A.1 THE SUMMATION OPERATOR AND DESCRIPTIVE
STATISTICS
The
summation operator
is a useful shorthand for manipulating expressions involving
the sums of many numbers, and it plays a key role in statistics and econometric analy
sis. If {
x
i
:
i
±
1,…,
n
} denotes a sequence of
n
numbers, then we write the sum of these
numbers as
±
n
i
±
1
x
i
²
x
1
²
x
2
²
…
²
x
n
.
(A.1)
With this definition, the summation operator is easily shown to have the following prop
erties:
PROPERTY SUM. 1:
For any constant
c
,
±
n
i
±
1
c
±
nc
.
(A.2)
PROPERTY SUM. 2:
For any constant
c
,
±
n
i
±
1
cx
i
±
c
±
n
i
±
1
x
i
.
(A.3)
643
Appendix
A
Basic Mathematical Tools
xd
7/14/99 8:51 PM
Page 643
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentPROPERTY SUM. 3:
If {(
x
i
,
y
i
):
i
±
1,2,…,
n
} is a set of
n
pairs of numbers, and
a
and
b
are constants, then
±
n
i
±
1
(
ax
i
²
by
i
)
±
a
±
n
i
±
1
x
i
²
b
±
n
i
±
1
y
i
.
(A.4)
It is also important to be aware of some things that
cannot
be done with the sum
mation operator. Let {(
x
i
,
y
i
):
i
±
n
} again be a set of
n
pairs of numbers with
y
i
³
0 for each
i
. Then,
±
n
i
±
1
(
x
i
/
y
i
)
³
²
±
n
i
±
1
x
i
³´²
±
n
i
±
1
y
i
³
.
In other words, the sum of the ratios is not the ratio of the sums. In the
n
±
2 case, the
application of familiar elementary algebra also reveals this lack of equality:
x
1
/
y
1
²
x
2
/
y
2
³
(
x
1
²
x
2
)/(
y
1
²
y
2
). Similarly, the sum of the squares is not the square of the
sum:
±
n
i
±
1
x
2
i
³
²
±
n
i
±
1
x
i
³
2
, except in special cases. That these two quantities are not gener
ally equal is easiest to see when
n
±
2:
x
2
1
²
x
2
2
³
(
x
1
²
x
2
)
2
±
x
2
1
²
2
x
1
x
2
²
x
2
2
.
Given
n
numbers {
x
i
:
i
±
1,…,
n
}, we compute their
average
or
mean
by adding
them up and dividing by
n
:
x
¯
±
(1/
n
±
n
i
±
1
x
i
.
(A.5)
When the
x
i
are a sample of data on a particular variable (such as years of education),
we often call this the
sample average
(or
sample mean
) to emphasize that it is com
puted from a particular set of data. The sample average is an example of a
descriptive
statistic
; in this case, the statistic describes the central tendency of the set of points
x
i
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 VANDERWAAL
 Derivative, Yi, exper, basic mathematical tools

Click to edit the document details