Post on 26-Dec-2015
Appendix A
Review of Math Essentials
Prepared by Vera Tabakova, East Carolina University
Appendix A: Review of Math Essentials
A.1 Summation
A.2 Some Basics
A.3 Linear Relationships
A.4 Nonlinear Relationships
Slide A-2Principles of Econometrics, 3rd Edition
A.1 Summation
The symbol Σ is the capital Greek letter sigma, and means “the sum of.”
The letter i is called the index of summation. This letter is arbitrary and may also appear as t, j, or k.
The expression is read “the sum of the terms xi, from i equal one to n.”
The numbers 1 and n are the lower limit and upper limit of summation.
Slide A-3Principles of Econometrics, 3rd Edition
1 21
n
i ni
x x x x
1
n
ii
x
A.1 Summation
Rules of summation operation:
Slide A-4Principles of Econometrics, 3rd Edition
1 21
n
i ni
x x x x
1 1
n n
i ii i
ax a x
1
n
i
a a a a na
1 1 1
( )n n n
i i i ii i i
x y x y
A.1 Summation
Slide A-5Principles of Econometrics, 3rd Edition
1 1 1
1 1 2
1 1 1 1 1 1
( )
( ) 0
n n n
i i i ii i i
n
ii n
n n n n n n
i i i i ii i i i i i
ax by a x b y
xx x x
xn n
x x x x x nx x x
A.1 Summation
Slide A-6Principles of Econometrics, 3rd Edition
1 21
( ) ( ) ( ) ( )
( ) ("Sum over all values of the index ")
( ) ("Sum over all possible values of ")
n
i ni
ii
x
f x f x f x f x
f x i
f x X
A.1 Summation
Slide A-7Principles of Econometrics, 3rd Edition
1 1 1 1
1 21 1 1
1 1 1 1
( , ) ( )
( , ) [ ( , ) ( , ) ( , )]
( , ) ( , )
m n m n
i j i ji j i j
m n m
i j i i i ni j i
m n n m
i j i ji j j i
f x y x y
f x y f x y f x y f x y
f x y f x y
A.2 Some Basics
A.2.1Numbers
Integers are the whole numbers, 0, ±1, ±2, ±3, …. .
Rational numbers can be written as a/b, where a and b are integers, and b ≠ 0.
The real numbers can be represented by points on a line. There are an uncountable number of real numbers and they are not all rational. Numbers such as are said to be irrational since they cannot be expressed as ratios, and have only decimal representations. Numbers like are not real numbers.
Slide A-8Principles of Econometrics, 3rd Edition
3.1415927 and 2
2
A.2 Some Basics
The absolute value of a number is denoted . It is the positive part
of the number, so that
Basic rules about Inequalities:
Slide A-9Principles of Econometrics, 3rd Edition
a
3 3 and 3 3.
If , then
if 0If , then
if 0
If and , then
a b a c b c
ac bc ca b
ac bc c
a b b c a c
A.2 Some Basics
A.2.2Exponents
(n terms) if n is a positive integer
x0 = 1 if x ≠ 0. 00 is not defined
Rules for working with exponents, assuming x and y are real, m and n are integers, and a and b are rational:
Slide A-10Principles of Econometrics, 3rd Edition
nx xx x
1 if 0n
nx x
x
A.2 Some Basics
Slide A-11Principles of Econometrics, 3rd Edition
1/
/ 1/
/ 1/
n n
mm n n
mm n n
a b a b
a a
a
x x
x x
x x
x x x
x x
y y
A.2.3 Scientific Notation
Slide A-12Principles of Econometrics, 3rd Edition
5 7
5 7
2
510,000 .00000034 (5.1 10 ) (3.4 10 )
(5.1 3.4) (10 10 )
17.34 10
.1734
5 512
7 7
510,000 5.1 10 5.1 101.5 10
.00000034 3.4 10 3.4 10
A.2.4 Logarithms and the number e
Slide A-13Principles of Econometrics, 3rd Edition
ln ln
Rules:
ln( ) ln( ) ln( )
ln( / ) ln( ) ln( )
ln( ) ln( )
b
a
x e b
xy x y
x y x y
x a x
A.2.4 Logarithms and the number e
Slide A-14Principles of Econometrics, 3rd Edition
A.2.4 Logarithms and the number e
The exponential function is the antilogarithm because we can recover
the value of x using it.
Slide A-15Principles of Econometrics, 3rd Edition
ln exp lnxx e x
A.3 Linear Relationships
Slide A-16Principles of Econometrics, 3rd Edition
(A.1)
(A.2)
1 2y x
2y x
1 2 1 2 10y x
A.3 Linear Relationships
Figure A.1 A linear relationship
Slide A-17Principles of Econometrics, 3rd Edition
A.3 Linear Relationships
Slide A-18Principles of Econometrics, 3rd Edition
(A.3)
(A.4)
2
dy
dx
1 2 2 3 3y x x
(A.5)2 3
2
3 23
given that is held constant
given that is held constant
yx
xy
xx
A.3 Linear Relationships
Slide A-19Principles of Econometrics, 3rd Edition
(A.6)
1 2 3Q L K
2 given that capital is held constant
, the marginal product of labor inputL
QK
L
MP
2 32 3
,y y
x x
A.3.1 Elasticity
Slide A-20Principles of Econometrics, 3rd Edition
(A.7)
∆y/y = the relative change in y, which is a decimal
%∆y = percentage change in y
(A.8a)
(A.8b)
(A.8c)
yx
y y y x xslope
x x x y y
% 100y
yy
A.4 Nonlinear Relationships
Figure A.2 A nonlinear relationship
Slide A-21Principles of Econometrics, 3rd Edition
A.4 Nonlinear Relationships
Slide A-22Principles of Econometrics, 3rd Edition
(A.9)
(A.10)
2dy dx
yx
dy y dy x xslope
dx x dx y y
A.4 Nonlinear Relationships
Slide A-23Principles of Econometrics, 3rd Edition
A.4 Nonlinear Relationships
Figure A.3 Alternative Functional FormsSlide A-24Principles of Econometrics, 3rd Edition
A.4.1 Quadratic Function
If β3 > 0, then the curve is U-shaped, and representative of average or
marginal cost functions, with increasing marginal effects. If β3 < 0,
then the curve is an inverted-U shape, useful for total product curves,
total revenue curves, and curves that exhibit diminishing marginal
effects.
Slide A-25Principles of Econometrics, 3rd Edition
21 2 3y x x
2 3 2 32 0, or / (2 )dy dx x x
A.4.2 Cubic Function
Cubic functions can have two inflection points, where the function crosses its tangent line, and changes from concave to convex, or vice versa.
Cubic functions can be used for total cost and total product curves in economics. The derivative of total cost is marginal cost, and the derivative of total product is marginal product.
If the “total” curves are cubic then the “marginal” curves are quadratic functions, a U-shaped curve for marginal cost, and an inverted-U shape for marginal product.
Slide A-26Principles of Econometrics, 3rd Edition
2 31 2 3 4y x x x
A.4.3 Reciprocal Function
Example: the Phillips Curve
Slide A-27Principles of Econometrics, 3rd Edition
11 2 1 2
1y x
x
1
1
1 2
% 100
1%
t tt
t
tt
w ww
w
wu
A.4.4 Log-Log Function
In order to use this model all values of y and x must be positive. The slopes of these curves change at every point, but the elasticity is constant and equal to β2.
Slide A-28Principles of Econometrics, 3rd Edition
1 2ln( ) ln( )y x
A.4.5 Log-Linear Function
Both its slope and elasticity change at each point and are the same sign as β2.
The slope at any point is β2y, which for β2 > 0 means that the marginal
effect increases for larger values of y.
Slide A-29Principles of Econometrics, 3rd Edition
1 2ln( )y x
1exp ln( ) exp( )y y x
A.4.6 Approximating Logarithms
Slide A-30Principles of Econometrics, 3rd Edition
(A.11) 1 0 1 00
1ln ln( ) ( )y y y y
y
1 0 1 00 0
ln(1 )
1ln ln( ) ln ( ) = relative change in
x x
yy y y y y y
y y
A.4.6 Approximating Logarithms
Slide A-31Principles of Econometrics, 3rd Edition
(A.12)
1 0
0
100 ln( ) 100 ln( ) ln( )
100
% percentage change in
y y y
y
y
y y
A.4.6 Approximating Logarithms
Slide A-32Principles of Econometrics, 3rd Edition
10
1 1
1
% 100 100( 1)
1 ln% 100 ln( )% approximation error 100 100
100 ln( ) ln( )
yy y
y
y yy y
y y
A.4.6 Approximating Logarithms
Slide A-33Principles of Econometrics, 3rd Edition
A.4.7 Approximating Logarithms in the Log-Linear Model
Slide A-34Principles of Econometrics, 3rd Edition
(A.13)
1 0
2 1 0
2
100 ln( ) ln( ) %
100
(100 )
y y y
x x
x
A.4.8 Linear-Log Function
Slide A-35Principles of Econometrics, 3rd Edition
0 1 2 0
1 1 2 1
ln( )
ln( )
y x
y x
1 0 2 1 0
21 0
2
ln( ) ln( )
100 ln( ) ln( )100
%100
y y y x x
x x
x
A.4.8 Linear-Log Function
Slide A-36Principles of Econometrics, 3rd Edition
1 2
2
ln( ) 0 500ln( )
500% 10 50
100 100
y x x
y x
A.4.8 Linear-Log Function
Slide A-37Principles of Econometrics, 3rd Edition
1 2
2
ln( ) 0 500ln( )
500% 10 50
100 100
y x x
y x
Keywords
Slide A-38Principles of Econometrics, 3rd Edition
absolute value antilogarithm asymptote ceteris paribus cubic function derivative double summation e elasticity exponential function exponents inequalities integers intercept irrational numbers linear relationship logarithm
log-linear function log-log function marginal effect natural logarithm nonlinear relationship partial derivative percentage change Phillips curve quadratic function rational numbers real numbers reciprocal function relative change scientific notation slope summation sign