english
math
statistics
data
knowledge
THE SCIENTIFIC METHOD
ENGLISH TO MATH
HYPOTHESIS IN ENGLISH: Revenues are
related to the economy
 HYPOTHESIS IN MATH: Revenues (R)
are related to income (Y), interest rates (I),
prices (P), and time (T):
 R = a + b*Y + c*I + d*P + e*T
 Assumptions on coefficients: eg. b>0

CRITICAL ASSUMPTIONS
REPRODUCIBILITY
 CORRECT SPECIFICATION
 ALL INFLUENCES THAT ARE NOT
INCLUDED, HAVE NO EFFECT
 ALL INFLUENCES THAT ARE
INCLUDED HAVE PRECISE, RIGID
EFFECT
 CETERIS PARIBUS

ADVERTISING AND CHANGE IN MARKET SHARE
Change in Market
Share
(%) 0.6
0.5
0.4
0.3
Estimated
Regression
Line
0.2
0.1
0
0
-0.1
-0.2
-0.3
-0.4
5
10
15
20
25
30
35
40
45
50Ad
Spending($mil)
MATH TO STATISTICS
NULL HYPOTHESES: State the opposite
of what you wish to prove and find a
counterexample.
 CRITICAL VALUES: You reject the null
hypothesis when you jump the hurdle
(critical value)

CRITICAL ASSUMPTIONS
CORRECT STATISTICAL METHOD
CHOSEN (eg. Regression)
 STATIONARITY (NO TREND EFFECTS)
 LEAST SUM SQUARED ERROR IS THE
APPROPRIATE CRITERION
 RANDOMNESS OF OUTSIDE
INFLUENCES (No autocorrelation or
heteroscedasticity)
 STATISTICAL DISCRIMINATION
POSSIBLE (No Multicollinearity)

FITTING THE REGRESSION LINE
M.Share = a + b* (Advt. Spending)
M.Share = .858 + .2246 * (Advt. Spending)
x
x
x
x
x
a={
x-
}=b
x
x
x
x
Advt. Spending
ADVERTISING AND MARKET SHARE: CIGARETTES
Market
Share
(%)25
UNexplained
error
20
TOTAL
error
15
explained
error
10
MEAN
5
REGRESSION LINE
0
0
3
6
9 12 15 18 21 24 27 30 33 36 39 42 45 48 51
Ad Spending($mil)
R-SQUARED
= EXPLAINED SUM SQUARED ERROR
 TOTAL SUM SQUARED ERROR
 FOR EXAMPLE: An R-squared value of
.90 means that ninety percent of the
variation in your dependent variable is
explained by the independent variables.

F-statistic
EXPLAINED MEAN SQUARE ERROR
 UNEXPLAINED MEAN SQ. ERROR
 Null Hypothesis: The dependent variable
is not explained by a combination of all of
the independent variables together.
 Go to F-tables (.05) to find the critical
values for rejecting the null hypothesis

t-statistic
The critical value tests the significance of
each variable (rejects the null hypothesis
on each variable).
 Null Hypothesis: The dependent variable
is not related to the independent variable.
 Go to t-tables (.05) to find the critical
values for rejecting the null hypothesis in
a two-tail test. Go to the .10 column for
one-tail tests.

HETEROSCEDASTICITY
x
}
x
x
x
x
x-
x
x
}
}
x
SMALL
ERROR
ELSEx
WHERE
}
LARGE
ERROR
AT THIS
END
HETEROSCEDASTIC PATTERNS OF ERROR
·
·
·
·
·
·
· ·
··
···· ·
·
··· ·
·
·
Scattered at one end
·
·
·
·· · ·
·
·
·
·
·
· ··
· ·· ·
Scattered in the middle
··
·
·
· ····
·
·
·
·
·
Scattered at both ends
AUTOCORRELATION
POSITIVE AUTOCORRELATION
(eg. curvilinear pattern or other
nonlinear pattern)
NEGATIVE AUTOCORRELATION
(eg. alternation above and below the
regression line)
·
··
· ·
·
·
·
· · ·
· · ··
·
·
·
·
·
·
·
·
·
DURBIN-WATSON TEST FOR AUTOCORRELATION
POSITIVE AUTOCORRELATION
0
|
.72
|
Reject the
null hypothesis that
there is no
POSITIVE
autocorrelation
Uncertain
region for
POSITIVE
autocorrelation
NEGATIVE AUTOCORRELATION
1.74 2.00 2.26
|
|
|
No
No
PONESIGATIVE TIVE
auto- autocorcorrerelation lation
3.28
|
Uncertain
region for
NEGATIVE
autocorrelation
Reject the
null hypothesis that
there is no
NEGATIVE
autocorrelation
4.00
|
STATISTICS TO DATA
How is data defined and collected?
 Is the data consistently collected across all
units?
 How should the data be transformed for
your particular use?

DATA COLLECTION
TIME SERIES: measures variation of a unit
or variable over several time periods
 CROSS SECTION: measures variation
during a given time period over several
different units
 POOLED CROSS SECTION- TIME
SERIES: measures variation of different
units over different time periods.

TIME SERIES
TRANSFORMATIONS
SAMPLE SIZE
 AGGREGATION OF TIME (YEAR? DAY
 AGGREGATION OF UNIT (FIRM,
MARKET, INDUSTRY)
 SPECIAL EVENT (DUMMY VARIABLE)
 MATH TRANSFORMATIONS

MATH TRANSFORMATIONS
 LOGARITHMS
 INVERSE
 PERCENTAGE
CHANGES
 INFLATION, SEASONALITY
STATISTICAL PROCEDURES REQUIRED FOR
DIFFERENT KINDS OF PROBLEM SOLVING
Simultaneous equation esimation
procedures should be used.
ARE THERE
MANY
EQUATIONS?
yes
no
DO THEY
INVOLVE
LINEAR
FUNCTIONS?
Is there
more than
one independent
variable?
yes
no
Apply Multiple
Linear Regression.
yes
no
Use Simple
Linear
Regression.
Use NON linear
regression or
other NON linear
estimation
techniques.
Global Maximum
25
OBJECTIVE FUNCTION
Local Maximum
20
15
Local
Minimum
10
5
Global Minimum
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Global Maximum
Local Maximum
25
OBJECTIVE FUNCTION
20
15
Local
Minimum
10
Global Minimum
5
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Marginal Contribution
10
5
0
-5
-10
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
WATCH OUT!! THE MARGINAL CURVE DOESN’T HAVE
THE SAME MINIMUMS AND MAXIMUMS !!!!!
MAXIMUM
SADDLE POINT
LEVEL
+
+
+
+
+
+
CHANGE
+
0
+
+
+
0
going up
flat
going up
-
0
+
CHANGE of
CHANGE
getting easier
0
getting harder
flat
getting easier
-
25
OBJECTIVE FUNCTION
20
15
10
at MAXIMUM
5
0
0
1
2
marginal
15
contribution is
10 and
zero
DECLINING!!!
5
3
4
5
6
7
8
9
10
11
12
13
14
Marginal Contribution
0
-5
-10
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
MAXIMUMS AND MINIMUMS
25
OBJECTIVE FUNCTION
20
15
10
at MAXIMUM
5
at MINIMUM
0
0
1
2
marginal
15
contribution is
10 and
zero
FALLING!!!
5
3
4
5
6
7
8
9
10
11
12
13
14
marginal
contribution is
zero and
RISING!!!
0
-5
-10
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Marginal Contribution
INFLECTION POINTS
25
OBJECTIVE
FUNCTION
20
15
an INFLECTION
10
on a downward
5
sloping
line
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15makes
the
10MARGINAL
curve bounce
5
back downward
0
-5
-10
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Marginal Contribution
INFLECTION POINTS
25
OBJECTIVE
FUNCTION
20
15
an INFLECTION
10
on a downward
5
sloping
line
an INFLECTION
on an upward
sloping line
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15makes
the
10MARGINAL
curve bounce
5
back downward
makes the
MARGINAL
curve bounce
back upward
0
-5
-10
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Marginal Contribution
100
TOTAL PROFIT FUNCTION
80
60
40
20
0
-20 0
-40
1
2
3
4
5
6
AVERAGE & MARGINAL PROFIT FUNCTION
100
Marginal profit
50
Average Profit
0
0
-50
-100
1
2
3
4
5
6
STOCK MARKET: Level, Change, & Change of change
LEVEL
+ ++ ++++++ + + + + ++
CHANGE
+
CHANGE of
CHANGE
-
STOCK MARKET: Level, Change, & Change of change
LEVEL
+ ++ ++++++ + + + + ++
CHANGE
+0
CHANGE of
CHANGE
- -
STOCK MARKET: Level, Change, & Change of change
LEVEL
+ ++ ++++++ + + + + ++
CHANGE
+0 -
CHANGE of
CHANGE
- - -
STOCK MARKET: Level, Change, & Change of change
LEVEL
+ ++ ++++++ + + + + ++
CHANGE
+0 - -
CHANGE of
CHANGE
- - - 0
STOCK MARKET: Level, Change, & Change of change
AT WHAT CHANGE & CHANGE OF CHANGE DO
WE WANT TO INVEST IN THE STOCK MARKET?
LEVEL
+ ++ ++++++ + + + + ++
CHANGE
+0 - - - 0+++ 0
-
- - - 0+ ++0- -
- 0 + ++
CHANGE of
CHANGE
-
- 0+
STOCK MARKET: Level, Change, & Change of change
AT WHAT CHANGE & CHANGE OF CHANGE DO
WE WANT TO INVEST IN THE STOCK MARKET?
LEVEL
+
+ +
+ + + + +
+ +
+
+
+
+
+
CHANGE
+
0
-
-
-
0 + +
+ 0
-
-
-
0
+
CHANGE of
CHANGE
-
-
-
0 +
+ + 0
-
-
0
+
+ +
-