Statistics and Mathematics for Economics

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Statistics and Mathematics for Economics

• Statistics Component Lecture One

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Objectives of the Lecture

• To explain why a student of Economics should be
  required to take a unit in Statistics or
  Quantitative Methods
• To introduce some concepts relating to
  probability theory

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Question
Why should a student of Economics be required to
have a knowledge Statistics or Quantitative
Methods? In order to answer this question,
consideration is going to be given to the
different activities which an economist performs.
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Where are economists employed?

• Private sector working for a large company,
  maybe responsible for producing forecasts
• Government sector perhaps assessing the
  implications of implementing a new macroeconomic
  policy
• Academic sector sometimes seeking to test the
  validity of an economic theory

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Forecasting the Future Price of Oil
An economist may be employed by a large
private company, such as B.P.. The economist may
be asked to forecast the future value of the
price of oil. Possible model Price of Oilt1
a Price of Oilt
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An Alternative Mathematical Model
Price of Oilt1 b(Price of Oilt) These
equations can be described as univariate models.
The equations can also be classified as
linear. A key responsibility is to estimate the
value of the unknown parameter.
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Assessing the impact of a government policy
What is the effect on the rate of price inflation
of a change in the rate of interest? Mathematical
Model Inflation a b(Rate of
Interest) A key responsibility of the economist
is to estimate the value of the unknown
parameter, b.
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Testing the validity of an economic theory
Possible Theory An increase (decrease) in
household income promotes an increase (decrease)
in household consumption expenditure. Mathematica
l model Consumption a b(Income) If b gt
0 then the model accords with the theory. If b ?
0 then the model contradicts the theory. The key
issue is obtaining an estimate of b and a measure
of its reliability.
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A Limitation of the Models
Unfortunately, all of the models which have
been constructed must be recognised as being
deficient. The essential problem is that they
maintain that the relationships between the
respective variables are exact.
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Implications of the Model for Price Inflation
Price Inflation a b(Rate of
Interest) The rate of interest is the only
determinant of price inflation. For a given
value of the rate of interest, there can be only
one value of price inflation. If a scatter
diagram was produced, founded upon pairs of
values of the rate of interest and price
inflation, then all of the observation points
would lie on the same straight line.
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Extending the Model
Price Inflation a b(Rate of Interest)
u u is referred to as a disturbance term for
the simple reason that, when its value is not
equal to zero, this has the effect of disturbing
an exact linear relationship between Price
Inflation and the Rate of Interest.
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Scatter Diagram
Price Inflation
a
x
x
x
slope b
Rate of Interest
0
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Interpretations of the Disturbance Term

• Diagrammatic interpretation is the vertical
  distance between the observation point and the
  straight line
• Verbal interpretation is the collective effect on
  price inflation of all of those variables which
  influence the latter but which have not been
  explicitly included in the equation

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The Potential for the Value of the Disturbance
Term to Vary

• Recall that, in practice, different values of the
  rate of price inflation can accompany a single
  value of the rate of interest
• So, for a given value of the rate of interest,
  there can occur different values of the
  disturbance term
• However, some values are more likely than others,
  which will be reflected in the probability
  distribution of the disturbance term
• A probability distribution indicates the possible
  values of a variable, together with the
  associated probabilities of occurrence

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Statistical Properties of a Disturbance Term

• The assumptions which are made about the
  statistical properties of the disturbance terms
  will govern the method which is relied upon for
  estimating values of parameters
• Also, the assumptions which are made about the
  probability distributions of the disturbance
  terms will decide the approach which is adopted
  towards testing an economic hypothesis

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The Practising Economist Requires a Knowledge of

• Probability Theory
• Specific probability distributions
• Hypothesis Testing/Statistical Inference
• Methods of Estimation
• Desirable features of estimators

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Probability Theory
Introduction to concepts relating to
probability theory. Reference will frequently be
made to an example in which a six-sided dice is
thrown and the number which is obtained is
recorded.
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Question
What is the probability of throwing a six? The
most obvious answer is 1/6. This value follows
from adopting either the empirical approach or
the classical approach.
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The Empirical Approach
The empirical approach involves throwing the
dice a large number of times (n). The number of
occasions on which a six is obtained is recorded
(m). P(6) m/n The probability equates with
a relative frequency.
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The Classical or A Priori Approach
The probability is produced using
deductive reasoning. P(6) the number of
sides of the dice showing a six ? the total
number of sides on the dice The assumption is
being made that the dice is not loaded, i.e.,
the game is fair. Also, it is recognised that
not more than one number can occur at a time.
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Subjective Approach
Another approach towards calculating a
probability is to use personal judgement. This
may be suitable if, for example, calculating the
probability that a team is going to win a
football match or that a political party is going
to win an election.
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An Experiment
The game of throwing a dice and observing the
number which is obtained is an example of an
experiment. In the context of probability
theory, an experiment is defined as an activity
which is performed, following which an
observation is made or a measurement is recorded.
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The Sample Space
A list of the potential outcomes of an experiment
is known as the sample space. In the
dice-throwing example, then, the sample space
is S 1, 2, 3, 4, 5, 6 The individual
entries are the elements or the sample
points. In this example, the sample space is
discrete.
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An Event
An event can be defined as a subset of the
sample space. For example, if E is the event of
throwing an odd number then E 1, 3, 5
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The Complement of an Event
The complement of event E is denoted by E. E
includes all of the elements which are
contained within the sample space which do not
feature in E. E 2, 4, 6
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Mutually Exclusive Events
If two events are mutually exclusive then the
occurrence of one of the events precludes the
possibility of the other event taking
place. Mutually exclusive events cannot occur
simultaneously. Mutually exclusive events will
not have any common elements.
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Event F
Define F to be the event of obtaining a number
which is greater than 3. Therefore F 4,
5, 6 Recalling that E 1, 3, 5, are E and F
mutually exclusive? The answer is no, since E
and F share a common element.
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Venn Diagram for Two Non-Mutually Exclusive Events
E
F
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Postulates of Probability Theory I
If E is an event then 0 ? P(E) ? 1. If E and F
are mutually exclusive events then P(E or F)
P(E ? F) P(E) P(F). If S S1, S2, ,
Sn Then P(S) P(S1 ? S2 ? ? Sn)
P(S1) P(S2) P(Sn) 1
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Postulates of Probability Theory II
If E is the complement of event E then P(E)
1 P(E). If E and F are not mutually
exclusive events then P(E ? F) P(E) P(F)
P(E ? F)
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An Application of the Previous Rule
E is the event of obtaining an odd number. F is
the event of obtaining a number which is greater
than 3. The events are not mutually exclusive,
and so P(E?F) P(E) P(F) - P(E?F)
½ ½ - 1/6 5/6
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One Event Containing Another
One event can be described as containing another
if it includes all of the elements which comprise
the other event, as well as one or more
additional elements which enter the sample
space. Example E is the event of obtaining an
even number G is the event of obtaining a number
which is divisible by 4. E 2, 4, 6 G
4 So G ? E.
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Venn diagram showing E and G
E
G
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Independent Events
If two events (A and B) are independent then the
occurrence of one event does not affect the
probability of the other event happening. A
necessary and sufficient condition for
statistical independence P(A ? B) P(A) x
P(B).
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An Example of Independent Events
Assume that there are two dice, which are
coloured blue and red. A is the event of
obtaining a 6 from throwing the blue dice. B is
the event of obtaining a 6 from throwing the red
dice. Demonstration that the events are
independent P(A ? B) 1/36 (classical
approach) P(A) 1/6 P(B) 1/6.
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Two events which are not independent
Assume that the blue dice is thrown before the
red dice. Whatever is the number which is
displayed on the blue dice must not be seen on
the red dice. Events A and B are no longer
statistically independent. The probability of
event B does depend upon whether event A
occurs. Demonstration P(A ? B) 0 P(A) 1/6
P(B) 1/6.
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Probability rule for when events are not
independent
If events A and B are not statistically
independent then P(A and B) P(A) x
P(B\A), where P(B\A) is a conditional
probability, i.e., the probability of B, given
that A has already occurred.