Transcript Intro to Metrics - Colorado College

Pre-regression Basics
•
•
•
•
•
•
•
•
Random Vs. Non-random variables
Stochastic Vs. Deterministic Relations
Correlation Vs. Causation
Regression Vs. Causation
Types of Data
Types of Variables
The Scientific Method
Necessary & Sufficient Conditions
1
Random Vs. Non-random
Variables
• A random (stochastic, non-deterministic)
variable is one whose value is not known
ahead of time.
• EX: Your final grade, tomorrow’s
temperature, Wednesday’s lecture topics
• What’s random to Jill may not be random to
Joe.
2
Non-random Variables
• A non-random (deterministic, nonstochastic variable) is one whose value is
known ahead of time or one whose past
value is known.
• EX: Tomorrow’s date, yesterday’s
temperature.
• Randomness & Time are linked
3
Probability
• Probability is the likelihood that a random
variable will take on a certain value.
• EX: There is an 85% chance of snow
tomorrow. Variable: Weather, Possible
values: Snow, No snow.
• Probability Distribution: The set of all
possible values of a random variable with
the associated probabilities of each.
4
Probability Distribution
Event
Prob
SNOW
85%
NO SNOW
15%
5
Continuous VS. Discrete
Distributions
• A continuous distribution shows the
probability of the different outcomes for a
variable that can take one of several
different values along a continuous scale.
• EX: Future inflation may be 3.001%, 3.002
% …50% etc. (The different possible values
are close to each other along a smooth
continuous scale)
6
Continuous Distribution
Inflation Rate
3.001
3.002
3.003
3.004
3.004
.
.
.
50
Prob
0.005
0.0025
0.34
0.45
.
.
.
.
0.002
7
Discrete Distribution
• A discrete distribution shows the probability of the
different outcomes for a variable that can take one
of several different values along a discrete scale.
• EX: The number of students in class next time
may be 1, 2, 3 etc.
• In reality most distributions (in Econ) are discrete
but we sometimes assume continuity for
theoretical & analytical ease.
8
Discrete Distribution
STUDENTS
PROB.
1
0.005
2
0.05
3
0.5
9
Subjective & Objective
Distributions
• A subjective distribution is when a person
has some idea of what the probabilities of
the different outcomes (for a RV) are but
does not have the exact numbers.
• EX: I have a pretty good guess that I will do
well in this class.
10
Objective Distributions
• An objective distribution is when the probabilities
of each outcome are based on the number of times
the outcome occurs divided by the total number of
outcomes.
• EX: The probability of drawing a red ball from a
jar with 5 red balls and a total of 50 balls is 5/50
or 1 chance in 10.
• Should all probabilities of an event sum to one?
11
Intellectual Doubletalk
• A non-random variable is a random variable
with a degenerate distribution.
• Translation: Any certain event can be
expressed as random event that happens
with probability one.
12
Stochastic Vs. Deterministic
Relations
• Deterministic relationships are exact formulas
where the dependent and independent variables
are non-random.
• EX: Ohm’s Law
Current = k*Voltage
• Stochastic relationships are not exact formulas that
relate dependent and independent variables.
• EX: Quantity demanded = f(Price, Random Term)
• Sources of Randomness: Measurement error,
unobservable variables etc.
13
Correlation Vs. Causation
• Loosely speaking correlation is the phenomenon
of two (or more) given variables exhibiting a
roughly systematic pattern of movement.
– Ex: Most of the time when stock prices fall the bond
market rallies.
• Causation is when one of the variables actually
causes the other variable to change.
• Correlation does not imply correlation.
• Causation implies correlation.
• Causation that is not supported by correlation
needs to be examined carefully.
14
Regression Vs. Causation
• A significant sign on a regression coefficient does
not imply causation.
• However if you suspect causation between X & Y
and the regression does not support this you must
proceed with caution. What is causing the lack of
significance? Experimental design flaw,
unobservable variables or poor theory?
15
Types of Data
• Time Series Data: The data are gathered over the
same set of variables in different time periods.
– EX: Price and Quantity of Summit Pale Ale Beer for a
ten year period.
• Cross Sectional Data: The data are gathered over
the same set of variables at a point in time over
different cross-sections.
– Ex: Quantity & Price of beer in ’02 across the fifty
states.
– EX2: Advertising and sales data across different firms
in MN in ‘02
16
Types of Data
• Pooled Data: The dataset is essentially a
cross-sectional dataset collected over the
same variables in each of several different
time periods.
• EX: Cigarette Price & Quantity data in each
of 50 states from 1955 – 1994.
17
Types of Variables
•
•
•
•
•
Dependent (Endogenous)
Independent(Exogenous)
Discrete
Continuous
Categorical
18
Dependent Vs. Independent
• The determination of a dependent variable
is explained by the theory.
• Independent variables come from outside
the theory. We do not know what causes
these variables but use the independent
variables to study the dependent variable.
19
Simultaneity
• Simultaneity: A theory may have more than one
dependent variable such that two or more
dependent variables influence each other. Such a
situation is referred to as a simultaneous
relationship.
• EX: Equilibrium price and equilibrium quantity
influence each other. Both are endogenous
variables explained by price theory.
20
Discrete Vs. Continuous
• A discrete variable is one that takes on
finitely many values. They do not have to
be integers such as 1, 2, 3 etc.
• A continuous variable can take on infinitely
many values.
• Dependent & Independent variables can be
either discrete or continuous.
21
Categorical
• Some variables may be either discrete or
continuous but may be grouped into
categories for ease of analysis.
• EX: Age 0 – 10 yrs, 11 – 20 yrs etc.
22
Historical Origin of Regression
• Regression is the process of finding the line
or curve that ‘best’ fit a given set of data
points.
• Francis Galton “Family Likeness in
Stature”, Proceedings of Royal Society
London, vol. 40, 1886.
23
The Scientific Method
Observe a
Phenomenon
Confirm / Re-examine
Not
Prove or Disprove
Carefully study it.
Systematic Observation
& Measurement
Develop a theory to
explain the data
Check the implications of your
theory against new data from
similar
circumstances
24
Necessary & Sufficient
Conditions
• A is said to be a sufficient condition for B.
If A happens B will be guaranteed to occur.
• EX: Ceteris Paribus, if it rains then the
football field will be wet. Necessary &
Sufficient Conditions.
A B
25
Testing Causality
• If A is observed and ceteris paribus B does
not occur then the idea that A causes B is
called into question.
• EX: Theory: C.P. Price is negatively related
to quantity demanded.
– We observe price falling and ceteris paribus
quantity demanded also falls. Does the data
support the theory?
26
Testing Causality
• Econometrically we can estimate an
equation for demand.
• Q = f(Price, Income, Other Variables)
• What is the predicted sign on the coefficient
of price? (Is it significant?)
27
Fallacies
• Denying the antecedent:
~ A ~ B
It did not rain therefore the football field cannot
be wet (How about a sprinkler system?)
• Affirming the consequent:
B A
The field is wet therefore it must have rained.
(Sprinklers may have been on)
28
Contrapositive
• The only logical equivalent to A=> B is the
contrapositive statement ~B => ~A.
• EX1: If it rains then the field will be wet.
(Contrapositive) The field is dry therefore it did not rain.
• EX2: If cigarettes are addictive then past
consumption influences present consumption.
(Contrapositive) If past consumption does not influence
present consumption then cigarettes are not addictive.
29