Pre regression basics
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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. Random Vs. Non-random Variables.

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Pre-regression Basics

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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


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.


Non-random Variables

  • A non-random (deterministic, non-stochastic 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


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.


Probability Distribution


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)


Continuous Distribution


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.


Discrete Distribution


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.


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?


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.


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.


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.


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?


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


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.


Types of Variables

  • Dependent (Endogenous)

  • Independent(Exogenous)

  • Discrete

  • Continuous

  • Categorical


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.


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.


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.


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.


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.


The Scientific Method


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.


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?


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?)


Fallacies

  • Denying the antecedent:

    It did not rain therefore the football field cannot be wet (How about a sprinkler system?)

  • Affirming the consequent:

    The field is wet therefore it must have rained.

    (Sprinklers may have been on)


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.


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