Preregression Basics Random Vs Nonrandom variables Stochastic Vs

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, 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

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 Event Prob SNOW 85% NO SNOW 15%

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 Inflation Rate 3. 001 3. 002 3. 003 3. 004. . . 50 Prob 0. 005 0. 0025 0. 34 0. 45. . 0. 002

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 STUDENTS PROB. 1 0. 005 2 0. 05 3 0. 5

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. – EX 2: 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 theory. • Independent variables come from outside 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 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. • EX 1: If it rains then the field will be wet. (Contrapositive) The field is dry therefore it did not rain. • EX 2: If cigarettes are addictive then past consumption influences present consumption. (Contrapositive) If past consumption does not influence present consumption then cigarettes are not addictive.