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Pre-regression Basics

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- 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

- 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.

- 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 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.

- 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)

- 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.

- 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.

- 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?

- 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.

- 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.

- 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.

- 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?

- 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

- 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.

- Dependent (Endogenous)
- Independent(Exogenous)
- Discrete
- Continuous
- Categorical

- 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: 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.

- 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.

- 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.

- 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.

- 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.

- 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?

- 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?)

- 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)

- 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.