Exam 3 Flashcards
OLS Assumptions Violated by Grouped Data and Repeated Observations + Consequence
Errors are correlated with each other and/or with predictors
Leads to biased coefficients and p-values
Simpson’s Paradox
Normal OLS may show one relationship between X and Y, but grouping can reverse or change this relationship
OLS Fixes for Grouped Data and Repeated Observations (2)
- Fixed Effects
- Clustered Standard Errors
Purpose of the fixed effect and clustered SE for repeated observations
Fixed Effect: accounts for the fact that individuals are different from one another
Clustered SE: accounts for the fact that individuals are similar to themselves
Fixed effects adjust ______
the coefficients
Clustered standard errors adjust ______
the p-values / confidence intervals
Problem of having both grouping and repeated observations
If knowing one variable (ex. student ID) guarantees you know another variable (ex. grade level), you cannot include both as fixed effects because they are perfectly correlated/redundant.
Do we always need both fixed effects and clustered SEs?
Fixed effects are always needed for both grouping structures and repeated observations
Clustered SEs are not always needed for grouping structures (but it’s good practice)
Clusterd SEs are always needed for repeated observations
OLS Assumptions Violated by Price v. Demand Models
Linearity
How can a price v. demand model be transformed to become linear?
Take the natural log of both price and demand
Transforming Price v. Demand Models into a linear relationship
Take the natural log of both price and demand
Own-Price Elasticity of Demand
% change in quantity sold / % change in price
Elastic v. Inelastic
Elastic (OPE < -1): Demand changes more than price.
Inelastic (-1 < OPE < 0): Demand changes less than price.
Cross-Price Elasticity of Demand
% change in quantity sold of X / % change in price of Y
Substitutes v. Compliments
Substitutes (CPE > 0): Increase in price of one product increases demand for the other
Complements (CPE < 0): Increase in price of one product decreases demand for the other.
Income Elasticity of Demand
% change in quantity sold / % change in income
Inferior v. Normal + Necessities v. Luxuries
Inferior Goods (IE < 0): Demand decreases as income rises
Normal Goods (IE > 0): Demand increases as income rises.
Necessities (0 < IE < 1): Demand grows slowly with income.
Luxuries (IE > 1): Demand grows faster than income.
OLS Assumptions Violated by a Binary Dependent Variable
Non-Linear and errors are correlated
Probability
successes ÷ total events
Odds
P ÷ (1 - P)
Interpreting Odds Ratios (Intercept)
Intercept = baseline odds
OR > 1: when X goes up by 1 unit, Y =1 is more likely
OR < 1: when x goes up by 1 unit, Y = 1 is less likely
Interpreting Odds Ratios (Other Coefficients)
“a one unit change in X/this category increases/decrease the odds that Y = 1 by a factor of β”
Significance Testing for a Logistic Regression
Null Hypothesis: OR = 1
If OR = 1 (high P-value), a change in x does not significantly affect the likelihood of Y = 1
If OR ≠ 1 (low P-value), a change in x does significantly affect the likelihood of Y = 1
Tjur’s R-Squared
Formula P1 - P0
P1 = Mean predicted probability for all points at which Y = 1
P0 = Mean predicted probability for all points at which Y = 1
Higher number signifies a better model fit.
Average Marginal Effect (Logic and Interpretation)
Logic: Since a logistic regression does not have a constant marginal effect, we calculate the average along the curve.
Interpretation: A 1-unit change in X changes the probability that Y = 1 by β% (on average).
