ECON306FINAL

Bernoulli random variable

A binary variable that takes values 1 or 0 only.

Bernoulli model

A model for binary outcomes where P(Y=1)=p.

Bernoulli MLE (no regressors)

The MLE of P(Y=1) equals the sample mean of Y.

Bernoulli MLE intuition

The estimated probability equals the fraction of observations where Y=1.

Bernoulli log-likelihood

The likelihood function for binary data based on Bernoulli probabilities.

Logit model

A binary response model where the error follows a logistic distribution.

Probit model

A binary response model where the error follows a standard normal distribution.

Difference between logit and probit

Logit uses a logistic distribution; probit uses a normal distribution.

Latent variable in probit/logit

An unobserved index that determines the probability of Y=1.

Interpretation of probit coefficients

They represent changes in the z-value, not direct probability changes.

Interpretation of logit coefficients

They represent changes in log-odds, not direct probability changes.

Odds ratio in logit

The exponentiated coefficient e^β representing the change in odds.

Marginal effects in binary models

The change in predicted probability from a small change in X.

Pseudo R-squared

A goodness-of-fit measure based on log-likelihood values.

When pseudo R-squared increases

When model fit improves and the log-likelihood increases.

Log-likelihood

The value of the likelihood function evaluated at estimated parameters.

Likelihood ratio (LR) test

A test comparing the fit of a full model to a restricted model.

LR chi-square statistic

Twice the difference in log-likelihoods between two models.

Prob > chi2

The p-value for the LR test of joint significance.

Z-statistic

Coefficient divided by its standard error.

P>|z|

Two-sided p-value testing whether a coefficient equals zero.

95% confidence interval

β̂ ± 1.96 × standard error.

When 1.96 is used

For 95% confidence intervals with z-statistics.

Perfect multicollinearity

An exact linear relationship among regressors.

Imperfect multicollinearity

High correlation among regressors that increases standard errors.

Dummy variable trap

Perfect multicollinearity caused by including all category dummies and an intercept.

Reference group

The omitted dummy category used as the baseline.

How to avoid dummy variable trap

Omit one dummy category or drop the intercept.

Measurement error

A difference between the true value and observed value of a variable.

Classical measurement error in Y

Increases variance but does not bias coefficients.

Classical measurement error in X

Causes attenuation bias toward zero.

Attenuation bias

Bias of estimated coefficients toward zero due to measurement error in X.

Selection bias

Bias arising from non-randomly selected samples.

Omitted variable bias

Bias from excluding a relevant variable correlated with X.

Simultaneous causality

When X affects Y and Y affects X.

Panel data

Data that tracks multiple entities over time.

Entity fixed effects

Controls for time-invariant characteristics of entities.

Time fixed effects

Controls for shocks common to all entities over time.

Clustered standard errors

Standard errors adjusted for within-entity correlation.

Why cluster by entity

Because errors may be correlated within the same entity over time.

Difference-in-differences

A method comparing changes over time between treated and control groups.

Difference-in-differences intuition

Identifies treatment effects using parallel trends.

Log-log model interpretation

Coefficients represent elasticities.

Linear-log model interpretation

A 1% change in X changes Y by 0.01β units.

Log-linear model interpretation

A one-unit change in X causes a percentage change in Y.