🎳Intro to Econometrics Unit 11 – Limited Dependent Variable Models

Key Concepts and Definitions

• Limited dependent variable models analyze outcomes with a restricted range or discrete values
• Dependent variable is not continuous or unlimited in its possible values
• Includes binary, categorical, ordinal, truncated, and censored variables
• Violate assumptions of linear regression models like normality and homoscedasticity
• Require specialized estimation techniques to account for the limited nature of the dependent variable
  • Maximum likelihood estimation is commonly used instead of ordinary least squares
• Interpretation differs from standard linear regression coefficients
  • Focus on probabilities, odds ratios, or marginal effects rather than direct impact on dependent variable
• Examples include modeling consumer purchase decisions (buy or not buy) or credit ratings (AAA, AA, A, etc.)

Types of Limited Dependent Variables

• Binary variables have only two possible outcomes (0 or 1, yes or no)
  • Indicates the occurrence or non-occurrence of an event or characteristic
• Categorical variables have more than two unordered categories
  • No inherent ranking or order among the categories (race, religion, political party)
• Ordinal variables have more than two ordered categories
  • Categories have a meaningful order or ranking (education level, income brackets, Likert scales)
• Truncated variables have a restricted range due to sample selection or data collection process
  • Observations outside a certain range are systematically excluded (income data above or below a threshold)
• Censored variables have a limited observable range, but true values exist beyond the limit
  • Observations at or above/below a certain threshold are recorded at the threshold value (top-coded income data)

Binary Choice Models

• Model the probability of an event occurring or a choice being made
• Dependent variable takes the value of 0 or 1
  • 1 indicates the event occurred or the choice was made, 0 otherwise
• Common binary choice models include logit and probit models
  • Differ in the assumed probability distribution (logistic vs. normal)
• Estimate the relationship between explanatory variables and the probability of the outcome
• Coefficients represent the change in the log odds (logit) or z-score (probit) of the outcome for a unit change in the explanatory variable
• Interpretation focuses on odds ratios, probability changes, or marginal effects
• Examples include predicting the likelihood of a customer churning or a student graduating

Multinomial and Ordered Choice Models

• Extend binary choice models to handle categorical or ordinal dependent variables
• Multinomial logit and probit models are used for unordered categories
  • Estimate the probability of an observation falling into each category relative to a base category
• Ordered logit and probit models are used for ordered categories
  • Estimate the probability of an observation falling into or below each category
• Coefficients represent the change in the log odds (logit) or z-score (probit) of being in a particular category or at or below a certain level
• Interpretation involves comparing probabilities across categories or levels
• Examples include modeling transportation mode choice (car, bus, train) or survey responses (strongly agree to strongly disagree)

Truncated and Censored Regression Models

• Account for the limited range of the dependent variable due to truncation or censoring
• Truncated regression models exclude observations outside the observable range
  • Estimate the relationship between explanatory variables and the dependent variable conditional on being within the observable range
• Censored regression models (Tobit models) include all observations but treat those at or beyond the limit as censored
  • Estimate the relationship between explanatory variables and the latent (unobserved) dependent variable
• Coefficients represent the marginal effect on the latent dependent variable, not the observed one
• Interpretation requires accounting for the probability of being uncensored and the expected value of the dependent variable conditional on being uncensored
• Examples include modeling household expenditure on a particular good (truncated at zero) or willingness to pay (censored at a maximum bid)

Maximum Likelihood Estimation

• Estimation technique used for limited dependent variable models
• Finds the parameter values that maximize the likelihood of observing the sample data given the model
• Likelihood function is based on the assumed probability distribution of the dependent variable
  • Bernoulli distribution for binary models, multinomial distribution for categorical models, truncated or censored normal distribution for truncated and censored models
• Maximizes the log-likelihood function using iterative optimization algorithms
  • Newton-Raphson, Fisher scoring, or gradient descent methods
• Produces consistent and asymptotically efficient estimates under correct model specification
• Standard errors are obtained from the inverse of the Hessian matrix evaluated at the maximum likelihood estimates
• Allows for hypothesis testing and inference using likelihood ratio, Wald, or Lagrange multiplier tests

Model Interpretation and Marginal Effects

• Interpretation of limited dependent variable models differs from standard linear regression
• Coefficients do not directly represent the marginal effect on the dependent variable
• Marginal effects measure the change in the expected value or probability of the dependent variable for a unit change in an explanatory variable
  • Computed as the partial derivative of the expected value or probability with respect to the explanatory variable
• Average marginal effects calculate the marginal effect at each observation and take the average across the sample
• Marginal effects at representative values compute the marginal effect at specific values of the explanatory variables (means or medians)
• Odds ratios (for logit models) represent the multiplicative change in the odds of the outcome for a unit change in an explanatory variable
• Predicted probabilities can be computed for specific combinations of explanatory variable values
• Statistical significance and confidence intervals can be obtained for marginal effects and odds ratios using delta method or bootstrapping

Applications and Real-World Examples

• Credit scoring and loan approval decisions
  • Binary logit or probit models to predict the probability of default based on applicant characteristics
• Consumer choice modeling and market segmentation
  • Multinomial logit models to analyze preferences for different product attributes or brands
• Educational attainment and grade progression
  • Ordered probit models to examine the factors influencing the highest level of education completed or the probability of advancing to the next grade
• Labor force participation and employment outcomes
  • Probit models to study the determinants of labor market entry or employment status
• Healthcare utilization and insurance coverage
  • Truncated regression models to analyze medical expenditures or the number of doctor visits, accounting for the presence of zero values
• Environmental and natural resource economics
  • Tobit models to estimate willingness to pay for environmental amenities or the value of recreational activities, considering censoring at zero or a maximum bid
• Transportation and travel behavior analysis
  • Multinomial logit models to examine mode choice decisions or destination preferences based on individual and alternative-specific attributes
• Agricultural and development economics
  • Probit models to investigate technology adoption decisions or participation in agricultural programs, considering the binary nature of the choice