What are the most effective methods for estimating a binary choice model?

The Linear Probability Model (LPM) is commonly estimated using Ordinary Least Squares (OLS), but it has several limitations that make it less effective for binary choice models. These limitations include the possibility of predicting probabilities outside the 0, 1 range, heteroscedasticity, and challenges in interpreting coefficient estimates. As a more effective alternative, models like the Logit and Probit models are widely used for binary choice models. The Logit model, in particular, addresses the limitations of the LPM by ensuring predicted probabilities remain within the valid range of 0 to 1 and allows for interpretable coefficients in terms of log odds. De la Torre, R., Onggo, B. S., Corlu, C. G., Nogal, M., & Juan, A. A. (2021).

- Logistic Regression: Widely used method assuming a logistic function to model the relationship between independent variables and the probability of the event. - Probit Regression: Another popular method assuming an underlying normal distribution for the latent variable and using the cumulative distribution function of the standard normal distribution. - Linear Probability Model: Simple method assuming a linear relationship between independent variables and the probability of the event. - Logit Model: Variant of logistic regression using the natural logarithm of the odds ratio as the dependent variable. - Poisson Regression: Method used for modeling count data, adaptable for binary choice modeling by assuming a Poisson distribution.

The best technique for your research will rely on a number of variables, including the type of data you have, the reliability of your assumptions, the existence of endogeneity, and your research objectives. First carry out sensitivity assessments, and then decide on whether to use Logistic regression, Probit Model, Bayesian Estimation, GMM or IV

I start by stating that probit and Logit models are very common in econometrics for binary outcomes. Probit uses the standard normal distribution while as Logit employs the logistic distribution. Logit is preferred for its simplicity but Probit may be chosen when extreme probabilities matter. Use likelihood values for empirical comparison and transform the binary outcome into a continuous variable with the appropriate link function before modeling.

The Logit and Probit models are statistical tools for analyzing binary choice data, with the Logit model assuming errors follow a logistic distribution and using the logit function to transform independent variables into predicted probabilities between 0 and 1. On the other hand, the Probit model assumes errors follow a standard normal distribution and utilizes the cumulative distribution function of the standard normal distribution. Interpretation of coefficients varies, with Logit focusing on changes in log odds and Probit on standard normal deviates. Ai, C., & Norton, E. C. (2003). Interaction terms in logit and probit models. Economics letters, 80(1), 123-129.

Instrumental variables (IV) methods address endogeneity by introducing a third variable, the instrument (Z), which must satisfy conditions of relevance, exogeneity, and exclusion restriction. Relevance ensures that the instrument is correlated with the explanatory variable, while exogeneity requires that the instrument is uncorrelated with the error term. The exclusion restriction mandates that the instrument affects the dependent variable only through its influence on the explanatory variable. Greenland, S. (2000). An introduction to instrumental variables for epidemiologists. International journal of epidemiology, 29(4), 722-729.

Semiparametric and nonparametric methods in econometrics offer valuable alternatives to traditional approaches by relaxing assumptions about functional forms and data distributions. Nonparametric methods require minimal assumptions, allowing data-driven insights, making them flexible for exploratory analysis but less efficient with smaller datasets. Semiparametric methods combine parametric and nonparametric elements, offering flexibility while maintaining efficiency. They can address non-normality and heteroscedasticity but may require complex estimation procedures. Horowitz, J. L. (2009). Semiparametric and nonparametric methods in econometrics (Vol. 12). New York: Springer.

Machine Learning methods are gaining traction in Discrete Choice Analysis (DCA) as they offer advantages such as capturing complex relationships, handling high-dimensional data, and integrating diverse data types. Random Forests, Gradient Boosting, Neural Networks, and Support Vector Machines are among the ML algorithms increasingly applied in DCA, each with unique strengths and weaknesses. However, using ML in DCA presents challenges like interpretability, overfitting, and data requirements. The paper "Discrete choice analysis with machine learning capabilities" by Aboutaleb et al. Aboutaleb, Y. M., Danaf, M., Xie, Y., & Ben-Akiva, M. (2021). Discrete choice analysis with machine learning capabilities. arXiv preprint arXiv:2101.10261.

Also you can try the Generalized Estimating Equations(GEEs) for estimating binary variables. GEEs handle correlated data in repeated measurements by offering a unique approach with a working correlation matrix. They focus on population that is averaged effects that may extend to multivariate binary data and also handle missing data effectively.