IVCLOGLOG: Stata module to estimate a complementary log-log model with endogenous covariates, instrumented via the control function approach (i.e., 2SRI)

-ivcloglog- is essentially the same thing as -ivprobit, twostep- but for the -cloglog- model. -ivcloglog- estimates a complementary log-log model with instrumenting of endogenous variables via the control function approach (also known in statistics as 2SRI – two-stage residuals inclusion). This just means that transformed versions of the residuals of the first stage regressions (i.e., the "auxiliary models") are included as regressors in the second stage (i.e., the "primary model"). In the case of -ivcloglog-, a linear first stage and polynomial control functions are used – the latter just means that powers of the residuals are used as generated second-stage regressors. An important use case is that -ivcloglog- allows users to estimate a Prentice and Gloeckler (1978) model with endogenous covariates (either continuous or functions of observed continuous covariates), just like how the basic version of the model (i.e., with all covariates exogenous) can be estimated using -cloglog-. For the guide to the theory behind this, see “A Theory Guide: Instrumenting a Discrete-Data Proportional Hazards Model with Control Functions” (Liu, 2023). For the complementary empirical guide with a real-data application, see “An IV Hazard Model of Loan Default with an Application to Subprime Mortgage Cohorts” (Palmer, 2023).