Quasi-Maximum Likelihood for Estimating Structural Models

The estimation of the structural model poses a major challenge because its underlying asset (the firm asset value) is not directly observable. We consider an extended structural model that accommodates alternative underlying Markov processes, arbitrary debt payment schedules, several seniority classes, multiple intangible assets, and various intangible corporate securities. We derive the likelihood function given the observed time series of the firm equity values. Then, we use dynamic programming to solve the model and, simultaneously, extract the associated time series of the firm asset values (the pseudo-observations). Finally, the likelihood function is approximated and optimized, which results in the quasi-maximum likelihood (QML) estimates of the model’s unknown parameters. QML is highly flexible and effective. To assess our construction, we perform an empirical investigation, highlight the credit-spread puzzle, and discuss a partial remedy via jumps and bankruptcy costs.