12.3: Conclusion

This chapter has outlined the fundamental process of maximum likelihood estimation (MLE) for linear models, following the established steps of formulating a likelihood function and optimizing it using calculus. Notably, for the classical linear model under the normality assumption, the MLEs for the regression coefficients coincide with the ordinary least squares (OLS) estimators. Consequently, these coefficient estimators share the desirable property of unbiasedness.

However, a key distinction arises in the estimation of the error variance: the MLE of this parameter is biased, which we can see from its divergence from the unbiased mean squared error (MSE). This may initially suggest that MLE offers no advantage over the simpler OLS approach. Yet, such a conclusion would be premature.

MLE provides significant theoretical and practical benefits. First, it is supported by a powerful large-sample (asymptotic) theory, which guarantees that, under general regularity conditions, maximum likelihood estimators are consistent and asymptotically unbiased. This means that for any model where we derive an MLE, its desirable large-sample properties are automatically established, whereas estimators like OLS often require separate, case-by-case proofs.

Second, and more importantly, the MLE framework is remarkably flexible. While the specific results in this chapter relied on the classical assumption of normally distributed errors, the MLE principle itself imposes no such restriction. This allows us to extend our modeling far beyond the confines of the classical linear model. We can specify likelihood functions based on a wide variety of distributions (e.g., binary, count, or duration data), thereby unlocking a vast toolkit for empirical analysis. It is precisely this flexibility that we will leverage in the next segment of the course, as we explore applications of MLE to models that violate the strict assumptions of the classical linear framework.