# Assumptions of MLE

I am currently reading up on Maximum Likelihood Estimation in Studies in Econometric Method. When describing the requirements for MLE to be consistent, they described it as the following:

A number of results have been obtained regarding he consistency of genuine and quasi-maximum-likelihood estimates. Mann and Wald  have proved the consistency of such estimates in linear models in which the equation system is stable (1), there are no exogenous variables (2), all parameters are identifiable(3) and all moments of the distribution function of the disturbance exist and are finite(4).... (page 166 in the pdf)

Are these the requirements of MLE? If they are how would I represent each one of these assumptions mathematically?

• If your intention is to learn about the MLE method, then this is a thoroughly bizarre book to be reading. If, on the other hand, you are specifically interested in this book, then you could at least tell us which article and which page number the quote comes from. Otherwise it is all thoroughly out of context. The quote makes it clear that method is not in fact MLE but quasi MLE, and that has no meaning except in the context of a specific model and likelihood approximation. – Gordon Smyth Jan 19 '18 at 5:14
• @GordonSmyth im open to recommendations, I haven't found a book that gives a good treatment on the topic. I forgot to add the page number, my bad. – EconJohn Jan 19 '18 at 5:29
• I'm a bit surprised that you haven't found any treatments, because it is a fundamental concept in statistics that is covered in most sufficiently advanced math stat textbooks. Have you tried starting with the Wikipedia page?en.wikipedia.org/wiki/Maximum_likelihood_estimation The introductions won't of course be in the context of economics or complex models. – Gordon Smyth Jan 19 '18 at 7:58
• @EconJohn I haven't found any mention of exogeneity in the paper by Mann and Wald. – Glen_b Jan 19 '18 at 13:05
• If the y-variate is normally distributed, then MLE and least squares are the same for estimating the regression coefficients. If it's not, then ML is generally better than least squares. MLE is never going to help you unless you can write down a probabilistic model, in which case you will already know whether the data is normal or not. – Gordon Smyth Jan 20 '18 at 0:54

In Boos & Stefanski "Essential Statistical Inference" (pp. 278-281), four conditions on the sampling density $f(\cdot;\theta)$ are given as sufficient for consistency:
1. Identifiability of the parameter $\theta$, i.e., when $\theta_1\ne\theta_2$, the set$$\{x;f(x;\theta_2)=f(x;\theta_1)\}$$is not of measure one;
2. Boundedness of the expected likelihood, i.e., $$\int\log\{f(x;\theta)\}\text{d}F(x;\theta_0)<\infty$$in a neighbourhood of the true parameter value $\theta_0$;
3. Differentiability of the log-likelihood, i.e., $\log\{f(x;\theta)\}$ is continuously differentiable in a neighbourhood of the true parameter value $\theta_0$ for almost every $x$ in the support of $F(x;\theta_0)$;
4. Uniform integrability, i.e., $\log\{f(x;\theta)\}$ is bounded uniformly in $\theta$ by an integrable function $h(x)$ [integrable against $F(x;\theta_0)$]