# Accuracy of MLE

Hi I have a question related to MLE.

Consider a simple linear regression model: y = a + b*x + u

Here, if the regressor X has small variability, the OLE estimator could be inaccurate.

My question is whether MLE also sufferes from the small variation in regressors. In a sentence, I want to know whether if the variability in regressors is small, the accuracy of MLE is also small.

Thank you for your time spending to read this question.

• "Here, if the regressor X has small variability, the OLE estimator could be inaccurate." relative to? Commented Nov 9, 2020 at 20:42

## 1 Answer

You are mistaken. Accuracy is not an operating characteristic of an estimator. Bias and variance are the precise terms you are after, or MSE. MLEs are asymptotically unbiased.

In a normal model, $$X$$ (despite its name) is not a random variable. The variance of the parameters is expressed as a function of $$X$$. Even still, the "variance" as an empirical property of a sample functions the opposite of what you say. Consider the normal model (for $$\epsilon$$). OLS is the MLE. The variance of $$\hat{\beta}$$ is proportional to 1/($$X^TX$$) or the inverse of the "variance" of $$X$$.

• Do you mean $(X^TX)^{-1}$ where you have $1/(X^TX)$? (For the model the OP mentions $X$ would have two columns.) Commented Nov 9, 2020 at 21:11