1
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ "Here, if the regressor X has small variability, the OLE estimator could be inaccurate." relative to? $\endgroup$
    – AdamO
    Commented Nov 9, 2020 at 20:42

1 Answer 1

0
$\begingroup$

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 $X$.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.