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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.

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

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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 $X$.

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    $\begingroup$ 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
    Nov 9, 2020 at 21:11

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