# Consistency of least absolute deviation estimator (LAD) vs OLS

If we know that OLS estimator of beta in linear regression model is unbiased and consistent and we don't have any further assumptions (on errors or anything), will LAD estimator of beta be also consistent?

• Hint: consider the model $E[Y_i]=\mu$ for fixed unknown $\mu.$ The OLS estimator is the mean of a sample but the LAD estimator is its median.
– whuber
Nov 19, 2019 at 21:39
• But I am specifically asking about a linear regression model y_i = \alpha + \beta*x_i + e_i. If I know that the OLS estimator of (alpha, beta) is consistent and unbiased, does it imply that my LAD estimator of (alpha, beta) is also consistent? Nov 20, 2019 at 9:46
• The model I posited above is linear regression. Its simplicity helps you understand the issues, which will translate readily to the more complicated models you have in mind.
– whuber
Nov 20, 2019 at 15:23
• @whuber I don't fully understand your first comment. Would you be able to expand a little?
– user285478
Jul 12, 2020 at 19:05
• I don't know. I'm curious myself. My intuition is "no, it does not mean the LAD estimator is consistent" because the LAD estimator does not necessarily have a unique solution. So at a minimum you would need an assumption for the solution being unique. Apr 4, 2021 at 12:01