# 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 '19 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? – doremi Nov 20 '19 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 '19 at 15:23