Existence of unbiased estimate of regression parameters in simple linear EV models

It is well known that for one-dimensional normal EV regression modelX = x + u,Y = α + βx +e, wherex,u,e are mutually independent normal variables andEu =Ee = 0, the regression parameters α and β are not identifiable without some restriction imposed on the parameters. This paper discusses the problem of existence of unbiased estimate for α and β under some restrictions commonly used in practice. It is proved that the unbiased estimate does not exist under many such restrictions. We also point out one important case in which the unbiased estimates of α and β exist, and the form of the MVUE of α and β are also given.