Contrary to accepted belief, the standard Tobit maximum likelihood estimator produces inconsistent parameter estimates, when the constant censoring threshold c is non-zero and unknown. Unfortunately the recording of a zero rather than the actual censoring threshold value is typical of economic data. Non-trivial minimum purchase prices for most goods, fixed cost for doing business or trading, social customs such as those involving charitable donations, and informal administrative recording practices represent common examples of non-zero censoring threshold where the threshold is not readily available to the econometrician. Monte Carlo results show that this bias can be extremely large in practice. A new estimator is proposed to estimate the unknown censoring threshold. It is shown that the estimator is superconsistent and follows an exponential distribution in large samples. Statistical tests for the censoring threshold are introduced. A simulation study shows that the finite sample size and power properties of the proposed tests are encouraging.