Classical methods for estimating additive faults are based on state-space models, e.g., moving horizon estimation (MHE) and unknown input observers (UIOs). This paper contributes new direct design methods from closed-loop I/O data for systems with stable inversion, which do not require building a state-space model by first principles, nor require identifying it. Inspired by subspace identification, we use the input and output (I/O) relationship of a plant in a Vector ARX (VARX) form to parameterize least-squares (LS) problems for estimating faults. We prove that with the order of the VARX descriptions tending to infinity, the fault estimates are unbiased. Under lower relative degrees, we prove that our new methods are equivalent to system-inversion-based estimation for both LTI and LTV systems. We will show more general unbiased estimation conditions for higher relative degrees. These require that the underlying inverted system from faults to outputs is stable. Algorithms of identifying unbiased fault estimation filters from data will be developed in this paper based on single LS. Moreover, covariance of the fault estimates can also be extracted from data.