A new pseudoinverse called the instrumental variable (IV) inverse is introduced. The inverse and the least-squares (LS) inverse are two special examples of the instrumental variable inverse. This paper presents order-recursive methods for the instrumental variable inverse and IV estimates of the AR and MA parameters of an ARM A process. In simple cases, this paper's methods provide directly the order-recursive algorithms for the least-squares inverse and LS estimates of parameters. Since the matrix form of parameter estimates for any basic or optimal IV method is the same, the order-recursive algorithm for the IV estimates is available for the offline forms of any basic or optimal IV methods. As regards programming the recursive method for IV estimates, such a subroutine is available for several purposes, for example, it can be used in the LS, generalized least-squares (GLS), IV and optimal IV estimations.