We propose a new method for approximating a matrix finite impulse response (FIR) filter by an infinite impulse response (IIR) filter of lower McMillan degree. This is based on a technique for approximating discrete-time descriptor systems and requires only standard linear algebraic routines, while avoiding altogether the solution of two matrix Lyapunov equations which is computationally expensive. Both the optimal and the suboptimal cases are addressed using a unified treatment. A detailed solution is developed in state-space or polynomial form, using only the Markov parameters of the FIR filter which is approximated. The method is finally applied to the design of scalar IIR filters with specified magnitude frequency-response tolerances and approximately linear-phase characteristics. A priori bounds on the magnitude and phase errors are obtained which may be used to select the reduced-order IIR filter order which satisfies the specified design tolerances. The effectiveness of the method is illustrated with a numerical example. Additional applications of the method are also briefly discussed.