Quick Stability Checks and Matrix Norms

Abstract

Consider the difference equation system yt=b+Ayt-i; where yt is an nXl variable vector, b is an nXl constant vector and A is an nXn constant matrix. The fundamental necessary and sufficient stability condition for the system is /x(^)< 1, where [J.(A) is the modulus of the characteristic root of A of largest modulus. That is, IJ.(A) is the spectral radius of A, so-called because all roots of A lie on or within a circle of radius ix(A) in the complex plane. There exists a class of scalar real-valued functions of matrices called matrix norms; and any matrix norm/(y4) has the property iJ^(A)<f(A). Hence a sufficient stability condition for yt=b+Ayi.i is f(A)<l, where/(.) is any matrix norm function. The condition is also sufficient for existence and uniqueness of the equilibrium yt=(I—A)~^b. The advantage of demonstrating existence, uniqueness, and stability through a norm f (A) rather than through the spectral radius IJI.(A) itself is that many norms are far easier to compute than iJi(A). The disadvantage is that the sufficient condition f(A) < 1 may be far stronger than necessary. Though the economic literature has exploited leading special cases of the f(A) < 1 condition, more emphasis on the general case seems appropriate. Section I reviews matrix norm theory, presents the basic f(A) < 1 condition, and relates it to existing stability results in the economic literature. The following sections extend the basic result to cover other contexts—structural systems, distributed lags and periodic coefficients.