8.13 The stability condition for the linear, autonomous, second-order difference equation is that the roots of the characteristic equation s2 − φ1s − φ2 = 0 lie inside the unit circle, i.e. have modulus strictly less than unity; recall Theorem 8.4.1. Show that each of the following is equivalent to this condition. (a) |φ1| 1 − φ2 and −φ2 1. (b) The roots of the lag polynomial associated with the difference equation, i.e. the solutions of φ(z) = 1 − φ1 z − φ2 z2 = 0, lie outside the unit circle, i.e. have modulus strictly greater than unity.
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