Can somebody please clarify whether the condition for a time series to be invertible is that modulus of beta is greater than one, or less than one, as the notes, questions in the notes, and assignment X4 are not consistant, with both being used. Thanks Jooser
The situation you are referring to only applies if the time series is of the form: X(t) = a string of X terms + Z(t) + bZ(t-1) ie there are only 2 Z terms. This is invertible either if |b| < 1 or if the root y of the characteristic equation: (1 + by) = 0 satisfies |y| > 1. You should be able to see that |b| < 1 implies |y| > 1 and vice versa from the characteristic equation. As soon as there are more Z terms, you cannot just look at the b terms, you need to do the characteristic equation, for example: X(t) = a string of X terms + Z(t) + bZ(t-1) + cZ(t-2). Characteristic equation = (1 + by + cy^2) = 0. And you want both roots y to satisfy |y| > 1.
