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# Stability conditions of discrete dynamic systems

I have a system of 6 difference equations with three state variables. Let X_t = [x_{1t}, x_{2t}, x_{3t}, x_{4t}, x_{5t}, x_{6t}]^T. The general form of the problem is g(X_{t+1}, X_t) = 0. The log-linear form of this system around the steady state X^* is:

X_{t+1} - X^* = M \cdot (X_t - X^*)

The eigenvalues of M are -32.43, 0.56 \pm 0.94i, 0.98 \pm 0.12i and 0.94. The stability conditions of discrete time system (|\lambda|<1), which imply that there are three stable roots and the system is saddle-point stable.

If, however, I set up the system in a ‘differential form’:

X_{t+1} - X_t = (M - I) \cdot (X_t - X^*) = N \cdot (X_t - X^*)

The eigenvalues of N are -33.43, -0.44 \pm 0.94i, -0.02 \pm 0.12i and -0.06. Now, we can apply the stability conditions of differential system (Re (\lambda)<0), which implies that there are six stable roots and the system has multiple paths to the steady-state.

How do I explain the difference in the predictions of the two forms of the same model? Which is the right way to look at the problem?

The first approach sounds like the right one to me. It’s a bit hard to tell without seeing all the details but the overall aim is to infer the stability of the original nonlinear system from its linearization around the steady state. They theorem you are applying here is called the “Hartman-Grobman Theorem”, which works for both discrete and continuous time systems.

From what I can understand, your first approach is a correct linearization of the system around the steady state, so your prediction of saddle path stability sounds like the right one.

For the second approach, I would have to see the relevant theorem and its proof to comment. But I don’t see why you need to go beyond the first approach (assuming you’ve carefully checked the linearization steps).

Hope that helps.

John.