ABSTRACT
Higher-order difference equations It should be clear by now that we can keep adding lags of the Y variable, thereby raising the order of our difference equation. A third-order difference equation, for example, would have the general form:
Yt + β1Yt−1 + β2Yt−2 + β3Yt−3 = g (4.1)
with characteristic equation:
λ3 + β1λ2 + β2λ + β3 = 0 (4.2)
The equilibrium value would be:
Y ∗ = g/(1 + β1 + β2 + β3) (4.3)
and the general solution is of form:
Yt = A1λt1 + A2λt2 + A3λt3 + Y ∗ (4.4)
where we would need three initial conditions to solve for the A terms. Since Equation (4.2) has three roots, we now have the possibility of a wide range
of time paths – we could now, for example, have one real and two complex roots.1 Assuming the system was stable, Y would still converge on its equilibrium value over time, but the cyclical element could manifest as cycles around the convergent path generated by the monotonic (stable) root. Empirically, we could wind up with what looked like a very irregular, but still stable, cycle. We could also find ourselves dealing with more complicated saddlepoint behaviour, should we have two stable and one unstable root, for example, or two unstable and one stable.
