# Examples of Non-Linear Time Series?

Does anyone have an example of real world (ideally multivariate) time-series data that depends on its past in a non-linear, but additive way?

I understand that there are several examples of non-linear series in the literature on chaos and other literatures in the sciences, but I'm looking to find data in finance or economics, for example, where typical VARMA modeling does an okay (but not good enough) job of fully capturing the dependence relationship in the data. Hopefully, I'm not placing too many constraints here...

Specifically, has anyone come across data where a second-order stationary—by which I mean stationary either naturally or after some other transformation—series $z_t \in \mathbb{R}^n$ ($n \geq 2$) is such that $$z_t - \sum_i \hat{f}_i(z_{t-i})= \epsilon_t - \sum_j \hat{g}(\epsilon_{t-j})$$ for some true functions $f(\cdot)$ and $g(\cdot)$ that may perhaps be approximated by their first-order Taylor expansions, but which are not truly linear?

The motivation here is to find an opportunity to exercise use of Hastie and Tibshirani's (1989) generalized additive model, but on time-dependent data as opposed to cross-sectional data.

• Labour market data for the U.S. economy, such as the unemployment rate, may not be time reversible, so that's one possible place to look in economics. – Graeme Walsh Jun 8 '15 at 0:50

Contrived financial example:

Lets say you put away amount $M$ into your savings account every month. You also put $\frac{1}{2}$ of the total money you have in the bank into a 2 month CD each month. The CD pays interest into your main account, and the interest rate for it is some step function $q(D)$, where $D$ is the amount you put in. All the remaining money which isn't in one of the CDs gets an interest rate of $r$. Finally, lets say you have some random cost each month $\epsilon_t$.

At time $t$, your total savings would be: $$X_t = M + r\frac{1}{2}(X_{t-1}-X_{t-2}) + q\left(\frac{1}{2}X_{t-1}\right)\frac{1}{2}X_{t-1} + q\left(\frac{1}{2}X_{t-2}\right)\frac{1}{2}X_{t-2} - \epsilon_t$$

Here we have an additive model that depends on its past in a way which is neither cleanly linear nor log linear. Its a toy example, but in general ones rate of return on an investment grows with the amount they have saved, so assuming its linear or log linear doesn't strike me as safe.