# Nonlinear autoregressive exogenous model

In time series modeling, a nonlinear autoregressive exogenous model (NARX) is a nonlinear autoregressive model which has exogenous inputs. This means that the model relates the current value of a time series to both:

• past values of the same series; and
• current and past values of the driving (exogenous) series — that is, of the externally determined series that influences the series of interest.

In addition, the model contains:

• an "error" term

which relates to the fact that knowledge of the other terms will not enable the current value of the time series to be predicted exactly.

Such a model can be stated algebraically as

${\displaystyle y_{t}=F(y_{t-1},y_{t-2},y_{t-3},\ldots ,u_{t},u_{t-1},u_{t-2},u_{t-3},\ldots )+\varepsilon _{t}}$

Here y is the variable of interest, and u is the externally determined variable. In this scheme, information about u helps predict y, as do previous values of y itself. Here ε is the error term (sometimes called noise). For example, y may be air temperature at noon, and u may be the day of the year (day-number within year).

The function F is some nonlinear function, such as a polynomial. F can be a neural network, a wavelet network, a sigmoid network and so on. To test for non-linearity in a time series, the BDS test (Brock-Dechert-Scheinkman test) developed for econometrics can be used.

## References

• S. A. Billings. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains, Wiley, ISBN 978-1-1199-4359-4, 2013.
• I.J. Leontaritis and S.A. Billings. "Input-output parametric models for non-linear systems. Part I: deterministic non-linear systems". Int'l J of Control 41:303-328, 1985.
• I.J. Leontaritis and S.A. Billings. "Input-output parametric models for non-linear systems. Part II: stochastic non-linear systems". Int'l J of Control 41:329-344, 1985.
• O. Nelles. "Nonlinear System Identification". Springer Berlin, ISBN 3-540-67369-5, 2000.
• W.A. Brock, J.A. Scheinkman, W.D. Dechert and B. LeBaron. "A Test for Independence based on the Correlation Dimension". Econometric Reviews 15:197-235, 1996.