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 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.