Nonlinear autoregressive exogenous model
In time series modeling, a nonlinear autoregressive exogenous model 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 series — that is, of the externally determined series that influences the series of interest.
In addition, the model contains:
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
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. For example, y may be air temperature at noon, and u may be the day of the 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 developed for econometrics can be used.
