# AR model driven by non-iid input

I want to use an AR(1) model to describe a time series, but I know that the driving variable at each timestep is not iid. The autocorrelation function of the time series suggests an AR(1) process, but is there a way for me to modify the statistical formulation of an AR(1) process in the case where the driving noise violates the iid assumption?

• What kind of non-iid'nes do you have? – Richard Hardy Sep 20 '16 at 16:13
• "Driving variable" -- is that an econometrics term? (I'm coming from stats) – Jon Sep 20 '16 at 16:32
• @Jon, that must be the shocks/innovations/errors that are "driving" the system. – Richard Hardy Sep 20 '16 at 18:58
• @RichardHardy, thanks for your comments. The non-iid'ness is a series of inputs - these tend to be a series of impulses. So the input is always positive, and is most likely correlated in time. Does that help? – hydrologist Sep 20 '16 at 20:07
• Let me guess: if hydrologist posts his data, the answer by IrishStat will be a complete model for the data created using commercial software. The actual question that is being asked will not be answered in there, and the attention will be drawn away from the actual question. That happens regularly and offers an example of a rather surprising phenomenon: posting data need not always be helpful. – Richard Hardy Sep 21 '16 at 5:31

## 1 Answer

Technically it should be possible to define a model like $$y_t = \beta_0 + \beta_1 y_{t-1} + u_t$$ (which would be AR(1) if $u_t$ were i.i.d.) with $u_t$ being, e.g., ARIMA(p,d,q) or some other process. You would simply add an equation for $u_t$ right under the equation for $y_t$ when you formulate the model. This could be implemented in R using

arima(y[-1], order=c(p,d,q), xreg=cbind(y[-length(y)]))


Perhaps the model has a simpler representation that could be found by playing with the lag polynomials for $y_t$ and $u_t$, but the current one should do the job, too (i.e. its parameters could be estimated and it could be used for forecasting).