# Forecasting time series data using auxiliary information and associated questions

Suppose I want to forecast MSFT time series, using MSFT history as well as SPY history. Are there good time series forecasting methods that permit auxiliary data to be used? Perhaps you should just model the entire time series (MSFT and SPY jointly) as a single vector time series?

Also, suppose we impose an autoregressive model (with lag $h$) $$\hat x_{t+1} = f(x_{t-h + 1}, \dots, x_t) := \theta_1 x_{t} + \cdots + \theta_{h} x_{t- h +1} + v.$$ Why not predict $\hat x_{t + 2} = f(x_{t-h+2}, \dots, x_{t}, \hat{x}_{t+1})$? It seems that most predictive models don't do this for some reason or another.

Also, I'm wondering if there are good explanations of numerical methods to forecast time series that don't rely so much on statistics. Not that I don't understand statistics, but I prefer distribution-free assumptions and weaker assumptions on the underlying structure (similar to how least squares can be thought of as an MLE estimate or as the solution to the squared-error minimization problem).

• I think you have 3 questions there. I can answer some of it.. The assumption you regarding squared error is also defined as the projection. Pretty much all time-series methods are based on the projection concept so, atleast with respect to linear models, not really parametric, aside from the distribution on the error terms. For the middle question, that 2 step ahead predition is often done but it needs to use a forecast rather than an actual value so the variance changes.and often becomes way more complex. More assumptions on $x_{t}$ are needed for the resulting forecast variance. Sep 18 '18 at 7:54
• the model defined by $f$ is definitely an autoregressive model. Sep 18 '18 at 7:58
• It is but I find the use of $x_{t}$ in AR(h) to be rare. In fact, I'd never seen it before so I got confused. Sep 18 '18 at 8:01