# How to estimate a continuous analog of the (discrete) vector autoregression (VAR) model

I have some ten to 100 thousand observations on each of around 500 entities. I have good reason to believe that these observations all mutually influence one another, in possibly complicated ways, or perhaps are driven by shared hidden variables. I would like to estimate these interrelationships, initially imposing minimal structure, and then testing for various restrictions that I believe may hold.

One framework that I understand tolerably well and wish I could use to explore these relations is the vector autoregression. However I can not do so, because the data arrives at random intervals, not controlled by me, and different for each variable. In essence the observations occur in continuous time. The individual series are strongly autocorrelated and generally trended over long intervals of time. I have not yet determined whether they are better modeled as trend stationary, random walks with drift, or maybe fractionally integrated.

I am looking for a technique that treats the usual approach to VARs as a discrete difference equation approximation of an underlying continuous stochastic differential process, and that treats this problem as simply a differently structured set of observations on that same process. The idea is that a VAR is a special case of a more general technique, where the observations of all variables are conveniently taken simultaneously and spaced apart at equal time intervals.

But I have never done any previous work in continuous time. Usually I work with economic data sets with, at most, monthly frequencies. Is there a name for the type of process I describe, that would help me locate the literature? Is there an estimation technique analogous to autoregression which is commonly used for problems of this sort?

In case it makes a difference, I am more interested in forecasting, estimates of forecast reliability, and the impulse response function than in hypothesis testing or parameter estimation per se. For empirical work I usually use R.

• I tried to make your title more informative, but feel free to revert to the original one or change further. – Richard Hardy May 14 '17 at 9:02