Time series factor model with one series more frequent

Question

Let's say I have two time series, one of which updates more frequently than the other:

$x_0,x_1,x_2,\dots,x_t,\dots$

$y_0,y_{10},y_{20},\dots,y_{10t},\dots$

I want to fit a model to this that predicts $y$ from $x$ (and possibly from previous values of $y$) at each of the values $1,2,3,\dots$, i.e. it gives a prediction even for values of $y$ for which we won't make an observation (equivalently, assume that there are true values for $y$ at every value of $t$, but we only observe it at $t=0,10,20,\dots$)

Is there a canonical way to do this?

Answer 1

The cannonical way is probably MIDAS regression. There is a Matlab toolbox for estimating, available upon request from the author Eric Ghysels. You might look into user guide of this toolbox, since it has a review of all literature on MIDAS.

The wikipedia page also talks about connection with Kalman filters, so @F. Tussel observation is spot on.

Update There is now also an R package midasr to estimate MIDAS regression.

Answer 2

I would cast the model in state-space form. Then there is no problem if one of the variables is observed more frequently than the other, or the observation times are irregular: the Kalman filter deals with missing and partially observed variables gracefully.

Without details on the exact kind of relationships you aim to model it is difficult to be more specific.