# Regression with different frequency

I am trying to run a simple regression but my Y variables is observed on a monthly frequency and x variables are observed on an annual frequency. I will really appreciate some guidance on a suitable approach that may be used for regressions with different frequencies.

Thank you very much

• If you conceive of the relationship as causal, it may be worth pondering how, exactly, you see the X leading to the Y - it will often then make a potential strategy more clear. How does your annual-thing lead to an outcome on your monthly thing? Is X a proxy for something else, or does Y really depend on annual-X? Feb 24, 2015 at 0:27

Three possibilities follow. Depending on the situation, any one could be suitable.

1. Time aggregation or dis-aggregation.

This is perhaps the simplest approach in which you convert the high-frequency data (monthly) into annual data by, say, taking sums, averages, or end of period values. The low frequency (annual) data could, of course, be converted into monthly data by using some interpolation technique; for example, using the Chow-Lin procedure. It might be useful to refer to the tempdisagg package for this: http://cran.r-project.org/web/packages/tempdisagg/index.html.

1. Mi(xed) da(ta) s(ampling) (MIDAS).

Midas regressions, popularized by Eric Ghysels, are a second option. There are two main ideas here. The first is frequency alignment. The second is to tackle the curse of dimensionality by specifying an appropriate polynomial. The unrestricted MIDAS model is the simplest from within the class of models and can be estimated by ordinary least squares. Further details and how to implement these models in R using the midasr package can be found here: http://mpiktas.github.io/midasr/. For MATLAB, refer to Ghysels' page: http://www.unc.edu/~eghysels/.

1. Kalman filter methods.

This is a state-space modelling approach, which involves treating the low-frequency data as containing NAs and filling them in using a Kalman filter. This is my personal preference, but it does have the difficulty of specifying the correct state-space model.

For a more in-depth look at the pros and cons of these methods, refer to State Space Models and MIDAS Regressions by Jennie Bai, Eric Ghysels and Jonathan H. Wright (2013).