Mixed-frequency methods typically involve using the higher frequency data (stock ticks, etc) to forecast the lower-frequency (GDP, in an example exaggerated to stress frequency disparity). This may be a very simple/obvious answer, but because I haven't been able to find the opposite goal addressed in any of the literature (a great guide to this literature is: http://www.uclouvain.be/cps/ucl/doc/ssh-ilsm/images/MIDAS_Course_Syllabus_NBB.pdf), I'm just not sure what I can get away with when it comes to combining two datasets at different frequencies before any time series work. It's possible that this is addressed in the literature listed in the above link, and I just haven't seen it. But the goal always seems to be in one direction: forecast lower-frequency data with higher-frequency.
What if I want to allow for the possibility of some endogenous two-way effects, and try to forecast the higher-frequency based on the lower-frequency?
"At a general level, the interest in MIDAS regressions addresses a situation often encountered in practice where the relevant information is high frequency data, whereas the variable of interest is sampled at a lower frequency. One example pertains to models of stock market volatility. The low frequency variable is for instance the quadratic variation or other volatility process over some long future horizon corresponding to the time to maturity of an option, whereas the high frequency data set is past market information potentially at the tick-by-tick level."
See also from same paper: "Take for instance the relationship between inflation and growth. Instead of aggregating the inflation series to a quarterly sampling frequency to match GDP data, one can run a MIDAS regression combining monthly and quarterly data."
What about the opposite? IE, if I have a dataset of inflation:
YearQ Inflation 2006Q1 3 2006Q2 3.5 2006Q3 3 2006Q4 3.5 2007Q1 3.5 2007Q2 3.4 2007Q3 3.4 2007Q4 3.4
And a dataset of GDP:
Year GDP 2006 3 2007 3.1
Can I simply combine the 2 like so? (really easy as it's just a simple join, so appealing to me for that reason when it comes to more complicated examples of same)
YearQ Inflation GDP 2006Q1 3 3 3 2006Q2 3.5 3 2006Q3 3 3 3 2006Q4 3.5 3 2007Q1 3.5 3.1 2007Q2 3.4 3.1 2007Q3 3.4 3.1 2007Q4 3.4 3.1
Can I now try to model inflation using RU-MIDAS/VAR/ETS/ARIMA/GARCH/VECM/whatever? That still feels clumsy to me. It would even be helpful if someone could just verify for me, "yes, it's a) covered in the literature, and b) it's not a conceptually worthless pursuit to try to predict high-frequency with lower-frequency data."
So, from the same basic intro paper I quoted before, after it introduces basic MIDAS equation: "The annual/quarterly example would imply that the above equation is a projection of yearly Yt onto quarterly data X (m) t using up to j max quarterly lags.4" That sounds like what I want - but the formulation is to have the independent variable (ignoring time for a moment) still be the higher-frequency. Do I simply invert the equation at this point for what I want ("solve for x," literally)?
Edit: Found a very recent working paper from Norges Bank (I can't post more than 2 links, google "Using low frequency information for predicting high frequency variables"), that is pretty good. The lit review on benchmarked approaches like MF-VAR should be useful too. It looks from this paper that my approach is theoretically acceptable in the abstract, although I should use more intelligent interpolation of the low-frequency data to the higher-frequency dataset, like cubic splines in SAS "PROC EXPAND" function. There are some people, like Marc Wildi, who use this same exaggerated example of GDP as the low-frequency variable to make the argument that there is less usefulness from the use case I'm proposing than the traditional opposite. He does have some cool MDFA signal extraction methods that I don't understand, so that's something else to read up about.