What can I do with these two time series? 
Info:
Blue line is the official inflation statistics by US Bureau of Labour Stats
Red line is by independent researchers who claim to have created better way to measure inflation
Blue line has been sampled once every month
Red line has been sampled daily
It can be seen that the red line is tracking inflation ahead of the blue line.

Basically these are two time series, with different sampling frequencies, that are tracking the same phenomenon, i.e. change in general price levels. I need advice on how I should go about analysing these processes. What are some possible hypothesis I could be testing for?
UPDATE: After mpiktas' suggestion, my current plan is to a MIDAS regression with some polynomial function for the weights and report the forecasting capabilities of the model. (and perhaps contrast this to a model using a simple-time-averaged red line)
Some of you might have done something similar to this, is there anything I should keep in mind or anything additional I could be doing that would complement this 6000 word project?
Additional: 


*

*If you can link me to some very relevant papers that have done similar analysis that would help.


Any input will be apreciated
 A: For forecasting low frequency variables with high frequency you can use MIDAS regression. The idea behind this regression is quite simple, average the high-frequency variable and then use it as a regressor. The key is to use custom weights. Suppose we have $Y_t$ which is sampled monthly and $X_\tau$, which is sampled daily. Then MIDAS regression is defined as follows:
$$Y_t=\sum_{h=0}^k\beta_hX_{tm-h}+\varepsilon_t$$
where we assume that for observation $t=s$ we have $m$ observations $\tau=sm-m+1,...,sm$. We also assume that $\beta_h=g(h,\theta)$, for some function $g$ and hyperparameter $\theta$. 
So if you want to test out whether the $X_\tau$ is a good predictor for $Y_t$, fit a MIDAS regression with different weight functions and inspect the results. 
If you enter MIDAS regression into google you'll find many articles. Forecasting monthly CPI with daily variable was investigated in the article "Forecasting with mixed frequencies" by Armesto, Engemann and Owyang. 
The MIDAS regression idea was introduced by Eric Ghysels, you can look into his articles. 
There are two software packages for fitting MIDAS regression: MIDAS Matlab toolbox and midasr R package. They both have user guides, where you can find more detailed examples and links to other literature. 
Note, this is only one possible way of solving your problem. Others surely exist too, but as I am the developer of midasr R package, I am biased in my suggestions.
A: For starters, I would check if the draws from list A are two standard deviations outside of the draws from list B.
Secondly, I would check if a 1 month moving average of list A is significantly different from the more slowly sampled list B.
A: Fit seasonal time series models to both and then compare the seasonality.
The TBATS model can handle daily data. It is available in the forecast package in R.
