Is this a valid application for a Kalman filter? I have 2 time series;

Both the series are tracking inflation.(they have different sampling frequencies)
Blue is the official CPI released by the US government.
Red is an independent group's measure of inflation
If I let the Blue line be the underlying system state, and the red line be the noisy input data, can I estimate the system state using the red line(and perhaps some other explanatory variables that I think are valid)
I am already fitting a Mixed Data Sampling (MIDAS) regression model to this data, but I want some other models so that i can compare forecasts.
 A: In order use a Kalman Filter you need a linear model of the system whose state you are estimating. If you have a model but it is not linear then you can use an extended Kalman filter. If, on the other hand, you do not have a model then no variation of the Kalman filter will be useable.
A: As the answer from DaemonMaker points out, you need a model for the state, and a state with jumps such as your blue line is inconvenient. On the other hand, an inflation rate with instantaneous jumps and constant everywhere else doesn't make much sense.
I think a better model would be to treat both the red and blue line as observations of an underlying, not directly observable state, which you could endow with a simple dynamic: a random walk, a local linear trend, autoregressive, etc. 
Note that if the CPI index is released only once per month and the red line is observed, for instance, once per week, there is no problem at all: the weeks in which you have no new observation of the CPI you would fill in a NA.
