What to do for missing points in CPI time series? I am looking at CPI datasets for developing countries which have gaps in them. 

For each country I have two time-series with annual averages for years 2000-2013: i) General/Overall CPI and ii) Food CPI. I'm also assuming that Food CPI must have some relationship with the General/Overall CPI since the Food category has its own weight in the General CPI.

Now, I have two types of cases, some such as: http://imgur.com/a/9z7o8#1 where gaps are between values. I'm assuming I can interpolate here, if so, how would I go forward with this? I also have to deal with more complicated cases such as: http://imgur.com/a/9z7o8#0, any suggestions in this case? Would a simple extrapolation even make sense here?

An option for my first case that I read (on BLS) is taking the geometric mean of the year immediately before and after of the missing value. Other people have suggested I should predict the missing values by a simple regression model of the CPI on the GDP deflator for that year (which I do have). 

Also, in some cases, gaps in annual averages exist because the monthly data needed to calculate these averages is incomplete. So say I only have 2006 data for Russia for months Jan-June, then the annual average data point is missing in the data series. I assume I can just take a simple average of the available months and impute that for 2006? 

Thanks in advance
Statistical Analysis with Missing Data by Little and Rubin is the go-to reference for working with missing data, at least if nothing state of the art is required. In general, this is a complex problem that remains an area of active research. The relatively easy cases are when the data is missing-completely-at-random or missing-at-random. Even among the most basic single imputation methods you have a lot of choices (list from Little and Rubin):


Mean imputation (replace with mean values)
Regression imputation
Stochastic regression imputation
Hot deck imputation (substitute individual values drawn from "similar" response units)
Substitution (not relevant in your context)
Cold deck imputation (replace missing value with constant value from external source like last value)
Composites of the above methods 


However, if all the the general price level is what's of interest and not the dynamics of the price level (e.g. because you want a deflator instead of studying inflation dynamics) linear interpolation / extrapolation may be just fine. Fundamentally, since deflation is rare, if prices are 100 at time t and 110 at t+2, realistically prices at t+1 are going to be somewhere in $[100,110]$ and lots of models can get you there.  

You can check the within and out of sample prediction quality to asses if your method is a good predictor of the missing prices. Within sample testing could be as simple as asking if the $R^2$ is high of predictive model. Obviously you can do much richer analysis than that. For out of sample testing, consider splitting the sample and calibrating the model only on the first half of the data, then evaluating prediction quality on the second half of the data.