Interpolating principal component In my thesis, I use PCA from a bunch of WVS responses to measure the social capital of a country (aggregating principal components to country averages). However, WVS provides a quite low frequency of data (1 wave in 4-6 years). To compare it with annual World Bank data I intend to interpolate principal components (linearly or quadratically).
Is that statistically correct approach? I suspect that this might be not OK since the principal component is sort of synthetic feature and interpolation would be a kind of synthetic procedure over an already synthetic variable. The point is to save the economic interpretation of this feature.
 A: Yes, you may treat principal components as you would any other "ordinary" factor. That is, if you have $D$ factors describing any one entry in your data table, then the same data will be describable by $D$ principal components that represent linear mixtures of the original factors with the added features 1) the first few components will have the largest variance, and 2) the covariance matrix is diagonal, so there are no "confounding factors" among principal components. You are free to regress on them as you would the original factors.
Which components: Wave1, Wave2 or Average?
The complication arises when you have two sets of data from Wave1 and Wave2, say. The Wave1 and Wave2 principal components will not in general be the same, although the first few (with the largest eigenvalues) should be pretty close. You have choices here.
1) Do the PCA analysis on Wave1 and use the principal components to describe the Wave2 data. That is, calculate Wave2 weights on the Wave1 components.
2) Do the PCA analysis on Wave2 and use the principal components to describe the Wave1 data. That is, calculate Wave1 weights on the Wave2 components.
3) Add the data together and do the PCA analysis on Wave1+Wave2. This will automatically generate weights for the Wave1 and Wave2 data sets in the same basis. This is the easiest strategy from a technician's point of view. Scientifically speaking, choosing Wave1 or Wave2 as a "base wave" is akin to using "constant 2001 dollars" in calculating adjusted CPI. Your data will be more generalizable to other studies if you use "constant Wave1 components" or some such. 
Interpolation
Once you've picked, say Year 2001 Components (Y2001C), then the weights on each component can be treated transparently. You may have several waves worth of data. You are at liberty to model the weights as a with linear ( quadratic, whatev.) function of wave number, and then use fractional wave numbers to interpolate to intermediate years. For example Wave 2.5 would be 2 years between wave 2 and wave 3, assuming a 4 year wave cycle. 
Your referees will like you more if . . .
If you use Y2001C for several waves, it would be good to show that the first few components remain nearly collinear through the several waves. That is the cosine of the angle between them is close to 1.  You can use a dot product for that. If they seem to "wander off," please write back. There is more that can be done here! :)
An important caveat
The entire PCA approach loses its dimensional reduction appeal if the data are spherical. That is, there is no opportunity for dimensional reduction if the eigenvalue spectrum of the covariance matrix results in a set of nearly identical eigenvalues. This is the same as saying that all of the principal components carry the same variance, or weight. The only reason to use PCA at that point is to take advantage of the zero-covariance, but must people don't bother.
