What can be inferred from a short multivariate time series? I have annual observations of 24 variables over 10 years, and I would like to identify evidence of structural change (regime shift). The data pertain to university enrollment & spending, so I expect the variables to be trend stationary and I'm looking for shifts in the linear trends.
My concern is the number of observations seems far too small to identify a regime shift with any certitude, but I hope there's a way to compensate using the large number of variables. Many of the variables are highly correlated so I've looked into PCA for dimensionality reduction, but then I'm unsure how to evaluate statistical certainty (for example with the Chow test).
A standardized version of the data can be found here as a CSV file:
https://www.dropbox.com/s/7z9b5n5exq2j6t5/standardized.csv?dl=0
Edit: I'm looking for general methods which could be useful in such a case, not an analysis of this data file specifically (which was included only upon request).
 A: The problem you face is twofold: 1 . you have to find that subset of 19 predictors which best "relates" to the series of interest. and 2. for that subset of series find out what contemporaneous and lag structure are needed for each of the selected candidates AND then find out (discover) via Intervention Detection procedures if any Pulses/Level Shifts/Trend Changes can be identified . Why don't you post your data and I fill try and help you. Others might also try using their own expertise/software. Your 10 observations will present a challenge both to me and I am sure other readers.
UPON RECEIPT OF YOUR DATA:
These 23 characteristics. each with 10 values were submitted to AUTOBOX for expert time series analysis to determine  the underlying equation and exceptional activity (structural changes). Your post requesting structural change/regime shift can only be answered by forming an ARIMA equation and then identifying four kinds of interventions (Pulses/Level Shifts/Seasonal Pulses and Local Time trends). With 10 values AUTOBOX was restricted to just searching for Pulses. I have saved a zip file (containing just 18 of the 23 as 5 had no Interventions Detected ) named  http://www.autobox.com/stack/23.zip . In general structural shifts can be due to a  possible changes in  model parameters over time ,  changes in error variance over time or a changes in either level or slope. A pulse is an instantaneous change in level. 
A: I downloaded your data and ran PCA. It came back with 9 components. Note, that the rank of your matrix is 9, not 10. There's got to be some linear dependencies, which you didn't reveal to us. We don't know what are your columns, maybe some columns are totals of subsets of other columns etc.
Anyhow, take look at the charts below. The top one shows the series over time, and the bottom shows 9 PCA components. From the top chart you can see that several series seem to be growing, while a few are stationary.
The bottom shart shows that only the first PCA component is growing, and it explains 60% of the variation. The others don't seem to be changing much. You can run statistical tests if you wish, but it's pointless in my opinion given the sample size. You won't be able to squeeze out more information than from the plots. There is no sign of structural break. You will not be able to prove it for overall data set. You may try to go after one or two series, based on the domain knowledge, of course. However, I doubt that you'll get anything. None of the PCA components are obviously breaking here.

MATLAB Code
[coeff,score,latent,tsquared,explained,mu] = pca(standardized);

subplot(2,1,1)
plot(standardized)
title 'Series'
legend('location','Best')

subplot(2,1,2)
plot(score)
title 'PCA components'
legend('location','Best')

rank(standardized)
explained

OUTPUT
ans =

     9

explained =

   60.8074
   13.8883
    7.9533
    6.0521
    4.5944
    3.5775
    1.8449
    0.8761
    0.4060

