What to do when the intersection of some time series doesn't contain enough data point? I have to do some studies on time series, which involve computing the covariance matrix. However, my time series span on different time intervals and their intersection is too short to provide me a good estimator. 
I guess I have to remove some time series but I don't know how to select the ones to keep. 
I can come up with my own methodology (it could be something like keeping the data that have the most recent points) but I would like to know first if there is a standard procedure for this kind of things.
 A: If you use R you can maybe 
1/ merge the 2 time series
2/ carry forward the values except if the delay is too long (kind of enhanced na.locf)
A: You can use an EM (expectation-maximization) algorithm to fill in the missing values.
M-step: Use the common data points to estimate an initial covariance matrix.
E-step: Use that covariance matrix to fill in the missing values. For example, say you have 3 timeseries X, Y and Z, and you are missing the Z value for some time t. You would fill in the conditional expectation for Z(t) = E(Z|X=X(t), Y=Y(t)) . See Wikipedia The formula for \mu is the one.
After you fill in, you can repeat the M-step to compute a new estimate of covariance matrix, and then the e-step to fill in all the orginally missing values using the updated covariance estimate.
You can repeat the two steps until either a fixed number of iterations or when the changes to covariance matrix are below some threshold.
I am not statistician, but a programmer who has implemented this.
There are some caveats I know of. This method assumes your data are wel l approximated by a multivariate normal, data from one timestep are uncorrelated with that from other timesteps, and the missing data are uncorrelated to missingness (this would be violated if the data are measurments taken by an instrument that fails to pick up very high or very low values, or if the data come from a survey where respondents with high or low values are more likely to conceal the information).
A: I assume that you have $n$ time series, 
with no temporal dependencies, 
with enough data to compute the covariance of any pair of variables,
but not enough data to compute the whole variance matrix.
For instance, you could have 3 time series of daily values, 
A with values in 2009 and 2010,
B with values in 2009 and 2011 and 
C with values in 2010 and 2011:
to compute the variance matrix, 
you would have to remove 2011 (to include A),
2010 (to include B) and 2009 (to include C) -- 
there would be no data left.
Some people just compute the matrix $C$ of pairwise
correlations and, since it is not guaranteed to be
positive semi-definite, try to fix it afterwards.
For instance, one could diagonalize the matrix 
and set the negative eigenvalues to zero (or some small positive value).
One could also shrink the matrix towards some positive definite matrix, 
e.g., find the smallest $\lambda$ so that $(1-\lambda)C + \lambda I$ 
be positive definite.
It is possible to solve the problem rigorously, i.e.,
find the closest symmetric positive semi-definite matrix,
for the $L^2$ norm, but it is more complicated:
see this question.
However, this assumes that there is enough overlap between your time series:
if it is not the case, the EM approach mentionned by @Maverick1 should still work,
provided the algorithm is implemented properly.
