# How to estimate variance-covariance matrix of assets with different length of historical data? [duplicate]

Consider you have 4 assets A, B, C and D, where

• Asset A started trading on 2 Jan 1990 (i.e. data is available since that point in time for every trading day until today)
• Asset B started trading on 2 Jan 1995
• Asset C also started trading on 2 Jan 1995
• Asset D started trading on 2 Jan 2010

In the simplest method, you would just use the joint history of all assets beginning on 2 Jan 2010, maybe fill missing data due to different holidays on different exchanges and compute the sample variance-covariance (VCV) matrix.

But this way you would throw away the longer history of the assets A, B and C resulting in a less stable VCV matrix.

Is there another way to come around the problem of estimating the VCV matrix for differing length financial time series? Can you e.g. construct the VCV matrix from pairwise covariance?

• How is this an EM algorithm? I can see that the estimation of the covariance matrix is the second step in the M part (i.e. Maximization), but doesn't the first step E should involve the expectation (hence the E) of the likelihood function conditional on the available observations, i.e. $E_{Y|X, \theta^{(t)}}[L(\theta; X,Y)]$? Or is the simulation of data used as a numerical approximation for this expectation? But if it is to serve as an approximation, shouldn't the number of simulated data points be large? And how does this relate to the number of missing observations? – Confounded Dec 2 at 11:08
I agree on Richard. the simpler you choose, the better it is so as to get reliable estimates. What's your data frequency? purpose? For model construction as far as I am concerned, daily data from 2010 is enough. Otherwise, you could use a proxy asset for asset D depending on its nature. To clarify, if D is an ETF let's say CAC 40 ETF, concatenate its return series up to the last available data point with the return series of its underlying - here CAC 40 - since your starting date. Same process if D is a derivatives. By, using this approach (concatenating return series) always make sure that the asset and its underlying are strongly correlated. Hope it helps. Cheers.