Suppose you have two simple Ar[1] series of the form $y_n=y_{n-1}+e_n$ and $x_n=x_{n-1}+m_n$, where $e_n$ and $m_n$ are normal white noise processes with no auto-correlation and $Corr(e_n,m_n)=p$. Then suppose we have possibly non-overlapping data for Y and X (IE, observation 10 exists for Y but not for X), and to avoid data generating process issues, assume that the distribution of missing data is random.
Is there any way to estimate p?
As a follow-up question, is there a way to easily generalize to a situation where $y_n$ and $x_n$ are observed with known normally distributed measurement error?
Best Answer
The paper "Application of Two-Directional Time Series Models to Replace Missing Data" offers two methods (not necessarily under your precise model), one that minimizes "the average error associated with the missing value" (the other I can't understand from the abstract).
Edit. I have changed my (old) answer to community wiki. Would someone please vote this up so that the bot that reposts those questions for which there are no upvoted answers stops recycling this one? Thanks.