Use Available Pairs Method for Missing Data in OLS I have renewed interest in handling missing covariate data in OLS using the pairwise covariance matrix estimator, i.e., using all available pairs of variables in computing variances and covariances.  This is thanks to a recent paper: http://heather.cs.ucdavis.edu/Missing.pdf .  The "use all available pairs" approach is far faster than multiple imputation on huge datasets.
There are several possible permutations when deciding how to do the calculations based on


*

*whether data are centered and the intercept is estimated after the fact  (as done in the above paper) vs. estimating the intercept simultaneously with all the slopes (which is easier to deal with)

*whether to use $n-1$ in demoninators rather than $n$


I prefer to build up the calculations using the raw data when computing the upweighted $X'X$ and $X'Y$ then using the standard $(X'X)^{-1}X'Y$ where $X$ includes a column of $1$s for the intercept.  This approach is more general and could be used in other models such as the logistic.
Does anyone know of a reason that one approach would be preferred over the other all-pairs approaches?  In other words what are the details of the best all-pairs method?
 A: It turns out there is a brief discussion of these methods in the book Missing Data Analysis by Little & Rubin chapter 3 section 4.
Matthai (1951) and Wilks (1932) discussed the available cases covariance estimator. They both suggest using the $n_{jk}-1$ degrees of freedom correction to covariance estimates where $n_{jk}$ is the number of available pairs for the $j$, $k$-th covariance. Research through the 60s and 80s seems to suggest available-pairs methods only work reasonably well under a rather narrow set of conditions: in particular, the data ought to be MCAR and the correlations should be modest-to-small in size. Kim & Curry 1977, Van Praag Dijkstra and Van Velzen 1985, Haitovsky 1968, Van Guilder 1981). 
Some of the undesirable side effects when ideal conditions are not met are having correlation coefficients which exceed the -1, 1 range, singular covariance matrices. If that is possible, then it may not be possible to calculate $\left( \mathbf{X}^T\mathbf{X} \right)^{-1}$ without some ad hoc corrections. The counterexample that Little & Rubin give is the following:
y1 y1 y3
 1  1 NA
 2  2 NA
 3  3 NA
 4  4 NA
 1 NA  1
 2 NA  2
 3 NA  3
 4 NA  4
NA  1  4
NA  2  3
NA  3  2
NA  4  1

The available pair analysis suggests the correlation from y1 to y2 is 1, and from y1 to y3 is 1. Intuitively that would mean the correlation between y2 and y3 is 1 but instead it is estimated to be -1.
Based on the relatively simple mathematics, I would imagine that omitting the intercept and performing available pairs regression on centered predictor/outcome data would have no net effect on the estimates and their SEs. 
