  In the era of data science, it is common to encounter data with different
subsets of variables obtained for different cases. An example is the split
questionnaire design (SQD), which is adopted to reduce respondent fatigue and
improve response rates by assigning different subsets of the questionnaire to
different sampled respondents. A general question then is how to estimate the
regression function based on such block-wise observed data. Currently, this is
often carried out with the aid of missing data methods, which may unfortunately
suffer intensive computational cost, high variability, and possible large
modeling biases in real applications. In this article, we develop a novel
approach for estimating the regression function for SQD-type data. We first
construct a list of candidate models using available data-blocks separately,
and then combine the estimates properly to make an efficient use of all the
information. We show the resulting averaged model is asymptotically optimal in
the sense that the squared loss and risk are asymptotically equivalent to those
of the best but infeasible averaged estimator. Both simulated examples and an
application to the SQD dataset from the European Social Survey show the promise
of the proposed method.
