  Blockwise missing data occurs frequently when we integrate multisource or
multimodality data where different sources or modalities contain complementary
information. In this paper, we consider a high-dimensional linear regression
model with blockwise missing covariates and a partially observed response
variable. Under this framework, we propose a computationally efficient
estimator for the regression coefficient vector based on carefully constructed
unbiased estimating equations and a blockwise imputation procedure, and obtain
its rate of convergence. Furthermore, building upon an innovative projected
estimating equation technique that intrinsically achieves bias-correction of
the initial estimator, we propose a nearly unbiased estimator for each
individual regression coefficient, which is asymptotically normally distributed
under mild conditions. Based on these debiased estimators, asymptotically valid
confidence intervals and statistical tests about each regression coefficient
are constructed. Numerical studies and application analysis of the Alzheimer's
Disease Neuroimaging Initiative data show that the proposed method performs
better and benefits more from unsupervised samples than existing methods.
