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We consider linear regression with missing responses as well as missing covariate data. When the missing data mechanism is ignorable, we show that regression parameters and the response mean can be estimated using standard methods and treating imputed values as observed data. We also show that the same procedure results in biased and inconsistent estimators when missing response mechanism depends on covariates that also have missing values and thus is nonignorable. Efficient estimation and imputation under nonignorable missingness is a challenge problem. Under some conditions, we derive some asymptotically unbiased and consistent estimators via direct estimation or imputation. Some simulation results are presented to examine the finite sample performance of various estimators.

Estimation and imputation in linear regression with missing values in both response and covariate