  International comparisons of hierarchical time series data sets based on
survey data, such as annual country-level estimates of school enrollment rates,
can suffer from large amounts of missing data due to differing coverage of
surveys across countries and across times. A popular approach to handling
missing data in these settings is through multiple imputation, which can be
especially effective when there is an auxiliary variable that is strongly
predictive of and has a smaller amount of missing data than the variable of
interest. However, standard methods for multiple imputation of hierarchical
time series data can perform poorly when the auxiliary variable and the
variable of interest are have a nonlinear relationship. Performance of standard
multiple imputation methods can also suffer if the substantive analysis model
of interest is uncongenial to the imputation model, which can be a common
occurrence for social science data if the imputation phase is conducted
independently of the analysis phase. We propose a Bayesian method for multiple
imputation of hierarchical nonlinear time series data that uses a sequential
decomposition of the joint distribution and incorporates smoothing splines to
account for nonlinear relationships between variables. We compare the proposed
method with existing multiple imputation methods through a simulation study and
an application to secondary school enrollment data. We find that the proposed
method can lead to substantial performance increases for estimation of
parameters in uncongenial analysis models and for prediction of individual
missing values.
