  Efficient estimation methods for simultaneous autoregressive (SAR) models
with missing data in the response variable have been well-developed in the
literature. It is common practice to introduce a measurement error into SAR
models. The measurement error serves to distinguish the noise component from
the spatial process. However, the previous literature has not considered adding
a measurement error to the SAR models with missing data. The maximum likelihood
estimation for such models with large datasets is challenging and
computationally expensive. This paper proposes two efficient likelihood-based
estimation methods: the marginal maximum likelihood (ML) and
expectation-maximisation (EM) algorithms for estimating SAR models with both
measurement errors and missing data in the response variable. The spatial error
model (SEM) and the spatial autoregressive model (SAM), two popular SAR model
types, are considered. The missing data mechanism is assumed to follow missing
at random (MAR). While naive calculation approaches lead to computational
complexities of $O(n^3)$, where n is the total number of observations, our
computational approaches for both the marginal ML and EM algorithms are
designed to reduce the computational complexity. The performance of the
proposed methods is investigated empirically using simulated and real datasets.
