This paper considers a spatial panel data regression model with serial correlation on each spatial
unit over time as well as spatial dependence between the spatial units at each point in time. In
addition, the model allows for heterogeneity across the spatial units using random effects. The paper
then derives several Lagrange multiplier tests for this panel data regression model including a joint
test for serial correlation, spatial autocorrelation and random effects. These tests draw upon two
strands of earlier work. The first is the LM tests for the spatial error correlation model discussed in
Anselin and Bera [1998. Spatial dependence in linear regression models with an introduction to
spatial econometrics. In: Ullah, A., Giles, D.E.A. (Eds.), Handbook of Applied Economic Statistics.
Marcel Dekker, New York] and in the panel data context by Baltagi et al. [2003. Testing panel data
regression models with spatial error correlation. Journal of Econometrics 117, 123–150]. The second
is the LM tests for the error component panel data model with serial correlation derived by Baltagi
and Li [1995. Testing AR(1) against MA(1) disturbances in an error component model. Journal of
Econometrics 68, 133–151]. Hence, the joint LM test derived in this paper encompasses those derived
in both strands of earlier works. In fact, in the context of our general model, the earlier LM tests
become marginal LM tests that ignore either serial correlation over time or spatial error correlation.
The paper then derives conditional LM and LR tests that do not ignore these correlations and
contrast them with their marginal LM and LR counterparts. The small sample performance of thesetests is investigated using Monte Carlo experiments. As expected, ignoring any correlation when it is
significant can lead to misleading inference.