with error terms. Provided the availability of valid instruments, RE and FE methods extend to the case where some of the explanatory variables are allowed to be endogenous. As in the exogenous setting, RE model with Instrumental Variables (REIV) requires more stringent assumptions than FE model with Instrumental Variables (FEIV) but it tends to be more efficient under appropriate conditions. To fix ideas, consider the following model: where is unobserved unit-specific time-invariant effect (call it unobserved effect) and can be correlated with for s possibly different from t. Suppose there exists a set of valid instruments . In REIV setting, key assumptions include that is uncorrelated with as well as for . In fact, for REIV estimator to be efficient, conditions stronger than uncorrelatedness between instruments and unobserved effect are necessary. On the other hand, FEIV estimator only requires that instruments be exogenous with error terms after conditioning on unobserved effect i.e. . The FEIV condition allows for arbitrary correlation between instruments and unobserved effect. However, this generality does not come for free: time-invariant explanatory and instrumental variables are not allowed. As in the usual FE method, the estimator uses time-demeaned variables to remove unobserved effect. Therefore, FEIV estimator would be of limited use if variables of interest include time-invariant ones. The above discussion has parallel to the exogenous case of RE and FE models. In the exogenous case, RE assumes uncorrelatedness between explanatory variables and unobserved effect, and FE allows for arbitrary correlation between the two. Similar to the standard case, REIV tends to be more efficient than FEIV provided that appropriate assumptions hold. Dynamic panel models In contrast to the standard panel data model, a dynamic panel model also includes lagged values of the dependent variable as regressors. For example, including one lag of the dependent variable generates: The assumptions of the fixed effect and random effect models are violated in this setting. Instead, practitioners use a technique like the Arellano–Bond estimator. See also Panel study Hausman test Factor analysis References Multivariate time series Panel data