1. There is admittedly an ambiguitiy in the way the term "exogenous" is used in the context of these dynamic panel models. Conventionally, a variable is said to be strictly exogenous if it is uncorrelated with the idiosyncratic error term from any time period. In the traditional sense, the variable would still be endogenous if it is correlated with the unit-specific effect, which is also part of the combined error term. For iv(x, model(level)), we need x to be truly exogenous in the traditional sense, i.e. it should not be correlated with either component of the error term. If the latter holds, then you would not specify the option in any other way because this way you maximize the correlation between the instrument and the regressor. If the variable is only uncorrelated with the error term but not with the unit-specific effect, then you would need to either specify instruments for a transformed model, or you could specify iv(x, diff model(level)), which would require the additional assumption that the first difference of x is uncorrelated with the unit-specific effect.
2. This statement corresponds to any type of variable. Usually, you would also specify instruments with lags for a transformed model. The additional lags for the level model are then typically redundant. If you specify instruments for the level model only, which is rarely done in the context of dynamic panel models but generally possible, then additional lags might still be useful.
3. See point 2. You need to start with lag 1 because the variable is not strictly exogenous (with respect to the idiosyncratic error component). And you would typically stick to this first lag if you are also using instruments for the transformed model. Note: Most of the time you would again want to add the diff option if you do not want to assume that x is uncorrelated with the unit-specific error component.
4. With gmmiv(x, model(diff) lag(a b)), you are only using the instruments for the differenced model. You would need to add a separate option for the level model if desired. Similarly the other way round. Whether to use a transformation (and if so, which) depends on the arguments in point 1 and further arguments set out in my 2019 London Stata Conference presentation.
2. This statement corresponds to any type of variable. Usually, you would also specify instruments with lags for a transformed model. The additional lags for the level model are then typically redundant. If you specify instruments for the level model only, which is rarely done in the context of dynamic panel models but generally possible, then additional lags might still be useful.
3. See point 2. You need to start with lag 1 because the variable is not strictly exogenous (with respect to the idiosyncratic error component). And you would typically stick to this first lag if you are also using instruments for the transformed model. Note: Most of the time you would again want to add the diff option if you do not want to assume that x is uncorrelated with the unit-specific error component.
4. With gmmiv(x, model(diff) lag(a b)), you are only using the instruments for the differenced model. You would need to add a separate option for the level model if desired. Similarly the other way round. Whether to use a transformation (and if so, which) depends on the arguments in point 1 and further arguments set out in my 2019 London Stata Conference presentation.
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