[Sebastian Kripfganz]

1. You are right that Arellano and Bover (1995) also propose a system-GMM estimator. In addition to a transformed model equation (first differences or forward-orthogonal deviations), they add a model equation for the average levels. This is essentially just a level equation with time-invariant instruments. What is often unnoticed: A similar system-GMM estimator was already proposed by Arellano and Bond (1991). When I refer to Arellano and Bover (1995) in the xtdpdgmm help file, the focus is on their proposal to use forward-orthogonal deviations for the transformed model equation (whether or not a level equation is added).
2. Thank you for this valuable feedback. I believe part of the confusion stems from the fact that in xtabond2 the same option is called equation(). I deliberately chose to replace equation() with model() because eventually there is just a single equation estimated. Actually, your initial thought is correct. The model() option eventually refers to a transformation of the instruments, although not in a straightforward way. I tried to illustrate this on slide 33 of my presentation. If you specify level instruments for the first-differenced model equation, those instruments essentially become transformed instruments for the level model equation. (I do not have a good name for this transformation. You might call it transposed differencing because the instruments are not multiplied with a first-difference transformation matrix D but with its transpose D'.) I will have to think about whether I can improve the help file in that regard.

[Joseph L. Staats]

Sebastian,

Thanks again for your help.

I have two more things to bring into the conversation, if I may, neither of which relates to my specific project.

1. At the top of Slide 36 of your 2019 London Stata Conference presentation you give an example of xtdpdgmm System GMM code that contains two model equations, one where the variables are transformed to first-differences and the instruments are at levels and the other where the variables are at levels and the instruments are transformed to differences. But in the help file that accompanies xtdpdgmm, you give two examples of code, one which you say represents Arellano-Bover two-step and the other Blundell-Bond two-step. As I understand it, Arellano-Bover and Blundell-Bond are both forms of System-GMM. But I only see one equation, not two, in your help-file examples. Am I misunderstanding or misreading something here?

2. The help-file discusses specifying a model, as follows: "model(model) specifies if the instruments apply to the model in levels, model(level), in first differences, model(difference), in deviations from within-group means, model(mdev), or in forward-orthogonal deviations, model(fodev). The default is model(level) unless otherwise specified with the global option model(model)." For quite a while when I first started trying xtdpdgmm, I thought this language was referring to transforming the instruments, not the variables themselves. I arrived at a correct understanding of this definition only after reading your 2019 London Stata Conference presentation, which has, in my opinion, a much clearer discussion/definition than the help-file. I only bring this up because other users and readers of Statalist may also have struggled with understanding how to use the model specification versus the instrument sub-option.

[Sebastian Kripfganz]

I am afraid this becomes a bit too application-specific. I do not see a problem with GMM in general. It is more a question whether your underlying model is correctly specified, which is something I cannot answer.

[Joseph L. Staats]

Sebastian,

Thanks for your thoughtful and informative answers to my questions.

1. It seems that I may not have a stationarity issue after all. In looking over my work, I see that I made an error in doing unit-root tests for M-2 and that it actually passes the xtunitroot fisher, dfuller test. And Var-A is a dichotomous categorical variable, so, as I understand it (after looking online for an answer), a unit-root test for this variable doesn't make any sense. By the way, I have 17 years of data, which I suppose is somewhere in between small-t and large-t.

2. Thanks for the suggestion to use a DAG to show the hypothesized causal path.

3. You said you had difficulty imagining using a dynamic model in the context of mediation analysis. I am doing research on the so-called "democratic advantage" for sovereign bond ratings (Moody's and S&P). My hypothesis is that democracy does not directly affect bond ratings but instead (sometimes) sets in motion other changes in the political system that lead to an increase in bond ratings. The particular mediation path is democracy->political competition->judicial power and independence->rule of law->bond rating (Moody's or S&P). I use xtdpdgmm for GMM FOD for each step in the process, and I include a lag of each outcome variable in each step (M-1, M-2, M-3, and Var-B). Of course I have control variables in each step, although most of them only appear in the topmost step where Var-B is the outcome variable. I get good coefficient, standard-error and instrument results using GMM FOB, although I have to use a third-degree polynomial transformation of political completion when it is acting on M-2 (there is theory to support the idea that increased political competition can be good up to a point where it thereafter becomes detrimental). Is there anything about all of this that strikes you as inappropriate for using GMM?

Thanks again for your help.

[Sebastian Kripfganz]

1. If GMM in all of the four steps correctly identifies the respective coefficients, then in principle the product of coefficients methods should be applicable. You would probably need to argue with the help of a DAG that this is the case. The answer depends very much on the specific application and cannot be answered in general, I believe.
2. Unit-root tests typically require a large time dimension to be reliable. If you believe in the results of your tests, this would raise questions whether a nonstationary variable can cause a stationary variable in your proposed sequence of mediator variables. Models with different integration orders of the dependent and independent variable are typically misspecified unless there are further variables in the model, e.g. lagged dependent variable and lagged independent variable, that help to obtain a stationary error term.
3. Are we talking about dynamic models with a lagged dependent variable? (I have some difficulties imagining such dynamic models in the context of a mediator analysis as you described in 1.) In such dynamic models, the stationarity condition for system GMM is effectively a condition on the initial observations. In static models, this initial observations problem does not occur. However, the first differences of nonstationary variables may generally be poor instruments for the levels (and vice versa). Because these system GMM estimators are discussed almost entirely in the context of dynamic models with a short time dimension, unit-root tests are usually not considered as they would require a large time dimension.
4. In the case of a FOD-transformed model only, we would not call it a system GMM estimator. You could possibly call it an FOD-GMM estimator. Nonstationarity could again lead to a poor performance of the estimator as lagged levels likely become weak instruments in the FOD-transformed model.