[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.

[Joseph L. Staats]

Dear Sebastian,

I am using xtdpdgmm in connection with an international political economy project and have some questions for which I don't have answers.

1. I hypothesize that Var-A has an indirect causal effect on Var-B operating successively through three mediator variables, M-1, M-2, and M-3. I have four separate levels of GMM models, the first estimating the effects of Var-A on M-1, then the effects of M-1 on M-2, then M-2 on M-3, and finally M-3 on Var-B. To estimate the coefficient for the indirect effect of Var-A on Var-B, I use the "product of coefficients method," which involves taking the product of the coefficients of interest obtained at each level of the mediation pathway. I arrive at a confidence interval for the indirect-effect coefficient using Monte Carlo simulations as suggested by Selig and Preacher (2008) and Preacher and Selig (2012). I had a manuscript reviewer question whether it was appropriate to use the product of coefficients method in connection with GMM. Can you think of anything in particular about GMM results that would make them inappropriate for using the product of coefficients method in the manner just described?

2. The reviewer also suggested I run unit-root tests on all my predictor and control variables to determine stationarity. I found two variables that failed unit-root tests. One of these is Var-A and the other is M-2. Do I need to do something in or outside of xtdpdgmm to account for this lack of stationarity?

3. I note from an earlier posting you made in Statalist https://www.statalist.org/forums/for...-arellano-bond, and in your 2019 London Stata Conference presentation (Slide 30), you say that lack of stationarity may be a sign that System GMM is not appropriate. Does that have anything to do with the lack of stationarity in my two variables as noted in No. 2 above? I ask because I could not find any recommendation by you or others who have written on stationarity and GMM, most especially Roodman (2009) and Kiviet (2012), suggesting that unit-root tests should be conducted to determine whether to use Difference or System GMM.

4. Also, there doesn't seem to be a bright line between what is meant by Difference and System GMM. I understand that System GMM comes into play if there are two equations, one with outcome and predictor variables transformed to first differences and using untransformed instruments (level) and the other using untransformed outcome and predictor variables (level) with first-differenced instruments. But what about the situation where there is only one equation, outcome and predictor variables that are transformed using the model(fod) option and instruments are untransformed? Is this considered System GMM, and does this trigger the concern with stationarity in GMM I asked about in No. 3 above?

Thanks in advance for your help on this.

[Sebastian Kripfganz]

1. The negative degrees of freedom indicate that the model is no longer identified after removing the moment conditions under question. To obtain a just-identified model, you would need at least 2 or 11 further moment conditions (i.e. instruments). In other words, you currently cannot test the validity of the overidentifying restrictions implied by the moment conditions 5 or 6.

2. You could possibly use the model and moment selection criteria, estat mmsc. See slides 91 and following of my 2019 London Stata Conference presentation.

[Prateek Bedi]

Hi,

I have two doubts as mentioned below.

1. From the output of difference-in-Hansen test shown below, what can we infer? Specially with regard to the last two moment conditions where the degrees of freedom are negative.

2-step weighting matrix from full model

                  | Excluding                   | Difference                  
Moment conditions |       chi2     df         p |        chi2     df         p
------------------+-----------------------------+-----------------------------
  1, model(fodev) |     0.4063      2    0.8162 |      0.2431      1    0.6220
  2, model(fodev) |     0.0360      1    0.8496 |      0.6134      2    0.7359
  3, model(fodev) |     0.0146      1    0.9038 |      0.6348      2    0.7281
  4, model(fodev) |     0.0654      1    0.7981 |      0.5839      2    0.7468
  5, model(level) |          .     -2         . |           .      .         .
  6, model(level) |          .    -11         . |           .      .         .

2. If there are two competing models, both of which satisfy AR(2) and Sargan-Hansen tests, how do decide which one is better? What are the criteria which we can use to choose one model over the other?

Thanks and Regards

[Nishant Kathuria]

Thank you so much for the explanation It is indeed very helpful. And for #4 above, my apologies as c.x1##c.x2##c.x3##c.t gives same results with separate interaction terms.... I realize that I was missing an interaction term, so the results were different..