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  • xtabond2 ivstyle vs compound ivstyle

    I am reading the paper by Roodman titled "How to do xtabond2: An introduction to difference and system GMM in Stata". I want to clarify a few things with respect to the sub-options associated with the option ivstyle in xtabond2. If I want to do a System GMM and use ivstyle() with xtabond2 WITHOUT specifying equation(diff|level|both) inside the parenthesis of ivstyle, is this going to produce a one-column vector of instruments like equation (24) in Roodman's paper?

    On page 41, the following is mentioned:

    The patterns in (25) can be requested with the equation() suboption, as in ivstyle(w1 w2, equation(level)) and the compound ivstyle(w1 w2, equation(diff)) ivstyle(w1 w2, equation(level)).
    Unfortunately, I don't understand the difference between ivstyle (w1 w2, equation(level)) and the compound ivstyle (w1 w2, equation(diff)) ivstyle (w1 w2, equation(level)). Which of these two actually produces the instruments matrices as those in equation (25)? Any insight will be highly appreciated.

  • #2
    Yes, not specifying the equation() suboption for the ivstyle() option creates a one-column vector of instruments as in Roodman's equation (24). There is hardly any reason why you would want to do that. In that regard, my recommendation is to always use the equation() suboption. This will then indeed create an instrument matrix as in Roodman's equation (25). The suboption equation(diff) creates the submatrix W* (upper-right block) and the suboption equation(level) the submatrix W (lower-right block).

    I have discussed this matter in my presentation (slide 20) at the 2017 UK Stata Users Group Meeting in London.
    https://twitter.com/Kripfganz

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    • #3
      Sebastian Kripfganz Thank you for the clarification. I read your presentation slides. However, it is still not clear why you recommend using the equation() suboption. Would you please explain what is the problem with not using it which produces the one-column vector of instruments like equation (24)? Thank you for your time.

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      • #4
        In principal, there is nothing wrong when you are not using the equation() option and you know what it is doing. My recommendation to not use it stems from the observation that many, if not most, people are not aware of the difference between iv(w1 w2) and the "compound" iv(w1 w2, equation(diff)) iv(w1 w2, equation(level)). They often specify this option in the first way, while they actually intend the instruments to be constructed as implied by the second way.

        Moreover, it is an odd way to specify the moment conditions as the sum of the moments for the first-difference and the level equation (see the formula just after equation (24) in Roodman's paper), even though this condition would of course be valid if the moment conditions for the two equations are valid separately.

        There is another argument that is underappreciated in the discussion of the system GMM estimator: If you have all the instruments specified in the block-diagonal form (equation (25) in Roodman's paper), then you can actually rewrite the instruments for the first-differenced equation such that they become transformed instruments for the level equation. This means that the system GMM estimator could actually be regarded as a GMM estimator with some instruments for the level equation only. This is nice because it highlights that you are not actually considering two separate equations but effectively only one. The first-differenced equation is a transformation of the level equation. You can either transform the variables in the model or you can appropriately transform the instruments instead. (This is not a first-difference transformation of the instruments which makes it a bit less instructive.) In any case, this equivalence no longer holds if you construct the instruments in the one-vector form (equation (24) in Roodman's paper).

        The last argument could also be used to justify why it is more reasonable to always specify the time-dummy instruments for the level equation and not the first-difference equation.
        https://twitter.com/Kripfganz

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        • #5
          Sebastian Kripfganz Thank you, Dr. Kripfganz for the explanation. Could you please refer me to some other readings which elaborate on the underappreciated argument you mentioned above? It was a little difficult for me to comprehend it completely.

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          • #6
            See for example:
            • Arellano, M., and O. Bover. 1995. Another look at the instrumental variable estimation of error-components models. Journal of Econometrics 68: 29-51.
            • Kripfganz, S., and C. Schwarz. 2015. Estimation of linear dynamic panel data models with time-invariant regressors. ECB Working Paper 1838. European Central Bank.
            The key idea is that the combined moment conditions can be written as:
            \[E [\mathbf{Z}' \mathbf{H} \mathbf{u}] = \mathbf{0}\]
            with
            \[\mathbf{Z} = \begin{pmatrix}\mathbf{Z}_d & \mathbf{0}\\\mathbf{0} & \mathbf{Z}_l\end{pmatrix} ,\quad \mathbf{H} = \begin{pmatrix}\mathbf{D}\\\mathbf{I}\end{pmatrix} \]
            where \(\mathbf{Z}_d\) and \(\mathbf{Z}_l\) are the usual instrument matrices for the first-differenced and the level equation, and \(\mathbf{D}\) is a first-difference transformation matrix and \(\mathbf{I}\) the identity matrix (no transformation for the level equation). Thus,
            \[\mathbf{Z}' \mathbf{H} = \begin{pmatrix}\mathbf{Z}_d' \mathbf{D}\\\mathbf{Z}_l'\end{pmatrix}\]
            where the product \(\tilde{\mathbf{Z}}_d' = \mathbf{Z}_d' \mathbf{D}\) constitutes a transformed set of instruments such that the moment conditions can be written in terms of some instruments for the level errors \(\mathbf{u}\):
            \[E [\tilde{\mathbf{Z}}_d' \mathbf{u}] = \mathbf{0} ,\quad E [\mathbf{Z}_l' \mathbf{u}] = \mathbf{0}\]
            https://twitter.com/Kripfganz

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            • #7
              Sebastian Kripfganz Can't thank you enough, sir.

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