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A Semi-Structural Methodology for Policy Counterfactuals∗

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A Semi-Structural Methodology for Policy Counterfactuals∗ A Semi-structural Methodology for Policy Counterfactuals∗ Martin Berajay Massachusetts Institute of Technology and NBER November 26, 2020 Abstract I propose a methodology for constructing counterfactuals with respect to changes in policy rules which does not require fully specifying a structural model, yet is not subject to Lucas Critique. It applies to a class of dynamic stochastic models whose equilibria are well approx- imated by a linear representation. It rests on the insight that many such models satisfy a principle of counterfactual equivalence: they are observationally equivalent under a bench- mark policy rule and yield an identical counterfactual equilibrium under an alternative one. As an application, I use the methodology to quantify how US fiscal integration contributes to regional stabilization. ∗I am indebted to my advisors Fernando Alvarez, Erik Hurst, and Veronica Guerrieri, for their encouragement, support, and guidance during my stay in Chicago, for our many insightful conversations, and for their patience and skill in distilling the important points from my presentations. For their valuable suggestions, I owe George- Marios Angeletos, Andy Atkeson, Philip Haile, Ricardo Lagos, John Leahy, Martin Schneider, Gianluca Violante, and Ivan Werning as well as my classmates Jonathan Adams, Philip Barret, and Chanont Banternghansa. I thank those at the Review of Economic Studies Tour, Minneapolis Fed Junior Scholar Conference, SED, “Pizzanomics,” NBER SI Dynamic Equilibrium Models, and Macro Seminars at MIT, Harvard, Yale, Wharton, UCLA, GWU, Stanford, Princeton, NYU, Michigan, Northwestern, Cornell, Brown, LSE, BU, Berkeley, Fed Board, Atlanta Fed, Chicago Fed, BCRA, and Norges Bank. I thank all participants of the Capital Theory Working Group, especially Harald Uhlig and Nancy Stokey. Finally, I thank Juliette Caminade for her support and comments on preliminary versions of this paper. yE-mail: [email protected]. Web: economics.mit.edu/faculty/maberaja 1 Introduction Economists commonly follow either structural or reduced-form (and semi-structural) approaches in order to answer counterfactual questions. The former relies on specifying primitives of a par- ticular model and identifying its parameters. This is a daunting task whenever researchers are uncertain about features of alternative models that are both difficult to distinguish with avail- able data and reasonable a-priori. If counterfactuals differ under these alternative models, their credibility is undermined because they lack robustness. The latter approaches require less com- mitment to particular model assumptions and have proven very useful to identify consequences of observed policy changes that are robust across models, as in Sims (1980). However, the struc- tural approach is the leading paradigm for studying unobserved, potential changes in policy rules because structural counterfactuals are not subject to the critique in Lucas (1976). Are there then reasonable circumstances when we can analyze the consequences of a counterfactual policy rule change without knowing the full structure of a model? And, relatedly, can we tell which features of disparate models crucially determine such consequences? In this paper, I propose a novel semi-structural method for constructing counterfactuals with respect to policy rule changes. It is useful in circumstances when a linearization approach is reasonable and we do not wish to fully specify a structural model, yet we are worried about using approaches that contain too little information for the analysis to be immune to Lucas Critique — like those based on structural VARs.1 The method hinges on an insight about dynamic stochastic models with equilibria that are well approximated by a linear, stable recursive representation. The insight is that many models that can match an economy’s recursive equilibrium under a benchmark policy rule—i.e., observationally equivalent models—will also generate an identical counterfactual equilibrium under an alternative policy rule. Thus, regardless of their specific microfoundations, such models will satisfy this Principle of Counterfactual Equivalence. The method has three steps. The first is using data from an economy under a benchmark policy rule to estimate a recursive representation of the equilibrium of dynamic stochastic models (e.g., a SVAR representation). The second is describing assumptions that lead to linear restrictions on these models’ structural equilibrium equations (e.g., exclusion restrictions). They need to be enough to identify all combinations of policy-invariant parameters in these equations — what I call a structure — given the benchmark recursive representation. The last step is solving for the counterfactual recursive representation under an alternative policy rule, given the identified structure. As such, the method is in the spirit of the literature on “sufficient statistics” (e.g., Chetty (2009), Alvarez, Le Bihan, and Lippi (2016)) as well as earlier ideas advanced at the Cowles Commission (e.g., Koopmans (1950), Marschak (1974), Hurwicz (1962)).2 Together, the 1Bernanke, Gertler, and Watson (1997) and Sims and Zha (2006) propose to analyze counterfactual policy rules in SVARs by “zeroing-out” policy responses to shocks. Despite being subject to Lucas Critique, this methodology is com- monly used in many policy institutions, as are macroeconometric models like FRB/US. Del Negro and Schorfheide (2009) studies interest rate counterfactuals in New Keynesian models and asses the sensitivity of results to model misspecification by considering perturbations around their equilibrium SVAR representation. 2For an excellent discussion of these ideas see Heckman and Vytlacil (2007). As they explain, Marschak’s Maxim 1 benchmark and counterfactual representations define a set of structures — which is larger than a singleton containing the identified structure — characterizing models that can generate them, and, therefore, are counterfactually equivalent. The counterfactual is immune to Lucas Critique if the data was generated by a model whose structure is in this set too. I begin the paper by formally describing the principle of counterfactual equivalence, and give necessary and sufficient conditions for a model’s structure to belong to a counterfactually equiv- alent set (Section 2). Then, I present the semi-structural method to construct counterfactuals with respect to policy rule changes (Section 3), which I call Robust Policy Counterfactuals. Parallel to the general treatment, I present an example in the context of fiscal union models where members receive federal transfers according to a transfer policy rule which summarizes how resources are redistributed in a state-contingent manner. In the Online Supplementary Material S.4, I present additional examples in the context of New Keynesian and DMP-style search and matching mod- els. Finally, in order to show how to construct Robust Policy Counterfactuals in practice, I use US state-level data to quantify how fiscal unions contribute to regional stabilization (Section 4) — a question for which, despite plenty of empirical and theoretical research (e.g., Sala-i Martin and Sachs (1991), Asdrubali, Sorensen, and Yosha (1996), Feyrer and Sacerdote (2013), Farhi and Werning (2014)), quantitative work is surprisingly scarce (see Evers (2015) for an exception). There are two main findings. First, in a counterfactual US economy without a transfer policy rule, the standard deviation of employment across states increases by about 1 percent in the Great Recession and 1.5 percent in the long-run. This is because the rule redistributes resources from well to poorly performing states, thus stabilizing regional economies.3 Moreover, had we used the “zeroing-out” methodology in Sims and Zha (2006) to construct the long-run counterfactual, we would have found an increase of only 0.5 percent — thus underestimating the stabilizations gains by ignoring changes in agents’ behavior and expectation-formation as they internalize the policy change. These results can therefore help inform policy-makers when weighing the stabilization benefits against the costs of future fiscal arrangements for the European Monetary Union. Second, exogenous shifters of the Euler equation are a key feature of fiscal union models that generate large stabilization gains, whereas other features like the particular form of wage stickiness or adjustment costs, whether there is discounting due to behavioral agents, or how hand-to-mouth agents are introduced seem to be less crucial. The counterfactually equivalent set I study includes the structures of many models with such features. This finding can thus help organize and rationalize previous quantitative results coming from seemingly disparate models which implied small stabilization gains from fiscal integration (Evers (2015)) and risk-sharing arrangements more generally (as in Backus, Kehoe, and Kydland (1992) and the subsequent literature). says that for many questions of policy analysis it may not be necessary to fully identify a structural model, but only combinations of subsets of the structural parameters. Hurwicz (1962) noted that these should also be policy invariant. 3For a sense of magnitudes, aggregate output volatility declined by about 0.7 percent during the “Great Modera- tion” (Clarida, Gali, and Gertler (2000)). The stabilization gains from fiscal integration are of similar magnitude. 2 2 The Principle of Counterfactual Equivalence This section describes how to characterize a set of models that are counterfactually
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