关于“非线性世界中的动态因果效应：好的、坏的与丑陋的”的讨论

Federal Reserve Board, Washington, D.C.ISSN 1936-2854 (Print)ISSN 2767-3898 (Online) Discussion of “Dynamic Causal Effects in a Nonlinear World: theGood, the Bad, and the Ugly” Edward P. Herbst, Benjamin K. Johannsen 2025-058 Please cite this paper as:Herbst, Edward P., and Benjamin K. Johannsen (2025). “Discussion of “Dynamic CausalEffects in a Nonlinear World: the Good, the Bad, and the Ugly”,” Finance and EconomicsDiscussion Series 2025-058. Washington: Board of Governors of the Federal Reserve System,https://doi.org/10.17016/FEDS.2025.058. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminarymaterials circulated to stimulate discussion and critical comment.The analysis and conclusions set forthare those of the authors and do not indicate concurrence by other members of the research staff or theBoard of Governors. References in publications to the Finance and Economics Discussion Series (other thanacknowledgement) should be cleared with the author(s) to protect the tentative character of these papers. Discussion of “Dynamic Causal Effects in a NonlinearWorld: the Good, the Bad, and the Ugly” Edward P. Herbst and Benjamin K. Johannsen∗ March 2025 Abstract This comment discusses Kolesár and Plagbord-Møller (2025) finding that the stan-dard linear local projection (LP) estimator recovers the average marginal effect (AME)even in nonlinear settings. We apply and discuss a subset their results using a simplenonlinear time series model, emphasizing the role of the weighting function and theimpact of nonlinearities on small-sample properties. 1Introduction Kolesár and Plagbord-Møller (2025) (hereafter, KP) is an exciting, important advance inthe literature on the estimation of dynamic causal effects in the context of local projections(LPs) (see Jordà (2005)). The paper establishes that the “standard” linear LP of an outcomeyt+honto a shockxt(and possibly a vector of controls) estimates anaverage marginal effect(AME) of the shock on the outcome. This result holds under suitable assumptions even—and perhaps especially—in the case of a nonlinear data generating process foryt. Derivingthe result requires connecting and extending a large literature in microeconometrics. This comment aims to provide an accessible discussion of some of the results reportedin KP that is tailored to macroeconomists. We begin by considering some of the theoreti-cal results in KP under common assumptions in the macroeconomics literature. We devoteparticular attention to the weighting function,ω, that is used to compute the average in theAM E. We then analyze theAM Eand its LP estimation in the context of thequadratic au-toregressive model(QAR(1,1)) model of Aruoba et al. (2017). This is a stationary, nonlinear time series model designed to mimic the statistical structure of a second-order approximationto the solution of a dynamic stochastic general equilibrium (DSGE) model. In the contextof theQAR(1,1)model, we relate the populationAM Eto a population nonlinear impulseresponse function (N IRF) defined in Koop et al. (1996). We also discuss small-sample prop-erties of the LP estimator of theAM Ewith a focus on how nonlinearities in theQAR(1,1)model affect those properties. 2What does the standard LP estimate? In this section, we discuss the LP estimator of theAM E. To establish notation, letyt+hbethe observed outcome of interest at timet+h, and letxtbe the observed shock of interestat timet.Collect all other variables that determineyt+hinto a vectorUh,t+h, which mayinclude past values ofyt, past (and future) values ofxt, and other controls. We require thatthe vectorUh,t+his independent ofxt.A representation ofyt+hbased onxtandUh,t+hiscalled thestructural functionand is given by The representation that is used to define the notion of dynamic causal effect used in KP istheaverage structural function, which is given by This function describes the expected outcomeyt+hgiven a specific value of the shockxt,integrating out all other sources of randomness. Note that because we have assumed thatxtis independent of all other factors affectingyt+h, the average structural function is equalto—and hence can be recovered from—the conditional expectation ofyt+hgivenxt.Thisquantity can in principle be estimated from the data. In macroeconomics, it can be difficult to estimate the average structural function due tosmall sample sizes. One approach is to impose strong assumptions about the data-generatingprocess foryt.For example, a researcher could assume thatytfollows anAR(1)process.Another option in nonlinear time series analysis is to estimate theAM E, defined as HereΨ′h(xt)represents the derivative of the average structural function.This derivativecaptures the effect of an infinitesimal change inxtonyt+h. The weighting function,ω(xt), determines how different values ofxtcontribute to the AME, defining the sense in which theAM Eis an average. KP study local projections that are indexed byhand