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Cited by 201Spatial and Panel Data AnalysisGlobal trade and economicsEconomic Growth and Productivity

Bias and consistency in three-way gravity models

Open access
Weidner, Martin, Zylkin, Thomas·2021·Journal of International Economics
DOI 10.1016/j.jinteco.2021.103513
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Abstract

We study the incidental parameter problem for the “three-way” Poisson Pseudo-Maximum Likelihood (“PPML”) estimator recently recommended for identifying the effects of trade policies and in other panel data gravity settings. Despite the number and variety of fixed effects involved, we confirm PPML is consistent for fixed T and we show it is in fact the only estimator among a wide range of PML gravity estimators that is generally consistent in this context when T is fixed. At the same time, asymptotic confidence intervals in fixed-T panels are not correctly centered at the true parameter values, and cluster-robust variance estimates used to construct standard errors are generally biased as well. We characterize each of these biases analytically and show both numerically and empirically that they are salient even for real-data settings with a large number of countries. We also offer practical remedies that can be used to obtain more reliable inferences of the effects of trade policies and other time-varying gravity variables, which we make available via an accompanying Stata package called ppml_fe_bias.

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  • Following the Weidner and Zylkin (2021) adjustment, the standard error is between the corresponding estimates with pair versus 3-way clustering.

    “We follow the procedure of Weidner and Zylkin (2021) in column (14) of Table 1, where we see that the standard error is between the corresponding estimates with the pair vs. 3-way clustering.”

    From Estimating the effects of trade agreements: Lessons from 60 years of methods and data (2024) · cites this paper as Weidner and Zylkin (2021)

  • IPP leads to an asymptotic bias that affects the validity of inferences in PPML specifications with three-way fixed effects.

    “However, Weidner and Zylkin (2021) show that, while still consistent, IPP leads to an asymptotic bias that affects the validity of inferences in PPML specifications with three-way fixed effects, e.g., such as ours, and propose an adjustment based on Taylor expansions to correct for such biases.”

    From Estimating the effects of trade agreements: Lessons from 60 years of methods and data (2024) · cites this paper as Weidner and Zylkin (2021)

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Research works printed but not linked: 44

  1. Alvarez, J., Arellano, M., 2003. The time series and cross-section Asymptotics of dynamic panel data estimators. Econometrica 71 (4), 1121-1159.
  2. Andersen, E.B., 1970. Asymptotic properties of conditional maximum-likelihood estimators. J. R. Stat. Soc. Ser. B Methodol. 32 (2), 283-301.
  3. Anderson, J.E., van Wincoop, E., 2003. Gravity with gravitas: a solution to the border puzzle. Am. Econ. Rev. 93 (1), 170-192.
  4. Arellano, M., Bonhomme, S., 2009. Robust priors in nonlinear panel data models. Econometrica 77 (2), 489-536.
  5. Bai, J., 2009. Panel data models with interactive fixed effects. Econometrica 77 (4), 1229-1279.
  6. Baier, S.L., Bergstrand, J.H., 2007. Do free trade agreements actually increase Members' International trade? J. Int. Econ. 71 (1), 72-95.
  7. Baier, S.L., Bergstrand, J.H., Feng, M., 2014. Economic integration agreements and the margins of international trade. J. Int. Econ. 93 (2), 339-350.
  8. Cameron, A.C., Gelbach, J.B., Miller, D.L., 2008. Bootstrap-based improvements for inference with clustered errors. Rev. Econ. Stat. 90 (3), 414-427.
  9. Carro, J.M., 2007. Estimating dynamic panel data discrete choice models with fixed effects. J. Econ. 140 (2), 503-528.
  10. Charbonneau, K.B., 2017. Multiple fixed effects in binary response panel data models. Econ. J. 20 (3), S1-S13.
  11. Chen, M., Fernández-Val, I., Weidner, M., 2019. Nonlinear factor models for network and panel data. arXiv preprint. arXiv:1412.5647. · arXiv
  12. Cheng, I.H., Wall, H.J., 2005. Controlling for Heterogeneity in gravity models of trade. Fed. Reserve Bank St. Louis Rev 87 (1), 49-63.
  13. Correia, S., Guimarães, P., Zylkin, T., 2020. Fast Poisson estimation with high-dimensional fixed effects. Stata J. 20 (1), 95-115.
  14. Dhaene, G., Jochmans, K., 2015. Split-panel jackknife estimation of fixed-effect models. Rev. Econ. Stud. 82 (3), 991-1030.
  15. Dzemski, A., 2019. An empirical model of dyadic link formation in a network with unobserved heterogeneity. Rev. Econ. Stat. 101 (5), 763-776.
  16. Eaton, J., Kortum, S., 2002. Technology, geography, and trade. Econometrica 70 (5), 1741-1779.
  17. Egger, P.H., Staub, K.E., 2015. GLM estimation of trade gravity models with fixed effects. Empir. Econ. 50 (1), 137-175.
  18. Egger, P., Larch, M., Staub, K.E., Winkelmann, R., 2011. The trade effects of endogenous preferential trade agreements. Am. Econ. J. Econ. Pol. 3 (3), 113-143.
  19. Fally, T., 2015. Structural gravity and fixed effects. J. Int. Econ. 97 (1), 76-85.
  20. Fernández-Val, I., Vella, F., 2011. Bias corrections for two-step fixed effects panel data estimators. J. Econ. 163 (2), 144-162.
  21. Fernández-Val, I., Weidner, M., 2016. Individual and time effects in nonlinear panel models with large N, T. J. Econ. 192 (1), 291-312.
  22. Fernández-Val, I., Weidner, M., 2018. Fixed Effect Estimation of Large T Panel Data Models. Economics, Annual Review of.
  23. Graham, B.S., 2017. An econometric model of network formation with degree heterogeneity. Econometrica 85 (4), 1033-1063.
  24. Hahn, J., Kuersteiner, G., 2002. Asymptotically unbiased inference for a dynamic panel model with fixed effects when both N and T are large. Econometrica 70 (4), 1639-1657.
  25. Hahn, J., Moon, H.R., 2006. Reducing bias of MLE in a dynamic panel model. Econom. Theory 22 (3), 499-512.
  26. Hausman, J., Hall, B.H., Griliches, Z., 1984. Econometric models for count data with an application to the Patents-R&D Relationship. Econometrica 52 (4), 909-938.
  27. Hinz, J., Stammann, A., Wanner, J., 2020. State dependence and unobserved heterogeneity in the extensive margin of trade arXiv preprint arXiv:2004.12655. · arXiv
  28. Imbens, G.W., Kolesar, M., 2016. Robust standard errors in small samples: some practical advice. Rev. Econ. Stat. 98 (4), 701-712.
  29. Jochmans, K., 2017. Two-way models for gravity. Rev. Econ. Stat. 99 (3), 478-485.
  30. Kato, K., Galvao Jr., A.F., Montes-Rojas, G.V., 2012. Asymptotics for panel Quantile regression models with individual effects. J. Econ. 170 (1), 76-91.
  31. Kauermann, G., Carroll, R.J., 2001. A note on the efficiency of sandwich covariance matrix estimation. J. Am. Stat. Assoc. 96 (456), 1387-1396.
  32. Lancaster, T., 2002. Orthogonal parameters and panel data. Rev. Econ. Stud. 69 (3), 647-666.
  33. Larch, M., Wanner, J., Yotov, Y.V., Zylkin, T., 2019. Currency unions and trade: a PPML re-assessment with high-dimensional fixed effects. Oxf. Bull. Econ. Stat. 81 (3), 487-510.
  34. Mackinnon, J.G., White, H., 1985. Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties. J. Econ. 29 (3), 305-325.
  35. Moon, H.R., Weidner, M., 2017. Dynamic linear panel regression models with interactive fixed effects. Econom. Theory 33 (1), 158-195.
  36. Neyman, J., Scott, E.L., 1948. Consistent estimates based on partially consistent observations. Econometrica 16 (1), 1-32.
  37. Palmgren, J., 1981. The fisher information matrix for log linear models arguing conditionally on observed explanatory variables. Biometrika 68 (2), 563-566.
  38. Pesaran, M.H., 2006. Estimation and inference in large heterogeneous panels with a multifactor error structure. Econometrica 74 (4), 967-1012.
  39. Phillips, P.C.B., Moon, H.R., 1999. Linear regression limit theory for nonstationary panel data. Econometrica 67 (5), 1057-1111.
  40. Rose, A.K., 2019. Soft power and exports. Rev. Int. Econ. 27 (5), 1573-1590.
  41. Santos Silva, J.M.C., Tenreyro, S., 2006. The log of gravity. Rev. Econ. Stat. 88 (4), 641-658.
  42. Stammann, A., 2018. Fast and feasible estimation of generalized linear models with high-dimensional k-way fixed effects. arXiv preprint. arXiv:1707.01815. · arXiv
  43. Verdier, V., 2020. Estimation and inference for linear models with two-way fixed effects and sparsely matched data. Rev. Econ. Stat. 102 (1), 1-16 (forthcoming).
  44. Wooldridge, J.M., 1999. Distribution-free estimation of some nonlinear panel data models. J. Econ. 90 (1), 77-97.

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Authors
Weidner, M., & Zylkin, T.
Affiliations
  • University of Oxford (GB)
  • University of Richmond (US)
Corresponding author
Zylkin, Thomas
ORCID
Venue
Journal of International Economics
Publisher
Elsevier BV
Published
2021-09
Volume
132
Pages
103513
ISSN
0022-1996 (ISSN-L)
DOI
10.1016/j.jinteco.2021.103513
OpenAlex
W4210367679
Cited by
201
References
4
Citations by year
2026: 39 · 2025: 48 · 2024: 43 · 2023: 35 · 2022: 25 · 2021: 5 · 2020: 4 · 2019: 1 · 2018: 1
FWCI
19.25
Type
journal article
Language
en
Open access
Licence
Locations
  • Journal of International Economics (publishedVersion, cc-by)
  • arXiv (Cornell University) (submittedVersion)
  • Oxford University Research Archive (ORA) (University of Oxford) (submittedVersion, cc-by)
  • arXiv (Cornell University)
Funders
  • Economic and Social Research Council (EST013567/1, RES-589-28-0001)
  • European Research Council (ERC-2014-CoG-646917-ROMIA)
  • National University of Singapore (R-109-000-183-646)
Topics
  • Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Spatial and Panel Data Analysis
  • Social Sciences › Economics, Econometrics and Finance › General Economics, Econometrics and Finance › Global trade and economics
  • Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Economic Growth and Productivity
Keywords
incidental parameter problem, estimation bias, cluster-robust variance estimation, asymptotic confidence intervals, trade policy effects

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