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Measurement error, fixed effects, and false positives in accounting research

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Abstract

We show theoretically and empirically that measurement error can bias in favor of falsely rejecting a true null hypothesis (i.e., a “false positive”) and that regression models with high-dimensional fixed effects can exacerbate measurement error bias and increase the likelihood of false positives. We replicate inferences from prior work in a setting where we can directly observe the amount of measurement error and show that the combination of measurement error and fixed effects materially inflates coefficients and distorts inferences. We provide researchers with a simple diagnostic tool to assess the possibility that the combination of measurement error and fixed effects might give rise to a false positive, and encourage researchers to triangulate inferences across multiple empirical proxies and multiple fixed effect structures.

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Notes

  1. Throughout the paper we use the term “measurement error” to refer to the gap between a theoretical construct and its measurement. For example, using log of total assets to measure the construct of firm size.

  2. As anecdotal evidence, we point to the Gow et al. (2016) survey of causal inference in accounting research. In the survey, the phrase “measurement error” never appears; “exogen-” appears nine times, and “endogen-” appears seven times. Examples of the earlier literature include the work of Christie (1987), Barth (1991), and Collins et al. (1994). See Kothari (2001) for a review.

  3. The term “high-dimensional fixed effects” refers to the practice of including a large number of fixed effects in the regression specification to absorb as much variation as possible while still allowing for estimation of the coefficient of interest (e.g., Correia 2015). This practice is also colloquially known as the “kitchen sink approach to fixed effects” or “saturating the model with fixed effects.” Supporting the growing popularity of this practice, deHaan (2021) reports that 96% of empirical papers published in 2019 in The Accounting Review, Journal of Accounting and Economics, and Journal of Accounting Research have a model that features a fixed effect structure, 84% have a model that features two fixed effect structures, and 29% have a model with three or more fixed effect structures.

  4. Khan, Serafeim, and Yoon (2016, p. 1699) best summarize the conventional wisdom on fixed effects: “The inclusion of both time and firm fixed effects in the panel regressions is a generalization of the difference-in-differences approach that allows a causal interpretation in a regression setting … The fixed effects soak up unobserved firm-specific and economy-wide factors that could otherwise cloud identification.”

  5. See also Armstrong et al. (2022) and Donelson et al. (2022), who document similarly high absorption rates in the context of state income tax rates and universal demand laws respectively.

  6. To illustrate this distinction, consider the following passage from Loughran and McDonald (2014, p. 1646), which advocates for measuring 10-K readability using file size (in megabytes) over the Fog Index: “10-K file size is exceptionally easy to determine and is not prone to the substantial measurement errors of other textual procedures requiring parsing of the 10-K documents.” While it may be the case that file size is easy to compute and is measured without error (i.e., we know the number of megabytes with absolute certainty), the measurement error for which file size measures the theoretical construct––readability––could be substantial and is difficult to assess without experimental data (e.g., Bonsall IV et al. 2017).

  7. This data is available on Josh Lee’s website.

  8. We use the Garmaise (2011) index of state noncompete laws updated by Ertimur et al. (2018). For examples of studies using the Garmaise (2011) index as either a control variable or a treatment variable, see among others, Aobdia (2018), Glaeser (2018), Ertimur et al. (2018), and Chen et al. (2018).

  9. Note that inferences would be stronger if the control variable were omitted from the regression equation. In this case, because the control is correlated with measurement error in the included variable of interest, the omission of the control would create classic omitted variable bias.

  10. Recall that there is measurement error in \(\hat{Z}\), which attenuates the coefficient on \(\hat{Z}\) per the classic attenuation bias described in Section 2.1.

  11. For readers interested in a more detailed treatment of the various strengths and weaknesses of fixed effects, see the following literature: (i) Angrist and Pischke (2009) discuss issues when both fixed effects and lag values of the dependent variable are included in the same model; (ii) Grieser and Hadlock (2019) discuss the strict exogeneity assumption of fixed effects; (iii) Berg et al. (2021) discuss how fixed effects exacerbate coefficient bias when seeking to estimate spillover effects; and (iv) Whited et al. (2022) discuss fixed effects in the more general context of the “bad controls” problem discussed by Angrist and Pischke (2009).

  12. See Donelson et al. (2022) for an application of this point in the context of universal demand laws.

  13. We selected these measures from among a variety of others, because they are simple, standard, and entail fewer “researcher degrees of freedom” (Loken and Gelman 2017). For example, there are a myriad of accruals models we could choose from, but it is well known that inferences in the literature are often sensitive to the choice of accrual model (e.g., Armstrong et al. 2022).

  14. We do not use the logistics/probit regression due to the “incidental parameters problem” associated with fixed effects. Specifically, Arellano and Hahn (2007) suggest that nonlinear maximum likelihood models can be inconsistent and biased when fixed effects are included. Avoiding maximum likelihood models allows us to pin the bias introduced by the fixed effects to measurement error rather than the “incidental parameters problem.” Using linear models with discrete outcome variables is common practice in accounting (e.g., Guay et al. 2016).

  15. For Restate, |0.011–0.002| / 0.002 = 4.5.

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Acknowledgements

We thank Beth Blankespoor, Ted Christensen, Ed deHaan, Patricia Dechow (editor), Ian Gow, Wayne Landsman, Alexander Ljungqvist, Nathan Marshall, Brian Miller, Robbie Moon, Robert Resutek, Cathy Schrand, Sarah Zechman, Frank Zhou, Christina Zhu, two anonymous reviewers, and seminar participants at the JFR mini-method conference, the City University of Hong Kong, University of Georgia, University of Iowa, University of Maryland, Michigan State University, New York University, University of Washington, and The Wharton School for helpful comments. We thank our schools for financial support. Headquarters data used in the paper are available on Josh Lee’s website.

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Appendices

Appendix 1

Table 11 Data on corporate headquarters locations
Table 12 Error rates in alternative data sources

Appendix 2 Alternative data sources for state of corporate headquarters

This appendix discusses Heider and Ljungqvist (2015) and the SEC header file as two alternative sources of data on state of corporate headquarters.

2.1 Heider and Ljungqvist

Heider and Ljungqvist (2015) make manual corrections to the 2013 Compustat file which they have shared with other researchers. They provide a list of corrections that would make the 2013 Compustat file accurate. While the corrections themselves are not problematic, it is problematic to apply these corrections to more recent Compustat files for two reasons.

  1. 1)

    These corrections were derived only on the specific sample used by Heider and Ljungqvist (2015), a sample that excluded financials and utilities and focuses only on NYSE/NASDAQ/AMEX listed companies. Applying these corrections to the broader universe of firms on Compustat will not correct the state of headquarters for firms that were not in their sample.

  2. 2)

    These corrections are to the 2013 Compustat file. As a result, they will miss any subsequent changes in corporate headquarters and associated back-filling. For example, consider the firm that moved from Texas to California in 2015. In 2015, Compustat would have back-filled all of the firm’s observations back to 1968 with a California headquarters. Applying the corrections would not correct for this, because the data from 1968 onward was correct as of 2013 and no corrections were recorded. In 2015, all observations for this firm between 1968 and 2014 would be incorrect and would indicate California headquarters.

Given these issues, Heider and Ljungqvist share their corrections with the following caveats:

figure a

We clarify it is not sufficient to simply delete post-2011 observations. Because of back-filling, all observations, even those before 2011, will be incorrect if the firm moved after 2013. The literature seems to be unaware of these issues––these issues not withstanding––a large and growing number of published and unpublished papers continues to apply the Heider and Ljungqvist corrections to recent Compustat files (e.g., Hanlon et al. 2021; Lai et al. 2020).

We assess error rates in the state of corporate headquarters if the Heider and Ljungqvist corrections were applied to the current Compustat file. We repeat the analysis in Table 6, comparing the listed state of headquarters on each file against the firm’s annual 10-K filing. Table 12 presents results. Column (1) presents error rates for the 2019 Compustat file after applying the Heider and Ljungqvist corrections. Consistent with the corrections missing any change in headquarters and associated backfilling subsequent to 2013, we find error rates of 10.8% in 1996 that decline to 4.8% in 2013. The findings in Appendix 2 Table 12 make clear that simply deleting observations after 2011 will not rectify the problem.

2.2 SEC Header

The SEC maintains a header file for each corporate filing, and this file contains the state of headquarters and business address. However, these header files are not always current and are sometimes updated with a significant lag. For example, Bemis Company moved its headquarters from Minnesota to Wisconsin in 2006, but the SEC header file continues to show Minnesota as the state of headquarters through 2010. Column (2) of Appendix 2 Table 12 presents error rates for the SEC header file. Of all data sources considered, the SEC header best replicates the state of headquarters listed on the firm’s annual 10-K filing. Column (2) shows error rates for the SEC header data are consistently around 2%.

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Jennings, J., Kim, J.M., Lee, J. et al. Measurement error, fixed effects, and false positives in accounting research. Rev Account Stud 29, 959–995 (2024). https://doi.org/10.1007/s11142-023-09754-z

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