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Evidence that the accuracy of self-reported lead emissions data improved: A puzzle and discussion


We investigate the accuracy of facility-reported data both within and across emissions and off-site transfer inventories of lead (Pb) in time. We build on recent work using Benford’s Law to detect statistical anomalies in large data sets. Our application exploits a regulatory experiment to test for systematic changes in firm behavior triggered by the 2001 implementation of the Final Rule, a major regulatory change governing the U.S. Environmental Protection Agency’s (EPA) oversight of lead emissions. Statistical results show that the EPA’s Final Rule functioned to significantly improve the accuracy of facility-reported lead data. This finding is surprising because abatement requirements increased and both the probability of firm audit and expected penalties for misreporting apparently decreased in the post-Final Rule period. To explain this counterintuitive result we develop a reporting model for the firm. We argue that organizational investments made in response to specific requirements of the Final Rule, as well as rising public awareness of the risks of lead, may have induced firms to report more accurately.

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  1. 1.

    Goodness-of-fit statistics show that first significant digit distribution of air lead monitor data is indistinguishable from Benford’s distribution: N = 1, 613 MAD = 0.372, χ 2 = 3.39 (p = 0.91), G 2 = 3.42 (p = 91). De Marchi and Hamilton (2006) find similar evidence in an earlier time period for a variety of toxic pollutants, including lead.

  2. 2.

    See Cohen (1999) and Heyes (2000) for useful reviews.

  3. 3.

    Modern Casting, the official publication of the American Foundry Society, for instance, published articles on how the Final Rule will affect facilities melting leaded-alloys or contaminated scrap (May 2001), and years later in a regulatory brief warned that the EPA was cross-referencing TRI reports that included lead, and that some metal casting facilities received inquiries from EPA on the use of lead and lead compounds.

  4. 4.

    Between 1990 and 2010, the percentage of firms using industry standard estimation technologies increased from 27.8 to 53.7%; meanwhile, the fraction of firms with technically-trained environmental, health and safety officers responsible for certifying emissions reports increased from 6.9 to 22%.

  5. 5.

    The probability of a FSD being d is: \( P(d) = \log {}_{10}\left(d+1\right)- \log {}_{10}(d)= \log {}_{10}\left(1+\frac{1}{d}\right),\kern0.5em d=1,\ 2, \dots,\ 9. \) Thus, the probability that the FSD is 1 is \( \log {}_{10}\left(1+\frac{1}{1}\right)=0.301, \) and the probability that the FSD is 2 is \( \log {}_{10}\left(1+\frac{1}{2}\right)=0.176. \) Provided a variable spans many orders of magnitude and is uniformly distributed on a logarithmic scale, the FSD of an observed variable is more likely to be 1 than 9 because the interval width of d and d + 1 declines in d on a logarithmic scale. The distance in interval [log10(1), log10(2)], for instance, is greater than the distance in interval [log10(8), log10(9)], 0.301 and 0.0512 respectively. The value of P(d) is therefore approximately proportional to the distance between d and d + 1 on a logarithmic scale.

  6. 6.

    In addition to Pearson’s χ 2 and D*, Table 1 reports mean absolute deviation (MAD) and log likelihood ratio (G 2) test statistics. \( MAD = \frac{{\displaystyle {\sum}_{i=1}^9}\left|{O}_i-{E}_i\right|}{9} \), and \( {G}^2=N\left(2{\displaystyle {\sum}_{i=1}^9{O}_i \ln}\left(\frac{O_i}{E_i}\right)\right) \).

  7. 7.

    Statistically, estimates of fugitive emissions incorporate uncertainty from stack emissions. In terms of accuracy, by definition reported fugitive emissions ≤ stack emissions. Fugitive emissions (E f ) are equal to total emissions (E t ) minus stack emissions (E f ). Stack emissions are calculated by: E s  = (E t  − E f ) × (100 % − α/100 %), where α is the collection efficiency of hoods.

  8. 8.

    The probability that a significant digit (d) is observed in the n th(n > 1) position is: FSD being d is: \( {\displaystyle {\sum}_{k={10}^{n-2}}^{10^{n-{1}_{-1}}}{ \log}_{10}}\left(1+\frac{1}{10k+d}\right),\ d=0,\ 1, \dots,\ 9. \)

  9. 9.

    In TRI data documentation, the technology used by a facility to estimate emissions is indicated. Four principal methods are coded: 1) mass balance calculations; 2) emission factors; 3) periodic, random, or continuous monitoring; or 4) other, unspecified. Over the period 1990–2010, facilities using one of the three specified estimation technologies—mass balance calculation, emission factors, or monitoring—report fugitive emissions that more closely match Benford expectation. Goodness-of-fit statistics for estimation method specified facilities, N = 8, 859, MAD = 1.940, χ 2 = 410.59, suggest greater accuracy than for estimation unspecified facilities, N = 8, 385, MAD = 4.456, χ 2 = 2, 432.07.

  10. 10.

    We exploit information on the title of the certifying official in TRI data documentation. An official with “environmental,” “health,” or “safety” in their professional title are coded as technical personnel. All other officials are coded as non-technical personnel. Common titles include: Environmental Supervisor; Director of Environmental Services; Plant Environmental Manager; Director of Environmental Restoration Division; and Director of Environmental Health and Safety, to name a few. Non-technical certifying officers are commonly titled as: President; Executive Vice-President; Plant Manager; and Chief Financial Officer. Goodness-of-fit statistics show that firms with technical personnel in charge of reporting differ significantly in reporting accuracy of fugitive emissions. For non-technical personnel firms: N = 15, 096 MAD = 3.369, χ 2 = 2, 498, D* = 0.122. For technical personnel firms: N = 505, MAD = 1.977, χ 2 = 102.7, D* = 0.065. In the post-Final Rule period, the proportion of facilities with technical personnel increased 77.7% (t = − 5.14, p <.01).

  11. 11.

    EPCRA also authorizes citizen enforcement suits: “any person may commence a civil action on his own behalf against . . . [a]n owner or operator of a facility for failure to . . . [c]omplete and submit a toxic chemical release form under section 11023(a) of this title.” 42 U.S.C. § 11046(a)(1)(A)(iv).

  12. 12.

    While not the purpose of our investigation, it is worth noting that data also show that the Final Rule functioned to reduce the environmental burden of industrial Pb. As compared to the pre-Final Rule period, the average annual sum of fugitive emissions decreased significantly (t = 2.573, p <.01) from 185,996lbs to 73,399lbs in the post-Final Rule period. This decline in the average annual sum of fugitive emissions is especially remarkable given the average number of facilities reporting more than tripled (from 355 to 1,241) in the post-Final Rule period. The average annual sum of stack emissions of Pb declined similarly, from 314,063lbs to 165,742lbs (t = 4.542, p <.01). Perhaps most remarkably, the average annual per firm emissions (fugitive + stack) declined from 1,397 to 194lbs (t = 7.778, p <.01). The decline in Pb emissions is an expected outcome. The TRI is an information-based strategy to address market failure (Stephan 2002), configuring market incentives to reduce the use of listed toxins in four important ways. First, by informing consumers of the external diseconomies of their purchasing decisions, the TRI may function to reduce consumption of products that impose high uncompensated costs on society. Second, the TRI may induce consumers to pressure producers to reduce use of listed toxins. Under threat of adverse public reaction, firms may have sufficient motivation to reduce emissions. Santos et al. (1996) find that facilities subject to TRI reporting requirements do, in fact, significantly reduce emissions in time, and significantly increase public outreach activities. Third, the TRI provides corporate lenders and stockholders with information to better calibrate investment decisions regarding a firm’s potential environmental liabilities, abatement expenditures, and fines. Investors may retreat from companies that traffic in listed chemicals. Hamilton (1995), for instance, finds that firm stock value declines by $236,000 for each additional TRI chemical reported to the EPA. Fourth, the TRI can inform facilities of effective pollution control strategies used by market competitors (Tietenberg and Wheeler 2001; Cohen 1999).


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We wish to thank the Robert Wood Johnson Foundation for providing its financial support.

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Corresponding author

Correspondence to Sammy Zahran.



Fig. 4

Fig. 4

Distribution of first digits of lead emissions from EPA monitors, 2005. Note: Observed distribution of first digits in bars and Benford expectation in connected dots

Table 5 EPA National enforcement data on inspections, administrative penalty orders, administrative dollar penalties, and judicial referrals

Proof of proposition 1

The objective function is convex in both variables, so the first-order conditions are sufficient. These are

$$ S^{\prime}\left({e}^T-\varDelta \right)+P^{\prime}\left({e}^T-\varDelta \right)=A^{\prime}\left(\overline{e}-{e}^T\right) $$


$$ S^{\prime}\left({e}^T-\varDelta \right)+P^{\prime}\left({e}^T-\varDelta \right)={q}_c\left( pen+ lie\right). $$

Combining, gives

$$ A^{\prime}\left(\overline{e}-{e}^T\right)={q}_c\left( pen+ lie\right), $$

which pins down e T. e T is unaffected by changes to S(⋅), but we do have \( \frac{\partial {e}^T}{\partial {q}_c}<0 \). Given this, we can use the second first order condition to establish \( \frac{\partial \varDelta }{\partial {q}_c}<0 \). Given e T, we can also see from the second first-order condition that increasing S ′ (⋅) causes Δ* to increase.

The latter result is illustrated in Fig. 2. Holding e T fixed—given by the last equation above— Δ* solves the second equation above. Because both the sanction function and the publicity function are increasing and convex, it follows that the left-hand side is decreasing in ∆. The figure is drawn in a way that assumes marginal costs are zero when reported emissions are zero (i.e., when Δ = e T).

Fig. 5

Sanctioning and underreporting

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Zahran, S., Iverson, T., Weiler, S. et al. Evidence that the accuracy of self-reported lead emissions data improved: A puzzle and discussion. J Risk Uncertain 49, 235–257 (2014).

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  • Lead emissions
  • Final Rule 2001
  • Benford’s law
  • Toxic release inventory
  • Self-regulation

JEL Classifications

  • Q53
  • Q58
  • K32
  • K42