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[ Journal of Labor Economics, 2009, vol. 27, no. 2] � 2009 by The University of Chicago. All rights reserved. 0734-306X/2009/2702-0001$10.00
When to Start a Fight and When to Fight Back: Liability Disputes in the
Workers’ Compensation System
David Card, University of California, Berkeley
Brian P. McCall, University of Michigan
Contrary to the original intention of no-fault workers’ compensation laws, employers deny liability for a substantial fraction of on-the- job injuries. We develop and estimate a simple structural model that explains the high rate of litigation as a consequence of asymmetric information. We estimate the model using data for a large sample of back injuries in Minnesota. Simulations under the counterfactual as- sumption that all denied workers pursue their claims suggest that the strategic incentive accounts for 30%–40% of observed liability dis- putes.
No-fault workers’ compensation (WC) was adopted in most states in the early twentieth century to eliminate costly litigation over liability for work-related injuries (Somers and Somers 1954; Fishback and Kantor 2000). Under an ideal no-fault system, employers agree to pay WC ben- efits for all work-related injuries, and employees forfeit their right to sue in the event of an accident. While the majority of WC claims are settled without a dispute, in a surprising fraction of cases—10% or more of injury claims in Minnesota, for example—employers refuse to accept liability
We are grateful to Brian Zaidman and the Minnesota Department of Labor and Industry for assistance in obtaining the data used in this article. We also thank John Kennan for very helpful comments on earlier drafts. Contact the corre- sponding author, Brian McCall, at [email protected].
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for the injury.1 The associated litigation costs are often blamed for con- tributing to the rapid rise in WC premiums over the past 2 decades (e.g., Long 2004).
In this article we propose and estimate a simple asymmetric information model of WC disputes using data on injury claims in Minnesota.2 We assume that otherwise identical injured workers differ in their costs of pursuing a disputed claim. Knowing this, employers have a strategic in- centive to deny liability for high-cost claims, even if the probability of being found liable is relatively high. Injured workers with higher costs of fighting back will then be induced to drop their cases, saving the firm some fraction of the cost of their claim.
Our data analysis uses a large sample of back injuries drawn from Minnesota WC administrative files from the late 1980s. We focus on these injuries because of their relatively high cost and because of the uncertainty over employers’ liability for many back-related injury claims (Burton 1992). Our structural model includes a rich set of observed characteristics of the injured worker and the injury and also allows for a flexible spec- ification of unobserved heterogeneity across dispute pairs. Consistent with the basic insights of our theoretical model, we find that employers are more likely to deny liability for an injury when there is a bigger expected payoff to “starting a fight” and that workers are more likely to respond by filing a claim petition (CP) when there is a bigger expected payoff to fighting back. Simulations of the model under the counterfactual as- sumption that denied claimants have no cost of fighting back—but face the same distribution of expected payoffs conditional on pursuing the dispute—suggest that the strategic incentive accounts for a significant share (30%–40%) of all liability disputes in our sample.
I. Institutional Background: The Minnesota WC System
Employees who incur a work-related injury in Minnesota are entitled to WC benefits for any injury that results in permanent disability or more
1 Berry (2002) reports that the rate of denial of liability for indemnity claims ranged from 14% to 16% in the 1990s. Hyatt and Kralj (2000) estimate a 9.5% denial rate for claims in Ontario in the late 1980s. Barth and Hunt (1980) report that, in the 1970s, 5%–10% of all WC claims nationwide were formally contested.
2 See Cooter and Rubinfeld (1989) for a review of the literature on disputes in a variety of legal settings. Roberts (1992), Thomason (1994), and Falaris, Link, and Staten (1995) have examined disputes in WC. Our analysis differs in that it takes a structural estimation approach that explicitly models the decision-making process. See Kreider (1999) for an example of such an approach applied to the disability insurance application process. Sieg (2000) estimates a structural model of medical malpractice disputes, based on the asymmetric information model of Bebchuk (1984) and Nalebuff (1987).
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than 3 lost workdays.3 Injured workers are also entitled to full reim- bursement of their medical treatment costs. In the event of a lost-time injury, the employer (or the employer’s insurance carrier) has 14 days to either begin paying benefits or deny liability by filing a “notice of denial” with the appropriate administrative body (the Department of Labor and Industry). During our sample period, about 10% of all lost-time back injury claims in the state were initially denied.
By filing a notice of denial, the employer may be disputing the existence of an injury, denying that it arose in the course of employment, or chal- lenging the compensability of the injury.4 A denied claimant who wishes to pursue the case normally consults with an attorney and then may initiate the dispute resolution process by filing a CP with the Department of Labor and Industry. Just over one-quarter of all denied back injury claims in the 1980s resulted in the subsequent filing of a CP.
Once a CP is filed, the dispute can be referred to nonbinding mediation by the Department of Labor and Industry or scheduled for an admin- istrative conference conducted by a departmental settlement judge.5 Fail- ing settlement at this stage, the dispute moves to the state Office of Ad- ministrative Hearings, where cases are heard in a formal setting by an administrative law judge. The judge’s decision can be appealed to the Workers’ Compensation Court of Appeals.
Figure 1 gives some basic information on the first two stages of the dispute process involving issues of primary liability for back injuries in Minnesota. The data in this figure are based on a 10% random sample of “first reports” of injuries that occurred between 1985 and 1989. Em- ployers routinely file such a report for any injury that might result in an indemnity claim, and a first report is legally required for any injury that actually leads to an indemnity payment (i.e., a payment for lost work time or as compensation for permanent disability). Thus, the sample frame includes all back injuries with positive indemnity benefits, as well as
3 The main type of benefits are “temporary total” benefits paid according to a statutory formula (based on the preinjury wage) for each lost workday. Partial benefits are also available for injured workers who can return to work on a reduced work schedule. These and other features of the Minnesota system are described in Minnesota House of Representatives Research Department (1988).
4 According to a study conducted by the Minnesota House of Representatives Research Department (1988), the majority of denials arise over basic factual issues such as whether the injury occurred at work. An important minority of denials arise over more subtle issues such as whether WC benefits are payable for stress- related diseases or occupational injuries like carpal tunnel syndrome.
5 The formal dispute resolution process is described in much more detail in Minnesota House of Representatives Research Department (1988).
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Fig. 1.—Denials and claim petition filings in Minnesota Workers’ Compensation System, 1985–89.
injuries that were filed on a first report but never generated an indemnity claim.6
Figure 1 shows the fraction of injury cases with positive payments and the mean payment amounts conditional on a positive payment for each of three possible denial/CP states: injuries for which the employer ac- cepted liability, injuries for which liability was denied and no CP was filed, and injuries that were denied and for which a CP was filed.7 Overall, injured workers receive indemnity payments in 87% of nondenied cases and 40% of denied claims. The probability of a payment is relatively high for denied cases with a CP (74%) but is significant (25%) even for denied cases with no CP. Judging by the payment frequencies and amounts, a significant fraction of denied cases are eventually revealed to be valid injury claims. Note that average payment amounts (conditional on a pos- itive payment) are much higher for denied and contested cases ($18,956)
6 The sample excludes some minor injuries that resulted in medical costs but less than 3 days of lost work time. Employers can (and sometimes do) dispute their liability for medical costs in such “medical only” claims. These denials are missing from our sample.
7 The payment amounts in the figure include all forms of benefits, including lump sum amounts paid to resolve certain cases, as well as temporary, total, and permanent partial benefits.
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than for nondenied cases ($7,991), suggesting that disputes tend to involve costly injuries.8
II. A Theoretical Model of Denials and Disputes
A. Basic Model
We model disputes over liability as the outcome of a two-party se- quential game between risk-neutral players, in which the employer first decides whether to deny liability, and the injured worker then decides whether to fight this decision by filing a CP. Once a petition is filed, the case moves to the dispute resolution stage. Although it would be useful and interesting to model the decisions of the parties in the dispute res- olution process (including the decision to settle out of court), our data set only includes information on the indemnity payments received by the injured worker once a CP is filed—not at the stage at which the case was settled. We therefore treat the post-CP process as a black box characterized by a distribution of indemnity payments. Any particular injury claim is characterized by a variety of facts, including observed attributes like the preinjury wage and characteristics that are known to the parties but un- observed by us, such as the absentee record of the injured worker. As discussed below, our empirical model includes a relatively flexible param- eterization for these observed and unobserved facts. For now, we treat these characteristics as fixed and known by the parties.
A worker whose injury claim has been denied has to decide whether to consult an attorney and file a CP or to give up. Assume that if the claim is denied and not contested, the worker receives no indemnity benefits or other compensation. (We discuss the reasoning behind this assumption, and an alternative, in the next subsection.) If a CP is filed, the worker faces a random payoff of y1, where the distribution of y1 incorporates the entire future sequence of potential outcomes (including payments determined at all stages of the dispute resolution process). As- sume further that the claimant has an expected cost of pursuing the dispute (c) that is unknown to the employer but distributed across the population with a distribution function F. Assuming risk neutrality, a denied claimant with expected litigation cost c will fight back by filing a CP if .E(y ) 1 c1 The probability that a denial is contested is therefore
Pr (c) p F(E(y )). (1)1
As noted above, most of the actual legal costs of fighting a denial are
8 WC payment amounts tend to be highly skewed. Thus, the median payment amount in each denial/CP category is far below the mean. The median payment for nondenied cases with positive payments is $632. The median for denied and uncontested cases with positive payments is $1,273. The median for denied and contested cases with positive payments is $10,847.
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borne by attorneys, who work on a contingent fee basis and receive a fraction of any settlement amount. Thus, the cost c should be interpreted as reflecting the combination of psychic and out-of-pocket costs borne directly by the injured worker.9 In our empirical implementation, we assume that c depends on the worker’s age, marital status, and weekly wage.
In the first stage of the game, the employer has to decide whether to accept or to deny liability for an injury. In the event of a denied claim, the employer’s expected indemnity payment is . Let y2 representPr (c)E(y )1 the indemnity payment conditional on not denying the claim. At the time the denial decision is made, y2 is uncertain since the extent of the worker’s lost time and medical costs can take many weeks to be resolved. Assume that the firm faces an expected litigation cost d of pursuing a denial. A cost-minimizing firm will deny liability for the claim if
d ! d* � E(y ) � Pr (c)E(y ). (2)2 1
Assuming that d is distributed across employers with a distribution func- tion G, the probability of a denial is
Pr (d) p G(d*). (3)
Note that it might be reasonable to model expected dispute costs as vary- ing with the size of likely indemnity payments with or without a denial. Nevertheless, in our empirical specification we assume that d depends on only the weekly wage of the injured worker and the type of insurance carrier. As discussed in more detail below, the implicit exclusion restric- tions are testable by a generalization of the conventional overidentification test used in linear simultaneous equations models.
Given distributions for the costs c and d and for the indemnity payments if a claim is accepted (y2) or denied and ultimately contested (y1), the model of equations (1) and (3) yields a relatively simple likelihood func- tion for the observed litigation outcome (not denied, denied and contested, or denied and not contested) and for the observed indemnity payment amount conditional on the litigation outcome. A key feature of this model is that it allows us to quantify the impact of the strategic incentive to deny liability, arising from the fact that workers with higher costs of pursuing a denied claim will drop their case, rather than fight back. In
9 An alternative assumption is that the decision process is determined by the incentives of lawyers, who will be willing to take the case if the expected contingent fee—which is just a fraction of —exceeds the expected litigation cost. ThisE( y )1 will generate a similar decision rule, although the interpretation of c will be slightly different.
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particular, if workers could pursue a denied claim costlessly, then , and firms would only deny liability for cases withPr (c) p 1
†d ! d ∼ E(y ) � E(y ).2 1
The excess fraction of denials arising from the strategic incentive is , which depends on the gap†G(d*) � G(d )
†d* � d p [1 � Pr (c)]E(y ).1
This gap will be larger, the larger the mean cost of contesting a dispute and the larger the firm’s expected liability in the dispute resolution phase.
B. Payments for Denied and Uncontested Claims
The simple model described by equations (1)–(3) is based on the as- sumption that a worker whose claim is denied receives nothing unless he or she decides to contest the denial. As we noted in the discussion of figure 1, however, about one-quarter of denied and uncontested claims generate positive indemnity payments. We believe that the vast majority of these cases represent denials that were subsequently withdrawn by the employer (or the insurer) once the facts of the case were established. Firms (or their insurers) have only 14 days from the occurrence of an injury to decide whether to begin paying the claim or to deny liability. Discussions with WC practitioners suggest that in cases in which a first report of injury form is incomplete, or in which the facts of an injury are still under investigation, insurers often issue a denial to preserve the option of dis- puting the claim. As the required information becomes available, the in- surer may decide to pay the claim, effectively withdrawing their objection. According to this interpretation, the roughly 2% of claims that are denied and not contested and have positive indemnity payments should be re- classified as nondenied claims.10 We follow this approach for our baseline empirical model.
As a simple alternative, we posit that there is some probability of re- ceiving a positive indemnity payment for a denied claim, even if no CP is filed, and that injured workers take the expected value of this payment stream into consideration when deciding whether to contest a denial. Specifically, let y3 represent indemnity payments for a denied but uncon- tested claim. A denied claimant with expected litigation costs c will fight
10 Some support for this interpretation comes from an analysis of denied WC claims in Virginia during the late 1990s. The Virginia administrative files include an indicator for whether claim payments were delayed by the insurance carrier: over 90% of claims that were denied but not formally contested and have positive indemnity payments are coded as having a carrier delay.
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back by filing a CP if . The probability that the denialE(y ) � c 1 E(y )1 3 is contested is then
Pr (c) p F(E(y ) � E(y )). (1′ )1 2
The employer’s expected indemnity payment, if it decides to start a fight by denying liability for the injury, is . A cost-Pr (c)E(y ) � [1 � Pr (c)]E(y )1 3 minimizing firm will deny liability if
d ! d** ∼ E(y ) � Pr (c)E(y ) � [1 � Pr (c)]E(y ), (2 ′ )2 1 3 implying that the probability of a denial is
Pr (d) p G(d**). (3 ′ )
As in the baseline case, it is possible to quantify the excess fraction of denials arising from the strategic incentive to deny liability in this alter- native model. If the firm expected that all denied workers would fight back (i.e., ), it would only deny liability for cases withPr (c) p 1
††d ! d p E(y ) � E(y ).2 1
Thus, the fraction of denials arising from strategic incentive is G(d**) � , which depends on the gap††G(d )
††d** � d p [1 � Pr (c)][E(y ) � E(y )].1 3
III. Empirical Implementation
To take either version of our model to the data, we need to develop a stochastic specification for the payment amounts y1 and y2 (or y1, y2, and y3) and for the expected litigation costs c and d of workers and employers. As illustrated in figure 1, the payment distributions at all stages of the dispute process have significant mass at zero. In view of this, we adopt a two-part parameterization for the probability of a positive payment at the jth node of the game and for the density of payments conditional on a positive payment:
y p D z , for j p 1, 2 (or j p 1, 2, 3 in the extended model), (4)j j j
where Dj is a random variable taking on values of zero and one, and zj is a strictly positive random variable that is independent of Dj.
11 We assume that
Pr (D p 1) p p p F(g x), (5)j j j
where F(z) is the Gaussian distribution function evaluated at z, and x represents a set of observed characteristics of the injury claim. In addition,
11 Such two-part models are widely used in the health economics literature, for example, to characterize the distribution of medical expenditures.
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we assume that the zj’s are log-normally distributed, conditional on the observed x’s and an unobserved heterogeneity component:
z p exp (xb � v � e ), (6)j j j j
where vj represents the contribution of case-specific facts that are observed by the parties but unobserved by us, and ej represents a normally dis- tributed error component. Under these assumptions, the expected pay- ment amount at the jth node is
1E(y Fx, v ) p F(g x) exp (xb � v � j 2), (7)j j j j j j2
where jj represents the standard deviation of ej. The two other ingredients of the theoretical model are the worker’s
expected cost of contesting a claim c and the firm’s expected cost of denying a claim d. We assume that these are given by
c p xb � e (8a)4 4
and
d p xb � e , (8b)5 5
where e4 and e5 are normally distributed with mean 0 and standard de- viations j4 and j5, respectively. We assume that the error terms (e1, e2, e3, e4, e5) are mutually uncorrelated.
Combining equations (7), (8a), and 8(b) with equations (1)–(3), it is straightforward to derive expressions for the probability of contesting a denial and for the probability that a case is denied. In our baseline model, an injured worker with expected litigation costs c will contest a denied claim if . Given the observed and unobserved case characteristicsE(y ) 1 c1 (x, v1, v2), this occurs with
E(y Fx, v ) � xb1 1 4 Pr (c) p F . (9)[ ]j4
Note that expected payments conditional on contesting the denial enter this equation with a coefficient of —thus, we can identify the standard1/j4 deviation of the unobserved component of worker’s litigation costs.12
Likewise, a firm with expected litigation costs d will deny a claim if , which occurs withPr (c)E(y ) � E(y ) 1 d1 2
E(y Fx, v ) � Pr (c)E(y Fx, v ) � xb2 2 1 1 5 Pr (d) p F . (10)[ ]j5
The difference in expected costs between accepting and denying liability
12 Given that j4 is identified, it is also possible to identify the coefficient vector b4.
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enters this equation with coefficient —thus, we can also identify the1/j5 standard deviation of the unobserved component of the firm’s expected litigation costs.
The likelihood function for the observed data, conditional on x and (v1, v2), consists of five parts, where Fj(z) is the density function for a log-normally distributed variable with mean , and var-log (z) p xb � vj j iance :2log (z) p jj
Denial/CP Status/Payment Likelihood
Not denied, no payment [1 � Pr (d)](1 � p )3 Not denied, payment ($y) [1 � Pr (d)]p F (y)3 3 Denied, no CP Pr (d)[1 � Pr (c)] Denied, CP, no payment Pr (d) Pr (c)(1 � p )1 Denied, CP, payment ($y) Pr (d) Pr (c)p F (y)1 1
Corresponding expressions for the extended model are very similar and are reported in appendix A.
As in other structural econometric models, a key issue is the param- eterization of the unobserved heterogeneity terms (v1, v2). We assume that the vector (v1, v2) has a point-mass distribution with a relatively small number of points of support:
Pr ((v , v ) p (v k, v k)) p qk, for k p 1, 2, … , K.1 2 1 2
This parameterization allows the unobserved heterogeneity components in the indemnity payout amounts (z1, z2) to be arbitrarily correlated across injuries. Each point of support for the bivariate distribution contributes three additional parameters: two location parameters (v1k, v2k) and a probability qk. The overall likelihood for the observed data is then ob- tained by taking a probability-weighted average of the likelihood for each point of support and jointly maximizing the likelihood with respect to the structural parameters (gj, bj, jj) and the heterogeneity parameters (v1k, v2k, qk).
13
Note that our parameterization of the model assumes that the parties observe the heterogeneity terms (v1, v2) and make their decisions accord- ingly. Effectively, we are assuming that the parties know a lot more about the facts of the cases than we do and that variation in the “unobserved facts” drives much of the variation in denial rates and the probability that injured workers whose claims are denied contest their case.
13 Similarly, for the generalized model we assume that the vector (v1, v2, v3) has a point-mass distribution. In this case, each point of support for the trivariate distribution contributes four additional parameters: three location parameters and a probability.
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Evaluating the Model
The structural model represented by equations (7)–(10) is relatively restrictive. For example, it ignores any unobserved heterogeneity in the probabilities of a positive indemnity payment (p1, p2) or in the expected litigation cost equations. Moreover, depending on the exclusion restric- tions embedded in the parameter vectors b1–b5, the model imposes a number of restrictions on the way the observed covariates affect the prob- abilities of positive indemnity payments, the mean payment amount (con- ditional on positive payments), and the probabilities and . AsPr (d) Pr (c) we discuss in more detail below, we include a relatively rich set of co- variates in the models for the probabilities of a positive payment, and for the conditional payment amounts, but adopt relatively parsimonious spec- ifications for the expected litigation cost equations (8a) and (8b). Under these assumptions, variables that are excluded from the employee cost equation only effect the probability that a denied claim is contested to the extent that they shift the expected payment amount conditional on fighting back ( ). Likewise, variables that are excluded from the firm’sE(y )1 dispute cost equation (8b) only effect the probability of denial to the extent that they shift the differential in expected costs of a nondenied and a denied claim ( ).E(y ) � Pr (c)E(y )2 1
These exclusion restrictions can be tested formally by a Lagrange mul- tiplier (LM) test. Essentially, the test involves computing the correlation between the excluded variables and the generalized residuals (Gourieroux et al. 1987) of the deny and contest equations.14 A related test can be derived by comparing the predicted and actual probabilities of denying a claim (or contesting a denied claim) by characteristics that are not di- rectly included in xb4 and xb5. For example, if the costs c and d are assumed to be independent of the cause and type of injury, then predicted differ- ences in denial rates by cause and type of injury can only arise through systematic differences in the way that these characteristics affect the ex- pected indemnity payments if the injury was denied and contested or was not denied. A high correlation between the actual and predicted denial rates for different causes and types of injuries, therefore, provides support for the assumed structure of the model.
A third test of our structural model arises from the fact that the prob- ability of denial depends on the difference in expected indemnity costs between accepting and denying liability. In particular, notice from equa- tion (10) that the terms and enter the proba-E(y Fx, v ) Pr (c)E(y Fx, v )2 2 1 1
14 Appendix A describes the implementation of the test in more detail.
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bility of denying a claim with equal and opposite coefficients. Consider an alternative unrestricted specification:
b5 Pr (d) p F[E(y Fx, v )k � Pr (c)E(y Fx, v )k � x ].( )2 2 1 1 1 2
j5
The functional form of our structural model can be tested by estimating this alternative version and testing that the coefficients k1 and k2 are equal in magnitude and opposite in sign.15
IV. Data Description and Preliminary Analysis
Before turning to a formal econometric analysis of the models described in the previous section, we present an overview of the Minnesota WC data and briefly summarize some results from a descriptive analysis of the decision processes of the two parties. As noted earlier, our main data source is a 10% random sample of the first reports of injuries filed with the Minnesota Department of Labor and Industry between 1985 and 1989. We also have information on 100% of the denied claims from the same time period, which we combine with the 10% sample of nondenied claims and weight appropriately.16 Throughout this article, we limit attention to injury claims with valid (nonmissing) data on the date of the injury and the injured worker’s gender and preinjury wage.17 Characteristics of the resulting sample of overall claims are presented in table 1. We also present data for the subsample of claims involving a back injury and for back injuries by denial status. In this table we reclassify denied and uncontested injury claims with positive indemnity payments as “nondenied,” under the assumption that these represent denials that were subsequently with- drawn by the employer.
The means in table 1 suggest that injured workers are relatively young, predominantly male, and typically employed in blue-collar occupations. The characteristics of workers with back injuries are fairly similar to those of the overall sample, as are the characteristics of workers with a back injury whose claims were either denied or not, although denied workers tend to have lower average wages than do those whose claims are accepted. Most of the monetary cost of worker injuries arises from the so-called temporary total benefits paid to workers who are off work and recovering from their injury (see table 1). During our sample period, the temporary
15 A similar test is available for the more general model. In this case, however, there are three coefficient restrictions—see appendix A for details.
16 To estimate the model, we use the weighted maximum likelihood procedure suggested by Manski and Lerman (1977).
17 These requirements eliminate about 10% of the sample. The main missing variable is the injured worker’s weekly wage. Claims were followed until 1994, and claim costs reflect accumulated costs up until that point.
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Table 1 Characteristics of Worker Compensation Claims
All Claims
Back Injuries
Back Injuries by Denial Status
Not Denied Denied
Claimant characteristics: Average age 35.2 35.0 35.0 35.1 Percent female 31.0 32.9 32.5 36.7 Percent married 52.4 54.5 55.0 49.0 Percent blue collar 60.6 58.0 58.3 54.5 Average weekly wage 358.1 360.6 362.1 344.4
Replacement rate: Percent with replacement rate ≥ 1 16.8 15.8 15.5 18.9 Percent with replacement rate from 2/3 to 1 66.6 67.5 67.5 67.8 Percent with replacement rate ! 2/3 16.6 16.6 16.9 13.3
Industry: Percent in construction 11.9 11.9 12.2 9.4 Percent in manufacturing 31.5 28.9 28.5 32.7 Percent in trade 19.2 18.2 18.4 16.4 Percent in services 22.8 27.1 27.0 28.6
Insurance carrier: Percent self-insured 20.6 21.5 21.4 23.0 Percent state fund 3.9 3.7 3.6 3.9 Percent assigned risk pool 7.4 6.5 6.1 10.2
Claim outcomes: Percent denied 9.0 8.6 .0 1.0 Percent of denials contested 25.8 33.3 . . . 33.3 Percent with payments 1 0 75.7 81.4 86.8 24.6 Average payment 4,773.7 6,722.7 6,917.2 4,657.7 Average payment if payment 1 0 6,306.1 8,258.8 7,969.1 18,933.7 Percent of payments from temporary total
benefits 77.8 80.8 81.3 48.2 Sample size 23,755 13,264 6,797 6,467
Note.—Data for all claims are derived from 10% sample of workers’ compensation claims filed in Minnesota. Data for back claims are derived from 10% of all claims, pooled with 100% sample of denied claims. Denied and uncontested claims with positive payments have been reclassified as nondenied claims. Payment amounts include all forms of benefits and lump sum amounts.
total disability benefit schedule in Minnesota set a standard wage-replace- ment rate of two-thirds for many workers, but because of minimum and maximum benefit rates and other features, replacement rates could range from substantially below this rate (for high-wage workers) to over 100% (for low-wage workers). As shown in table 1, about two-thirds of injured workers in the sample had a replacement rate between two-thirds and one, while 16% had a replacement rate above 100%, and a similar-sized group had replacement rates below the standard rate. Interestingly, workers with high replacement rates are overrepresented in the denied claims sub- sample, while those with low-replacement rates are underrepresented.18
18 Krueger (1990) analyzes the effect of WC benefit rates on claim frequencies and finds a positive effect, suggesting that workers with higher benefits are more likely to file fraudulent claims. Similar reasoning suggests that workers with higher replacement rates may be more likely to file fraudulent claims, leading to higher denial rates.
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The industry distributions for the various subgroups are shown in table 1. Again, these are not too different, although service workers account for a larger fraction of back injuries than they do of overall injuries, and injured construction workers appear to be less likely to have their claim denied than do those from other industries. Table 1 shows the fraction of claims arising from firms with three key types of insurance arrange- ments: self-insurance, insurance through the state-run competitive fund, and coverage through the state’s assigned risk plan. (The remainder of firms are insured with private insurers.) Comparing the percentages across the final three columns, it appears that self-insured firms have slightly higher denial rates than other insurers, while denial rates at firms covered by the assigned risk plan are nearly twice as high as average.
Finally, the bottom rows of table 1 show the denial rate for injury liability, the fraction of denials in which a CP was filed by the worker, and information on the payments associated with different types of in- juries. The (reclassified) denial rate for liability of all injury claims in our sample is 9%, while the denial rate for back injury claims is 8.6%. The fraction of denied injuries that are contested by injured workers is higher for back injuries, however, as is the fraction of injuries with a positive payment.
As we noted in the discussion of figure 1, the probability of receiving WC payments and the mean level of payments differ significantly between denied and nondenied claims. Although denied claims are less likely to generate payments, the mean level of payments conditional on a positive amount is much higher for denied than for accepted claims ($19,000 vs. $8,000). Figure 2 shows smoothed estimates of the frequency distributions of log indemnity payments, conditional on a positive payment, by denial status.19 Both distributions are bimodal, with modes at around $500 ( ) and $15,000 ( ). Relative to the distri-log (500) p 6.2 log (15,000) p 9.6 bution for accepted claims, however, the distribution for denied claims has far more mass around the upper mode.
Table 2 documents some of the variation in dispute rates and dispute outcomes across different subgroups of back injury claimants. We focus on four key dimensions: the wage-replacement rate of the injured worker, gender, the cause of the injury, and the type of injury.20 For each subgroup,
19 Note that with our reclassification of withdrawn denials, only denied and contested claims have positive payments. Thus, the conditional distribution of payments for denied claims is the conditional distribution for denied and contested claims.
20 Our database includes information on 14 injury types and 12 injury causes. The limited classifications shown in table 2 were selected after some experimen- tation and capture a high fraction of the variation in denial and claim-petition filing rates by type and cause. Falaris et al. (1995) have documented that litigation rates over lost-time injuries in Delaware differ by average weekly wages, gender, the cause of the injury, and the type of injury.
Liability in Workers’ Compensation 163
Fig. 2.—Log costs of back injury claims: reclassified data. Estimates are based on a 10% random sample of all workers’ compensation claims and a 100% sample of all denied claims for back injury filed in Minnesota from 1985 to 1989. Density estimates were produced using kernel density estimation with a Gaussian kernel and bandwidth equal to 0.30.
table 2 shows the average denial rate, the fraction of denied claims that are contested, and average payments for claims that are accepted, denied, and denied and contested. Several interesting patterns emerge. Claims from high replacement rate (i.e., low-wage) workers are more likely to be denied but have a lower contest rate if denied and generally lower costs. Claims from women are also more likely to be denied, less likely to be contested, and of lower average cost than claims from men. Claims caused by slips and falls are particularly likely to be denied (denial
), as are claims that are classified as dislocations (denialrate p 14.5% ) and those that are classified as unknown type (denialrate p 15.3% ).rate p 17.3%
A more detailed picture of the variation in denial and contest rates across claim subgroups is provided in figure 3. Here, we plot the con- ditional rate of contesting a denied claim against the denial rate for groups of workers classified by replacement rate, gender, cause of injury, and type of injury.21 We use different markers, depending on the range of replacement rates in the cell. The scatter of points suggests a positive correlation between denial and contest rates ( ). The correlationp p .11 between replacement rates and denial rates is illustrated by the tendency for the squares (representing cells with replacement ) to berates ! 2/3
21 There are 36 cells representing the cross-classification of two genders, three replacement rate ranges, two injury causes, and three injury types.
164 Card/McCall
Table 2 Claim Outcomes by Characteristics of Claimant and Cause/Type of Injury
Weighted Count of Injuries
Denial Rate (%)
Denials Contested
(%)
Average Total Payments ($)
Not Denied Denied
Denied and Contested
All back injuries 74,370 8.6 33.3 6,917 4,658 13,977 (.2) (.6) (241) (179) (477)
By replacement rate: Replacement
rate ≥ 1 11,781 10.3 30.2 2,829 2,462 8,168 (.6) (1.3) (327) (219) (634)
Replacement rate of 2/3–1 50,217 8.6 33.6 6,529 4,480 13,318
(.3) (.7) (258) (195) (506) Replacement
rate ≤ 2/3 12,372 6.9 36.2 12,216 8,683 23,963 (.6) (1.6) (913) (836) (2,037)
By gender: Men 49,923 8.1 34.9 8,014 5,327 15,264
(.3) (.7) (332) (248) (632) Women 24,447 9.6 30.6 4,641 3,503 11,445
(.4) (1.0) (266) (231) (668) By cause of injury:
Causes other than slip/fall 54,375 6.5 33.1 6,336 4,461 13,470
(.3) (.8) (258) (222) (590) Cause p slip/fall 19,995 14.5 33.6 8,646 4,896 14,583
(.5) (.9) (568) (289) (775) By type of injury:
Type other than dislocation or unknown/not reported 60,960 6.9 30.9 6,402 4,295 13,903
(.3) (.7) (259) (212) (609) Type p dislocation 4,946 15.3 43.1 10,490 6,365 14,781
(1.1) (1.8) (958) (536) (1,084) Type p unknown/
not reported 8,464 17.3 35.2 8,954 4,811 13,657 (.8) (1.2) (828) (407) (1,050)
Note.—Standard errors in parentheses. Based on 10% sample of claims and 100% sample of denied claims. Denied and uncontested claims with positive payments have been reclassified as nondenied claims.
concentrated on the left side of the plot and the diamonds (cells with replacement ) to be concentrated on the right.rates ≥ 1
Our theoretical model implies that denials are more likely to be con- tested when there is a higher expected payment for denied and contested injuries and that claims are more likely to be denied when there is a bigger gap between the expected payments for accepted claims and denied claims. To test these basic insights, we ran two simple descriptive regressions across the 36 cells illustrated in figure 3. The first regression related the fraction of contested denials to the average total payment for denied and contested claims in the cell. The fitted relationship shows a marginally significant positive effect, with $10,000 in higher expected payments lead-
Liability in Workers’ Compensation 165
Fig. 3.—Denial and contest rates by claimant/injury characteristics. Each marker rep- resents a claimant cell characterized by replacement rate, gender, cause of injury, and type of injury.
ing to a 3% increase in the fraction of contested denials ( ). Thet p 2.0 second regression relates the fraction of denied claims in a cell to the mean expected payments for claims that are accepted or denied. The fitted relationship is
0.06(expected payments if accepted) Pr (d) p 0.09 �
10,000
0.10(expected payments if denied) � ,
10,000
with t-statistics for the coefficients of 1.3 and 1.2, respectively. The pattern of the coefficients is supportive of our basic model, although the effects are quite imprecise. Of course, these estimates make no allowance for unobserved heterogeneity between the claims that are denied or contested within each cell and provide only suggestive evidence in favor of the basic structure of our model.
V. A Structural Model of Denials and Disputes
Estimation Results
We fit a series of versions of our baseline and extended structural models with different numbers of points of support for the unobserved hetero- geneity distribution, including a benchmark version with no heterogeneity
166 Card/McCall
Table 3 Structural Estimates: 9-Mass-Point Specification, Denial and Contest Models
Probability of Denial (1)
Probability of Contesting Denial (2)
( , 5)1/j j p 4j 31.410 (4.692) 1.149 (.098) Log weekly wage .226 (.091) �.292 (.091) In assigned risk plan .270 (.059) . . . Self-insured .144 (.042) . . . State fund .166 (.087) . . . Age . . . �.013 (.006) Age2/100 . . . .011 (.007) Married . . . .081 (.015)
Note.—Standard errors in parentheses.
and specifications with 2–9 mass points.22 For the sake of brevity, we will focus our discussion on the estimation results of the baseline model. Estimation results for the extended model are presented in appendix B. We briefly note some of the major differences between the models in the footnotes.
Using the Akaike (1973) information criterion, we found that the 9- mass-point model provides the best fit to the data. Following up on the discussion in Section II, we also compared the predicted and actual dis- tributions of log indemnity payments implied by the various models at each outcome node. The results of this comparison are summarized in figures 4 and 5, which show the predicted and actual log cost distributions for injury claims that were accepted without denial and denied and con- tested, respectively, from a selection of alternative model specifications.23
As can be seen from these figures, at least 6 mass points are needed to reproduce the bimodal log cost distribution for accepted claims.
Parameter estimates from the 9-mass-point specification are presented in table 3, which reports the estimates for equations (8a) and (8b)—the models for the probability of denying a claim and contesting a denied claim, respectively. Table 4, columns 1 and 2, presents the estimates of equation (6), which specify the conditional claim amounts z1 and z2 for claims that are denied and contested and not denied, respectively. Finally,
22 As in tables 1 and 2, we combine the 10% random sample of back claims with the 100% sample of denial claims and use Manski and Lerman’s (1977) weighted maximum likelihood procedure to correct for choice-based sampling. The data did not support the addition of a tenth mass point; the location param- eters associated with the tenth mass point always “collapsed” into one of the other sets of location points, and the log likelihood function was never increased over the model with 9 mass points.
23 The actual distributions presented in figs. 4 and 5 were smoothed using kernel density estimation with a Gaussian kernel and a bandwidth of 0.30.
Liability in Workers’ Compensation 167
Table 4 Structural Estimates: 9-Mass-Point Specification, Denied/Uncontested Cases with Positive Payments Treated as Accepted Claims
Log Cost Equations Probability of Positive
Payment Equations
Accepted (1)
Denied and Contested
(2) Accepted
(3)
Denied and Contested
(4)
Age .032 (.006) .040 (.007) .022 (.010) �.050 (.013) Age2/100 �.033 (.008) �.042 (.008) �.024 (.013) .058 (.016) Female .043 (.027) �.070 (.028) .127 (.045) .138 (.058) Blue collar .136 (.024) .085 (.025) .084 (.042) .027 (.048) Log weekly wage .503 (.050) .165 (.052) �.212 (.089) .529 (.099) Dislocation .220 (.039) �.032 (.039) .213 (.080) �.055 (.070) Unknown type .138 (.044) �.042 (.048) �.025 (.078) �.104 (.089) Slip .078 (.028) �.095 (.031) .038 (.047) �.058 (.056) Slip and type unknown .246 (.060) �.066 (.060) �.018 (.115) .031 (.114) Replacement rate 1 1 .247 (.131) �.646 (.166) �.635 (.252) .792 (.317) Replacement rate p 1 �.121 (.084) �.388 (.092) �.299 (.156) .359 (.190) .67 ! replacement rate ! 1 �.059 (.058) �.215 (.063) �.123 (.109) .243 (.136) Replacement rate p .67 �.112 (.034) �.083 (.038) �.117 (.067) �.034 (.090) Log likelihood �91,880.0
Note.—Standard errors in parentheses.
table 4, columns 3 and 4, presents estimates for the probability of a positive payment at the two nodes (i.e., the models for p1 and p2).
The specifications reported in table 4 include a total of 13 covariates in the models for zj and pj: a linear and quadratic term in age, a gender dummy, a dummy for blue-collar occupation, the log of the preinjury weekly wage, controls for four different causes/types of injuries, and a set of four dummies indicating the WC benefit replacement rate.24 In contrast, we include only a few selected control variables in the contest and denial models. Covariates should enter these equations only to the extent that they affect the mean of the firm’s expected litigation costs, d, or the mean of the worker’s expected litigation costs, c. On the employee side, we assume that the expected cost of contesting a claim depends on the worker’s wage and on age and marital status. We assume that the firm’s litigation costs depend on the worker’s wage and on the firm’s insurance carrier, reflecting the fact that the denial process is often initiated or mon- itored by the carrier. Based on some experimentation, we decided to group all private carriers into one category and include dummies for the other three possible types of insurance arrangements: self-insured firms, firms
24 We include dummies differentiating between the two main causes of injury identified in table 2 and the three main types of injury, as well as an interaction for injuries of unknown type caused by a slip/fall, which have a particularly high denial rate.
168
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170 Card/McCall
covered by the competitive state fund, and firms covered by the assigned risk pool (the insurer of last resort for new and high-risk firms).
Estimates for the denial equation (col. 1 in table 3) yield a positive and statistically significant estimate for , the coefficient associated with1/j5 the expected cost differential between accepting and denying the claim,
. The magnitude implies that a 10% reductionE(y Fx, v ) � Pr (c)E(y Fx, v )2 2 1 1 in (the probability that a claimant will contest a denied claim) raisesPr (c) the probability of denial, on average, by 13.3% (from 8.3% to 9.4%).25
The corresponding estimate of in the contest equation (col. 2) is also1/j5 positive and statistically significant. The magnitude of the estimate implies that a $1,000 increase in the expected benefits of contesting a denial, on average, increases the probability of fighting back by 7% (from 33.0% to 35.3%).26 The other parameter estimates in the denial equation suggest that self-insured employers and those in the assigned risk plan are more likely to deny a claim than are those with private coverage (holding con- stant the expected payment differences between accepting and denying liability). The parameter estimates in the contest decision model suggest that there is an inverted U-shaped age profile in the willingness to contest a denied injury. Married workers are also more likely to contest a denial, while higher-paid workers are less likely (controlling for the expected benefits of contesting).
Based on the parameter estimates for the conditional payment amounts and the payment probabilities, the effects of age and blue-collar occu- pation are similar across all four equations, with an inverse-U-shaped pattern of age effects and a systematic positive effect for blue-collar oc- cupations. Injured women have higher probabilities of a positive payment whether their claims are accepted or denied and contested, although the gender effects on the size of payments, conditional on a positive payment, are insignificant (for accepted claims) or negative (for denied and contested claims). Higher-wage workers have more costly claims (conditional on a positive payment), as would be expected given the fact that compensation payments tend to be proportional to wages, although the estimates imply that a higher-wage worker is less likely to get a positive payment from an accepted claim and more likely to get a positive payment from a denied and contested claim.
The coefficients associated with different ranges of the wage-replace- ment rate show an interesting pattern. For denied and contested claims, higher replacement rates are associated with lower payments (conditional on a positive payment) but higher probabilities of a positive payment.
25 For the extended model, the estimate of is also positive and significant1/j5 and implies that a 10% reduction in raises the probability of denial by 1%.Pr (c)
26 For the extended model, a $1,000 increase in the expected benefits of con- testing a denial, on average, increases the probability of fighting back by 11%.
Liability in Workers’ Compensation 171
Simulations of the two equations suggest the net impact on expected payments for a denied and contested claim is relatively small. Since the incentive to contest a denied claim in our model depends on the expected payment if a claim is contested, the structural estimates suggest that the net effect of replacement rates on the probability of contesting a denied claim is small. For accepted claims, the estimated replacement rate effects on payment amounts (conditional on a positive payment) are relatively small and unsystematic, whereas higher replacement rates have a negative effect on the probability of a positive payment.
To judge the net effect of replacement rates on dispute outcomes, we conducted a simple simulation, taking every claimant with a back injury and setting his or her replacement rate to either two-thirds or one. We then computed the probabilities that the injured worker’s claim would be denied and, if denied, contested. The results, shown in the following table, suggest that higher replacement rates are indeed associated with a higher probability of claim denial but with relatively little change in the probability of contesting a denied claim.
Replacement Rate p 2/3 Replacement Rate p 1
Pr (d ) .083 .098 .328 .295
Note that the implied effects of the replacement rate on denial and contest behavior work through the implicit cost calculations of the parties: our denial and contest models exclude any direct effect of the replacement rate.27
Evaluating the Model
To test the adequacy of our structural model, we conducted a series of specification tests. As mentioned above, several variables (a total of nine covariates, including age, gender, blue-collar status, the replacement rate dummies, and the cause and type of injury variables) only affect denial probabilities through their effects on the expected net benefit of denying a claim. An LM test for the exclusion of these variables has a value of 13.88, which does not reject at conventional significance levels. To gain further insights, we computed actual and predicted denial frequencies within replacement rate # gender # injury-cause # injury-type cells, using the same classifications as in table 2 (three replacement rate ranges, two genders, two injury causes, and three injury types). The predicted and actual numbers of denials in each of the resulting 36 cells are shown in table 5. At this level of aggregation, the model does a relatively good job of predicting the distribution of denials across cells: a simple x2 statistic
27 Results for simulations from the extended model are qualitatively similar.
172 Card/McCall
Table 5 Actual and Predicted Denial Frequencies: 9-Mass-Point Estimates
Replace Rate (RR) Gender Cause Type Actual Predicted
RR ≥ 1 Male Nonslip Other 211 196.71 Dislocation 22 23.91 Unknown 47 43.30
Slip Other 111 103.39 Dislocation 13 9.26 Unknown 81 72.30
Female Nonslip Other 321 380.27 Dislocation 37 31.33 Unknown 42 47.08
Slip Other 172 142.57 Dislocation 25 21.77 Unknown 116 104.69
2/3 ≤ RR ≤ 1 Male Nonslip Other 1,108 1,131.25 Dislocation 181 173.79 Unknown 190 176.45
Slip Other 702 681.35 Dislocation 170 174.89 Unknown 412 400.16
Female Nonslip Other 642 700.05 Dislocation 71 78.83 Unknown 129 114.19
Slip Other 353 314.43 Dislocation 66 65.70 Unknown 248 233.82
RR ! 2/3 Male Nonslip Other 277 290.35 Dislocation 76 70.04 Unknown 39 41.25
Slip Other 175 181.34 Dislocation 65 63.51 Unknown 117 107.53
Female Nonslip Other 39 40.36 Dislocation 7 5.71 Unknown 8 14.78
Slip Other 17 17.41 Dislocation 8 5.73 Unknown 15 17.64
Note.—Estimates based on model in tables 3 and 4.
for the goodness of fit is 46.0, which is not statistically significant at the 5% significance level ( ).28p p .101
As a second test of adequacy, we reestimated the structural model, al- lowing separate coefficients in the denial model for expected payments if the claim is accepted ( ) or denied ( ). The estimatedE(y Fx, v ) Pr (c)E(y Fx, v )2 2 1 1 coefficients are opposite in sign, as suggested by our model, but statistically different in magnitude, leading to a x2 statistic of 21.6 ( ) for thep ! .01 restriction implied by the model. A similar exercise for our extended model tests the set of three linear restrictions implied by the structure of equations (1′)–(3′). Again, the qualitative pattern of the unrestricted estimates is con-
28 For the extended model, the corresponding x2 statistic is 26.3, with p p ..855
Liability in Workers’ Compensation 173
sistent with the model, but the restrictions are rejected ( ,2x p 72.15 p ! ). Overall, we conclude that both variants of the structural model pro-.001
vide reasonably good qualitative descriptions of the data but impose overly strong restrictions to pass conventional overidentification style tests.
Quantifying the Strategic Incentive for Injury Denials
As noted in Section II, a key feature of our structural model is that we can use it to estimate the fraction of WC liability disputes that are at- tributable to the strategic incentive for employers (or insurers) to deny liability for high-cost claims. Specifically, we simulated the predicted probability of denied claims under the assumption that all denied workers file a CP. The average predicted probability of denial drops by 28%, from 8.4% to 6.0%.29 From this perspective, a substantial fraction of denials appears to be due to the strategic incentive of employers.
Finally, we used the parameter estimates from our model to evaluate the effect of imposing a penalty (or “tax”) on employers in the event that a denied claim was contested and subsequently found valid, where we define the latter outcome as cases in which a denied and contested claim resulted in a positive payout. Specifically, we assumed that if the claimant wins the claim dispute, then the employer must pay the claimant an additional $1,000. Assuming that this change has no effect on the other parameters of the model or on the distribution of injury claims filed by workers, our structural estimates imply that the imposition of such a penalty would cause the denial rate for back injury claims to decline from 8.3% to 6.4% and would also lead to a decline in the probability of a claimant contesting a denied claim, from 32.4% to 28.3%. The average costs of accepted claims would rise from $6,497 to $6,512, while the average costs of denied and contested claims would fall from $13,912 to $10,957 (excluding penalty costs).30 Thus, a tax on denials that are subsequently found to be valid would lower the rate of litigation over liability and cause a compositional shift that reduces the number of higher-cost injuries that are litigated.
VI. Conclusion
This article presents a model of liability disputes in the WC system. A simple sequential asymmetric information model is developed and struc- turally estimated using data from Minnesota. We find that employers are more likely to deny liability for injury claims when their expected gains from doing so are larger. In particular, they are more likely to deny liability when faced by an injury claimant who is less likely to fight back if denied.
29 For the extended model, the average predicted probability of denial drops by 37%, from 10.3% to 6.6%.
30 Qualitatively similar conclusions emerge from simulating the effect of a tax in the extended model.
174 Card/McCall
Claimants, however, were found to be more likely to contest denied claims when their expected returns from fighting back were larger.
For back claims, the structural model fit the data reasonably well and provides a relatively successful explanation for the distribution of liability disputes across different injury classes. The model was tailored explicitly for the dispute system in Minnesota. Since WC laws and dispute pro- cedures differ across states, it may not be directly applicable to other states. Moreover, because of data limitations, we only modeled the first two steps of the dispute process and made no attempt to disentangle the various factors governing subsequent stages of litigation. Nevertheless, the model offers an interesting perspective on the relatively high rate of liability disputes over back claims in our sample. In particular, the esti- mates suggest that a relatively high fraction of denials arise because of the strategic incentive that insurers have to deny liability when they know that a sizable fraction of injured workers will simply walk away from a dispute, rather than pursue their claim through the litigation process. In this setting, a policy such as a tax on denied claims could have a substantial effect on reducing the number of liability disputes in the WC system.
Appendix A
Likelihood for the Extended Model
In the extended model, an injured worker with expected litigation costs c will contest a denied claim if . Given the observed andE(y ) � E(y ) 1 c1 3 unobserved case characteristics (x, v1, v2, v3), this occurs with
E(y ) � E(y ) � xb1 3 4 Pr (c) p F . (9 ′ )( )j4
A firm with expected litigation costs d will deny a claim d ! E(y ) �2 , which occurs withPr (c)E(y ) � [1 � Pr (c)]E(y )1 3
E(y ) � Pr (c)E(y ) � [1 � Pr (c)]E(y ) � xb2 1 3 5 Pr (d) p F . (10 ′ )( )j5
The likelihood function for the observed data, conditional on x and (v1, v2, v3), consists of six parts:
Denial/CP Status/Payment Likelihood
Not denied, no payment [1 � Pr (d)](1 � p )2 Not denied, payment y [1 � Pr (d)]p F (y)2 2 Denied, no CP, no payment Pr (d)[1 � Pr (c)](1 � p )3 Denied, no CP, payment y Pr (d)[1 � Pr (c)]p F (y)3 3 Denied, CP, no payment Pr (d) Pr (c)(1 � p )1 Denied, CP, payment y Pr (d) Pr (c)p F (y)1 1
Liability in Workers’ Compensation 175
Specification Testing of Extended Model
Note that and enter equation (9′) with coefficients ofE(y ) E(y ) 1/j1 3 4 and , respectively, while , , and enter�1/j E(y ) Pr (c)E(y ) [1 � Pr (c)]E(y )4 2 1 3 equation (10′) with coefficients of , , and , respectively.1/j �1/j �1/j5 5 5 These constitute a set of three linear restrictions that can be tested by estimating the model without imposing the restrictions and comparing the fit to the restricted specification.
Testing Exclusion Restrictions in the Denial Equation
Equation (10) specifies that the probability of denial conditional on the observed covariates X and the value of the random effect v can be written as , where F is the normal cumulative probability func-Pr (d) p F[F(X, v)] tion, and
E(y FX, v ) � Pr (c)E(y Fx, v ) � xb2 2 1 1 5 F(X, v) p
j5
is a function that depends on X, v, and the parameters of the model. Suppose that the rows of b5 associated with the subset of covariates X2 are assumed to be zero (i.e., these covariates are excluded from the cost of denial equation). Consider the LM test associated with this restriction. It can be shown that this test statistic equals the product of the sample size (N) times the R2 from a regression of a vector of ones on the vector of variables defined for each individual ( ) asi p 1, … , N
X # j # q p # r (X , v ),2 j i i ji
where ri(Xi, vj) is the “generalized residual” from the denial model for observation i, evaluated with the random effect set to vj, and qjP is the posterior probability that the random effect takes on its jth value, given the likelihood for the observed data. The generalized residual is
r (X , v ) p {1(d p 1) � F[F(X, v)]}J[F(X, v)]i i j
and
F[F(X, v)]{1 � F[F(X, v)]},
where J[ ] is the normal density, and the parameters are evaluated at their maximum likelihood values.
176 Card/McCall
Appendix B
Table B1 Extended Structural Model Estimate: 9-Mass-Point Specification, Denial and Contest Models
Probability of Denial Probability of Contesting Denial
( , 5)1/j j p 4j 7.389 (.759) 1.097 (.075) Log weekly wage 1.353 (.177) �.233 (.047) In assigned risk plan .681 (.102) . . . Self-insured .325 (.083) . . . State fund .171 (.152) . . . Age . . . .017 (.015) Age2/100 . . . �.030 (.018) Married . . . .087 (.040)
Note.—Standard errors in parentheses.
Table B2 Extended Structural Model Estimate: 9-Mass-Point Specification; Accepted, Denied/Contested, and Denied/Uncontested Cases
Log Cost Equations Probability of Positive Cost
Equations
Accepted Denied and Contested
Denied and Not
Contested Accepted Denied and Contested
Denied and Not
Contested
Age .017 .056 �.040 .020 �.021 �.002 (.007) (.011) (.012) (.011) (.017) (.008)
Age2/100 �.014 �.061 .048 �.022 .020 3.002 (.009) (.013) (.015) (.014) (.021) (.011)
Female .097 �.106 .024 .161 .023 �.040 (.032) (.040) (.052) (.048) (.067) (.036)
Blue collar .156 .085 �.067 .121 .063 3.061 (.030) (.036) (.049) (.045) (.063) (.034)
Log weekly wage .654 .157 .522 �.320 .019 �.011 (.023) (.034) (.039) (.095) (.116) (.048)
Dislocation .102 .289 �.564 .163 .110 �.042 (.063) (.049) (.072) (.086) (.081) (.052)
Unknown type .132 .146 �.176 �.039 .023 �.229 (.054) (.063) (.091) (.082) (.111) (.052)
Slip �.020 .075 �.180 �.014 .063 �.223 (.033) (.038) (.055) (.049) (.067) (.037)
Slip and type unknown .203 .017 �.116 �.069 �.093 3.177 (.081) (.082) (.114) (.122) (.138) (.080)
Replication rate 1 1 .470 �.374 �.060 �.921 .011 �.175 (.095) (.138) (.170) (.270) (.343) (.137)
Replication rate p 1 �.064 �.082 .049 �.494 �.028 �.146 (.050) (.074) (.089) (.168) (.201) (.083)
.67 ! replication rate ! 1 �.034 �.018 .081 �.245 .012 �.120 (.042) (.074) (.075) (.117) (.149) (.066)
Replication rate p .67 �.047 �.024 .074 �.145 �.080 �.075 (.033) (.044) (.060) (.071) (.101) (.049)
Log likelihood �93,434.3
Note.—Standard errors in parentheses.
Liability in Workers’ Compensation 177
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