Moral Hazard in Monday Claim Filing: Evidence from Spanish Sick Leave Insurance

4.1 Stage 1: Definition of Main Aggregates

In order to link this theoretical section with the empirical work carried out later, first we define our variable of interest, which is the ratio of hard-to-diagnose injuries (HD) to total injuries, i.e. HD plus easy-to-diagnose injuries (ED), and we call it AR:

where the subscript j refers to Monday (M) or to the rest of the days (R). ^[10] At this point it is necessary to clarify that HD injuries have a triple nature and, for this reason, it is important to break them down into their three components. In formal terms: HDj=HDj1+HDj2+HDj3.

The first component, HDj1, captures those hard-to-diagnose injuries that a worker would suffer as a consequence of the performance of the normal routine in the workplace. Thus, there is no reason to expect any difference between Mondays and the rest of the days in this component and due to this HDM1=HDR1=HD‾1. The second element, HDj2, encompasses precisely the excess of reporting of HD injuries on Mondays compared to the rest of the days and as a consequence of the inactivity period before starting to work. As it has been stated in the literature, since that period is much longer on Mondays (because of the weekend) than on the rest of the days, the proneness of experiencing an injury of this type is higher on Monday than from Tuesday through Friday. This is why we assume HDM2>0 but HDR2=0. With this formulation we try to incorporate to the model the idea of the “physiological Monday effect”. ^[11]

However, the most important factor for the purposes of this article is the third one: HDj3. This term captures the number of HD injuries reported for “economic reasons”. At the beginning of every working day, there are a number of potential cheating situations from an aggregate point of view, as a result of injuries out of the workplace. We denote them by the Greek letter capital eta, Hj. As the amount of outside-of-work time is much higher before the Monday workday than on the Tuesday through Friday workdays, we assume that HM=σHR>HR, being σ>1 an increasing factor responsible for the “economic Monday effect” following, for instance, the Smith’s (1990) way of reasoning.

Nonetheless, it is worth mentioning that not all of those potential cheating situations will become false on-the-job accidents. Only a proportion of them π will be reported. This proportion π will play a key role in our strategy to disentangle the “economic Monday effect” from the other components of the Monday gap. We consider it to be duration dependant, i.e. πd. Where d stands for the duration of the injury. In the following subsections we devote some additional effort to model such a proportion and to determine its specific functional form. The basic notion we need to have in mind for this introductory theoretical subsection is that we can write HDj3=πdHj.

As regards the easy-to-diagnose injuries, we treat them the same way as HDj1, i.e. we consider them to be constant and evenly distributed throughout the working week. Putting it in mathematical terms: EDM=EDR=ED‾.

From the previous discussion it is clear that ARj is a function of duration because of πd. What is more, for a given duration, we might think that the ratio ARj would depend on the distribution of the different types of injuries. Nevertheless, we discard this consideration as important for the modelling process under a realistic assumption. As it will be useful for the purposes of this article later, we formalize this idea.

Let ζd be the likelihood that an injury has a given duration d. Furthermore, and for the sake of simplicity, let us assume that such likelihood is distributed similarly among easy-to-diagnose and hard-to-diagnose injuries, which seems to be a realistic assumption according to Figure 5 in Appendix A. In that case, we can write ARjd as:

From eq. [2], it is evident that if we define the Monday gap (M) as the difference between the ratio of HD to total injuries on Mondays and on the rest of days, such a gap is also a function of duration:

In eq. [2] there are some components that may be obtained easily from the data, e.g. ED‾. It is also feasible to measure directly the total amount of HD injuries, both HDM and HDR. By means of econometric techniques that will be explained in a later section, we are able to estimate the total Monday gap defined in eq. [3] for every duration as well.

However, the aim of this article is not only to measure Md but also to decompose it into its different components. If we knew HD‾1, HDj2, Hj and πd, the task would be trivial. Unfortunately those components are unknown, at least from a strictly empirical point of view. In the rest of this theoretical section what we do is to develop a strategy to achieve the above-mentioned goal. First we “rationalize” πd in the two following subsections with the help of a rather standard microeconomic model. Then, we break down HD injuries according to their triple nature by means of some theoretical assumptions and observed facts. Finally we decompose the Monday gap into three components, which we call economic Monday effect (ME), physiological Monday effect (MP) and statistical Monday effect (MS).

4.2 Stage 2: Individual Decision Process

After having described the main features of the sick leave insurance system in Spain, it is easy to understand, from a theoretical point of view, why problems of moral hazard (mentioned in the introductory section) may arise. The differences in percentages shown in Table 1 may cause a potential situation of substitution moral hazard (in terms of Fortin and Lanoie’s classification). There is an incentive for the worker to substitute sick leave insurance by workers’ compensation insurance due to the differences in replacement rates. In other words, some individuals might try to conceal out-of-work accidents and report them as work related.

To better understand this idea, let us construct a simple microeconomic model which captures two basic features ^[12]: first, in the Spanish case WC is more generous than sick leave compensation (SC), but the difference between both coverage rates depends on the length of the recovery spell. For this reason a worker could try to substitute SC by the WC. But by doing this, and in the second place, the worker has to face reputational risk. This would mean that if the worker were caught performing this opportunistic behaviour, he/she would be penalized with a lower labour income in the future. ^[13] This penalty will be imposed by the employer ^[14] because there is a cost for the firm of reporting high rates of accidents and injuries. ^[15]

To rationalize the empirical findings obtained in this paper, we are going to use a utilitarian economic framework. Let us assume that individuals maximize their expected discounted utility. Their planning horizon consists of two periods. ^[16] Period 1 comprises the present while period 2 refers to the future. Period 1 begins after the weekend in which an individual has suffered an accident out of the workplace. In that moment, the worker has to take the decision of reporting the accident as non-work related or, alternatively, of concealing the injury when arriving at the workplace on Monday and trying to notify the physical damage as work related. In the first case, the worker obtains SC and in the second he/she receives WC. Period 1 has a definite duration T and it is equal to the maximum work-related sick leave spell allowed by Spanish law. As a worker can be off work on sick leave for a maximum period of 18 months in Spain, period 1 would last a year and a half. On the other hand, period 2 would last since period 1 has finished until the worker reaches the retirement age. This means that period 2 has a variable size, depending on the workers’ age.

With these theoretical assumptions, we can formalize the workers’ decision as follows: if the worker chooses not to report a false work-related accident he or she will receive SC benefits in period 1 and a labour income in period 2, as it is established in expression [4]:

where U⋅ is the utility function (with U0=0, U′⋅>0 and U′(·)>0, for the sake of simplicity), δ is the discount factor,T is the number of days of which period 1 is composed, and de is the expected number of days in which the worker is off the job. We consider de as being an expectation because the decision has to be taken at the beginning of period 1. In any case, we will assume perfect foresight (i.e. de=d) in the empirical section, so in the rest of the paper we will use de and d interchangeably.

Moreover, w1 is the daily wage, which coincides with the daily contributory basis to assess the amount of both WC and SC in period 1, and w2 is the labour income for period 2. So as to better understand the concepts of w1 and w2, their order of magnitude and, in the end, the ultimate meaning of the periods 1 and 2 within our model, we construct Table 2. What Table 2 illustrates is that the amount of income related to period 2 is huge, so every little thing that could affect that period has a potentially very important effect within the model. This will be even clearer later.

The parameter α refers to the percentage of the contributory basis received as SC, and evidently it is comprised between 0 and 1. Furthermore, it should be noted that α is a function of de. More precisely:

On the other hand, if the worker chooses to report an out-of-the-work accident as work related, he/she will receive WC benefits in period 1 but in period 2 the worker will face a reputational risk. The employer may detect the opportunistic behaviour of the worker and in such a case the employer can take measures that lower the worker’s labour income in retaliation. With probability p, the employee escapes from the monitoring of the employer and faces no penalty, and with probability 1−p he/she is caught and, as a consequence, earns less in period 2. The discount expected utility for an opportunistic worker is:

In expression [5] β is the replacement rate of WC and λ is a parameter (0<λ<1) which reflects the fact that if the worker is caught shirking he/she will receive less income in period 2 than if he/she behaves non-opportunistically. ^[17]

It has to be noted that p is assumed to be a decreasing function of de (pde with ∂p/∂de<0). The reason for that is not difficult to understand: if the worker suffers a minor injury, it will be easier for him/her to conceal it at the workplace than if he/she suffers a serious one. Minor injuries are related to short out-of-work spells and serious injuries are associated with long absences. Thus, the shorter the expected recovery period is the higher the probability of not being caught. In more formal terms, we make use in our model of the following probability function:

The function in expression [6] has three important properties that must be discussed in some detail. The first one is that limde→1pde=1. This assumption is crucial to obtain some theoretical results in our model, but, at the same time, we strongly believe that it reflects the Spanish labour market in a realistic way. ^[18] In the second place, we assume a linear probability function (i.e. ∂2p/∂de2=0) for the sake of simplicity. Finally, and due to the linear form of pde, there will be an expected duration, which we call n, that makes zero the likelihood of concealing the injury. In other words, for de≥n, it is impossible to perform an opportunistic behaviour without being detected. Again, we deem that this is a rather realistic assumption. ^[19]

From the rationale described above we can conclude that if VA is larger than VB then the worker will choose to behave correctly and otherwise to behave opportunistically. To put it another way, if we define the indicator Zde=VA−VB, then the individual will perform an opportunistic behaviour when Z<0 and will behave as a non-shirking worker when Z>0. Consequently, Zdemay be written as follows:

4.3 Stage 3: Two Theoretical Effects at the Individual Level

The form and position of the Z function depend on the individual’s preferences and on the above-mentioned parameters. To begin with, due to the supposition that limde→1pde=1, we obtain that Z is negative for the lower values of de20. For the purposes of this paper, we are also interested in analysing how Z varies when de varies. It is worth noting that Z is a piecewise function of de. The first section would be defined for those durations between 1 and 3 days, the second one for durations between 4 and 20 and the last one for sick leaves over 20 days. When the section changes the slope of Z will change.

It is important to bear in mind that we are interested in isolating two theoretical effects: one related to period 1 and another related to period 2. So as to identify both of them clearly, let’s split Z into its two components Z1 and Z2.

Making use of the linearity of the utility function, we have:

where UC (the marginal utility) is a constant as a consequence of the assumption that U′(·)>0.

On the other hand, for period 2 we have:

From expression [8.2], it is easy to calculate the slope of Z1de in the space Z1, de:

Regarding the slope of Z2, it can be expressed in formal terms by means of the following expression:

As it is clear from such a Figure 1, the theoretical effect related to period 1 (and captured by Z1) would make the individual behave opportunistically as a consequence of the greater generosity of WC payments as compared to SC benefits. The theoretical effect associated with period 2 is a reputational effect that makes the individual behaves non-opportunistically. Putting together Z1 and Z2 we obtain Z, which is negative for low values of de. Nonetheless, if the reputational effect of period 2 is strong enough, Z will eventually become positive. This is what we represent in Figure 1 from dRe onwards, where dRe stands for “reservation duration” and it is a threshold from which the individual stops behaving opportunistically and begins to behave non-opportunistically. This is due to that Z1 is flat from day 20 onwards and we assume that the reputational effect would finally offset the economic incentives to behave opportunistically.

Figure 1:

Z, Z₁ and Z₂ as functions of the expected duration of the sick leave.

Why do we suppose that the reputational effect is strong enough to eventually offset period 1 effect? Based on Table 2 figures, we construct Table 3, in which we compare the potential monetary costs and benefits of cheating. Those figures (and their order of magnitude) make us think that assuming that eventually Z becomes positive is not unrealistic. The monetary amounts for period 2 are so huge that it is difficult to imagine a reasonable parameterization that makes Z not to be positive sooner or later. ^[20] In addition, this implies that there exists a dRe for every individual and the key point is to analyse how they are distributed from a statistical point of view.

4.4 Stage 4: The Aggregation Process

It is worth mentioning that the graphical representation showed in Figure 1 is only a particular case. It would be easy to prove that, depending on worker’s preferences and characteristics, the critical value dRe could be below 20 days. What is more, for those extremely job-committed workers the value of dRe could perfectly be lower than 3 days. To be more accurate, the reservation duration would be a function of individual’s preferences and characteristics for each worker. In more formal terms ^[21]:

Expression [12] reveals that, in order to reach valuable conclusions at the aggregate level, it is necessary to make some assumptions regarding the distribution of dRe23. Consistent with our microeconomic model, as d increases, the number of individuals carrying out an opportunistic behaviour reduces. This means that if the proportion of “well-behaved” workers follows a cumulative distribution function (CDF) ΦdRe, the proportion of “bad-behaved” workers is distributed according to 1−ΦdRe. At this point, it should be already clear that πd=1−ΦdRe. In principle, and from a theoretical standpoint, it is not critical to impose a specific CDF for ΦdRe, and consequently for πd, as long as the right tail of the distribution is decreasing and asymptotic to zero.

4.5 Stage 5: The Monday Gap

Once we have described the individual’s behaviour and the aggregation process, let us come back to expression [2]. If we now use small letters to denote a specific realization for a determined duration (e.g. hd‾1d=ζdHD‾1 or ηjd=ζdHj), we may rewrite the expression [2] in the following way:

With this compact notation we may define the statistical Monday effect MSd as:

As it is evident from expression [14], the statistical Monday effect is a consequence of the fact that in the denominator of the Monday ratios there are more injuries. Although it has no clear meaningful economic interpretation, if we did not consider this effect, we would be underestimating the other two effects. This is so because the excess of reporting is calculated as a ratio and on Mondays there are more injuries, independently of the underlying cause. In the top left-hand panel of Figure 2, we depict MS as a function of d. If letter s represents short durations (i.e. πs≈1) MSs is negative for two reasons: (1) because hdM2s>0 and (2) σ>1. As d increases πd becomes smaller and MSd increases. However, even for very long durations expression [14] is still negative. Putting it another way, if letter l represents long durations (i.e. πl≈0), MSl is negative because of the term hdM2l>0 in the denominator of the minuend in the right-hand side of the expression [14]. Furthermore, expression [14] has an asymptote due to the fact that is the difference between two asymptotic terms, as a consequence of πd=1−ΦdRe.

On the other hand, the formal expression that allows us to measure physiological Monday effect, that is eq. [15], is simpler because of our assumption of hdR2d=0:

The graphical representation of MPd is shown in the top right-hand panel of Figure 2. From eq. [15], it is clear that it is an increasing function of d with an asymptote at hdM2l/hd‾1l+hdM2l+ed‾l>0 for long durations.

Finally, following the same line of reasoning and consistent with our modelling layout, the corresponding expression for the economic Monday effect might be summarized through eq. [16]:

The behaviour of MEd, which is represented in the bottom left-hand corner of Figure 2, is more complex. First of all, it would be positive if and only if the excess of reporting for strictly economic reasons is strong enough to compensate the higher number of Monday injuries due to strictly physiological motives. In other words, expression [16] would be positive whether σ>1 is big enough to offset hdM2d>0 in the denominator of the first right-hand side term in eq. [16] above. ^[22] Anyhow, this is not the main concern because it is relatively easy to check out this point by inspecting our database, and as it will be shown later, this is the case. The main difficulty would emerge because πd appears both in the numerator and in the denominator of both the minuend and the subtrahend of the right-hand side of expression [16]. However, it is relatively easy to verify that eq. [16] has an asymptote at zero since both terms of the right-hand side tend to zero as duration increases. This is so because the numerators in both the first and the second terms of the right hand-side of eq. [16] tend to zero as duration expands, whereas the denominators tend to a positive value.

As a result of the previous discussion, and with eqs [14]–[16] in mind, it is now convenient to rewrite expression [3] in terms of MSd, MPd and MEd:

From eq. [17] it follows that the total Monday gap is the aggregation of its three components. That is what we represent in the bottom right-hand panel of Figure 2. It is worth noting that it does not have to be a decreasing and convex function of duration unambiguously, as it is shown in Figure 2. It could be an increasing and concave function or even show and mixed concave–convex profile. It all depends on the magnitude of MSd, MPd and MEd. However, Md has unambiguously an asymptote, which is the result of adding those of the statistical Monday effect and the physiological Monday effect. The reason for having depicted Md as a decreasing and convex function of duration is because this profile fits what we find in our data, as it will be shown later.

Figure 2:

Monday effect as a function of the expected duration of the sick leave.

Interestingly, our model is able to provide values analytically for the statistical Monday effect and the physiological Monday effect for long durations (i.e. for MSl and MPl). They can be obtained by making use of the assumptions of the model and taking into account that we can easily measure (in our database) ed‾d at any duration and the aggregate MSl+MPl for long durations, ^[23] due to the fact that MEl=0. Taking advantage of this information, then we can decompose, also analytically, the total Monday gap into its three components MSs, MPs and MEs for short durations by just doing an additional assumption about the size of the parameter σ. On the other hand, we can also assess the total Monday gap for every duration by means of econometric techniques that will be explained in a following section. Finally, putting all this together, and by means of a calibration procedure, with an appropriate CDF for πd, it is possible to break down the total Monday gap into its three components for every duration in line with expression [17], which in the end is the final objective of this research.

7.1 Monday Gap Estimation

Table 5 shows the results of the probit estimation of eq. [18]. In such a table, the coefficients and the Z-statistics associated with the Monday covariate are included. The Z-statistics were calculated from the standard errors of the Observed Information Matrix (robust and cluster methods have been used to obtain the standard errors, with the results proving quite similar). Those coefficients are obtained for six separate regressions ^[28] and for five econometric specifications. Model I only includes the Monday dummy as a covariate, taking value 1 if the accident occurs on Monday and 0 otherwise. Model II adds age, job tenure, sex, region and the time of the accident as control variables. Model III also includes which sector of industry the firm is involved in. Model IV takes into account covariates that describe the worker occupation. Model V includes sick leave duration as a proxy of expected duration. Finally, model VI uses model specification V, but takes into account potential endogeneity problems caused by including sick leave duration as an explanatory variable. According to the Wald and Hausman tests, there is an endogeneity problem in the model when the duration variable is included. To correct this problem, a two-step estimation is carried out where duration is instrumented by dummy variables related to the severity of the injury (two dummy variables that indicate whether the accident was serious or very serious, and another variable that takes 1 if primary care is provided by a hospital). ^[29] Finally, the Amemiya–Lee–Newey test shows the absence of over-identification in the instrumented model. ^[30]

With regard to the endogeneity of the duration variable, an additional consideration needs to be taken into account. Although hard-to-diagnose injuries might always appear to be only slight and of a short duration, this is not the case for Spain. The mean duration of injuries of this nature is close to that of all sick leave periods (and indeed above it in the case of accidents entailing less than 60 days off). ^[31] Moreover, if we analyse the distribution of accidents in terms of their duration, these injuries can be seen to under-represented in the shortest durations. ^[32]

From the high significance of the coefficients related to the Monday covariate appearing in Table 5, we can conclude that there is an important Monday effect. However, it is necessary to clarify that a positive sign on such a variable can only be found for lower back injuries as well as strains and sprains. In other words, these two types of injuries are overrepresented on the first day of the week. Researchers agree that it is in this kind of injuries where the problems of moral hazard are more acute. A good example of this research is the work of Campolieti (2006) which finds ex ante causality moral hazard in reports of back pain.

More specifically, the likelihood of suffering a lower back injury on Monday is 3 percentage points higher than during the rest of the week. For the case of strains and sprains, the gap is about 1.5 percentage points higher. Another aspect that might be stressed is that there are no sizeable differences in the estimated coefficients for different specifications. The only significant effects are those of the covariates included to control for observable worker characteristics on lower back injuries and strains and sprains, on the one hand, and the effect of sick leave duration on fractures, on the other hand. ^[33]

As previously discussed, model VI seeks to correct the endogeneity problem of model V. Results failed to reveal any major differences. Only a slight reduction in the Monday variable coefficient was observed when analysing sprains and strains.

7.2 Sensitivity and Robustness Analysis

The above results show differences in the composition of the injuries reported on Monday. However, prior to decomposing the Monday gap, it seems appropriate to perform a sensitivity analysis to check the stability of the results and the conclusions. To this end, different strategies may be adopted. First, we estimate an additional model which complements those already shown in Table 5. In this case, the duration variable in model V is replaced by a group of dummy variables that indicate changes in the replacement rates associated with non-work-related injuries (from 1 to 3 days, from 4 to 20 days, and 21 or more). In light of the results presented in Table 6, it is found that the failure to include a set of dummy variables related to changes in replacement rates does not alter the results. Coefficients maintain the magnitude, sign and the signification already observed in Table 5 (Model V).

Second, using an indicator of whether the worker reports a hard-to-diagnose injury as a dependent variable, model V in Table 5 is re-estimated for different worker characteristics. We can thus measure the sensitivity of the Monday effect to changes in occupation, age, or type of contract. Table 7 presents estimates where the Monday variable includes multiplying dummies that reflect different worker characteristics. Results show a greater effect among workers in occupations requiring greater skill. This is because they are more difficult to replace and therefore less exposed to a reputation effect. Another important conclusion is that the Monday effect is greater in older workers. This suggests a lower reputation effect (in terms of lost income) when retirement is closer. Finally, the higher coefficient observed for workers with open-ended contracts indicates greater protection against employer retaliation resulting from the cost of dismissal.

Table 8 summarizes the last two steps of the analysis. First, we estimate model VI using two different groups of instruments to correct for endogeneity of the duration variable, how the accident came about and the severity. The second is a placebo test where Monday accidents are removed and Tuesday becomes the first day of the week.

The first part examines whether model VI is robust to changes in the instruments used to correct endogeneity. Results are seen to be very similar regardless of whether the instrument used is severity or how the accident occurred. We can therefore say that the instrument used is correct. The placebo test checks the uniqueness of the first day of the week. In this case, it is estimated whether Tuesday has a significant effect on the likelihood of a reported injury being categorized as hard-to-diagnose. Results show that, in this case, the coefficient associated with the first day of the week is not significant and thus has no differential effect on the dependent variable.

7.3 Monday Gap Decomposition

Having seen that there are significant differences in the types of injuries reported on Mondays, we move on to the next stage of the analysis. Here we group all the hard-to-diagnose injuries (that is, strains, sprains and lower back pains) into one category and the remaining injuries into another. We then carry out two different econometric regressions, one for accidents occurring on Mondays and another for accidents occurring from Tuesday to Friday. In both cases the chosen specifications correspond to model VI, which takes into account the possible endogeneity problem, and is therefore performed using variables related to accident severity. The above-mentioned econometric regressions are also the basis for elaborating the counterfactual distribution used in the last part of our empirical analysis.

Table 9 reports the results of the estimates of the likelihood that a reported injury was categorized as “hard-to-diagnose” depending on the day the accident was reported (Monday or the rest of the week). The main conclusions to emerge from Table 9 are that the probability of reporting a hard-to-diagnose injury increases with age, with job tenure, with the amount of benefits, during the first working hours and in the morning shifts. Nonetheless, the probability is lower for males and for the longest sick leaves. We can also detect in Table 9 some differences in the magnitudes of the coefficients for Monday accidents and for Tuesday to Friday accidents which are responsible for the observed gap. Thus, an increase in sick leave duration reduces further the probability of reporting a soft-tissue injury if the accident occurs on Monday.

Nevertheless, we must not forget that one of the main aims of this paper is to identify traces of moral hazard in the Spanish workplace accident insurance system. With this objective in mind, we study the different effect of similar characteristics depending on which day of the week the accident takes place. At the same time, we analyse how such differences evolve while sick leave duration increases. With the estimated coefficients for Monday and for Tuesday through Friday accidents and the characteristics referring to the latter, we build the counterfactual distribution and the unjustified component described in expression [20]. By assessing this unjustified component for a number of sick leave spells we depict what we label the “actual Monday gap” in Figure 3. In this figure, we also represent the best econometric fit according to expression [21] and name it “adjusted Monday gap”. The length of the sick leave (expressed in days) is plotted in abscissas, ^[34] whereas the magnitudes of the “Monday gaps” are shown in ordinates.

To give a better sense of the order of magnitude of the relative importance of the alternative explanations, we focus on the results associated with the econometric adjustment of eq. [21], whose graphical representation is the “adjusted Monday gap” in Figure 3. The best result was obtained for θ=0.1 and the estimated values of κ0=0.0263 (t-statistic=23.48) and κ1=0.0181 (t-statistic=11.93). The R-squared of the regression was 0.7105.

Figure 3:

Actual and adjusted Monday gap.

The shape of the estimated unjustified component or “actual Monday gap” is strongly consistent with the model developed in Section 4. As a matter of fact, the “theoretical” Monday gap in Figure 3 was depicted that way since our “empirical” Monday gap shows a negative slope and a convex profile. Theoretically other profiles were possible, but once we have measured empirically the total Monday gap it is easy to decompose it into its three components according to the procedure described in Section 4.5 and Appendix B. First of all, we must define what short and long durations are. We made the decision of considering short durations those lasting only one day (the minimum possible) and long durations those with recovery periods over 90 days. ^[35] Taking into account these limits and proceeding in the manner described in Section 4.5 and Appendix B, we obtain the following figures: al=0.431; as=0.436; bl=0.468; bs=0.481; ed‾s=2,309; ed‾l=1,507; hd‾1l = 1,143; hd‾1s=1,572; hdM2l=184, hdM2s=283, and ηRs=36.

Another important decision is to provide a value for the parameter σ. Within the calibration process, a value for such a parameter that yielded good results was σ=3. This means that the number of potential situations causing the report of a false on-the-job accident is three times higher on Mondays than on the rest of the working days, which appears to be a quite reasonable order of magnitude. Finally, we need to choose a CDF for ΦdRe, so as to compute πd=1−ΦdRe. Although we tried several alternatives, due to the type of phenomenon we are studying (duration analysis), the Weibull distribution seemed to us an appropriate option. After the calibration process, our choice was a Weibull with parameters 1 and 10, which produced good outcomes.

As a result of all this calibration work, we may depict Figure 4. In this figure we show our calibrated Monday gap (purple line) together the adjusted Monday gap from Figure 3 (black dotted line). As can be observed, the two curves resemble each other remarkably well. To be more precise, the correlation coefficient is about 97%. Once we are confident that the calibrated curve captures reasonably well the reality measured through econometric techniques, we apply the theoretical reasoning pointed out in Section 4.5 and Appendix B so as to decompose the total Monday gap into its three components. Such components are also displayed in Figure 4.

Figure 4:

Calibrated and adjusted Monday gap.

There are several comments to be made about Figure 4. Firstly, we would like to highlight the importance of the statistical Monday gap, which is an effect neglected in previous research. For asymptotic values, it equals 2.8 (negative) percentage points. Although it does not have a meaningful economic interpretation and it is just a consequence of the fact that there are proportionally more accidents on Mondays, we need to take it into account in order to make accurate measures of the other two types of Monday gaps. Without taking the statistical Monday gap into consideration, we would be underestimating the other two effects.

Regarding the economic Monday gap, it reaches a value of 1.4 percentage points for those durations lasting only one day. As long as the duration increases this effect eventually vanishes, accordingly to our theoretical assumptions. For example, for a duration of 5 days, it represents 0.95 percentage points, and for a sick leave lasting 14 days it still remains at a level of 0.39 percentage points. From one month onwards, this effect is practically non-existent (less than 0.1 percentage points). Although, it is necessary to bear in mind that the bulk of the accidents have an associated duration much shorter than one month (see Appendix A), so it would be a mistake to discard the economic Monday gap as an important component to understand the whole phenomenon.

However, and without a doubt, the order of magnitude of the physiological Monday gap is outstanding. For durations of one day, it reaches a value of 6.37 percentage points, but rapidly converges to its asymptotic level of about 6.51 percentage points. With this size, it is uncontroversial to state that the physiological Monday gap is the most important factor to understand the Monday effect, and this result should be kept in mind when designing policy actions to cope with this fact.

As a result of all this, we might conclude this section by answering the two main questions posed in this paper. Firstly, it seems clear that there is a Monday effect in the Spanish WC. Secondly, to better understand such a Monday gap, it is necessary to take into consideration both the economic and physiological hypotheses, together with the statistical effect identified and explained in this research. Although the physiological explanation undoubtedly dominates from a quantitative standpoint, the economic view still plays a role.