Stay or flee? Hit-and-run accidents, darkness and probability of punishment

Empirical studies on the economic theory of crime have extensively analyzed the importance of the probability of punishment with regard to premeditated criminal activities. Unplanned crimes also occur, however, and this paper will focus on a very serious and widespread example: the hit-and-run road accident. Using police records for every road accident with injuries or mortalities that took place in Italy in the period 1996–2016, we rely on changes in daylight, both when switching between daylight saving time and winter time and across seasons, as an exogenous source of variation affecting the probability of apprehension and find that the likelihood of hit-and-run conditional on an accident taking place increases by around 20% with darkness. Our results suggest that policies increasing the likelihood of apprehension could be effective in reducing hit-and-run.


Introduction
Beginning with Becker's (1968) seminal paper, which first applied an expected utility model to criminal behavior, economists have extensively analyzed incentives to violate the law, both theoretically and -particularly in more recent times -empirically. Becker's theoretical framework, with its focus on weighing the expected benefit of committing a crime against the expected cost, is particularly well suited to the study of premeditated crimes, spanning from insider trading to pickpocketing.
The empirical literature testing the economic theory of crime has deeply studied the deterrent and incapacitation effects of prison (Drago et al., 2009;Buonanno and Raphael, 2013;Barbarino and Mastrobuoni, 2014) as well as the impact of the probability of detection (Kleven et al., 2011;Doleac and Sanders, 2015;Mastrobuoni and Rivers, 2019). 1 These studies have typically focused on general crime or on specific premeditated felonies, e.g., bank robberies or tax evasion. These examples of criminal activity do not, however, include spontaneous, unplanned crimes. It is therefore of interest to investigate whether the predictions of the economic model of crime hold in these circumstances. While some types of crime encompass both planned and unplanned cases (e.g., in the case of homicides, murders vs. involuntary manslaughter), in other instances criminal activity is characteristically unplanned.
Such is the case for the crime analyzed in this paper, namely hit-and-run road accidents with injured or dead victims. This crime is by its nature unplanned, as it follows an unforeseen event, a car accident involving the death or injury of another person, and the decision is adopted under strict time constraints and dramatic psychological conditions which could compromise the agent's judgment (Hammond, 2000). Therefore, we contribute to a better understanding of behavior in extreme and high-pressure situations (see, for instance, Frey et al., 2011;Elinder and Erixson, 2012), when planning is not feasible. Under these conditions, the agent should be less likely to have a rational response to incentives. Another peculiarity of hitand-run is that, compared to other types of crimes like burglaries, it is more likely to be committed by citizens without criminal records. 2 Therefore, the evidence we provide is representative to a broader population and not only individuals that are willing to commit a crime. Ordinary citizens have a higher discount factor (Åkerlund et al., 2016;Mastrobuoni and Rivers, 2016) and higher risk aversion (Block and Gerety, 1995) compared to criminals. Economic theory predicts that agents with such a psychological profile should react more acutely to the threat of penalties (Becker, 1968;Chalfin and McCrary, 2017). Therefore, if a rational response to incentives does exist, it should be easier to detect. Here, we study the likelihood of 1 Freeman (1999) provides an overview of the economics of crime literature, while Chalfin and McCrary (2017), Dobb and Webster (2003), Durlauf andNagin (2011), andNagin (2013) focus on crime deterrence. 2 As pointed out by MacLeod et al. (2012) hit-and-run drivers are more likely to have prior traffic code violations or prior suspensions of the driving license. Indeed, in instances where the driver is identified, invalid license, positive blood alcohol content, and Driving-While-Intoxicated within the past three years significantly increased the risk of leaving the scene. To that extent, hit-and-run drivers may actually have lawbreaking tendencies, but it seems still reasonable to assume that they are more similar to "standard" citizens compared to people committing other crimes that have been studied in the literature, e.g., bank robberies.
hit-and-run using Italian data. Figure 1 reports the scatterplot of the daily share of hit-and-run accidents from January 1996 to December 2016, together with a time trend obtained through a smooth local linear polynomial. What emerges is a sharp increase over the last fifteen years, from the minimum of 0.4% in the early 2000s to almost 1% in 2016, an increase of around 150% in relative terms. Even if Italy has lower percentages compared with other developed countries, the problem is becoming more serious over time, a general trend observed also in the US.
We show that fleeing increases with darkness and argue that this is related to a decrease in the probability of punishment. In particular, we use data on the universe of Italian road accidents with injured or dead people from 1996 to 2016. Building on a recently developed methodology (Doleac and Sanders, 2015;Taulbee, 2017;Domìnguez and Asahi, 2017;Chalfin et al., 2019) and on evidence that the likelihood of apprehension in hit-and-run accidents increases with daylight (MacLeod et al., 2012), we use variation in daylight, both when switching between daylight saving time and winter time and across seasons, to argue that the likelihood of hitand-run conditional on an accident taking place increases when the probability of being identified decreases. Through an extensive series of tests, we show the robustness of this result. To the best of our knowledge, this is the first paper using this identification strategy to study hit-and-run accidents.
In terms of the decision problem faced by a driver, if after an accident the driver does not flee, he or she may get a punishment because of having caused an accident. If instead the driver flees but does not get caught, there is no punishment. If he or she flees and gets caught, the punishment is at the highest, including also the additional penalty for running away. Increasing the probability of detection for hit-and-run would imply, other things being equal, a lower likelihood of fleeing. In the institutional setting we study, Italy, according to the best available source the overall probability to be identified after a hit-and-run is rather high, at over 50%, in line with what reported for the US (MacLeod et al., 2012). This probability is even higher, at 70%, in case of fatal accidents. 3 While for some types of crimes like tax evasion for incomes without information reporting, the probability of getting caught is so low that the literature sometimes wonders why people comply at all (Andreoni, Erard, Feinstein, 1998), this is not the case for hit-and-run. Fleeing after causing an accident implies a very concrete prospect of getting caught and being punished.
Beyond these aspects related to the economics of crime literature, hit-andrun road accidents with injured or dead people are a very serious crime worthy of investigation per se. In fact, around 35% of victims die within 1-2 h of the accident; therefore, a delay in emergency response dramatically reduces survival rates (Tay et al., 2009). Hit-and-run is also very common around the world. According to the AAA Foundation for Traffic Safety, in the US more than one hit-and-run crash -with or without serious consequences -happens every minute, while in 2015 these types of accident were responsible for 1819 fatalities (5.1% of total in road accidents) and 138,500 serious injuries (5.9%), with a steady increase in recent years in both absolute and relative terms. 4 In selected EU countries, the share of fatal accidents involving hit-and-run over the period 2009-2014 ranged between 1 and 6%, while that of injuries between 2 and 14%, with the United Kingdom presenting the highest incidence in both cases. 5 This paper contributes to the economics literature on crime by studying a felony, thus far neglected, of interest both for its dire societal consequences as well as for its unplanned nature. One further advantage of studying hit-and-run road accidents is the absence of the issue of crime displacement which typically hinder the identification of the effects of an increased probability of detection (see, for instance, Pricks, 2015). Having a serious road accident and fleeing the scene are undesired and unplanned events; therefore, a policy successful in decreasing the share of hit-andrun accidents would not cause a spatial displacement or a parallel increase in other offences, which could potentially leave the overall number of victims unchanged. In other words, while it is possible that a higher probability of detection for bank robberies could increase security van robberies, for instance, this consequence is unlikely to be a concern for hit-and-run.
In addition to the economics of crime literature, this paper is also related to the literature studying driving behavior. For instance, De Paola et al. (2013) and Abay (2018) investigate the effect on accidents of the penalty point system, while De Angelo and Hansen (2014) focus on the effect of policing, and Traxler et al. (2018) look at the impact of penalties on speeding. The focus of our study, however, is not driving behavior or car accidents per se, but rather the decision whether to stay or flee after the accident has already occurred.
The remainder of the paper is structured as follows. Section 2 discusses the identification strategy. Section 3 describes the dataset and reports descriptive statistics. Section 4 analyzes the impact of daylight on hit-and-run, while Sect. 5 performs a number of robustness checks. The final section discusses the policy implications of the empirical estimates and offers a brief conclusion.

Identification strategy
Here, we first present two different identification strategies based on daylight. Then, we discuss the link between daylight and the probability of detection.

Winter and daylight saving time transitions
We first consider the 7 days (one whole week) or 5 days (excluding weekends) before and after moving to Winter Time (WT, at the end of October) and to Daylight Saving Time (DST, at the end of March). Due to the change to WT, we gain one hour of daylight in the morning and lose one hour in the afternoon, and vice versa due to the change to DST. Specifically, when moving to WT, Italy loses one hour of darkness (− 1) from 7 a.m. to 9 a.m., while gaining one hour (+ 1) from 4 p.m. to 6 p.m.; when moving to Daylight Saving Time, Italy gains one hour of darkness (+ 1) from 6 a.m. to 8 a.m., while losing one hour (− 1) from 6 p.m. to 8 p.m. (see Table 1 for a summary). We exploit the changes in terms of daylight around these dates to identify its effect on the likelihood of hit-and-run.
In particular, we compare the incidence of hit-and-run -that is, the likelihood of flight conditional on having had an accident -during the same two-hours periods in two adjacent weeks. The type of driving-related activities conducted in these twohour periods is presumably very similar across the two weeks in terms, for instance, of commuting or drinking behavior. This should be particularly so when looking only at weekdays, when people have typically less flexibility regarding the scheduling of their day due to their working activity. Of course, in these specific times of the year, these two-hours periods differ in terms of daylight, and therefore a change in the incidence of hit-and-run can be attributed to differences in daylight. One disadvantage of looking at time changes is that, in the short term, sleeping patterns may be altered, as people haven't had time to adapt, and this alteration may affect, for instance, the decision-making process (Harrison and Horne, 2000;Smith, 2016) and, therefore, the propensity to hit-and-run. However, gaining or losing one hour of sleep is not collinear with the variable described in Table 1. Indeed, each transition from WT to DST or vice versa involves either one hour more or less of sleep, but it always involves both a time slot with more darkness and a time slot with less darkness, so it is possible to disentangle the effect of darkness from that of having one hour more or less of sleep. Finally, to confirm that the difference is indeed due to daylight, we also conduct placebo tests, applying the very same methodology to the earlier and later twohours periods, where there is no effect due to daylight, but where effects due to other concurrent factors, like the change in sleeping patterns, should be present. If the identifying assumptions are satisfied, we expect these placebo tests to produce null results.
This identification strategy has long been used in medicine and engineering to test the effect of daylight on road accidents with mixed results (for a review, see Carey and Sarma, 2017;see Laliotis et al., 2019, for a recent contribution in economics). More recently, economists have used the time change to show the negative impact of daylight on criminal behavior. Doleac and Sanders (2015) use Regression Discontinuity 6 and Differences-in-Differences applied to US data to show the negative impact of DST on robberies. This conclusion is confirmed with similar methodologies by Taulbee (2017) with other US data, by Domìnguez and Asahi (2017) with Chilean data, and by Tealde (2020) using data from Uruguay. In a completely 6 Unfortunately, incorporating an RD design on the likelihood of hit-and-run in the spirit of Doleac and Sanders (2015) would reduce our dependent variable to statistical noise. In their regressions, Doleac and Sanders (2015) use 558 jurisdictions and look at crimes like robbery, rape, aggravated assault and murder. These are far more common than hit-and-run. According to data from the FBI (https:// ucr. fbi. gov/ crime-in-the-u. s/ 2019/ crime-in-the-u. s.-2019/ topic-pages/ tables/ table-1), in 2019 there have been in the US 821,182 aggravated assaults, 267,988 robberies, 139,815 rapes, and 16,425 murders. Also notice that for these two crimes with lower incidence, Doleac and Sanders (2015, page 1094) find "suggestive evidence of impacts for rape and murder, though results are more sensitive to time-of-year controls than robbery". On the other hand, according to the AAA Foundation for Traffic Safety (based on the Fatality Analysis Reporting System), in the US in 2016 there have been 1980 fatal hit-and-run crashes. Of course, there are many hit-and-run that do not result in fatalities, but in this case in the US the reporting is based only on sampling (National Automotive Sampling System General Estimates System). Notice that in Italy systematic reporting is available not only for accidents resulting in fatalities, but also for accidents resulting in injuries. As mentioned in the introduction, however, in Italy hit-and-run road accidents are a lower percentage than in the US (0.6% of accidents with injuries and/or mortalities in Italy, see Table 2, compared to 5.5% of all traffic fatalities in the US), making them a rare event. To be more specific, in the time windows of interest in the 7 + 7 days around the change to DST and WT the number for the whole period under consideration amounts to 361 hit-and-run accidents (179 + 89 + 93, see upper part of Table 4). Spreading these observations among 20 Italian regions and 14 days would imply no statistical power.

3
European Journal of Law and Economics (2023) 55:117-144 Table 2 Accidents with injuries and/or mortalities in Italy (2017) uses daylight as a source of exogenous variation for monitoring to show temporal displacement of illegal discharges of oil from shipping in the North Sea. To the best of our knowledge, this paper is the first to use this identification strategy in the context of hit-and-run accidents.

Winter vs summer
We complement the analysis with a second approach that exploits more long-term changes in daylight. We exclude mid-seasons and consider only winter months (from November to February) and summer months (from May to August) and focus on three time windows, namely 2 p.m.-4 p.m., 6 p.m.-8 p.m., and 10 p.m.-midnight. In Italy, in both seasons, during the hours 2 p.m.-4 p.m. it is bright, 10 p.m.-midnight is dark, while 6 p.m.-8 p.m. is dark during wintertime and bright during summertime. 7 Therefore, we can identify the impact of daylight in a differences-in-differences framework.
This focus on the longer term allows drivers to adapt to time changes and acclimate to darkness/brightness, while the previous approach focuses on the short term, when people are more likely to maintain the same driving habits (Doleac and Sanders, 2015). The risk of this second approach is, of course, that people may have systematically different behavior in terms of driving-relevant activities between winter and summer, but an analysis of the two time slots adjacent to 6 p.m.-8 p.m. will allow us to discern whether this is indeed the case.

Daylight and probability of detection
The link between daylight and probability of detection relies on the idea that daylight makes identification of the car by law enforcement more likely, for instance by making it easier for victims or witnesses to read the car plate or to identify the make or color of the vehicle, therefore increasing the probability of apprehension. The importance of daylight for the likelihood of driver identification in hit-and-run road accidents has been shown by MacLeod et al. (2012) with US data on single pedestrian-motor vehicle fatal crashes over the period -2007. In particular, MacLeod et al. (2012 show that hit-and-run driver identification happens in 53% of the cases with daylight, 41% with some light (dawn, dusk, and dark but under streetlights), and 42% with darkness, with a statistically significant difference between daylight and the other conditions. It thus appears that, in this context, it is daylight that plays an important role, while streetlights do not. This may not necessarily be the case for other types of crimes. Indeed, the effect of light on crime is very clearly identified by Chalfin et al. (2022), who use random variation in streetlights in New York City to show a sizeable reduction in outdoor crimes connected with increased lightning. Using exogenous changes in daylight to study the effect of a higher probability of detection on criminal activity solves the issue of reverse causality that would be present, for instance, when using police presence as a measure of enforcement. 8 While we can credibly identify the effect of daylight on hit-and-run, it is more problematic to attribute this effect entirely to a change in the probability of detection. There is a legitimate concern that other things may change with daylight that impact the likelihood of hit-and-run, like the characteristics of accidents or of drivers. For instance, MacLeod et al. (2012) show that aggravating crimes such as a positive blood alcohol content or an invalid driver's license significantly and strongly increase the chances of a hit-and-run. If fewer intoxicated drivers circulate when it is lightful, then at least part of the effect of daylight on hit-and-run could be due to this. The focus on the very same time slots one week apart, employed in our first empirical approach, should minimize this concern. Moreover, several tests we conduct in Sect. 5 suggest that these possible confounding factors are not a major concern. Notice that even random variation in enforcement (e.g., randomly occurring budget cuts reducing police on the roads) would be problematic to identify the impact of enforcement on hit-and-run, as this could possibly also affect the characteristics of accidents or of drivers involved in an accident. Absent a (natural) experiment randomly varying the probability of detecting hit-and-run only and no other traffic violations, we believe the evidence we provide is a valuable contribution.

Dataset and summary statistics
The data come from the Italian Institute of Statistics (ISTAT), which provided us with the records of every road accident with injuries or mortalities from 1996 to 2016, with over 4.5 million observations. The database contains information on the time (hour) and location (province) of the accident; on some characteristics of drivers, passengers, pedestrians and bikers involved (gender, age) and of their transport vehicles (e.g. whether it is a car or a motorbike); on the road type and characteristics (e.g. rural or urban, number of lanes, conditions); on climate conditions; on the total number of injured and dead individuals. We know whether an individual has fled after an accident, but we have no information about the subsequent identification of the driver and his or her characteristics. 9 Our outcome variable is a dummy variable taking the value of one if the accident was a hit-and-run. Control variables include dummy variables for region and for day of the week, month, year, and national holiday, which capture factors like the seasonality connected to traffic, patterns in alcohol consumption related to weekends, and other unobservables that are invariant to region or time. Furthermore, to take into account the effect of macroeconomic variables which might affect traffic and accidents, we added real monthly per-capita income (in 2016 constant Euros), regional monthly unemployment rate (obtained from linear interpolation of quarterly data) 10 , and the real daily Brent oil price (in 2016 constant Euros). Table 2 describes the variables and reports the summary statistics. The share of hit-and-runs is equal to 0.6% of all accidents, corresponding to a total of 26,106. June and July are the months with the highest share of accidents, while the distribution over the week is quite constant apart from Sunday, with less accidents. Most of the accidents occur on urban roads, then on regional roads, and seldom on highways. The number of victims can vary dramatically, with an average of 0.02 dead and 1.42 injured per accident.
There is no clear geographical pattern in terms of the incidence of hit-and-run, with high and low incidence regions present in both the North and the South of Italy (see Fig. 2 in the Appendix).
Within a single day, the distribution of hit-and-run is not uniform. Table 3 displays by two-hour time windows the total number of road accidents and of hit-andruns, plus the relative share of hit-and-runs over the total number of accidents. The evidence shows that both the number of accidents and of hit-and-runs is higher between 8 a.m.-8 p.m., while the share of hit-and-runs is higher between 8 p.m.-8 a.m. Table 3 Mean and confidence intervals of the share of hit-and-run by time window in Italy, 1996Italy, -2016 The share of hit-and-run has been multiplied by 100. The sample is the same as in Table 2

3 4 Hit-and-run accidents and daylight
In this section, we study the effect of exogenous variation in daylight on hit-and-run accidents. We first present the econometric method and analyze the correlates to hitand-run accidents. Then, we present our results.

Method and correlates to hit-and-run
The analysis relies on logit regressions with robust standard errors. The dependent variable is a dummy equal to 1 if the accident was a hit-and-run. We multiply the coefficients, standard errors and marginal effects by 100 to make the results easier to read, so they should be interpreted as the change in the likelihood of a hit-and-run in percent. 11 As a descriptive exercise on the correlates to hit-and-run, Table 7 in the Appendix reports the marginal effects of the logit regressions for the full sample (Column 1), for accidents between a vehicle and a pedestrian (Column 2) and for accidents involving at least one motorbike (Column 3). In particular, in Column 1, we estimate the following equation: We use dummy variables to control for the region where the accident took place, time when the accident occurred (year, month, day of the week, national holiday, night), environmental conditions (weather, road characteristics -e.g., ruined road; two lanes road -and type of road -urban, regional or highway). We also control for characteristics of the accident itself through dummies for involvement of pedestrians or motorbikes and through variables with the number of vehicles involved in the accident and the numbers of dead and injured people caused by the accident. Finally, we control for the log of the real per capita income at the country level in the month the accident took place, unemployment rate in the quarter and region where the accident took place, and the real oil price on the day the accident took place. (1) 11 Hit-and-run accidents are relatively rare. To understand whether this represents an issue for the estimation, we implemented the R package "bgeva" (Marra et al., 2013) that fits regression models for binary rare events where the link function is the quantile function of the generalized extreme value (GEV) random variable. The fully parametric univariate GEV model is coherent with the logistic analysis and does not improve the goodness of fit to the data (results available upon request). Therefore, we have implemented the more commonly used logistic regression.
Looking at the full database (Column 1), in line with the literature (see Tay et al., 2009;MacLeod et al., 2012;Jiang et al., 2016), the likelihood of hit-and-run increases during national holidays, during the weekends, at night and during wintertime, when presumably the consumption of alcohol and leisure time are higher and when it is dark. Drivers flee more often in urban areas and when the counterpart cannot pursue (accidents involving a motorbike and, mostly, a pedestrian).
Results are similar when we restrict the analysis to the two subsets of accidents with pedestrians and motorbikes involved (Columns 2 and 3). The coefficient on night (defined as 10PM-5AM) is positive, but it may capture not only changes in the probability of apprehension due to the absence of daylight, but also, for instance, changes in alcohol and drug consumption that are correlated with time of the day.

The effect of daylight on hit-and-run
To identify the effect of daylight we use the two strategies described in Sect. 2. Before moving to the regression analysis, we first report some descriptive statistics on the share of hit-and-runs. Looking at the periods around the change to WT and DST (Table 4), we can see that, compared to the seven days before the time change, there is a lower incidence of hit-and-run when there is more daylight (0.47% instead Table 4 Descriptive statistics for the share of hit-and-run over selected days The share of hit-and-run has been multiplied by 100. The sample is restricted to only the 7 + 7 (upper panel) or 5 + 5 (lower panel) days around the time change Share of hit and run around the change to WT and DST (7 + 7) Change in hours of darkness 1 3 of 0.54%) and a higher incidence when there is more darkness (0.70%, difference significant at the 5% level). Results are even stronger in the panel just below, where we exclude the weekend (5 days). Studying accidents in the three selected time windows during winter and summer (in Table 8 in the Appendix), we see that we have the largest difference and the strongest statistical significance in the 6-8 p.m. window, with hit-and-run accidents rising from 0.46% in summer to 0.79% in winter.
Our main result is reported in Table 5 that shows the regression results of the short-term effects of darkness on the probability to hit-and-run. Notice that with different time windows and number of days (5 or 7 in a week) the number of observations changes considerably.
In Column 1 we identify the short-term effects of daylight by considering the 7 days before and 7 days after the spring and fall equinoxes. We consider only the two-hour time window when there is an increase or decrease in darkness. We exclude all the observations outside this window and run a logit regression on the hit-and-run dummy variable over the same standard set of controls of Table 7, Column 1. Compared to the specification in Eq. (1), there is obviously no dummy for accidents happening at night, while we add as control variable "change in hours of darkness", which is equal to 0 for accidents happening in the 7 days before the Table 5 Short-term effect of daylight Robust standard errors in parentheses. The dependent variable is a dummy equal to 1 in case of hitand-run. Regressions include DVs for year, day of week, holiday, region, night, good weather; road type and conditions; number of vehicles and of dead and injured people; DVs for motorbikes and pedestrians involved; logarithm of real per capita income; regional unemployment rate; daily real price of oil. The sample is restricted to only the 7 + 7 (upper panel) or 5 + 5 (lower panel) days around the time change Marginal effects and standard errors have been multiplied by 100 to make the results easier to read ***p < 0.01, **p < 0.05, *p < 0.1

Variables
(1) time change and equal to + 1 or − 1 for accidents happening after the time change, depending on whether we are moving to WT or DST and whether the accident happens in the morning or evening time slot (see Table 1 for a summary). Results in Column 1 of Table 5 show that there is a significant effect of one additional hour of darkness on this type of crime, with a 0.11% higher probability to flee after an accident which, applied to the 0.57% average, implies a 19% shortterm increase. In absolute terms, considering that the total number of hit-andrun accidents is 26,106, this 19% increase corresponds to almost 5000 additional cases over 21 years, i.e., more than 230 per year. The 19% increase is smaller to what emerges in the associational study focusing on pedestrian fatalities in the US by McLeod (2012), where the odd ratio of "dark" vs the referent category "light" is 1.32-1.36, depending on the model. The magnitude of our finding is in between what has been identified in the literature investigating the link between darkness and crime. Indeed, Doleac and Sanders (2015) estimate a 7% increase in robberies following the shift to DST in the US; Chalfin et al. (2022) find that the NYC street lightning intervention they study reduced outdoor nighttime index crimes by approximately 35%; very close to our result, Domınguez and Asahi (2017) find in the Chilean context a 20% decrease (increase) in crimes when the DST transition increases (decreases) the amount of sunlight by one hour during the 7-9 p.m. period, a result mostly driven by robbery in residential areas, while Taulbee (2017) uses data on US counties and finds that an extra hour of evening daylight is associated with an estimated 18% fewer total violent crimes.
To better understand the source of this effect and see whether or not it is concentrated in a particular time period, we also run a specification (not reported) with dummies for the four different periods indicated in Table 1. From this specification, it emerges that the effects of the two periods with an additional hour of light (morning in WT and evening in DST) are not statistically different from one another. The same is the case for the two periods with one additional hour of darkness (evening in WT and morning in DST).
The effect is also present when controlling separately for one additional hour of sleep. The results in Table 9 in the Appendix show that the variable capturing one additional hour of sleep is not significant, while the marginal effect of darkness increases, thus showing that our findings are not due to changes in sleeping hours. In Columns 2 and 3 of Table 5, we perform two placebo tests by applying the very same methodology to the earlier and later two-hours slots, when there is no variation in daylight. The fact that the coefficients in these cases are insignificant confirms that it is indeed daylight affecting the likelihood of hit-and-run and not some other concurrent event that affects differentially the weeks before or after the switch to WT or DST. Finally, in the lower panel, we repeat the analysis by excluding the weekends from the sample and confirm the previous findings for a subsample of days in which there is less room for people to alter their driving behavior.
In the Appendix, we show that this result holds also when applying the second methodology described in Sect. 2, comparing winter and summer. In particular, Table 10 shows that when we consider time windows for which there is no change in darkness between summer and winter, there is no differential trend in how the incidence of hit-and-run changes with the time of day. When there is different brightness between the two seasons, in the 6 p.m.-8 p.m. time window, we find instead a statistically and economically significant coefficient.
Both methodologies thus confirm that an increase in darkness, associated with a decrease in the probability of detection, increases the likelihood of hit-and-run accidents.

Robustness checks
As mentioned in Sect. 2, while we can credibly identify the effect of daylight on hitand-run, there is a legitimate concern that things other than the probability of detection may also change with daylight, for instance the characteristics of accidents or of drivers. It is thus important to underline that in all regressions we control for the severity of accidents and for the subjects involved, that is, the number of dead and injured people, the number of other vehicles/motorbikes/pedestrians involved, the road type and conditions. Therefore, we are comparing, at least along these important dimensions, similar accidents occurring with and without daylight.
One specific concern may be that with darkness a motorist may not even notice that he/she hit a person; the motorist may incorrectly assume he/she hit an inanimate object or an animal. Therefore, the higher share of hit-and-run might be due to reduced awareness rather than to the lower probability of identification. To address this issue, we repeat regression 1 of Tables 5 and 9 by excluding accidents with pedestrians. The idea is that motor vehicles and motorbikes have lights and are more visible than pedestrians at night; furthermore, they are heavy, and a road accident implies a noisy and hard crash which is more difficult to overlook. Results (see Table 11 in the Appendix) are in line with previous findings, alleviating the concern that awareness may be behind the effect of daylight.
A change in the composition of drivers having accidents could also be an issue. The propensity to flee after an accident may be systematically different among different types of drivers (e.g. males vs. females or young vs. old). If this is indeed the case, then the effect we detect could be the sum of a behavioral effect and a compositional effect, that is, changes in the population of drivers having accidents, and these could be due to changes in the population of drivers or to changes in the likelihood of having an accident.
Reviewing the literature that uses data on hit-and-run accidents where the driver was subsequently identified, male and young drivers appear to have a higher propensity to flee after an accident than do female and elderly drivers. This is the case in the US (see Solnick and Hemenway, 1995, for accidents involving pedestrians;and Tay et al., 2008, for all accidents), as well as in the UK (Broughton, 2004). Albeit based on less systematic evidence, this appears to be the case in Italy as well. 12 To understand whether a compositional effect is driving our results, we explore the demographic characteristics of those involved in accidents, looking especially at gender and age. Unfortunately, additional demographic characteristics, for instance regarding education or socio-economic background, are not available in the dataset.
We focus first on the probability of detection identified through variation in daylight. In particular, we investigate whether potential compositional effects are affecting our estimates based on time shifts. Table 6 displays the statistics for the share of accidents with at least one male driver aged 18-45 -the typical profile of identified individuals in cases of hit-and-run -over the total in the two time windows and days of interest. It appears that, for any given time shift, the share of accidents involving this category of drivers does not change before or after the time shifts.
Also concerning the characteristics of drivers, in principle people have indeed some choice about whether they want to drive after dark and so the composition of drivers may be different. In order to see whether this is indeed a problem, the lower parts of Tables 3 and 9 restrict the analysis by excluding the weekend. The idea is that during weekdays people have typically less flexibility regarding the scheduling Table 6 Descriptive statistics for the share of accidents with 18-45 y.o. male drivers, before and after time change The table summarizes descriptive statistics for the days before and after the time shift with weekend (7 + 7) and without weekend (5 + 5) of their day due to their working activity, the need to bring kids to school or classes, etc. The fact that the results are qualitatively the same is reassuring. Next, we check whether there is heterogeneity in the effect across Italian regions. To this end, we create three dummy variables identifying region in the center, south and in the islands (north being the reference case), interacting them with the target variable (Change in hours of darkness). We repeat regression 1 of Table 5 with these slope dummy variables and none of the associated coefficients is statistically significant. Similarly, we create three dummy variables equal to one for the six regions with highest, for the six regions with lowest and for the eight regions with medium share of hit-and-run. Then, we interact the three dummy variables with the target variable (Change in hours of darkness) and repeat regression 1 of Table 5 with these three slope DVs. This decomposition shows that the effect of the target variable is weaker for the regions with medium share of hit-and-run. A test for difference in the coefficients confirms that one hour more darkness in regions with medium share of hit-and-run has a (weakly, 10% level) significantly smaller coefficient than in regions with high and low share of hit-and-run, while the difference between regions with high and low share of hit-and-run is not significant. Results are omitted for reasons of space but are available upon request. Finally, we check whether the results are robust to a linear probability model instead of Logit and how results differ when in the Logit we do not include control variables such as road characteristics, deadinjured, other vehicles, unemployment rate, and price of gasoline. Thus, we repeat regression 1 of Table 5 (i) with OLS instead of Logit and (ii) with Logit but excluding the aforementioned control variables. Results -which are omitted for reason of space but are available upon request -are robust, change in hours of darkness being strongly significant.

Policy implications and conclusions
The importance of incentives is a central tenet in economic analysis, also with regards to criminal activities. There are, however, instances in which we could expect incentives not to play a role. For instance, decisions that are unplanned, and taken under great time pressure and emotional stress may trigger what Kahneman (2011) has defined "System 1", which acts automatically and quickly, as opposed to the more controlled and deliberative "System 2". Hit-and-run is indeed not planned and follows another unplanned event: a road accident with serious injuries or mortalities. The decision of whether to stay or flee is therefore taken under intense emotional stress and time pressure. We have shown evidence pointing to a reaction to the probability of apprehension, thus deepening our understanding of decision making even under these extreme circumstances.
Moreover, hit-and-run is a felony which delays emergency response and thus further aggravates the life losses due to road accidents and our analysis suggests a role for enforcement in preventing a crime even in circumstances where one could have expected agents not to be responsive to incentives.
As mentioned in the introduction, an important feature of hit-and-run is that the probability of detection is quite high, at over 50%, both in the institutional context we study, Italy, and in other places like the US (MacLeod et al., 2012). Of course, this figure could be further enhanced through traditional policies like increased police manpower, policing intensity, and investigation technologies, 13 or by exploiting technological advances in automatic monitoring and identification systems, like Lojack (Ayres and Levitt, 1998) and CCTV (Priks, 2015) -especially with facial recognition. 14 An additional tool that builds on the evidence we provide would be to inform drivers about the very high likelihood of detection in case of hit-and-run. First, it would be necessary to determine what are the beliefs regarding the probability of being identified. If, as it seems plausible, these beliefs turn out to be relatively low compared to the actual figure, then there could be interventions to communicate the true probability, for instance by including the information in the study material to get a driving license or, more difficult to implement, through information campaigns. Evaluating the effectiveness of such a policy would be of great interest for future research.
dummy variable "DID-Winter-10 p.m.-12 a.m.". 15 The insignificant coefficient of this latter variable indicates that there is no differential trend in how the incidence of hit-and-run changes with the time of day between summer and winter, when we consider time windows for which there is no change in darkness. The significant coefficient of the former variable captures the effect of the different brightness of the 6 p.m.-8 p.m. windows between the two seasons.
The separate regressions of Columns 2-4 confirm the long-term effects of daylight; in these specifications, the variable of interest is "winter", which is significant only in the time window 6 p.m.-8 p.m. (Column 3), when it is bright during the summer and dark during the winter. The effect of daylight is both statistically and economically significant. The marginal effect for two additional hours of darkness ranges between 0.16% (Column 1) and 0.21% (Column 3), which for one hour is 0.08-0.10%. Since the average rate of hit-and-run is 0.57%, this result signifies that one additional hour of darkness generates a long-term increase of 14-18%.  Regarding the remaining tables in the Appendix, Table 8 reports descriptive statistics for the share of hit-and-run over selected days. Table 9 reports results controlling for changes in hours of sleep, while Table 11 reports results without pedestrians.   Robust standard errors in parentheses. The dependent variable is a dummy equal to 1 in case of hitand-run. Regressions include DVs for year, day of week, holiday, region, night, good weather; road type and conditions; number of dead and injured people; number of vehicles, motorbikes and pedestrians involved; logarithm of real per capita regional income; regional unemployment rate; daily real price of oil The analysis considers only winter (November-February) and summer (May-August) Marginal effects and standard errors have been multiplied by 100 to make the results easier to read ***p < 0.01, **p < 0.05, *p < 0.1 Marginal effects of the determinants of hit-and-run in Italy, 1996-2016 (Logit models)

Variables
(1) (2) (3) Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Robust standard errors in parentheses. The dependent variable is a dummy equal to 1 in case of hitand-run. Regression 1 repeats regression 1 of Table 5 and regression 2 repeats regression 1 of Table 10, without considering accidents which involve pedestrians. Regressions include DVs for year, month, day of week, holiday, region, night, good weather; road type and conditions; number of dead and injured people; number of vehicles, motorbikes and pedestrians involved; logarithm of real per capita regional income; regional unemployment rate; daily real price of oil. The sample is restricted to only the 7 + 7 days around the time change