Daniel L Chen
email: daniel.chen@iast.fr.

Markus Loecher

Andrew Schoenholtz

Phillip Schrag

Mood and the Malleability of Moral Reasoning: The Impact of Irrelevant Factors on Judicial Decisions

We detect intra-judge variation spanning three decades in 1.5 million judicial decisions driven by factors unrelated to case merits. U.S. immigration judges deny an additional 1.4% of asylum petitions-and U.S. district judges assign 0.6% longer prison sentences and 5% shorter probation sentences-on the day after their city's NFL team lost. Bad weather has a similar effect as a team loss. Unrepresented parties in asylum bear the brunt of NFL effects, and the effect on district judges appears larger for those likely to be following the NFL team. We employ double residualization for a "causal" importance score.

1 Introduction David Hume once argued that moral reasoning is "slave to the passions" [START_REF] Hume | A treatise of human nature[END_REF]. Jurist Jerome Frank proposed that "uniquely individual factors often are more important causes of judgments than anything which could be described as political, economic, or moral [views]" [START_REF] Hutcheson | The Judgment Intuitive: The Function of the "Hunch" in Judicial Decision[END_REF]Frank 1930[START_REF] Frank | Law & The Modern Mind[END_REF])), a perspective often caricatured as "what the judge had for breakfast" [START_REF] Schauer | Thinking Like a Lawyer[END_REF]. Recent work in cognitive science provides strong evidence for a link between emotion and moral judgment [START_REF] Prinz | The Emotional Construction of Morals[END_REF]. Moving beyond judgment, this article examines the decisions of judges. We study the link between judicial decisions and emotional cues associated with wins and losses by professional football teams and with the weather, using the universe of U.S. asylum decisions and U.S. district court sentencing decisions. Behavioral anomalies in these high-stakes settings highlight their fragility and the promise of AI to improve efficiency and fairness through personalized nudges of judges. Psychologists have documented many effects of moderate size in the lab, so "settings where people are closer to indifference among options are more likely to lead to detectable effects [of behavioral biases] outside of it" [START_REF] Simonsohn | Spurious? name similarity effects (implicit egotism) in marriage, job, and moving decisions[END_REF]. As the preferences over the legally relevant factors wanes, the influence of extraneous factors grows. Then to measure behavioral biases of judges is to document their revealed preference indifference, which raises the spectre of difference in indifference, i.e., privilege disparities when prominent legal actors are indifferent to certain societal groups.

What determines judicial decisions? We would like to believe it is "the law". To be sure, the law may be hard to determine or even indeterminate. The identity of the judge can thus make a large difference. This simple fact was statistically established at least a century ago [START_REF] Everson | The Human Element in Judging[END_REF]) and triggered policy responses such as the U.S. Federal Sentencing Guidelines. In particular, judicial decisions differ by judicial ideology, as surveyed in later studies [START_REF] Fischman | What Is Judicial Ideology, and How Should We Measure It?[END_REF]) and recognized in the frequent judicial confirmation battles. But while these inter-judge differences show that the meaning of "the law" is not unique in practice, they are consistent with each judge consistently applying his or her version of the law. Indeed, normative theory readily admits such variation because the correct interpretation of the law is not unique or at least not discernible for real world judges [START_REF] Dworkin | Law's Empire[END_REF], [START_REF] Kennedy | A Critique of Adjudication[END_REF]).

In this article, we provide evidence from a natural experiment for a different kind of variation. We detect intra-judge variation driven by factors completely unrelated to the merits of the case, or to any case characteristics whatsoever. Concretely, we show that asylum grant rates in U.S. immigration courts differ by the success of the court city's NFL team on the night before, and by the city's weather on the day of the decision. Our data include 1.5 million decisions spanning three decades -22,000 asylum decisions on Mondays after a game; a half million asylum decisions in total; and a million sentencing decisions -and allows exclusion of confounding factors, such as scheduling and seasonal effects. Most importantly, the design holds the identity of the judge constant. On average, U.S. immigration judges grant an additional 1.4% of the asylum petitions-and U.S. district judges assign 0.6% fewer prison sentences and 5% longer probation sentences (a substitute for imprisonment)-on the day after their city's NFL team won, relative to days after the team lost. Bad weather on the day of the decision has approximately the opposite effect. By way of comparison, the average grant rate is 39%, the average imprisonment rate is 88%, and the average probation length is 40 days. Effects are larger with upset losses (defeats when the team was predicted to win by four or more points), but not upset wins (victories when the team was predicted to lose), consistent with asymmetry in the gain-loss utility function.

Further, the available data allow us to determine if sports outcomes and weather influence the judge directly or indirectly through lawyer behavior-we find that the effect of NFL outcomes on asylum decisions is entirely borne by unrepresented applicants. In this sample, there is no lawyer, and these applicants bear the brunt of judicial indifference. Moreover, we suspect that refugees are not avid fans of NFL games to be affected by their outcomes. In U.S. federal district courts, sentencing decisions are almost always made after a guilty plea, without a trial. Taken together, our results demonstrate that case outcomes depend on more than "the law," "the facts" of the case, judicial ideologies, or even constant judicial biases, for example with respect to race. While we suspect that many practitioners would not be surprised by that basic claim, the contribution of this article, however, is to provide clear causal identification of two such factors, and to measure their magnitude. The measured effects of 1.4%, 0.6%, and 5% appear large for two reasons. First, we can only measure one emotional influence out of many.

Presumably, other factors such as family problems or joys, traffic jams, or health fluctuations have an even greater influence on a judge's state of mind and thus plausibly case outcomes. If we had data on these, we would expect to find much larger effects.

Second, even the estimate of NFL effects is only a lower bound, for lack of better data on the diverging preferences of the judges only partly addressed by our using the birth state of the district court judges. We find that the effect of NFL outcomes on sentencing decisions is more salient for those judges born in the same state as the football team, which points towards a more direct effect on the judge rather than an indirect effect. If we had the true preferences of judges, the coefficients may be even larger.

Numerous field studies that have shown humans in many settings to be influenced by seemingly irrelevant factors in general and by sports outcomes and weather in particular. For example, sports results affect stock returns [START_REF] Edmans | Sports sentiment and stock returns[END_REF]), sports results influence voting in political elections [START_REF] Healy | Irrelevant events affect voters' evaluations of government performance[END_REF]), and disappointing NFL football results trigger domestic violence [START_REF] Card | Family violence and football: The effect of unexpected emotional cues on violent behavior[END_REF]). Many similar studies exist for weather, and the research on weather's effect on economics and finance has been summarized and experimentally traced to weather's effects on risk attitudes [START_REF] Bassi | O Sole Mio: An Experimental Analysis of Weather and Risk Attitudes in Financial Decisions[END_REF]). Moreover, bad weather on visiting days increases the chance that an admitted student will enroll [START_REF] Simonsohn | Spurious? name similarity effects (implicit egotism) in marriage, job, and moving decisions[END_REF]).

Such effects are manifestations of the broader point that weather strongly influences mood [START_REF] Connolly | Some Like It Mild and Not Too Wet: The Influence of Weather on Subjective Well-Being[END_REF]).

One article detects time-of-day patterns in Israeli judges' parole decisions (Danziger et al. [2011b]). Concretely, the article shows that parole approval rates drop with the time from the judges' last meal. One potential problem with this research design is that the order of prisoners' appearance before the judges and the exact time judges choose to take breaks may not be random [START_REF] Weinshall | Overlooked factors in the analysis of parole decisions[END_REF], Danziger et al. [2011a]). The sample size (N = 1, 112 and 8 judges) is also several orders of magnitude smaller than ours. There are also various papers showing clear judicial biases in the laboratory environment (e.g., [START_REF] Guthrie | Inside the judicial mind[END_REF], [START_REF] Guthrie | Blinking on the bench: How judges decide cases[END_REF]; [START_REF] Rachlinski | Does Unconscious Racial Bias Affect Trial Judges?[END_REF], [START_REF] Rachlinski | Altering Attention in Adjudication[END_REF]; cf. [START_REF] Simon | Doubt: The Psychology of the Criminal Justice Process[END_REF]). In particular, these experiments clearly identify racial bias (e.g., [START_REF] Rachlinski | Does Unconscious Racial Bias Affect Trial Judges?[END_REF]). Outside the lab, findings of racial bias are always subject to at least the theoretical possibility that different outcomes reflect unobserved case heterogeneity beyond race.

Massive inter-judge variation in asylum grants has been documented by [START_REF] Ramji-Nogales | Refugee Roulette: Disparities in Asylum Adjudication[END_REF], who introduced the legal literature to the asylum data. They showed that grant rates for the same applicant nationality in the same city could be anywhere between, e.g., 0 and 68% depending on the judge who heard the case. Our findings complement theirs. Their findings, while shocking, would be consistent with individual judges steadily applying the same legal philosophy -their own -, but legal philosophy differing across judges. By contrast, our finding shows that consistency is limited even within judge.

Asylum courts involve serious, potentially life-or-death decisions [START_REF] Ramji-Nogales | Refugee Roulette: Disparities in Asylum Adjudication[END_REF]). Their case load is very high, forcing immigration judges to make important decisions with little time (on average 7 minutes by one estimate2 ) and hence presumably with less deliberation and more of a "hunch" than other judges [START_REF] Hutcheson | The Judgment Intuitive: The Function of the "Hunch" in Judicial Decision[END_REF]). The Board of Immigration Appeals provides little guidance on the application of the broad standard for asylum petitions, namely "reasonable fear of persecution." This lack of time for deliberation coupled with a very open-ended decision standard may amplify emotional influences. For replication purposes, we consider criminal sentencing by federal district judges in 1971-2012 (primarily 1998-2011), the only other large data base of comparable judicial decisions that we are aware of. Like the asylum data, sentencing cases are numerous and relatively homogeneous, and the outcomes are easy to classify. With hundreds or even thousands of similar cases per judge, we can thus construct fairly precise baseline approval or sentencing rates for each judge from the judge's own decision record. While we may like to believe that sentencing by federal district judges is not susceptible to influence by extraneous factors (due to higher quality of the federal judges, more time for deliberation, or the constraining effects of federal sentencing guidelines), in fact, district judges are susceptible to the same influences as asylum judges.

Furthermore, an article finds that outcomes of games played by Louisiana State University football team affects judicial decisions handed down by judges in a Louisiana juvenile court [START_REF] Eren | Emotional judges and unlucky juveniles[END_REF]). 3 The article finds that unexpected losses increase sentence length on juvenile defendants imposed by the judges by around 6.4 percent. Like their analysis finding larger effects for judges who attended Louisiana State University, we find larger effects for judges who grew up in the area of the football team. One difference between their setting and ours is their sample size of 9,346 defendants and 207 judges is smaller than the 1.5 million decisions and 1,684 judges analyzed in this article. A second difference is that judges in Louisiana juvenile courts may be somewhat less professional than judges appointed by the U.S. President and confirmed by the Senate, which may amplify emotional influences. 4 A subsequent work that replicates our weather results finds that asylum denial rates monotonically increase with temperature [START_REF] Heyes | Temperature and decisions: evidence from 207,000 court cases[END_REF]). 5 Our complementary analysis also finds temperature effects for sentencing decisions. Their article attributes the channel to the judge (rather than the lawyer or defendant) because of the differences by gender of judge. 6 We also complement their evidence with a less ambiguous measure of bad weather (rain, winds, snow), and we find non-monotonic mood effects with temperature (possibly due to our larger sample). [START_REF] Baylis | Temperature and temperament: Evidence from twitter[END_REF] documents a clear U-shape between temperature and sentiment measured in twitter. We also complement these articles by using machine learning to estimate causal importance scores.

Unlike these other articles, our findings suggest a policy in that courts could impose a requirement that respondents are due free access to counsel in asylum cases. The positive effects of lawyers on outcomes for respondents in observational studies are large, and we present evidence for one kind of mechanism (reducing "indifference") that the presence of a lawyer helps sharpen judicial analysis, removing arbitrary things from the outcomes, undercutting the negative effects of mood swings (e.g., upset losses).

Data

The first empirical setting is U.S. asylum court decisions and the second is U.S. federal district court decisions.

Asylum Judges: Data Description and Institutional Context

The United States offers asylum to foreign nationals who can prove that (1) they have a well-founded fear of persecution in their own countries, and (2) their race, religion, nationality, political opinions, or membership in a particular social group is one central reason for the threatened persecution. Decisions to grant or deny asylum are potentially very high stakes for the asylum applicants. An applicant for asylum may reasonably fear imprisonment, torture, or death if forced to return to her home country (see [START_REF] Ramji-Nogales | Refugee Roulette: Disparities in Asylum Adjudication[END_REF] for a more detailed description of the asylum adjudication process in the U.S.). This article uses administrative data from 1993 to 2013 on U.S. refugee asylum cases considered in immigration courts.

Judges hear two types of cases: affirmative cases (where the applicant seeks asylum on her own initiative) and defensive cases (where the applicant applies for asylum after being apprehended by the Department of Homeland Security (DHS)). Defensive cases are referred directly to the immigration courts while affirmative cases pass a first round of review by asylum officers in the lower level Asylum Offices. See Appendix A for more details regarding the asylum application process and defensive vs. affirmative applications.

The court proceeding at the immigration court level is adversarial and typically lasts several hours. Asylum seekers may be represented by an attorney at their own expense. A DHS attorney cross-examines the asylum applicant and argues before the judge that asylum is not warranted. Those that are denied asylum are ordered deported, although in some cases applicants may further appeal to the Board of Immigration Appeals.

Judges have a high degree of discretion in deciding case outcomes. They are subject to the supervision of the Attorney General, but otherwise exercise independent judgment and discretion in considering and determining the cases before them. This discretion is evidenced by the wide disparities in grant rates among judges associated with the same immigration court.

Judges are appointed by the Attorney General and typically serve until retirement. Their base salaries are set by a federal pay scale and locality pay is capped at Level III of the Executive Schedule. In 2014, that rate was $167,000. Based upon conversations with the President of the National Association of Immigration Judges, no bonuses are granted.

We obtained the data directly from EOIR via a FOIA request (we also obtained a nearly identical data set via Transactional Records Access Clearinghouse (TRAC) and double-checked our results on those data). The data contains information on hearing dates, the completion date, whether the applicant was legally represented, whether the application was filed affirmatively or defensively (i.e., in defense of a removal proceeding), and the applicant's origin. We exclude non-asylum related immigration decisions and focus on applications for asylum, withholding of removal, or protection under the convention against torture (CAT). Applicants typically apply for all three types of asylum protection at the same time. As in Ramji-Nogales et al.

[2007], when an individual has multiple decisions on the same day on these three applications, we use the decision on the asylum application because a grant of asylum allows the applicant all the benefits of a grant of withholding of removal or protection under the withholding-convention against torture while the reverse does not hold. The two categories are almost always ancillary to the asylum application, in which case they are not independent data points. There are only 22,000 independent withholding of removal and protection under the convention against torture applications, far fewer than the 434,000 asylum applications. We keep withholding of removal and protection under the convention against torture applications while only marginally increasing sample size, but only keep those that constituted independent applications. 7 The main merit hearing is the hearing at which the case's substance is tried. Several practitioners have said that the judge will almost inevitably announce the final decision at the conclusion of the hearing. This is thus the relevant date for our purposes. However, the data does not explicitly flag the main merit hearing date. If the judge renders an oral decision at the hearing's conclusion, the main merit hearing date will coincide with the case completion date, which is in the data. We use the completion date, and drop from the data all cases for which the completion date does not coincide with a hearing date. 8 At this point, our data slims to 424,065 observations from the initial 456,686. For analyzing the impact of NFL games, 26,910 of the observations occur after a game and 88,456 are on Monday. The intersection of these data restrictions yields 22,294 observations. Over 89% of the decisions after an NFL game fall on Monday as opposed to the other days, so we restrict our baseline analysis to Mondays for the NFL analysis. 9 We later use all the Mondays to see if the wins increase grant rates or the 7 We keep applications with a unique idncase idnproceeding. 8 Sometimes, however, the judge reserves a written decision. In that case, the official completion date and the main hearing date do not coincide. Consistent with this, we find that this latter group of cases is more likely to involve a lawyer (94% vs. 90%), more likely to be a defensive case (46% vs. 38%), andperhaps because the proportion of defensive cases is higher-less likely to result in a grant (36% vs. 39%). This introduces the theoretical possibility that the effects we observe are not true effects on the ultimate decision, but rather case composition effects as judges are more or less prone to reserve a written decision after a game was won. We have two replies to this. Firstly, the basic point would still go through: extraneous factors influence judicial decisions, even if the decision is procedural rather than substantive. Second, the number of decisions per day given our sample restriction is not systematically greater or smaller after wins. For the same reasons, and because they are reportedly very rare anyway, we are not worried that a greater or lesser rate of continuances after wins biases our results. 9 The sample of Monday and Thursday night games is too small. Monday night games constitute a sample size one-tenth as large-and Thursday night games constitute a sample size of one-six hundredth as large-as the sample size of Sunday night games. [START_REF] Card | Family violence and football: The effect of unexpected emotional cues on violent behavior[END_REF] also exclude Monday and Thursday night games in their empirical analysis. The restricted sample has the advantage of observing judgments by a judge on the same day in different losses decrease grant rates (or both) relative to Mondays that do not fall after a game. Asylum seekers need to navigate complex legal challenges and those without access to representation will have to represent themselves pro se. Many asylum seekers cannot afford to retain private counsel, which can be both costly and difficult to obtain, especially for detained asylum seekers, who cannot work to pay legal counsel fees. Free or low-cost legal representation is scarce in rural areas where detention centers are sometimes located, and federal funding restrictions limit the availability of legal services for asylum seekers [START_REF] Ardalan | Access to justice for asylum seekers: Developing an effective model of holistic asylum representation[END_REF]).

Federal Sentencing Data

We obtain data on criminal sentencing by federal district judges from TRAC. Extensive description of these data is available elsewhere [START_REF] Yang | Have Interjudge Sentencing Disparities Increased in an Advisory Guidelines Regime? Evidence From Booker[END_REF]) and Appendix K. In brief, federal district judges hear cases involving federal law and cases prosecuted by federal agencies. The roughly 700 judges are appointed for life by the U.S. President and confirmed by the Senate. The district court judgeships are among the most prestigious and revered judicial posts, only below that of the roughly 180 circuit court judgeships and the 9 on the Supreme Court. Thus, it becomes more hopeful that these judges would be more experienced and less susceptible to behavioral biases.

Criminal cases are prosecuted by the US Attorney, also politically appointed by the President. According to statistics from a recent study, 96% of defendants plead guilty, so there is no jury and only the sentence remains to be determined; 32% of cases have federal public defenders and another 21% have private counsel [START_REF] Mcconnell | Ethnicity, sentencing and 9/11[END_REF]). The data span 1971 through 2012. For earlier years, we have only a selection of sentences, and very few before 1998. In total, there are approximately 900,000 cases.

The data contain information on prison sentences, probation sentences, fines, and the death penalty. The death penalty is exceedingly rare in federal cases (71 cases). Monetary fines are mostly very small relative to prison sentences. The median non-zero monetary fine is $2,000, and the 90th percentile is $15,000. We thus ignore them, and focus exclusively on prison sentences and probation.

The U.S. federal sentencing guidelines also help limit judicial discretion in sentencing. The guidelines specify a minimum and maximum sentence depending on offense severity and criminal history. However, judges can deviate from the guidelines if they find mitigating circumstances, such as family responsibilities, good work, prior rehabilitation, or diminished capacity.

Probation is another means with which a judge can mitigate discipline. Prior to the federal sentencing guidelines, probation would delay the imposition or execution of sentence. If a defendant violated a condition of probation, the court had the option to revoke probation and impose the prison sentence previously stayed. Probation as a means to "suspend" the sentence was abolished with the Sentencing Reform Act (1984), which recognized probation as a sentence in itself.

years, and judgments of different judges on the same day in a given year.

NFL Data

The article focuses on professional football because it is the most popular sport in the U.S. 10 We merged the asylum and sentencing data with NFL outcome data. As nearly all NFL games are played on Sundays, we dropped all other game days to keep the sample homogenous. We matched the courthouse of the judge to the NFL team most favored by the local community in 2013 according to Facebook likes.11 Likewise, [START_REF] Card | Family violence and football: The effect of unexpected emotional cues on violent behavior[END_REF] assign all residents of a state to their "local" NFL team. They argue that "Weaker emotional cues presumably lead to attenuated estimates of the effect of wins versus losses. We suspect that our assignment procedure is likely to lead to a conservative assessment of the effect of emotional cues on family violence." They use 6 NFL teams (our sample includes 28). We do not know the personal preference of any given judge. We considered surveying the immigration judges, but figured that asking the judges about their sports preferences would generate a near zero response rate. While it is reasonable to guess that a judge who cares about football would follow the local team, he or she may not, and in fact may not care about football at all. The lack of separate information on judges' preferences also prevents disentangling whether sports influence decisions directly through the judges' mood, or through their environment or the other court house participants. However, we can use the sample of asylum cases that are resolved without lawyer representation. Moreover, the birth state of district judges (but not asylum judges) are available from the Federal Judiciary Center. More salient effects for judges who are likely fans of the NFL team, proxied using the location of their birth, would be suggestive that the effects are due to judge decision-making as opposed to the game or weather outcomes affecting other court participants such as lawyer behavior.

Weather

We use weather data from the National Weather Service. To combine the weather data with the courts data, we merge on date and location. We used rainfall, high winds, and snow. This article does not claim that sports and weather are the main determinants of people's moods. But among the plausible influences on mood, they are ones we can actually measure for a large number of cases. The public has no access to data on judges' health status, family events, commuter traffic, etc. Other events, such as stock market crashes or terrorist attacks, are measurable and will likely have a much stronger effect on mood than weather or sports, but the sample size is (fortunately) much too small.

Power

Based on prior research on intra-judge differences, one important factor predicting case outcomes is the identity of the judge.

There are 340 immigration judges in the asylum data set, compared to 1,268 district judges in the sentencing data set. Moreover, all asylum cases have the same binary potential outcome, while sentencing cases present vastly differing potential sentence ranges. To appreciate the demands on sample size, consider the following numbers. The asylum and sentencing data sets are the largest case data sets we am aware of, at present. The relevant subset of comparable decisions after a football game, however, only comprises 58,000 sentencing and 22,000 asylum decisions, respectively. A 1% treatment effect is thus 110 additional grant decisions in the treatment group relative to the control group. If we had only a tenth of the overall sample size, a mere 10 such additional decisions in the control and treatment group, respectively, could create the misleading appearance of a 1% treatment effect and would prevent any reasonable inference from such a smaller sample. We would not be able to claim with any certainty that the 1% estimated effect is a true effect or mere noise. Comparability of the underlying cases greatly facilitates bounding the probability of a chance result. Similarly, if we had at least a fairly good estimate of what the decisions should be absent the treatment, the actual difference would provide a fairly good estimate of the treatment effect.

Table 1 presents summary statistics for all variables in the two datasets. Summary statistics for court cases and NFL outcomes are summarized over the data analysis sample restricted to Mondays after NFL games. The weather data is summarized for the entire data analysis frame. Appendix B presents distributions for cities, the teams, and over time. 12 Appendix C presents motivating bivariate tests of the data for interested readers. 12 The percent of decisions occurring after games predicted to win is higher for sentencing. This is because district court sentencing decisions occur in regions and time periods more enthusiastic of teams predicted to win. For example, Dallas Cowboys is matched to 29% of the sentencing data but only 4% of the asylum data.

(Quasi-)Random Assignment

Obviously, the outcomes and characteristics of asylum cases do not influence NFL outcomes (neither directly nor through scheduling) or the weather. It is conceivable that case scheduling adjusts to NFL scheduling, outcomes, or the weather. For a number of reasons, however, this is extremely unlikely. First, we have learned from conversations with practitioners that the main merit hearings in asylum cases are scheduled first-in first-out, leaving no role for discretionary adjustments. Second, even if there were such room, it seems implausible that NFL and the weather would enter the picture. In fact, many cases are scheduled so far in advance that not even the NFL schedule, let alone the result or the day's weather, would be known at the time of scheduling. The NFL schedule comes out in April 13 , while asylum cases may be scheduled over a year in advance. Third, once scheduled, the main hearing date is essentially set in stone, and decisions are rendered on the spot in almost all cases. Finally, we verify empirically that cases heard after NFL wins are not statistically different on observable case characteristics (other than the grant decision) from cases heard after losses.

It is not easy to conceive of third factors that might influence both (unobserved) asylum case characteristics and NFL outcomes, let alone the weather. Perhaps cities that become wealthier attract (or cultivate) both a better football team and a more sophisticated set of asylum petitioners. The latter would be attracted by higher wages (although one might also think that economic migrants are unlikely to obtain asylum). The former would be attracted by the higher purchasing power and the concomitantly higher advertisement revenue. We account for this possibility by controlling flexibly for city time trends.

As prima facie evidence, we present regression discontinuity plots of the data. NFL outcomes make it easy to present a discontinuity graphically, especially for outcomes that are easy to classify like the granting of asylum. Figure 1 shows the grant rate plotted against the point differential in NFL games. We present a local polynomial regression overlaid on the raw data that is jittered. 14 Losses occur to the left of a 0 and wins occur to the right. An increase in the grant rate occurs when the court city's NFL team on the night before wins. The confidence intervals are wider to the edges since very few games have high realized point differentials. 15 13 See, e.g., http://www.nfl.com/photoessays/0ap1000000161578. 14 The grant rate is jittered to more clearly present the mass of data (grant rates are usually 0 or 1 for any given judge on a given day) and thus will occasionally appear outside [0,1].

15 Interestingly, further away from 0, the effect is less clear. This could be related to expectations.

-.5 0 .5 1 -50 0 50

(mean) grant 90% CI lpoly smooth: (mean) grant lpoly smooth: (mean) grant

Wins vs. Losses

We begin with the effect of NFL football wins vs. losses. For the reasons mentioned, we restrict the sample to asylum cases decided on Mondays after Sunday games. The sample is further collapsed to judge-city-day 16 and not every judge sits on a case on a Monday after NFL. We then present the effect of wins vs. no game and losses vs. no game, and finally, the effects of unexpected losses.

Table 2 estimates a fixed effects regression for applicants we, judges j, cities c, and decision date t of the following form:

Grantratio jct = baserate jc + δT ct + β 1 X jct + β 2 calendar t + jct
Grantratio is the ratio of grants to the number of decisions handed down by the judge in a given court house on a given day (that is, this reduces the dataset to at most one observation per judge per day). Baserate jc is a fixed effect for judge j sitting in city c. T ct indicates the treatment and δ the coefficient of interest (e.g., win or loss). X jct is a vector of average applicant covariates for the applicants who appeared before the judge in that court on that day. In particular, it contains whether the claim was defensive or affirmative, and whether the applicant was legally represented. We also include the fraction of applicants who were of the most frequent nationality. 17 Calendar t is a collection of calendar dummies for each week of the year (1-52) and for each NFL season between 1992 and 2013. jct is an error term.

The case covariates X jct are not required for identification. In fact, as already mentioned, we test that they are randomly distributed across treatment and control groups identified by T ct . A separate issue is dependence of observations from the same city and, a fortiori, same judge. The standard way of dealing with dependence of observations is clustering. There are two levels at which cases are not independent, and they are not nested: the judge, and the city. we thus cluster either by city, judge, or both. 18 It hardly matters which way we cluster. In fact, we have found that the clustering surprisingly has only a small effect compared to no clustering. 16 For example, if judge Smith granted four applications and denied one on 4/15/2013 in Newark and granted one in New York City, we would collapse this into two data points: one data point Smith-Newark-4/15/2013 with value 0.8, and one data point Smith-NYC-4/15/2013 with value 1. 17 In the full sample, Chinese are over 20% of the applicants and by far the largest group. No subnational disaggregation is available. The next largest group is 7%. 18 The results are similar clustering by judge, so we just present clustering by both city and judge. Controlling for application characteristics, the estimated effect of an NFL win is 1.3%. With only judge fixed effects, the estimated effect is slightly larger, namely 1.9%. The difference between the two estimates is not statistically significant however. The estimated effect is stable with the gradual inclusion of controls, assuaging concerns of omitted variables.

Table 2 does not yet explicitly address the concern that applicant pools and NFL teams may develop in parallel. Table 3 explicitly addresses this possibility in two different ways. Models 1 and 2 include city-specific time trends, i.e., a separate time trend for each city. Here the coefficient stays at 1.4%. However, the city-specific polynomial trend is rather crude. 19 A more flexible way to account for unobserved common trends is to match a decision to its nearest neighbor. That is, rather than imposing a particular polynomial model, we compare each decision to the closest decision by the same judge in the same city after the opposite game result. For example, if the city's team lost on weekend 47, we compare the decision on the following Monday to decisions after the nearest win(s): weekend 46 and 48, if any; if not, weekend 45 and 49, if any; and so on. Technically, this is a matching estimator [START_REF] Abadie | Large Sample Properties of Matching Estimators for Average Treatment Effects[END_REF]). Model 3 requires an exact match on judge, city, and half-decade. Model 4 further requires the comparison be made to a match found within three months. This restriction hardly matters. The estimated effect is 1.9%. To address concerns of omitted variables another way, Appendix D presents the results of "placebo regressions" (balancing checks) using the application controls as the "outcome" variable. Appendix D also reports attenuation and anticipation estimates. The point estimates for the "effect" of Sunday night NFL games on the Friday before or on Tuesday decisions are very small with similar standard errors, which assuages concerns of the main estimated effects being due to statistical noise. Appendix E reports that no significant differences are found for whether the NFL team and the courthouse are in the same city 20 , whether the game is played in the home city, or whether the game is a playoff game. These results suggest the mechanism is not due to fans attending the game.

Impacts of Wins vs. Impacts of Losses

Table 4 asks the separate question of the effects of (1) a loss, and (2) a win, compared to Mondays after non-game Sundays.

We could then see if the effects of winning and losing are asymmetric. For example, if judges barely changed their decisions after a win as compared with an "untreated" Monday after non-game Sunday, but reacted negatively to losing (or vice versa), that might lead to a more complete understanding of the underlying psychology of mood. The estimation sample is the set of Mondays that occur through the NFL season. The results look largely due to losses. This is consistent with fans who experience loss aversion. 20 There are 56 cities and 24 teams matched to asylum data.

Impacts of Upset Losses

Table 5 investigates the effects of unexpected losses. [START_REF] Card | Family violence and football: The effect of unexpected emotional cues on violent behavior[END_REF] regressed domestic violence on indicators for upset loss, close loss, upset win, predicted win, predicted close, and predicted loss. [START_REF] Eren | Emotional judges and unlucky juveniles[END_REF] do the same with juvenile sentencing. In order to as close as possible to these prior and parallel work such that each counts as a replication of the other, using the same specification, we find that in asylum decisions, an upset loss leads to 2.5% decline in the grant ratio.

The point estimate of the effect of a loss when the game is predicted to be close is small with similar magnitude of standard errors. The estimated effects of an upset win are also small and not significantly different from 0. [START_REF] Card | Family violence and football: The effect of unexpected emotional cues on violent behavior[END_REF] and [START_REF] Eren | Emotional judges and unlucky juveniles[END_REF] also report significant effects of upset losses and no significant impacts of close losses or upset wins.

The coefficients associated with the range of the spread are significantly different from 0 and are potentially interesting, but less easily interpreted, since they may be correlated with other factors associated with the asylum grant rate. The coefficient is stable in more parsimonious models, which are presented in Appendix F.

Heterogeneity

This section examines whether the effects of NFL games are larger for unrepresented parties. This type of analysis would be suggestive that the effects are due to judge decision-making as opposed to the game outcomes affecting other court participants such as lawyer behavior.

The results are striking. NFL football games affect asylum cases more for unrepresented applicants. Table 6 shows that NFL outcomes affect the grant likelihood by 3.7% for defendants without lawyer representation. The effect of NFL win on unrepresented parties is statistically significant at the 1% level. When there is a lawyer, there is essentially no effect of the NFL outcome. The interaction term is statistically significant at the 10% level. Models 2 and 3 present the results for the sample with and without lawyers, which effectively fully interacts the controls with the presence of a lawyer. Appendix G

shows the estimated coefficient is stable across model specifications, which assuages concerns of omitted variables that vary with the presence of a lawyer and the NFL win. This finding is consistent with the presence of lawyers overcoming the behavioral biases of judges, for example, by increasing the judge's attention to the case. It is also consistent with behavioral biases playing a larger role when judges are nearly indifferent for more disadvantaged applicants [START_REF] Eren | Emotional judges and unlucky juveniles[END_REF]). This leads us to suspect the NFL effects are not due to the lawyer behavior.

Table 7 reports a similar finding with unexpected outcomes. Upset losses affect the grant likelihood by 6.6% for defendants without representation. Interestingly, close losses also affect the grant likelihood for defendants without representation, by 4.6%. When there is a lawyer, there is essentially no effect of the NFL outcome. The interaction terms are statistically significant at the 5% level. Models 2 and 3 present the results for the sample with and without lawyers, to fully interact the controls with the presence of a lawyer.

Weather

Table 8 looks at the effect of three types of bad weather on the day of the decision: rain, snow, and high winds. In each case, we have not only a dummy from the national weather service, but also a continuous variable measuring the intensity. We include city by week fixed effects so the weather variables are measured as a deviation from the norm for that week in that city. We include city by season fixed effects to control for trends in weather by city. We also include application controls, day of week fixed effects, and judge fixed effects. Thus, these effects capture intra-judge variation in asylum decisions. We again cluster the standard errors by city.

As can be seen, all three types of bad weather present reduce the grant rate. For example, the presence of snow reduces the grant rate by 1.0% and the effect is statistically significant at the 1% level. The presence of rain reduces grant rate by 0.2% but the effect is not statistically significant. The presence of high winds reduces grant rate by 2.3% and the effect is statistically significant at the 5% level. The intensity of bad weather does not have a statistically significant impact controlling for the presence of the bad weather. The F-test of joint significance rejects the null hypothesis of no effect in Columns 1, 3, and 4. Appendix H presents specifications to be comparable to the previous sections (judge by city and judge by season fixed effects). It hardly matters. Appendix H also presents a placebo regression with the lawyer representation. No effect is found for whether there is a lawyer. We also checked if decisions for unrepresented parties are more affected by the weather. There are no statistically significant different weather effects for the two groups. One reason could be that the impact of weather is not overcome by a lawyers' presence. Another is that asylum applicants are affected by the weather-in a manner that does not happen with NFL games, which may be less relevant to asylum applicants-regardless of whether the lawyer is present.

4 Sentencing Decisions

Linear Regression

A first question is if and to what extent the results generalize to other judicial settings. As already mentioned, immigration courts are rather special. They have an extremely high workload, the judges are not life-tenured judges, and the applicable legal standard is rather loose.

We thus ran similar tests with the federal sentencing decisions. The results are in Table 9. Two things are immediately apparent. First, the estimated coefficients are negative. That is, as with asylum decisions, judges appear to be, if anything, Case controls are whether or not the case was tried and-except in the drugs regression-the department of the offense classification. Regressions are restricted to Monday decisions and control for having an NFL game yesterday.

more lenient after a positive sports outcome. Controlling for defendant characteristics and flexible time trends, the estimated effect of an NFL win on imprisonment rates is a reduction of 0.6%, which is statistically significant at the 5% level. Judges also assign probation lengths that are 5% longer, also statistically significant at the 5% level. For federal felony convictions, probation is probably an indicator of lenience, as probation is often imposed as a substitute for imprisonment and there are functionally inexhaustible resources at the Federal level for incarceration.

We next run similar tests with expectations. Notably, we replicate the findings that upset losses drive the results. Estimates are present in Table 10. The estimated effect of an NFL upset loss on imprisonment rates is an increase of 1.6%, which is statistically significant at the 1% level. Judges also assign probation lengths that are 11% shorter, also statistically significant at the 1% level. The result is likely due to judges handing out more prison sentences and less probation sentences, since 99% of individuals with any prison sentence have zero probation sentence lengths, while 88% of individuals who do not receive a prison sentence have a positive probation sentence. In these regressions, case controls are whether or not the case was tried and the department of the offense classification. Regressions are restricted to Monday decisions after an NFL game. 21 Appendix I presents specifications to be comparable to the asylum analysis (judge by season fixed effects). It hardly matters. Appendix I also presents a placebo regression-no effect is found for whether the primary offense was for drugs. The point estimates are small and the standard errors similar in size to the first binary regression.

Turning to the mechanism, we cannot estimate a specification with unrepresented defendants (since lawyers are required). Instead, we exploit the fact that judges are more likely to be fans of an NFL team if the formative years of childhood are in the area of the NFL team. This type of analysis would further support the inference that the effects are due to judge decisionmaking as opposed to the game outcomes affecting other court participants. The results are again striking. NFL football games affect judicial decisions more saliently for judges born in the state of the courthouse. The effect is not statistically significant for those born in a different state. Table 11 shows the effects are statistically significant at the 1% level for judges born in the same state. Here we only present models that cluster standard errors at the district court level as the results are essentially identical also clustering at the judge level, as we see in the previous two tables. In the final replication of the asylum results, we assess the impact of bad weather on federal sentencing decisions. As can be seen in Table 12, the impact of bad weather on imprisonment is jointly significant at the 5% level. For example, the presence of rain increases imprisonment rate by 0.2% and the effect is statistically significant at the 10% level. The presence of high winds increases imprisonment rate by 0.9% and the effect is statistically significant at the 10% level. The F-test of joint significance rejects the null hypothesis of no effect on probation sentence length. The effects of bad weather are weaker in the federal district courts than in the asylum courts, perhaps because the federal district court judges are more professionalized.

Random Forest Modeling and Orthogonalized Machine Learning

In this section we explore whether machine learning can be used to detect extraneous factors, such as sports or weather. We look past the recommended sentencing range and predict the sentence length within this range. We investigate sentence length percentile relative to the sentence guideline range as a dependent variable. This standardization allows us to look at where within a guideline range a sentence falls. The interpretation of this percentile measure is described in Table 13 below. The details of the data sources and processing are given in Appendix K. To preview our results, Figure 3 shows marginal correlations between ML-selected weather features and the sentence percentile. The raw data suggests a U-shape pattern between maximum temperature for the day and judicial decisions. Such a dependency is also supported by Chen and Eagel [2017] who show that temperature is an important feature for asylum decisions. When it is too hot or too cold, asylum grant rates fall. [START_REF] Card | Family violence and football: The effect of unexpected emotional cues on violent behavior[END_REF] also report that domestic violence increases when the maximum temperature is over 80 degrees Fahrenheit. As the temperature and mood link seem to be validated, in Appendix J we check the effect of NFL outcomes and snow, rain, and winds on twitter mood data measured daily for 1 year across 8 cities using data from [START_REF] Mislove | Pulse of the nation: Us mood throughout the day inferred from twitter[END_REF]. NFL wins the day before improve mood, and the effect is statistically significant at the 10% level. The F-test of joint significance rejects the null hypothesis of no effect from bad weather. For example, the presence of high winds decreases mood, an impact that is statistically significantly at the 1% level. [START_REF] Baylis | Temperature and temperament: Evidence from twitter[END_REF] also documents a U-shape between temperature and sentiment measured in twitter.

For our machine learning exercise, we compared the performance of three models, Random Forests (RF), Linear Regression and Gradient Boosting, and found that RF performed the best. We utilized parameter tuning to choose the best model from this hypothesis space. The optimal hyperparameters we found were min-samples-leaf = 9 and max-features = 0.6 (60% of features used in each node split). Random forests have become a highly competitive modeling tool, that performs well in comparison with many standard methods. They are popular, because (i) they can handle large numbers of variables with relatively small numbers of observations, (ii) can be applied to a wide range of prediction problems, even if they are nonlinear and involve complex high-order interaction effects, and (iii) produce variable importance measures for each predictor variable.22

Variable Importance

We first introduce variable importance in the context of linear regression with p variables and n observations.

Variable importance is not very well defined as a concept. Even for the case of a linear model with n observations, p variables and the standard n >> p situation, there is no theoretically defined variable importance metric in the sense of a parametric quantity that a variable importance estimator should try to estimate [START_REF] Grömping | Variable importance assessment in regression: linear regression versus random forest[END_REF]). In the absence of a clearly agreed true value, ad hoc proposals for empirical assessment of variable importance have been made, and desirability criteria for these have been formulated, for example, decomposition of R 2 into nonnegative contributions attributable to each regressor has been postulated [START_REF] Grömping | Variable importance in regression models[END_REF]). An important distinction must be drawn between the two extremes of marginal importance, such as squared correlations versus conditional measures, e.g. squared standardized coefficients or sequential increase in R 2 , as critically discussed, for example, by [START_REF] Richard B Darlington | Multiple regression in psychological research and practice[END_REF].

A recurring theme in the literature is that relative importance should balance out conditional and marginal considerations, a requirement brought forward by [START_REF] David | Dominance analysis: A new approach to the problem of relative importance of predictors in multiple regression[END_REF] and later also by Johnson and [START_REF] Lebreton | History and use of relative importance indices in organizational research[END_REF].

Simulating Data

For the sake of illustrating these concepts we generate a simple "linear" data set (no interactions, no nonlinearities)

y = β 0 + β 1 x 1 + . . . + β 12 x 12 (1)
The predictor variables are sampled from a multivariate normal distribution X 1 , . . . , X 12 ∼ N (0, Σ) where the covariance structure Σ is chosen such that all variables have unit variance σ j,j = 1 and only the first four predictor variables are blockcorrelated with σ j,j = 0.9 for j = j ≤ 4, while the rest are independent with σ j,j = 0. Of the twelve predictor variables only six are influential, as indicated by their coefficients in Figure 4.

Figure 4: Population coefficients for simulated data in analogy to [START_REF] Strobl | Conditional variable importance for random forests[END_REF].

Notice the equal magnitudes (importance) of the set of coefficients x 1:4 and x 5:8 while only x 1:4 are correlated. The zero coefficients x 4,8:12 should get no weight which is confirmed by a linear regression. However, because of the imposed correlation structure, the variable x 4 might appear to be related to the dependent variable which could cause a high marginal variable importance. Generally speaking, for low values of mtry we would expect the correlated predictors to serve as replacements of the truly influential ones. Figure 5 confirms this expectation and also the diminishing stand-in behavior of

x 4 for increasing values of mtry. We further observe that the correlation structure of x 1:4 dampens their individual VI scores: 3). The color coding is: green for truly nonzero coefficients, orange for correlated zero-value coefficients and red for all other β j = 0. Note the slight negative values for the importance scores of the variables with no predictive information. variables x 5:6 are consistently assigned a variable importance which is almost 4 times as high as the one for x 1:2 .

Variable Importance in Random Forests We assume the reader is familiar with the basic construction of random forests which are averages of large numbers of individually grown regression/classification trees. The random nature stems from both "row and column subsampling": each tree is based on a random subset of the observations, and each split is based on a random subset of mtry candidate variables. The tuning parameter mtry -which for popular software implementations has the default p/3 for regression and √ p for classification trees -can have profound effects on prediction quality as well as the to be introduced variable importance measures.

Our main focus in this paper is the CART algorithm [START_REF] Breiman | Classification and regression trees[END_REF], [START_REF] Breiman | Random forests[END_REF]) which chooses the split for each node such that maximum reduction in overall node impurity is achieved. Alternatively, multiplicity-adjusted conditional tests could be used in the splitting process which avoid the known bias of the CART algorithm towards categorical variables with different numbers of categories, or differing numbers of missing values [START_REF] Hothorn | Unbiased recursive partitioning: A conditional inference framework[END_REF], Strobl et al. [2007a]).

These so called conditional inference (CI) trees replace the CART bootstrap row sampling by sampling without-replacement of size 0.632 • n. In either case, 36.8% of the observations are (on average) not used for an individual tree; those out of bag (OOB) samples can serve as a validation set to estimate the test error, e.g.:

E Y -Ŷ 2 ≈ OOB M SE = 1 n n i=1 y i -ŷi,OOB 2 (2)
where ŷi,OOB is the average prediction for the ith observation from those trees for which this observation was OOB.

The splitting bias that was mentioned above also affects the originally proposed so-called Gini importance for classification and its analogue, average impurity reduction, for regression forests (Strobl et al. [2007b]). We will adopt the widely used alternative reduction in MSE when permuting a variable as a measure of variable importance defined as follows:

VI = OOB M SE,perm -OOB M SE (3)
An attempt at a theoretical foundation of variable importance for binary regression trees and forests is given in [START_REF] Ishwaran | Variable importance in binary regression trees and forests[END_REF]. In related work [START_REF] Hemant Ishwaran | Random survival forests[END_REF]), the authors point out that VI measures do not attempt to directly estimate the change in prediction error for a forest grown with and without the variable in question. We further note that the variable importance measure as defined above, has been shown to be closer to a measure of marginal importance rather than conveying the conditional effect of each variable [START_REF] Kellie | Empirical characterization of random forest variable importance measures[END_REF], [START_REF] Strobl | Conditional variable importance for random forests[END_REF]). It can be shown that the permutation importance tests a joint hypothesis of independence between X j and both Y and the remaining predictors

Z : X 1 , . . . , X j-1 , X j+1 , . . . , X p : H 0 : X j ⊥Y ∧ X j ⊥Z.
Hence a nonzero importance measure can be caused by a violation of either part: the independence of X j and Y , or the independence of X j and Z. The distinction between conditional and marginal influence is highly relevant for disentangling causal effects of (groups of) variables. For example, in our case we would like to make sure that the high variable importances for weather and sports features are not simply due to geographic or temporal confounding. An alternative conditional permutation scheme is proposed in [START_REF] Strobl | Conditional variable importance for random forests[END_REF] which appears to mitigate the overestimation of the importance of correlated variables. We refer the reader to Appendix L for the potential shortcomings of variable importance measures when correlated variables are not taken into account. Next, we describe our proposed solution to mitigate the confounding effect of correlated variables.

Residualizing

The distinction between marginal and conditional variable importance in multiple linear regression is at the heart of the ceteris paribus interpretation of the estimated coefficients and covered in all introductory econometrics textbooks. The key insight we borrow is that the coefficient βj does not change when we residualize, i.e. regress x j on the remaining variables x i =j . The effects of this type of residualizing in linear models are well understood though [START_REF] Lee | What residualizing predictors in regression analyses does (and what it does not do)[END_REF] highlights some undesirable effects. We extend the idea of residualizing in order to uncover the conditional effects of covariates to nonlinear models in analogy to the recently proposed concept of "Double Machine Learning" [START_REF] Chernozhukov | Double/Debiased Machine Learning for Treatment and Causal Parameters[END_REF]). In particular, for each of the "seemingly unrelated" weather/sports variables x i,SU we train a random forest model using only the "appropriate features" as explanatory variables. We then replace the original x i,SU feature with the residuals rf Res -x i,SU from the respective auxiliary RF model. The main idea of this procedure is to remove existing correlations/dependencies between the two sets of variables and allow an interpretation of variable importance in the traditional sense of "controlling for XYZ". The results are promising for the simulated data. Figure 7 demonstrates that residualization appears to report conditional variable importances instead of marginal ones. We now apply the same idea to the court data in order to test whether the (3) after replacing variables x 4,8:12 with their respective residuals from random forest models using x 1:3,5:7 as features. The color coding is as before.

observed importance scores for seemingly unrelated variables in Figure 6 are robust under this "conditioning procedure". Figure 8: Normalized permutation importance after each "unrelated" variable is replaced by the corresponding residuals from a (left panel) random forest or (right panel) linear regression with the "appropriate" features as independent variables.The color coding is as in Figure 6 . from a random forest regression and confirms the robustness of the weather/sports feature influence.

Important Features Figure 6 displays the most important features based on a permutation scheme. While dummifying categorical variables in linear models is well understood, its profound effects on model performance [START_REF] Potts | Are categorical variables getting lost in your random forests?[END_REF])

and key measures such as variable importance in tree based models are often overlooked. In fact, the majority of software implementations of random forests require dummy coding of categorical variables23 which makes benchmarking difficult. We feel that a fair and honest evaluation of the true impact of variables in machine learning needs to communicate both modeling approaches and their different interpretations. The left panel of Figure 6 evaluates the factor levels individually while the right panel (no dummy coding) compares the overall contributions of the variables as a whole.

For the dummified model we found the most predictive feature the date of the sentencing decision included both as a continuous variable as well as a binary feature encoding the 2005 United States Supreme Court decision referred to as United States v. Booker (Wikipedia [2018]), see also Figure K.7 in Appendix K. For both models, location specific features capture high importance scores. For the left panel these would be specific states such as CA, NY, TX, AZ and districts 74, 7, 54 while the factor model yields high scores to state, district and location of probation office.

The most important feature related to the defendant was the crime type, which is a reassuring sign that the judge is using case specific information in their decision. The dummy model can be more specific w.r.t. the various types of crime and scores crimes involving drug trafficking, immigration and firearms highest.

We find characteristics of the defendant that arguably should not be important to be among the top 20 most predictive features, such as race and education level. It is beyond the scope of this paper to investigate whether these features truly influence the judge.

We found some weather features appear in our most predictive features. Temperature maximum and minimum were our 10th/11th as well as 8th/10th most predictive features, respectively. We further found that some sports features, do in fact predict criminal sentence length to a small degree. It is worth noting that they are due to games that happened the prior day, not games that are going to happen. In fact we included the same sports features for the decision day but those accumulated no rmse reduction in predictions. That by itself supports a causal interpretation and the findings reported without using machine learning.

Null Distribution of Importance Scores

Finally, it is not clear whether the observed smaller positive values of variable importance measures could be due to chance since there is no a well defined Null distribution for these scores. In order to identify cutoff values above which predictors would be considered to have a significant impact on model predictions, we have implemented a permutation test which generates a null distribution of importance scores for each predictor against which the observed importance scores are compared. This null distribution is created by randomly permuting the response variable (class assignments in a classification model, or independent continuous response in a regression model) among cases, running the same Random Forest model on the permuted data, and storing the resulting importance scores. (Note the computationally demanding nested permutations: an outer loop permuting the dependent variable and an inner loop shuffling the relevant predictors.) Under this procedure, a predictor that is not adding any significant information to the model will have an observed importance score that is similar to those generated by a random shuffling of the response, while a significant predictor will have an importance score much larger than the null. Significance p-values are then calculated as the fraction of replicates in the null distribution that are greater than or equal to the observed value.

Results: for the top 20 variables shown in Figure 6, the observed importance scores lie far above the extreme percentiles of the Null distributions, hence providing strong evidence for their significance.

Conclusion

This article has documented-using the universe of U.S. asylum and federal sentencing decisions spanning three decadesintra-judicial variation caused by extraneous factors, i.e., factors unrelated to the case merits. It is not the "judge's breakfast", but close.

We cannot directly determine whether judges intentionally or unintentionally behave harshly after NFL losses or bad weather. Nevertheless, the influence of behavioral bias on high-stakes decisions is consistent with revealed preference indifference. First, the effects of NFL games on asylum decisions are driven entirely by unrepresented parties. This means that the lawyer's behavior is not driving the results and that the presence of a lawyer is a strong behavioral nudge to a judge to pay attention to be less indifferent. Second, the effects of NFL games on sentencing decisions are larger for judges likely to be fans of the NFL team. This suggests that the judge rather than other courtroom actors are explaining these findings.

After presenting the causal evidence, we present methods to interpret and evaluate the causal importance score of features in random forests. We begin with non-parametric partial correlation between extraneous factors like temperature and precipitation with judicial decisions. We then residualize to account for geographic and temporal factors; we compare a linear residualizing approach with residualizing based on auxiliary random forests. This approach is inspired by the Frisch Waugh Lovell theorem and also reminiscent of the recent advances in double or orthogonal machine learning [START_REF] Chernozhukov | Double/Debiased Machine Learning for Treatment and Causal Parameters[END_REF]). We consistently find extraneous factors like weather and sports to predict judicial decisions. Notably, when NFL features other than those happening the day before the decision are used, they are not deemed important by the random forest (likewise, they were deemed statistically insignificant in the causal regressions). We also examined the Null distribution of the variable importance scores to make sure that they are not spurious effects.

A justice system reasonably aspires to be consistent in the application of law across cases and to account for the particulars of a case. In a prediction model of criminal sentence lengths that accounts for non-judicial factors such as weather and sports events, these features unrelated to crime were predictive of sentence length as well. Future work should investigate whether the behavioral influence on judicial decision-making is a way to detect judicial indifference and whether personalized nudges of judges can improve the production of justice.

A Additional Background on Asylum Judges Immigration Courts Overview

The immigration judges are part of the Executive Office for Immigration Review (EOIR), an agency of the Department of Justice Pol [2014]. At present, there are over 260 immigration judges in 59 immigration courts. In removal proceedings, immigration judges determine whether an individual from a foreign country (an alien) should be allowed to enter or remain in the United States or should be removed. Immigration judges are responsible for conducting formal court proceedings and act independently in deciding the matters before them. They also have jurisdiction to consider various forms of relief from removal. In a typical removal proceeding, the immigration judge may decide whether an alien is removable (formerly called deportable) or inadmissible under the law, then may consider whether that alien may avoid removal by accepting voluntary departure or by qualifying for asylum, cancellation of removal, adjustment of status, protection under the United Nations Convention Against Torture, or other forms of relief Executive Office for Immigration Review [2014].

Immigration Judges

The immigration judges are attorneys appointed by the Attorney General as administrative judges. They are subject to the supervision of the Attorney General, but otherwise exercise independent judgment and discretion in considering and determining

Proceedings before Immigration Courts

There are two ways an applicant arrives to the Immigration Court. First, the asylum seeker can affirmatively seek asylum by filing an application. In the event that the Asylum Office did not grant the asylum application24 and referred it to Immigration Court, the asylum seeker can now pursue his or her asylum claim as a defense to removal in Immigration Court. Second, if the asylum seeker never filed for asylum with the Asylum Office but rather the government started removal proceedings against him or her for some other reason, he or she can now pursue an asylum case in Immigration Court Pol [2014]. This latter group is classified as defensive applicants and includes defendants picked up in immigration raids. ), produces the sentencing guidelines for federal judges to use when they make their sentencing decisions. The judges are given a guideline range for the criminal sentence that is based upon the severity of the crime and the defendant's criminal history. Due to these guidelines, the largest factor determining sentence range is the criminal charges brought to the judge by the prosecutor.

The 2005 United States Supreme Court decision referred to as United States v. Booker court decision determined that only prior convictions, facts admitted by the defendant, and facts proved to the jury beyond reasonable doubt could be used to extend the criminal sentence longer than the mandatory maximum. In other words, it introduced situations in which a judge could prescribe a sentence outside the sentencing range. We believe that this formal decision on opportunities to vary sentence length encouraged judges to change the way they made this determination. Interesting, while the U.S v. Booker case questioned the judge's right to increase the sentence length past the maximum guideline sentence, we saw an overall decrease in the length of sentence term relative to guideline range. Additionally, the range of minimum and maximum sentences becomes more extreme, as shown in Appendix Figure K.7.

Discrepancies across choice of criminal charges do not fully explain these disparities. Judges are also known to, for example, give females a sentence nearer the guideline minimum, or prescribe criminal sentences outside of the guideline range for males [START_REF] Mustard | Racial, Ethnic, and Gender Disparities in Sentencing: Evidence from the U.S. Federal Courts[END_REF]). This motivates our decision to focus on sentence length relative to the recommended guideline range. For the USDC, the Federal Sentencing Commission writes recommended sentence minimum and maximum terms to help ensure that convicts who committed similar crimes are charged with similar sentences. As can be seen in the lookup tables in United States Sentencing Commission, 2018], the severity of the crime and the criminal history of the convict are used to determine the appropriate sentence range. The judge then determines or approves a sentence length, frequently, but not necessarily within this range.

following features: total daily sunshine and precipitation, maximum and minimum temperature, and average cloudiness.

Sports Data

Sports data available to us included data from MLB (Major League Baseball), NBA (National Basketball Association), NFL (National Football League), NHL (National Hockey League), and college football (CFB) for the years in which we had U.S. District Court Data. For the four professional leagues (MLB/NBA/NFL/NHL), there was an instance of each team in each game played (i.e. each game had two instances). While the features available were not identical across sports they were generally similar, and included information such as team name, field played on, score, and betting over/under. For the CFB data, there was one instance per game. Unlike the professional sports data, the CFB data is not as complete. This is understandable due to the organization of college football competitions. Teams typically play schools of the same size, budget, and quality of facilities. Due to this, some games played by smaller schools are not recorded. However, the games played by the Division I schools, the schools with the most developed football programs and likely the greatest regional following, are well represented. This data included team name, field played on, score, and so on. For each of the five sports datasets, we tabulated information about the each team per game on the same day as the trial, including the date, team name, whether a game occurred, and whether the game would be played at the home stadium, or away. We assumed that the judge would not know the result of the game before the end of the workday.

Our assumption is that the outcome of a game could influence a trial only by games played the day before and aligned the following features appropriately: whether the game occurred, whether the game would be played at the home stadium or away, the points scored by the team, the points scored by the opposing team, the score margin (difference between team's scores), and whether the team won or lost.

Data Processing

There were several challenges when pre-processing the sports data so that they could be organized into these dataframes. For example, for the score margin was not included in all data, and was calculated in these cases. The CFB data was organized differently from the professional sports data, so each instance of a game had to be split between the results per game per team. A lookup table between the team names and the district that would presumably be interested in that team was curated manually. For the lookup table to remain useful, several simplifying assumptions had to be made. In the first pass, each team was paired with the district where their home stadium was located. This meant that major cities such as Los Angeles, which is located in the Central California District Court district, were represented several times in the lookup table. New York City was challenging in that Brooklyn falls under the Eastern New York District Court, and the rest of New York City falls under the Southern New York District Court. In the majority of cases, New York City teams were represented by both districts, unless Brooklyn had its own team. After each team was paired with its "hometown" district, we induced spatial spread in the professional sports data. First, in states that have several districts but only one team, the team was paired with all districts in that state. If a state had several districts and several teams, fandom maps based on Facebook likes were used to determine the more popular team in the ambiguous districts in that state. Finally, Boston teams were assigned to all New England districts, assuming homogeneity of fandom. If a district had no team and no obvious way to induce spread, it was not assigned any team (e.g., Guam, Puerto Rico, Montana, etc.). Due to the number of CFB teams, and assumptions about college football fan followings, we did not feel that spreading data outside of the district the school is located in was appropriate or desired.

We choose not to use the betting over/under information included in the professional sports data, though that would be an interesting area of research worth pursuing. In the college football data, we choose not to include team ranking or whether the game was a special championship. An interesting future research aim would be to give a heavier weight to championship games and bowls, presuming that the lead up and results of the games would be more impactful on the community of fans invested in the game. Similarly, this information could be incorporated into the professional sports data.

Data Merge

To combine the weather data with district courts data, we merge on date and location. The features city and courthouse correspond to the location in the weather and district courts datasets, respectively. However, we found that the city names differ between the USDC and weather datasets. In other words, we found many courthouses for which there was no corresponding weather data. To avoid dropping criminal cases that do not have corresponding weather data, we created our own metadata to link courthouses in the district data to the nearest city in the weather data. Through this, we were able to precisely merge the two datasets without loss of information. The schema of this merge includes all district court features, along with weather features 0-4 in the weather table above.

To merge the sports data with the previously merged district court and weather data, we first dropped team name; we were interested to see if home-team games affected the judges' sentence, rather than particular teams. For each of the sports dataframes described above, we merge over date and district. Each sport is represented separately. If no sports data was available for any day-district combination, the sports data fields were filled with zeros.

Figure 1 :

 1 Figure 1: NFL & Asylum: Grant rates by point differences

Figure 2 :

 2 Figure 2: NFL & Sentencing: Prison and probation by point differences

Figure 3 :

 3 Figure 3: left panel We fit a generalized additive model to just the temperature to explore the marginal effects of the maximal temperature. Overlaid are the average values for the sentence percentile in bins of width 0.01, as long as the sample size is above n = 500. The histogram plays the equivalent of a rugplot and shows the distribution of the data on an arbitrary y scale. right panel Same for the maximum precipitation

Figure 5 :

 5 Figure 5: Permutation Variable Importance as defined in Eq. (3). The color coding is: green for truly nonzero coefficients, orange for correlated zero-value coefficients and red for all other β j = 0. Note the slight negative values for the importance scores of the variables with no predictive information.

Figure 6 :

 6 Figure6: Normalized permutation importance -defined as percent increase in prediction-rmse when randomly shuffling a variable -for the original dataset without residualizing. We color code "unrelated" variables such as sports and weather features in red and the remaining variables in blue. left panel Model with dummy coding of factors. right panel Model with no dummying of categorical variables.

Figure 7 :

 7 Figure7: Permutation Variable Importance as defined in Eq. (3) after replacing variables x 4,8:12 with their respective residuals from random forest models using x 1:3,5:7 as features. The color coding is as before.

  Figure 8 shows the normalized permutation importance after each "unrelated" variable is replaced by the corresponding residuals to ta l s u n s h in e R e s u lt _ G a m e N ig h tB e fo re G a m e T h a tN ig h t m a x im u m te m p e ra tu re G a m e N ig h tB e fo re m in im u m te m p e ra tu re s e n te n c in g m o n th p re c ip it a ti o n p o s t B o o k e r e d u c a ti o n p ro b a ti o n o ff ic e te n c in g m o n th m a x im u m te m p e ra tu re m in im u m te m p e ra tu re to ta l s u n s h in e c lo u d in e s s p re c ip it a ti o n p o s t B o o k e r e d u c a ti o n p ro b a ti o n o ff ic e

  the cases before them. See INA sec. 101(b)(4) (8 U.S.C. 1101(b)(4)); 8 CFR 1003.10(b), (d). Decisions of the immigration judges are subject to review by the Board pursuant to 8 CFR 1003.1(a)(1) and (d)(1); in turn, the Board's decisions can be reviewed by the Attorney General, as provided in 8 CFR 1003.1(g) and (h). Decisions of the Board and the Attorney General are subject to judicial review Executive Office for Immigration Review [2014]. Many previously worked as immigration lawyers or at the Immigration and Naturalization Service (INS) for some time before they were appointed.

Table 1 :

 1 Standard deviations only presented for continuous variables. 2 Summarized for positive values.

	Summary Statistics
		Asylum	Sentencing
		µ	σ 1	µ	σ 1
	Grant	39%
	Defensive	39%
	Lawyer	90%
	Any Prison			88%
	Probation Length in Days 2			40	28
	Drug			35%
	Trial			5%
	NFL Win	50%		51%
	Upset Loss	7%		8%
	Close Loss	25%		22%
	Upset Win	9%		7%
	Predicted Win	27%		32%
	Predicted Close	47%		43%
	Predicted Loss	26%		25%
	Snow present	4%		4%
	Snow amount in mm 2	40	50	40	51
	Rain (may include freezing rain) present	38%		32%
	Precipitation in mm 2	92	139	91	147
	Highwinds present	0.3%		0.4%
	Windspeed (tenths of meters per second) 2	40	18	35	16
	Notes: 1

Table 2

 2

			: Main NFL Regressions
	Dependent variable		Judge-City-Day Ratio of Granted Asylum
		(1)	(2)	(3)	(4)	(5)	(6)	(7)
	Yesterday's NFL Win	0.019** 0.017* 0.018* 0.016** 0.014* 0.013* 0.013*
		(0.010) (0.009) (0.009) (0.008) (0.008) (0.008) (0.007)
	Judge Fixed Effects	X	X
	City Fixed Effects		X
	JudgeXCity Fixed Effects			X	X	X	X	X
	Season Fixed Effects				X
	JudgeXSeason Fixed Effects					X	X	X
	Week Fixed Effects						X	X
	Application controls							X
	N	13504	13504	13504	13504	13504	13504	13504
	R 2	0.21	0.23	0.23	0.26	0.44	0.44	0.46
	Notes: Standard errors in parentheses (* p < 0.10; ** p < 0.05; *** p < 0.01) clustered at the judge and city level.

Table 3 :

 3 NFL Regressions with flexible time controls

	Estimation technique		OLS	Nearest-neighbor matching
	Dependent variable	Judge-City-Day Ratio of Granted Asylum
		(1)	(2)	(3)	(4)
	Yesterday's NFL Win	0.014*	0.014*	0.019*	0.019*
		(0.007)	(0.008)	(0.011)	(0.011)
	Fixed Effects / Exact Match	JudgeXCity	JudgeXCityXHalfDecade
	Time control	City-specific trends	Match on date
	Time restriction				Within 3 months
	Week Fixed Effects	X	X
	Season Fixed Effects	X	X
	Application controls	X	X
	N	13504	13504	7474	6832
	Clustering	City	+Judge
	Number of clusters	56	56x340

Notes: Standard errors in parentheses (* p < 0.10; ** p < 0.05; *** p < 0.01).

Table 4 :

 4 NFL Regressions with all Mondays

	Dependent variable	Judge-City-Day Ratio of Granted Asylum
		(1)	(2)
	Yesterday's NFL Win	0.001	0.001
		(0.005)	(0.006)
	Yesterday's NFL Loss	-0.014**	-0.014**
		(0.006)	(0.006)
	JudgeXCity Fixed Effects	X	X
	City-specific trends	X	X
	Week Fixed Effects	X	X
	Season Fixed Effects	X	X
	Application controls	X	X
	N	21468	21468
	Clustering	City	+Judge
	Number of clusters	56	56x340

Notes: Standard errors in parentheses (* p < 0.10; ** p < 0.05; *** p < 0.01).

Table 5 :

 5 NFL Regressions with Mondays after NFL games

	Dependent variable	Judge-City-Day Ratio of Granted Asylum
		(1)	(2)
	Loss X Predicted Win (Upset Loss)	-0.025**	-0.025**
		(0.011)	(0.012)
	Loss X Predicted Close (Close Loss)	0.002	0.002
		(0.011)	(0.012)
	Win X Predicted Loss (Upset Win)	0.002	0.002
		(0.011)	(0.013)
	Predicted Win	0.053***	0.053***
		(0.012)	(0.012)
	Predicted Close	0.027**	0.027**
		(0.012)	(0.013)
	JudgeXCity Fixed Effects	X	X
	City-specific trends	X	X
	Week Fixed Effects	X	X
	Season Fixed Effects	X	X
	Application controls	X	X
	N	21468	21468
	Clustering	City	+Judge
	Number of clusters	56	56x340

Notes: Standard errors in parentheses (* p < 0.10; ** p < 0.05; *** p < 0.01). Predicted Win indicates a point spread of -4 or less, Predicted Close indicates a point spread between -4 and 4 (exclusive), and Predicted Loss stands for a point spread of 4 or more. Predicted Loss is the omitted category.

Table 6 :

 6 Effect of NFL Outcomes by Lawyer Representation

	Dependent variable		Granted Asylum
		(1)	(2)	(3)
	Yesterday's NFL Win	0.037***	0.006	0.027**
		(0.014)	(0.008)	(0.012)
	Yesterday's NFL Win X Lawyer	-0.032*
		(0.017)
	Lawyer	0.186***
		(0.022)
	JudgeXCity Fixed Effects	X	X	X
	City-specific trends	X	X	X
	Week Fixed Effects	X	X	X
	Season Fixed Effects	X	X	X
	Application Controls	X	X	X
	N	22282	20058	2224
	Sample	All	With Lawyer Without Lawyer

Note: Standard errors in parentheses (* p < 0.10; ** p < 0.05; *** p < 0.01). Standard errors are clustered by city. Observations are at the decision level.

Table 7 :

 7 Effect of Unexpected NFL Outcomes by Lawyer Representation

	Dependent variable		Granted Asylum
		(1)	(2)	(3)
	Loss X Predicted Win (Upset Loss)	-0.066***	-0.007	-0.067**
		(0.022)	(0.011)	(0.030)
	Loss X Predicted Win (Upset Loss)	0.061**
	Lawyer	(0.023)
	Loss X Predicted Close (Close Loss) -0.046**	0.008	-0.045**
		(0.022)	(0.011)	(0.021)
	Loss X Predicted Close (Close Loss)	0.054**
	Lawyer	(0.024)
	Win X Predicted Loss (Upset Win)	-0.023	-0.001	-0.036
		(0.035)	(0.015)	(0.032)
	Win X Predicted Loss (Upset Win)	0.020
	Lawyer	(0.036)
	JudgeXCity Fixed Effects	X	X	X
	City-specific trends	X	X	X
	Week Fixed Effects	X	X	X
	Season Fixed Effects	X	X	X
	Application Controls	X	X	X
	N	22167	19948	2219
	Sample	All	With Lawyer Without Lawyer

Notes: Standard errors in parentheses (* p < 0.10; ** p < 0.05; *** p < 0.01). Standard errors are clustered by city. Observations are at the decision level. Predicted Win indicates a point spread of -4 or less, Predicted Close indicates a point spread between -4 and 4 (exclusive), and Predicted Loss stands for a point spread of 4 or more. Predicted Loss is the omitted category. All level terms, such as Predicted Win and Predicted Close, are included.

Table 8 :

 8 Judicial Decisions and Today's Weather

	Dependent variable		Judge-City-Day Ratio of
			Granted Asylum
		(1)	(2)	(3)	(4)
	Snow present	-0.010***			-0.010***
		(0.003)			(0.004)
	Snow amount in mm 1	0.003			0.003
		(0.002)			(0.002)
	Rain (may include freezing rain)		-0.002		-0.001
	present		(0.002)		(0.002)
	Precipitation in mm 1		0.001		0.001
			(0.001)		(0.001)
	Highwinds present			-0.023** -0.024**
				(0.010)	(0.010)
	Windspeed (tenths of meters			0.002	0.002
	per second) 1			(0.003)	(0.003)
	F-Test of Joint Significance	0.020	0.372	0.074	0.005
	Judge Fixed Effects	X	X	X	X
	CityXWeek Fixed Effects	X	X	X	X
	CityXSeason Fixed Effects	X	X	X	X
	Application Controls	X	X	X	X
	Day of Week Fixed Effects	X	X	X	X
	N	239741	239741 239741	239741
	R 2	0.29	0.29	0.29	0.29

Note: Standard errors in parentheses (* p < 0.10; ** p < 0.05; *** p < 0.01). Standard errors are clustered by city. Observations are at the judge x day x city level. 1 Log of the underlying value+1.

Table 9 :

 9 NFL and Sentencing Regressions with flexible time controls

	Dependent variable	Any Prison	Probation Length 1
		(1)	(2)	(3)	(4)
	Yesterday's NFL Win	-0.006** -0.006** 0.050** 0.050**
		(0.003)	(0.003)	(0.020)	(0.020)
	JudgeXCity Fixed Effects	X	X	X	X
	DistrictXSeason Fixed Effects	X	X	X	X
	Case controls	X	X	X	X
	Week Fixed Effects	X	X	X	X
	Yesterday's NFL Game	X	X	X	X
	N	208,126	208,126 208,125 208,125
	R 2	0.17	0.17	0.17	0.17
	Clustering	District	+Judge	District	+Judge
	Number of clusters	94	94x1344	94	94x1344
	Notes: Standard errors in parentheses (* p < 0.10; ** p < 0.05; *** p < 0.01). Dependent variables are any prison sentence,

log of probation sentence length, and whether the primary offense was for drugs (trafficking, communication, or possession).

Table 10 :

 10 NFL and Sentencing Regressions with flexible time controls

	Dependent variable	Any Prison	Probation Length 1
		(1)	(2)	(3)	(4)
	Loss X Predicted Win (Upset Loss)	0.016***	0.016*** -0.109*** -0.109***
		(0.005)	(0.005)	(0.039)	(0.039)
	Loss X Predicted Close (Close Loss)	-0.002	-0.002	0.008	0.008
		(0.004)	(0.004)	(0.028)	(0.028)
	Win X Predicted Loss (Upset Win)	-0.004	-0.004	0.050	0.050
		(0.008)	(0.009)	(0.047)	(0.047)
	Predicted Win	-0.012*** -0.012***	0.071**	0.071**
		(0.005)	(0.005)	(0.033)	(0.033)
	Predicted Close	-0.007	-0.007	0.059	0.059
		(0.005)	(0.005)	(0.037)	(0.037)
	JudgeXCity Fixed Effects	X	X	X	X
	DistrictXSeason Fixed Effects	X	X	X	X
	Case controls	X	X	X	X
	Week Fixed Effects	X	X	X	X
	N	57037	57037	57036	57036
	R 2	0.21	0.21	0.21	0.21
	Clustering	District	+Judge	District	+Judge
	Number of clusters	94	94x1344	94	94x1344

Notes: Standard errors in parentheses (* p < 0.10; ** p < 0.05; *** p < 0.01). Predicted Win indicates a point spread of -4 or less, Predicted Close indicates a point spread between -4 and 4 (exclusive), and Predicted Loss stands for a point spread of 4 or more. Predicted Loss is the omitted category. 1 Log of probation length in days+1.

Table 11 :

 11 NFL and Sentencing Regressions by Judge Born-in-State Predicted Close indicates a point spread between -4 and 4 (exclusive), and Predicted Loss stands for a point spread of 4 or more. Predicted Loss is the omitted category. Columns 1-2 are limited to the judges born in the same state and Columns 3-4 are limited to judges born out of the state. 1 Log of probation length in days+1.

	Dependent variable	Any Prison Probation Length 1 Any Prison Probation Length 1
		(1)	(2)	(3)	(4)
	Loss X Predicted Win (Upset Loss)	0.020**	-0.145***	0.011	-0.042
		(0.008)	(0.051)	(0.008)	(0.060)
	Loss X Predicted Close (Close Loss)	0.000	-0.004	-0.007	0.028
		(0.005)	(0.034)	(0.006)	(0.038)
	Win X Predicted Loss (Upset Win)	-0.004	0.038	-0.003	0.074
		(0.010)	(0.063)	(0.011)	(0.065)
	Predicted Win	-0.013	0.069	-0.010	0.058
		(0.008)	(0.053)	(0.008)	(0.059)
	Predicted Close	-0.009	0.062	-0.002	0.045
		(0.007)	(0.047)	(0.008)	(0.051)
	JudgeXCity Fixed Effects	X	X	X	X
	DistrictXSeason Fixed Effects	X	X	X	X
	Case controls	X	X	X	X
	Week Fixed Effects	X	X	X	X
	N	32654	32654	24383	24382
	R 2	0.223	0.221	0.245	0.232
	Clustering	District	District	District	District
	Number of clusters	94	94	94	94
	Sample		Born In State	Born Out-of-State
	Notes: Standard errors in parentheses (* p < 0.10; ** p < 0.05; *** p < 0.01). Predicted Win indicates a point spread of -4 or
	less,

Table 12 :

 12 Sentencing Decisions and Today's Weather

	Dependent variable		Any Prison		Probation Length 2
		(1)	(2)	(3)	(4)	(5)
	Snow present	-0.004			-0.004	0.035
		(0.003)			(0.003)	(0.022)
	Snow amount in mm 1	-0.001			-0.001	0.002
		(0.001)			(0.001)	(0.007)
	Rain (may include freezing rain)		0.002**		0.002*	-0.011
	present		(0.001)		(0.001)	(0.007)
	Precipitation in mm 1		-0.0005*		-0.0004	0.003
			(0.0003)		(0.0003)	(0.002)
	Highwinds present			0.008	0.009*	-0.043
				(0.005) (0.005)	(0.037)
	Windspeed (tenths of meters			0.001	0.001	-0.004
	per second) 1			(0.001) (0.001)	(0.006)
	F-Test of Joint Significance	0.114	0.108	0.199	0.022	0.093
	Judge Fixed Effects	X	X	X	X	X
	CityXWeek Fixed Effects	X	X	X	X	X
	DistrictXSeason Fixed Effects	X	X	X	X	X
	Case controls	X	X	X	X	X
	Day of Week Fixed Effects	X	X	X	X	X
	N	916129 916129 916129 916129	916129
	R 2	0.16	0.16	0.16	0.16	0.15

Note: Standard errors in parentheses (* p < 0.10; ** p < 0.05; *** p < 0.01). Standard errors are clustered by city. 1 Log of the underlying value+1. 2 Log of probation length in days+1.

Table 13 :

 13 Interpretation of Range Percentile Measure

		< 0%	0% -50%		50% -100%		> 100%
	sentence	below guideline min-	between	guide-	between	guide-	above guideline max-
	length	imum (rare)	line minimum and	line midpoint and	imum (rare)
			midpoint		maximum

  Asylum data restricted to Mondays after NFL games. Enlongated for readability.The United States District Courts (USDC) are the judicial backbone for hearing and sentencing federal crimes in the United States(United States Courts [a]). Federal crimes include illegal activity committed on federal land, crimes committed by or against federal employees in particular roles, matters involving federal government regulations (e.g., illegal immigration, federal tax fraud, counterfeiting), or crimes against the U.S. that occur outside of the United States, such as terrorism (United States Courts[b]). Among federal crimes, the most frequently heard cases involve immigration, drug trafficking, firearms, and fraud. Most frequently, the defendant in a case enters a plea agreement with the prosecutor, which is then approved of, or denied, by the judge. Otherwise, a sentencing trial is held and the judge determines the sentence for the criminal to serve: probation, federal prison, or both. In either situation, the judge has final say on the criminal sentence. There are 94 district courts in the United States. At least one district court is located in each state or U.S. territory. States that are large or have a large population have sub-state regional courts instead. The United States Sentencing Commission (USSC) (United States

	K Sentencing Data	Appendix Figure B.3: Distribution of Cities Appendix Figure B.6: Distribution of Districts
	United States District Court
	YORK, PENNSYLVANIA VARICK SPC, NEW YORK ULSTER -NEW YORK STATE DOC, NEW TUCSON, ARIZONA TACOMA, WASHINGTON STEWART DETENTION FACILITY, GEOR SEATTLE, WASHINGTON SAN FRANCISCO, CALIFORNIA SAN DIEGO, CALIFORNIA SAN ANTONIO, TEXAS SALT LAKE CITY, UTAH SAIPAN, NORTHERN MARIANAS ISLAND PORTLAND, OREGON PHOENIX, ARIZONA PHILADELPHIA, PENNSYLVANIA PEARSALL, TEXAS ORLANDO, FLORIDA OMAHA, NEBRASKA OAKDALE FEDERAL DETENTION CENTER NEWARK, NEW JERSEY NEW YORK CITY, NEW YORK NEW ORLEANS, LOUISIANA MIAMI, FLORIDA MEMPHIS, TENNESSEE LOS FRESNOS (PORT ISABEL SPC), T LOS ANGELES, CALIFORNIA LAS VEGAS, NEVADA KROME NORTH SPC, FLORIDA KANSAS CITY, MISSOURI IMPERIAL, CALIFORNIA HOUSTON, TEXAS HOUSTON SPC, TEXAS HONOLULU, HAWAII HARTFORD, CONNECTICUT HARLINGEN, TEXAS GUAYNABO (SAN JUAN), PUERTO RICO FLORENCE SPC, ARIZONA FISHKILL -NEW YORK STATE DOC, N ELOY, ARIZONA ELIZABETH DETENTION CENTER, NEW EL PASO, TEXAS EL PASO SPC, TEXAS EL CENTRO SPC, CALIFORNIA EAST MESA, CALIFORNIA DETROIT, MICHIGAN DENVER, COLORADO DALLAS, TEXAS CLEVELAND, OHIO CHICAGO, ILLINOIS CHARLOTTE, NORTH CAROLINA BUFFALO, NEW YORK BOSTON, MASSACHUSETTS BLOOMINGTON (ST. PAUL), MINNESOT BATAVIA SPC, NEW YORK BALTIMORE, MARYLAND ATLANTA, GEORGIA ARLINGTON, VIRGINIA ADELANTO, CALIFORNIA Sentencing Commission, 2018]Wyoming Wisc, W Wisc, E Wash, W Wash, E W Virg, S W Virg, N Virg, W Virg, E Vermont Utah Texas, W Texas, S Texas, N Texas, E Tenn, W Tenn, M Tenn, E S Dakota S Car R. I. Penn, W Penn, M Penn, E Oregon Okla, W Okla, N Okla, E Ohio, S Ohio, N New Hamp Nevada Nebraska N. Y., W N. Y., S N. Y., N N. Y., E N. J. N Mexico N Dakota N Car, W N Car, M N Car, E Montana Mo, W Mo, E Miss, S Miss, N Minnesota Mich, W Mich, E Ala, M Ala, N Ala, S Arizona Ark, E Ark, W Cal, C Cal, E Cal, N Cal, S Colorado Conn Delaware Fla, M Fla, N Fla, S Ga, M Ga, N Ga, S Hawaii Idaho Ill, C Ill, N Ill, S Ind, N Ind, S Iowa, N Iowa, S Kansas Ken, E Ken, W Maine Maryland Mass
	0	0 2,000	1,000 2,000 3,000 4,000 5,000 4,000 6,000	8,000
			frequency frequency

Notes:

Notes: Sentencing data restricted to Mondays after NFL games. Enlongated for readability.

Kathy Morris Wolf, California Courts and Judges (1996) p. 1020 (http://bit.ly/1k7goag, visited 3/29/2014).

Eli Saslow, "In a crowded immigration court, seven minutes to decide a family's future," The Washington Post, 2/2/2014.

Our article, first submitted in January 2014 and presented in May 2014 American Law and Economics Association (and a few universities before then) is antecedent to theirs.

Another difference is probation length, which is authorized by U.S. law as an alternative for imprisonment and viewed as an act of grace, delaying the imposition or execution of a sentence. However, in[START_REF] Eren | Emotional judges and unlucky juveniles[END_REF]'s setting, probation is a measure of severity. In the federal courts, it is not so clear. In general, probation can be interpreted as the judge viewing the criminal record of the defendant as not sufficient for imprisonment of a certain length, or as a form of rehabilitation. Historically, a defendant could be assigned a sentence and be placed on probation, with his or her sentence suspended. In Davis v. Parker, 293 F Supp 1388 (DC Del 1968), probation was "an act of grace". In United States v. Allen, 349 F Supp 749 (ND Cal 1972), the court ruled that "Probation's primary objective is to protect society by rehabilitating the offender".

[START_REF] Heyes | Temperature and decisions: evidence from 207,000 court cases[END_REF] use data from "a website run by an international consortium of agencies that helps asylum seekers in Australia, Canada, the United States and several countries in Europe." This data source reports a far lower average grant rate of 16%.

A potential alternative explanation could be that male judges are less affected by the different behaviors that lawyers and applicants exhibit on hot days. High temperature increases apathy and lowers effort[START_REF] Cao | Stock market returns: A note on temperature anomaly[END_REF],[START_REF] Wyndham | Adaptation to heat and cold[END_REF]).

This is confirmed by google search trends. https://trends.google.com/trends/explore?date=all&geo=US&q=NFL,NBA,MLB,NHL

Cf. http://www.facebook.com/notes/facebook-data-science/nfl-fans-on-facebook/10151298370823859. This method is reasonable since 94% of the data are between 1996-2013.

The city-specific seasonal trends include linear, quadratic, cubic, and quartic terms.

As before, the coefficients associated with the range of the spread are significantly different from 0 and are potentially interesting, but less easily interpreted, since they may be correlated with other factors associated with sentencing outcomes.

[START_REF] Daniel | Can Machine Learning Help Predict the Outcome of Asylum Adjudications? Artificial Intelligence and the Law[END_REF] report that extraneous factors, like weather, have roughly the same random forest importance weight as whether the asylum applicant has a lawyer or the applicant's nationality.

https://blog.hwr-berlin.de/codeandstats/categorical-variables-in-trees-i/

For application at the Asylum Office, see chapters 14-26 of: http://immigrationequality.org/get-legal-help/our-legal-resources/immigration-equalityasylum-manual/preface-and-acknowledgements/

project and numerous colleagues with helpful comments. The paper benefits from data or code University provided the sentencing data used in this paper as well as a separate copy of the asylum court data. This project was conducted while Chen received financial support from Alfred P. Sloan Foundation (Grant No. 2018-11245), the European Research Council (No. 614708), Swiss National Science Foundation (No. 100018-152768), and Agence Nationale de la Recherche.

"Judge Reid is best avoided on a Monday following a weekend in which the USC football team loses." 1

C Preliminary Bivariate Tests and Visualizations

This section presents some simple comparisons. Table C.1 compares mean grant rates on days after a win with grant rates after a loss. Looking at individual decisions (test 1), the average grant rate after a win is 3.7% higher than after a loss, or about 10% of the base grant rate. This difference is both economically large and, subject to the very important caveat in the next paragraph, highly statistically significant. Changing the unit of observation to individual judge-day grant rates (test 2) or city-day grant rates (test 3) barely changes this result. Similarly, grant rates are strongly positively correlated with wins, regardless of the level at which the data are pooled ( To be sure, the simple statistical tests treat each case or ratio, as the case may be, as independent. In reality, however, observations from the same city and even more so from the same judge are subject to many of the same influences from unobserved factors. Moreover, the argument that NFL wins are randomly assigned to cases becomes tenuous over long time periods. As cities get richer, their football teams and asylum applicant pools may both become systematically stronger.

D Placebo Regressions -Balancing Checks and Attenuation/Anticipation Regressions Table D.3 reports placebo regressions using the case covariates (i.e., lawyer, defensive, whether the defendant is from China).

The point estimates are all small and the standard errors similar in size to the main regressions.

Finally, we examine Tuesday decisions after and Friday decisions two days before Sunday NFL games. These regressions help assess the degree of attenuation or anticipation of the Sunday's NFL results. We report these results in Table D.4; the point estimates are all small and the standard errors similar in size as reported in Table 3.

Appendix Table D

(3) Table E.5 checks for and finds no significant differences depending on the location of the game and time of year. Note that the coefficient for "Same city as NFL team" and "Playoffs" are less interpretable as they are associated with, inter alia, factors associated with grant rates that vary by region or time.

Appendix

I Sentencing and NFL Regressions

Table I.10 reports the effect of NFL outcomes using a specification similar to the asylum analyses. The results are hardly affected.

Appendix Table I

J Twitter Regressions

The final analysis considers the hypothesized mechanism using a proxy for mood. We examine the effect of NFL outcomes and bad weather on twitter mood data measured daily for 1 year across 8 cities using data from [START_REF] Mislove | Pulse of the nation: Us mood throughout the day inferred from twitter[END_REF]. Table J.11 reports that NFL wins the day before improve mood, and the effect is statistically significant at the 10% level. 25 The F-test of joint significance rejects the null hypothesis of no effect from bad weather. For example, the presence of high winds decreases mood, an impact that is statistically significantly at the 1% level. [START_REF] Baylis | Temperature and temperament: Evidence from twitter[END_REF] also documents a U-shape between temperature and sentiment measured in twitter.

Appendix Table J 25 Because of the small number of clusters, We also present robust standard errors without clustering. In addition, we execute wild bootstrap for the weaker result. Following [START_REF] Cameron | Bootstrap-based improvements for inference with clustered errors[END_REF] renders 95% confidence intervals between .006 and .09 for NFL wins.

Data Sources

The United States District Court Federal Sentencing data was made available by the Office of Research and Data in the United States Sentencing Commission. This data spanned federal court cases from 1992 -2013. There are 35 features in this data, characterizing the defendant and crime. We keep 15 of these features due to their interpretability. For those models that cannot handle categorical features directly, dummy variables were created as needed for features including race/ethnicity, location and citizenship resulting in a total of 253 features. Our target variable was sentence length percentile relative to the range. We compute the value using standard normalization.

x new =

x -x min x max -x min As our target variable was defined with the minimum and maximum sentence range, we dropped the minimum and maximum sentence range features when fitting our model to prevent data leakage.

Weather Data

In order to properly account for the weather in each district on a given day, we used a dataset originating from the NOAA (National Oceanic and Atmospheric Administration) database. This dataset consists of daily weather for 96 cities from 1992 -2013. It includes over 90 features that depict various aspects of the weather conditions for each day. However, many of these features contain missing values, or are merely translations of other features. We chose to include only the

L Default Variable Importance

For completeness as well as a cautionary tale, in Figure L.8 we also provide the mean decrease in impurity (or gini importance) scores which happens to be the default choice in most software implementations of random forests. This mean decrease in impurity importance of a feature is computed as a (weighted) mean of the individual trees' improvement in the splitting criterion produced by each variable. A substantial shortcoming of this default measure is its evaluation on the in-bag samples which can lead to severe overfitting. It was also pointed out by Strobl et al. [2007b] that the variable importance measures of Breiman's original Random Forest method ... are not reliable in situations where potential predictor variables vary in their scale of measurement or their number of categories.