Seasonal Variation in Homicide and Assault Across Large U.S. Cities

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published online 3 June 2014Homicide StudiesDavid McDowall and Karise M. Curtis

Seasonal Variation in Homicide and Assault Across Large U.S. Cities

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Article

Seasonal Variation in Homicide and Assault Across Large U.S. Cities

David McDowall1 and Karise M. Curtis1

AbstractAlthough most crimes follow seasonal cycles, homicide is an apparent exception. The absence of homicide seasonality is surprising given that assault, a closely related offense, has an obvious annual pattern. Focusing on large U.S. cities, this article reevaluates seasonality in homicide rates using data with more extensive spatial and areal dimensions than in previous research. Panel decompositions reveal seasonal cycles in both homicide and assault rates. Seasonality stands out more clearly in assault, however, and the patterns differ somewhat in their details. The findings support the idea that assault and homicide have similar seasonal fluctuations, but they also suggest that the crimes are more distinct than criminologists often believe.

Keywordsseasonality, trends, homicide, assault, correlates

Seasonal patterns are helpful in understanding the mechanisms that underlie variations in crime rates, and they have been the subject of a long history of criminological research. A puzzling issue that arises from this work is the apparent absence of season-ality in homicide. Some studies do find that homicide exhibits a seasonal cycle, but these are exceptions to the more usual conclusion that the crime lacks any meaningful annual structure. The negative results are especially notable given that the closely related offense of aggravated assault clearly shows the presence of strong and well-defined seasonality.

The current article reconsiders whether homicide possesses a seasonal pattern, and if it does, whether this pattern is the same as the one for assault. The article evaluates

1University at Albany, State University of New York, USA

Corresponding Author:David McDowall, School of Criminal Justice, University at Albany, State University of New York, 135 Western Avenue, Albany, NY 12222, USA. Email: [email protected]

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2 Homicide Studies

the seasonal cycles in both crimes through three related analyses. The first of these examines seasonality in homicide and assault with panel data that include more areas and a much longer time dimension than previous efforts have used. Short time series and small geographical samples could have influenced findings about crime seasonal-ity overall, and about homicide seasonality in particular. If so, seasonal fluctuations should reveal themselves with greater clarity in a more extensive set of data.

The second analysis uses the large temporal and areal dimensions to assess the stability of seasonal patterns across time and space. If seasonality is not temporally and spatially constant, studies could reach different conclusions depending on the composition of the samples that they analyze. Much of the divergence in the findings of past research could then be due to variations in the choice of areal units and time intervals. This would especially be true if homicide seasonality is more sensitive to sample features than is seasonality in assault.

The third analysis considers if monthly temperature differences mediate the influ-ence of homicide and assault seasonality. Major criminological theories suggest that temperatures will carry the impact of seasonal changes, so that the strength of annual fluctuations varies with the extremity of environmental differences. Here again, find-ings about seasonality could depend on the geographical units under consideration.

In the following sections, the article first discusses past efforts to study seasonal variations in violent crime rates and reviews the mostly negative conclusions that they have reached about homicide. It then describes the current study’s data and methods and presents the findings of the three analyses. The results show that homicide rates possess a seasonal structure that is broadly similar to the pattern in assault. Homicide seasonality is weaker than is seasonality in assault, however, and the annual cycles for the offenses also diverge in other key respects. The final section considers the implica-tions of this outcome for research on seasonality and for the relationship between homicide and assault more generally.

Findings on Seasonality in Homicide and Assault

Systematic investigations of seasonality began early in criminology, and since Aldophe Quetelet’s (1842/1969) pioneering efforts, studies that examine seasonal crime fluc-tuations have appeared at a steady pace. A usual finding from this research is that seri-ous assault follows a seasonal cycle that peaks during the summer. This result so consistently appears across different time periods and geographical units that Rock, Judd, and Hallmayer (2008) suggest it is true universally. A few examples of analyses that find such a pattern are Block (1984); Cohen (1941); Cohn and Rotton (1997); Deutsch (1978); Dodge (1988); Harries and Stadler (1989); Hird and Ruparel (2007); McDowall, Loftin, and Pate (2012); Michael and Zumpe (1983); and Rock et al. (2008).

In contrast, existing research has produced mixed but largely negative results on seasonal fluctuations in homicide. Some past studies have been successful in detecting seasonality in homicide data, but by far the more frequent conclusion is that the crime contains no clear annual pattern. Studies that report seasonal cycles in homicide



McDowall and Curtis 3

include, among others, Hakko (2000); McDowall et al. (2012); Rock, Greenberg, and Hallmayer (2003); Tennenbaum and Fink (1994); and Warren, Smith, and Tyler (1983). Some of the more numerous studies that find no homicide seasonality are Abel, Strasburger, and Zeidenberg (1985); Block (1984, 1987); Brearley (1932); Cheatwood (1988); Deutsch (1978); Harries (1989); Landau and Fridman (1993); Michael and Zumpe (1983); Rock et al. (2008); Schmid (1926); and Wolfgang (1958). Comprehensive reviews by Block (1984), Cheatwood (1988), and Rock et al. (2008) all also conclude that the research literature offers little support for the existence of seasonality in homicide rates.

Rock et al. (2008) point out that most studies have lacked data from identical areas and time periods, and that they have therefore not examined assault and homi-cide side-by-side. This mismatch complicates attempts to compare the two crimes, since it opens the possibility that the differences in findings may be due to irrelevant features of the samples. The few studies that used the same settings and analysis periods have nevertheless yielded results much like those from other research. Rock et al. (2008) note, for example, that only a handful of Block’s (1984) analyses of identical units showed seasonality in both offenses. Studies of comparable areas by Michael and Zumpe (1983) also found seasonal variation only in assault, and research in England and Wales yielded the same result (Rock et al., 2008). In studies of city-level data, Deutsch (1978) and Harries (1989) similarly discovered evidence of assault seasonality only.

One of the exceptions to finding seasonal variation in assault but not in homicide was McDowall et al. (2012), who observed annual cycles in both crimes. This out-come is notable because McDowall et al. used city-level panel data that covered the largest set of areas and the longest time span of the existing research. Like their study, the current analysis uses data from a panel of cities. McDowall et al. focused on gen-eral characteristics that operated across many crimes, however, and they gave little attention to individual offenses. The present study more closely examines similarities and differences specific to seasonality in homicide and assault rates. It also uses a longer time series than did McDowall et al., and it undertakes analyses specifically tailored to the nature of the two offenses.

Explaining the Differences Between Homicide and Assault Seasonality

Given the similar circumstances surrounding assault and homicide, one might expect both crimes to have the same seasonal characteristics. The apparent absence of homi-cide seasonality has accordingly been a matter of concern to researchers, and they have proposed a variety of explanations to account for it. A notable example is Block (1984), who argues that neither assault nor homicide is in fact truly seasonal. Assaults are more likely to occur outdoors during the summer, however, increasing the chances that they will come to police attention then. Most studies use police-compiled Uniform Crime Report (UCR) data, and the summer assault peaks may therefore exist only as the result of a recording artifact.



4 Homicide Studies

Later analyses of victimization data did not strongly support Block’s hypothesis, at least for serious violence. Carbone-Lopez and Lauritsen’s (2013) study of the National Crime Victimization Survey (NCVS) found that a large seasonal cycle in aggravated assault remained after controlling for reporting behavior. Seasonality in less serious simple assaults did have a reporting component, and seasonal fluctuations in both aggravated and simple assaults differed across age groups. Overall, however, Carbone-Lopez and Lauritsen’s results suggest that crime reporting does not account for sea-sonality in aggravated assault. An earlier and less detailed analysis of victimization data by Dodge (1988) reached a similar conclusion.

Another explanation is that characteristics of assault offenders or their victims vary with the seasons, and that this operates to reduce seasonality in assault death rates. Although assaults may occur more often during the summer months, for example, summer assaults may less often result in serious injuries. The proportion of victims who die will then be lower during the summer, reducing any seasonal fluctuations in homicide.

Related to this possibility, assault victims could be physiologically more vulnerable to death during the winter season than is true during the summer. Fewer assaults may occur in the winter, but winter assault victims may be more likely to die of their wounds. In this case, identical injuries would lead to different outcomes depending on the time of the year.

In either of these situations, homicide-relevant characteristics of assault would fol-low a seasonal cycle that counterbalanced annual variations in the incidence of the crime. These offsetting forces would in turn work to flatten homicide rates across the year. Rock et al. (2008) find support for such a possibility, showing that the proportion of assaults resulting in death falls during the months in which the crime is most fre-quent. The months in which the fewest assaults occur, in contrast, are also the months in which they are most likely to be lethal.

Besides measurement artifacts and self-canceling cycles, a third possibility is that homicides display little apparent seasonality only because they occur less often than do assaults. The comparative rarity of the crime then makes homicide time series vulner-able to the influence of chance events and to random external noise. In a short series, the presence of these stochastic variations will obscure evidence of systematic structure and reduce the ability to identify seasonal patterns. Even long monthly time series cover relatively few complete cycles, and time series of crime rates are usually not long.

Examples of typical crime time series come from Block’s (1984) studies, which considered data spanning periods of between 4 years (California crime rates) and 20 years (Canadian crime rates). Other representative examples include Deutsch (1978), who analyzed 10 years of data; Hird and Ruparel (2007), who analyzed 6 years; and Schmid (1926), who analyzed 11 years. In all of these cases, the seasonal component is short enough that detecting a cycle in a noisy series would be a difficult task.

The amount of variation due to external sources may also differ across time or geography. Besides being affected by the length of the series, findings about seasonal-ity might therefore hinge in part on the specific periods and areal units under examina-tion. Seasonal patterns may stand out more sharply during some years than during




others, and some areas may have consistently larger seasonal cycles. Depending on the data under analysis, this variation could further reduce the chances of finding season-ality in homicide.

To address geographical variations in seasonal patterns and extend the available data series, several investigators have included more than one area in their studies. Block (1984), for example, examined an assortment of time series from cities, states, and nations.1 Deutsch (1978) similarly studied data from 10 cities, and McDowall et al. (2012) analyzed a multi-city panel. The larger number of areas provided oppor-tunities to replicate the results across diverse settings and helped make the conclusions less dependent on the features of a particular site.

Despite their advantages, additional areal units only affect cross-sectional varia-tion, and inferences about seasonal effects ultimately rest on the length of the temporal component. Increasing the number of geographical areas can be helpful, but it does not substitute for examining more repetitions of the seasonal cycle.

The Current Study

The major explanations of why homicides might appear non-seasonal are all reason-able on their face, and they are all worthwhile candidates for additional examination. In empirically evaluating these possibilities, however, the idea that previous research has overlooked the presence of homicide seasonality comes logically before the rest. The other explanations begin by assuming that homicides do not contain seasonal structure and they seek to account for why this is so. If seasonal homicide patterns do exist, the other explanations lose their force and become irrelevant. Before entertain-ing more complicated hypotheses, one would therefore want to give careful consider-ation to the possibility that in fact homicides are seasonal. The present study examines this possibility with three related analyses.

Seasonality in a Long Time Series

Among the existing research efforts, McDowall et al. (2012) used the largest combina-tion of spatial and temporal components, with 88 cities and 24 years. Their finding of a seasonal cycle in homicide rates is consistent with the idea that annual fluctuations in the crime will become apparent in larger sets of data. If one views each year as a single observation on seasonality, however, the McDowall et al.’s analysis amounted to an examination of only 24 cases.

The current study follows McDowall et al. in using panel data, but it expands the homicide analysis to 45 years. Longer time series will more effectively smooth envi-ronmental noise, making it easier to discern the shape and form of any seasonal pat-terns. Beyond increasing measurement precision, the additional temporal observations should also reduce variations unique to the particular years under study and so provide more stable estimates.

If the analysis finds that an annual homicide cycle exists, it can next compare homi-cide seasonality with seasonality in assault. As far as homicide and assault both follow



6 Homicide Studies

the same annual pattern, their similarity should be clear in time series as long as the ones here. A substantial divergence in the seasonal fluctuations would raise questions about whether the two crimes are as alike in their characteristics as theorists have usu-ally believed.

Variations in Seasonality Over Time and Space

Beyond examining the overall features of homicide and assault seasonality, the current study also considers the constancy of seasonal fluctuations across areas and years. Temporal or spatial differences in the seasonal patterns would provide evidence that sample composition can help account for the unsupportive results of previous homi-cide research.

Seasonal patterns might shift over time for many reasons, but technological change is perhaps the most obvious possibility. Air-conditioned housing units, for example, may reduce the frequency of serious assaults during hot weather periods (Harries & Stadler, 1989; Harries, Stadler, & Zdorkowski, 1984; also see Rotton & Cohn, 2004). The availability of air conditioning has expanded continuously since the 1950’s (Ackermann, 2002), and some authors suggest this may have imparted a downward trend on summertime violence (e.g., Arsenault, 1984). If so, more recent years might display less homicide seasonality than in the past.

A related but more speculative explanation is that improvements in medical tech-nology may have reduced the seasonal component in assault death rates. Harris, Thomas, Fisher, and Hirsch (2002) show that assault lethality has decreased since 1960, and they present evidence that this is due to advances in trauma care and related support services. Conceivably, these developments might have broken a link between the month in which an assault occurred and the chances that it would have a lethal outcome. As with the air-conditioning hypothesis, this would generate a downwardly sloping trend in the size of any seasonal homicide variations.

Besides changing over time, seasonal patterns might also differ across geographic areas. Fewer theories predict spatial differences in seasonality than is true for temporal ones, but spatial variation could arise from a variety of mechanisms. One possibility is that seasonal patterns are a non-linear function of temperatures, allowing unique rela-tionships to exist in different climate regions. Alternatively, crime seasonality might vary across areas depending on their social and demographic characteristics. Occupational structures might influence seasonal cycles, for example, so cities that are more reliant on manufacturing would have different patterns than would cities with larger educational or tourism sectors. In these cases, the specific cities in an analysis could affect findings about how seasonality operates.

The Impact of Temperatures on Violence Seasonality

Besides these two analyses, the article also examines the extent to which average monthly temperatures mediate the operation of seasonal patterns. The notion that tem-peratures influence human behavior has a long history in social thought, and Cheatwood




(1988) traces its origin back at least 5,000 years. Within criminology, Cheatwood calls attention to the importance of Quetelet’s (1842/1969) “thermic law” that linked warmer temperatures to higher rates of assaultive violence. Quetelet’s law had a con-siderable influence on scholars of his time, especially among members of the Italian School. The presumed impact of temperatures on offending continues to the present, and it is central to both major current explanations of crime seasonality, temperature-aggression theories, and routine activities theory.

Temperature-aggression theories appear in several variants, and disagreements exist between the competing versions (Anderson, Bushman, & Groom, 1997; Bell & Baron, 1976; Cohn, Rotton, Peterson, & Tarr, 2004). All temperature-aggression theo-ries nevertheless have in common the idea that human beings become more prone to violence as heat and humidity levels rise (e.g., Anderson, 1989). Rates of homicide and assault should therefore have summer peaks due to the frequency of uncomfort-ably warm temperatures during that time of the year.

Routine activities theory stresses the broader role of environmental conditions in structuring social interactions. For seasonality, it highlights how annual rhythms alter risks of violence by encouraging persons either to stay inside their homes or to venture outside into public areas (e.g., Cohn, 1990). Routine activities theory has somewhat more general applications to crime seasonality than do temperature-aggression expla-nations (Hipp, Bauer, Curran, & Bollen, 2004; McDowall et al., 2012). Still, like them, it predicts that violent crimes will rise during the warmer months.

An implication of both temperature-aggression and routine activities theories is that areas with more extreme monthly temperature differences should also experience more pronounced seasonal assault and homicide cycles. If temperatures mediate the influence of seasonality on assaultive violence, larger month-to-month changes in temperature should generate larger month-to-month variations in crime. Here again, conclusions about crime seasonality would partly depend on the characteristics of the areas under analysis.

Method

Data

The current study uses data from the 88 U.S. cities that had populations of 200,000 persons or more in the 2000 Census. It covers the periods between 1960 and 2004 for homicide and (due to data availability) between 1964 and 2004 for assault. The monthly crime counts were recorded by the UCR program, and the city populations for converting the counts to rates are interpolations of annual UCR estimates.

The data are from an agency-level UCR file constructed and distributed by Michael Maltz (2012). Although the UCR program collects information on monthly crime counts, it concentrates its resources on insuring the accuracy of the annual totals that it reports to the public. The program therefore devotes little attention to the monthly data, and it allows agencies to make intermittent submissions that aggregate multiple months into an overall total (Maltz, 1999, 2007). Distinguishing multi-period aggregates from



8 Homicide Studies

individual monthly counts can be difficult in practice, and a zero in the monthly file may mean either no offenses or no report.

For each of the more than 17,000 agencies that participate in the UCR program, the Maltz file identifies entries that are aggregated, missing, or apparently incorrect. This greatly reduces the problems in analyzing monthly UCR data, and it allows research-ers to avoid the inclusion of problematic submissions.

The current analysis examines only a fraction of the jurisdictions available in the Maltz file, and it might have expanded the cross-sectional dimension to encompass more cities. Areas with smaller populations have fewer crimes, however, and this will make seasonal patterns more difficult to detect. Even with a restriction to populations exceeding 200,000, some cities in the sample had zero homicides for multiple con-secutive months. Including less populous areas would make the homicide data even sparser, and continued relaxation of the size criterion would eventually affect the assault data as well. Although the population cutoff is somewhat arbitrary, it is consis-tent with the goal of maximizing the accuracy of the seasonal estimates.

Analytical Model

The analysis uses a panel data extension of the “classical” decomposition model, which divides a time series into trend, seasonal, and random components (e.g., Brockwell & Davis, 2002). The basic form of the decomposition is as follows:

Y Time Time January 1 1it i t n tn

t= + ( ) + + ( ) + ( ) +α β β γ. . . . .

November v11

.

,+ ( ) +γ t it

Here, the αis are coefficients for dummy variables (fixed effects) that separately index each city; β1 through βn are coefficients for a linear time trend and its powers; γ1 through γ11 are coefficients for a set of monthly dummy variables that measure the seasonal variations; and vit is an error specific to a year and city. December is the com-parison baseline for the other months.

The classical decomposition is only one of several reasonable approaches to extract-ing seasonal effects from a time series, but it has two distinct advantages in the present case. First, the model offers a simple way to incorporate panel data structures and to study differences in seasonality over space and time (Gorr, Olligschlaeger, & Thompson, 2003; McDowall et al., 2012). Most other available methods do not easily accommodate more than one geographical area, and most methods cannot consider both temporal and spatial variations.

Second, the classical decomposition provides flexibility in the form of the model coefficients, allowing many possible types of seasonal patterns.2 Some of the leading alternative approaches use only smooth curves to model seasonality, imposing a sym-metry that may be inconsistent with the data.

Related to the previous points, the classical decomposition separately estimates the size of the fluctuation for each month in a cycle. This is not true for other analytical




methods, which usually characterize seasonality more globally. A seasonal ARIMA model, for example, allows analysts to detect an annual pattern and to control its influ-ence (see, for example, Ghysels & Osborn, 2001). An ARIMA model nevertheless considers the entire yearly cycle as a whole, and it does not provide estimates of the differences between individual months.

Besides the city-specific fixed effects, seasonal dummy variables, and time trends, the basic model also includes a variable measuring the length of each month in days. Longer months have more opportunities for crime than do shorter ones, and this may produce a spurious cycle of its own. A related concern is that crime frequencies vary over the days of a week, and monthly differences between high-crime days and low-crime days will also create false seasonal patterns. One can allow for such effects by adding day-of-the-week variables to the decomposition, but these were consistently insignificant in the analyses and the models below omit them.

The basic decomposition model estimates the overall size and pattern of the sea-sonal variations in a time series. This addresses the questions of whether homicide seasonality will appear in data with large temporal and spatial dimensions, and of whether homicide and assault follow similar annual cycles. To assess the constancy of the effects across space, a second analysis adds a set of coefficients that allow interac-tions between the dummy variables for seasons and cities. To assess constancy over time, a third analysis similarly includes interactions between the seasonal dummy variables and the dummies for each year. Finally, to assess the mediating effects of temperature differences, an analysis adds to the models a measure of the average monthly temperature in each city.

Results

Overall Seasonal Patterns

Results for the basic homicide and assault models appear in Table 1. To avoid unneces-sary detail and to simplify the presentation, the table does not include the coefficients for the 87 cross-sectional fixed effects.3 The dependent variable in the equations is the logarithm of the relevant crime rate, which helps reduce heteroscedasticity arising from the positively skewed distributions. The standard errors of the coefficients are heteroscedasticity and autocorrelation consistent, controlling for serial dependence as well as for any remaining differences in the error variances.

Table 1 shows seasonal patterning in both homicide and assault. For each crime, the dummy variables indexing seasonality significantly improve the model fits after removing the influence of the other variables, F(11, 36832) = 12.77, p < .0001 for homicide; F(11, 40013) = 140.17, p < .0001 for assault. The finding for homicide helps support the idea that small areal samples and short time series can account for the absence of seasonality that has appeared in past research. Expressed in terms of substance, the homicide cycle has about a 10% difference between its July peak and its January trough. This is not strikingly large, and it is much less than the 29% peak-to-trough difference for assault. The contrast is nevertheless more consistent with



10 Homicide Studies

substantively meaningful fluctuations hidden in noise than with a trivial effect that is of little practical importance.

Figure 1 visually displays the seasonal curves for homicide and assault. For assault, the curve shows a rapid increase from winter to summer that is not as obvious in the curve for homicide. Figure 1 also calls attention to a December spike in homicide that

Table 1. Seasonal Decomposition Model for Homicides and Assaults.

Homicide coefficient (SE) Assault coefficient (SE)

January −0.08334* −0.03508*(.01321) (.00687)

February −0.07004 0.07834*(.06358) (.02197)

March −0.08332* 0.05426*(.01226) (.00629)

April −0.06679* 0.14278*(.02349) (.01070)

May −0.05870* 0.17115*(.01226) (.00855)

June −0.03957 0.22023*(.02512) (.01225)

July 0.01595 0.23127*(.01146) (.01225)

August 0.00922 0.21712*(.01241) (.00975)

September 0.00584 0.20850*(.02600) (.01161)

October −0.01412 0.11848*(.01238) (.00640)

November −0.04048 0.07044*(.02735) (.00932)

Time 0.02324 0.41584*(.01720) (.03300)

Time2 −0.07182* −0.04831*(.01275) (.02410)

Time3 0.01101* −0.01022(.00305) (.00640)

Time4 −0.00332* −0.01201*(.00170) (.00387)

Total days 0.03625 0.06102*(.02254) (.00761)

Constant −0.82878 1.19562*(.69933) (.23955)

*p < .05, two-tailed test.




is absent in assault. The spike makes the December homicide rate almost as high as its July peak, and several other studies have also observed this characteristic (e.g., Cheatwood, 1988; Deutsch, 1978; Lester, 1979). December is the only month where the assault and homicide patterns do not at least roughly correspond to each other, making it the largest difference between the two crimes.

At a broad level of comparison, homicide and assault have similar seasonal cycles. Both offenses peak in July, and both are lowest in January. Assault nevertheless dis-plays considerably more variability than homicide, and its monthly dummy variables are all significantly different from the December baseline. For homicide, the seasonal fluctuations are less extreme, and none of the months between June and November significantly differ from December. Both assault and homicide rates are seasonal over-all and both follow generally comparable patterns. Still, homicide is flatter over its yearly cycle than is assault, and the impact of seasonality is much smaller.

Spatial and Temporal Variation in Seasonality

The basic decomposition analysis assumes that homicide and assault seasonality do not vary across time or space. Tests of temporal invariance provide evidence that fluctua-tions in assault are, in fact, constant over time while seasonal homicide patterns are significantly variable. In particular, interaction terms allowing the seasonal coefficients to change from year to year improve the homicide model fit, F(484, 36303) = 1.11, p < .05, but do not affect the model fit for assault, F(451, 34520) = .84, p > .99. This

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Homicide Assault

Figure 1. Basic seasonal patterns for homicides and assaults.



12 Homicide Studies

outcome suggests that one could reach different conclusions about homicide depending on the period under study. Seasonal patterns in assault, in contrast, should have the same characteristics in any two equally long strings of data. Choice of time periods might then help account for the difference in previous findings about the crimes.

Although seasonal homicide fluctuations vary across time, the differences do not have a discernable structure. Figure 2 plots the variation in the seasonal coefficients for each study year, with vertical lines connecting the year’s highest and lowest months.4 The larger the variation that occurs within a year, the wider the line and the greater the degree of seasonality. If the seasonal fluctuations were unchanging, the spread of the coefficients would be identical over the study period. Consistent changes in the seasonal cycle would appear as trends in the monthly variations over a multi-year interval.

Figure 2 shows substantial dispersion in the month coefficients, but these differ-ences do not reveal any predictable features. Some years (such as 1984 or 1998) have relatively limited seasonal variation and other years (such as 1973 or 2002) have a great deal of it. The sizes of the differences nevertheless appear essentially random, and the graph does not show, for example, the steadily decreasing seasonality that a hypothesis about technological advances would require.

In contrast to the temporal differences for homicide, neither homicide nor assault displays evidence of cross-sectional variation in its seasonality. Interactions between cities and months do not significantly improve the fit of the original decomposition model for either crime, F(957, 35875) = .89, p > .99 for homicide; F(957, 39056) = .51,

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Figure 2. Relative variation in homicide seasonal patterns, 1960-2004.




p > .99 for assault. For large U.S. cities, at least, this supports the idea that seasonality operates identically over geographical units. Conclusions about the nature of seasonal-ity in assault and homicide should therefore not depend on the specific areas under analysis.

The Influence of Temperatures in Mediating Seasonality

The final analysis, reported in Table 2, adds to the models a measure of mean monthly temperatures in each city.5 Temperatures affect both homicide and assault rates, and including them in the models reduces the seasonal coefficients. Statistically significant amounts of seasonal variation nevertheless remain in both crimes, F(11, 36830) = 8.62, p < .01 for homicide; F(11, 40011) = 2.07, p = .02 for assault. Temperatures therefore only partially mediate seasonality, and meaningful components of the annual cycles are due to other sources.

To clarify how much the seasonal patterns depend on environmental differences, Figure 3 displays plots of homicide and assault before and after adding the tempera-ture variables.6 The figure shows that month-to-month temperature variations are a major source of seasonality in assault, and that its annual fluctuations are much flatter with temperatures held constant. This flattening is especially apparent during the sum-mer months, where the differences between assault and homicide were originally larg-est. Homicide also becomes slightly less seasonal after allowing for temperatures, but overall it has much the same pattern as in the original analysis.

Holding temperatures constant, homicide and assault are almost equal in their sea-sonal variations, with peak-to-trough differences of 11% for homicide and of 10% for assault. This corresponds to the original differences of 10% and 29%. Again, much of the seasonal fluctuation in assault appears to exist because of temperature variations over the year, but this is less true for homicide.

A notable change in the homicide pattern after removing temperature differences is that December replaces July as the highest month in the yearly cycle. Although the source of the December peak is not obvious, the Christmas holidays are a time of heightened alcohol consumption and informal socializing (Lemmens & Knibbe, 1993; Time Use Institute, 2010; Uitenbroek, 1996). These and other holiday activities may raise the potential for violent interactions, and they could help account for the higher December homicide rate. The most obvious mechanisms would require a similar peak in assault, however, and such a spike is not visible in the data.

The analysis of geographic constancy showed that the homicide and assault cycles operate identically across space, implying that the sample composition should not affect conclusions about seasonal variation. The results of the mediation analysis qual-ify this conclusion in a minor way. This qualification arises because areas with greater temperature differences will experience stronger seasonal cycles than will areas where temperatures are more constant. Even though seasonality operates identically across space, detecting it will be easier in cities that have larger annual fluctuations. The par-ticular areas under study can therefore ultimately affect the chances that an analysis will find the presence of a seasonal homicide pattern.



14 Homicide Studies

Routine activities and temperature-aggression theories both predict that tempera-tures will mediate the influence of violent crime seasonality, and the findings about mediation are partially consistent with their expectations. The logic of the theories

Table 2. Seasonal Decomposition Model for Homicides and Assaults, Controlling for Monthly Temperature Differences.

Homicide coefficient (SE) Assault coefficient (SE)

January −.07600* −.01897*(.01295) (.00665)

February −.07100 .07493*(.06379) (.02159)

March −.10265* .00758(.01337) (.00695)

April −.10627* .04956*(.02705) (.01229)

May −.11802* .03119*(.02365) (.01392)

June −.11775* .03615(.03619) (.01941)

July −.07204* .02380(.03039) (.02021)

August −.07626* .01509(.03084) (.02024)

September −.06549 .04024*(.03483) (.01803)

October −.06156* .00633(.01938) (.01133)

November −.06285* .01785(.02846) (.00982)

Time .02224 .41575*(.01719) (.03299)

Time2 −.07176* −.04826*(.01274) (.02410)

Time3 .01110* −.01021(.00306) (.00650)

Time4 −.00333* −.01202*(.00169) (.00387)

Total days .03619 .06095*(.02254) (.00759)

Monthly temperature .00250* .00585*(.00080) (.00058)

Constant −.95644 .89548*(.70008) (.24221)

*p < .05, two-tailed test.




suggests that temperatures will have identical effects on the seasonal patterns in both crimes, however, and the analyses show that this is not so.

Of the two theories, routine activities theory appears better-suited to explaining the mediation analysis results. Routine activities theory stresses the role of social vari-ables in generating crime, and an adaptation of it might allow for the differences between the assault and homicide findings. The December homicide peak, especially, might be the outcome of social interactions that are more common during the holidays. If these interactions often occur in settings with easy access to drugs and deadly weap-ons, holiday-period assaults might have a larger chance of becoming murders.

Temperature-aggression theories in contrast rely heavily on physiological responses to heat that should affect assault and homicide offenders identically. The prospects of changing the theory to allow for the differences in the seasonal patterns are therefore not as promising. This more positive assessment of routine activities explanations is consistent with the conclusions that Hipp et al. (2004) reached in their comparisons of the two theories.

Additional Analysis

Classical decomposition implicitly assumes that the observations in a time series are covariance stationary. Covariance stationarity requires a series to possess a constant mean and variance after allowing for the operation of any deterministic trends (see, for

December

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Janu

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June Ju

ly

Augus

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Novem

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Homicide Homicide controlling for monthly temperatureAssault Assault controlling for monthly temperature

Figure 3. Seasonal patterns for homicides and assaults, controlling for monthly temperature differences.



16 Homicide Studies

example, Enders, 2010, pp. 53-55). An alternative possibility is that a series is not stationary around a mean or deterministic trend, and that it instead contains stochastic trends or time-varying drifts. The classical decomposition is largely a descriptive device, and its results will hold over the study period even if the homicide and assault series are not stationary. Stochastic trends or drifts are nevertheless clearly possible, and one would have more confidence in the findings if they did not rest on the assump-tion of stationary series. A supplementary analysis therefore considered the potential impact of stochastic trends or drifts—more formally, “unit roots”—on the results.

A variety of panel unit root tests (Baltagi, 2008, pp. 278-282) rejected the presence of stochastic trends or drifts in both homicide and assault. Although this is consistent with stationarity, the tests will produce such a result if it is true even for a few panel members (see, for example, Pesaran, 2012). This was the case in the present situation, where unit root tests for individual cities showed that only a handful of areas accounted for the overall rejections.

A cautious approach would therefore be to analyze the data as though both the homicide and assault panels violated the stationarity assumption. One can make a unit root series stationary by converting it from its original values to its first-differences. The differencing operation removes linear stochastic trends and drifts, leaving a set of period-to-period changes. In repeating the analyses on differenced data, the spatial interaction coefficients for both homicide and assault became statistically significant. In other respects, the findings for the crimes were qualitatively similar to those from the main analysis. The two sets of models are not identical in their results, but they are similar enough to conclude that the major seasonal features do not depend on the ana-lytical method.

Discussion

The analyses in this article considered whether homicide follows a seasonal cycle, and if it does, whether its cycle resembles the one in assault. The answer to the first ques-tion is clearly affirmative, since the homicide analyses consistently found that the crime exhibited an annual pattern. Seasonal fluctuations in homicide are much smaller than those in assault, however, making it more difficult to separate them from back-ground variation.

The strength of homicide seasonality also varies over time, and temperatures help mediate the cycles in both offenses. The variability implies that differences in sample composition may influence findings about seasonal patterns, especially for homicide. Because homicide seasonality is weaker to begin with, its sensitivity to situational features further increases the chances of failing to observe seasonal fluctuations in it.

The variability in seasonal patterns opens the possibility that the findings in the current article are themselves in part characteristics unique to the study period. The analysis considered spans of 45 years for homicide and of 41 years for assault. These are much longer intervals than in previous research, but they are still brief enough that they might conceal broader patterns of relationship. In particular, seasonal features may shift or evolve over time in subtle ways that the current study cannot detect.




An example of evolving seasonal patterns comes from Ajdacic-Gross and col-leagues (2005), who studied Swiss suicide rates between 1896 and 2000. They found that suicide seasonality became steadily less pronounced over the 125-year period, enough so that its annual cycle appeared likely to disappear entirely in the not-distant future. Other researchers (e.g., Simkin, Hawton, Yip, & Yam, 2003) also report similar findings for suicide over less extended intervals and in other areas.

Progressively more muted seasonal patterns might conceivably also occur for homicide and assault rates. Ajdacic-Gross et al. attribute the Swiss reductions to increasing urbanism and the declining size of the agricultural workforce. These and similar technology-induced changes might possibly have similar effects on seasonal fluctuations in other types of violence. A flattening pattern was not apparent in the present analysis, but small or very gradual influences could require additional time to become detectable.

More generally, homicide and assault seasonality may have stood out with greater or lesser clarity in earlier periods than they do in the present. Future patterns may also differ from the ones that operate now. Stine, Huybers, and Fung (2009) present evi-dence that temperature seasons have shifted worldwide, so that by 2007 warm weather began about 2 days earlier than in 1954. This difference is too small to have a notice-able influence on crime, and the current findings should be stable short of an order-of-magnitude expansion in the study period. Still, the possibility of changes due to climate or social conditions points to the desirability of continued investigation of crime sea-sonality, adding to the series as more data become available.

While the results provide unambiguous support for homicide seasonality, the ques-tion of whether assault and homicide have the same pattern requires a more nuanced answer. The lower level of certainty about a shared cycle comes from the fact that any correspondence between the offenses is a matter of degree. Seasonality in both crimes has the same overall fluctuations, with summer peaks and winter low points, and the seasonal features mirror each other in additional ways as well. The larger assault rate variations nevertheless mean that the crimes are not identical in their patterns, and that homicides are not a constant proportion of assaults through the year. Assault and homi-cide also do not respond to monthly temperature differences in the same way, with fluctuations in assault being much more sensitive to temperatures than is true for homicide.

The current data do not allow a detailed investigation of why seasonality in assault and homicide might diverge like this, but the general outline of an explanation is straightforward. Given the pattern of results, the characteristics of either assault offenders or of assault victims must follow their own unique annual pattern. In both cases, the seasonality in assault outcomes would differ from the seasonality in assault incidents themselves.

In this connection, Rock et al. (2008) suggest that physiological variations make winter assault victims more vulnerable to fatal injury than is true in the summer. Such a victim-based possibility is consistent with the findings of the current study, espe-cially after controlling for temperatures. An alternative but structurally similar offender-based explanation is that motivations for assault differ over the year, and that



18 Homicide Studies

summer assailants have less desire to kill than do their wintertime counterparts. As with an explanation that focuses on victims, the mechanisms that link assaults and homicides would accordingly change with the seasons.

Both the victim-based and offender-based explanations begin from the idea that homicides are always the product of assaults. In different ways, both explanations then try to account for why assaults are not all equal in their lethality. The current data do not provide a way to distinguish between the two possibilities, but this is an issue worth future investigation.

Speculatively, annual changes in human biochemistry could in part underlie both victim-based and offender-based explanations. Rock et al. (2008) note that prothrom-bin time—the time required for blood to clot—has a seasonal component that might help explain the differences in the homicide and assault patterns. Blood clots faster in the summer than in the winter, and this makes it less likely that summer assault victims will die. The lethality difference is consistent with the higher summer peak and larger variations across seasons in assault.

In a different context, Jamison (1999) points out that seasonal variation exists in many of the brain chemicals that influence mood disorders. She suggests that these biological rhythms may account for the substantial seasonality in suicide rates. An obvious possibility is that related biological changes operate to help produce the sea-sonal structures in homicide and assault.

Suicide seasonality has itself been a longstanding topic of research, and similarities and differences between it and assaultive violence may be useful in understanding the annual cycles in both. Like homicide and assault, suicide is lowest during the winter months. Unlike the crimes, however, suicide peaks in the spring rather than in the sum-mer (see, for example, Chew & McCleary, 1995; Jamison, 1999). These timing differ-ences could help in identifying common and divergent features in the two types of violence, improving the understanding of each individually. They have relevance as well for assessing and elaborating theories that view self-directed and other-directed killings as manifestations of a common impulse (Unnithan, Huff-Corzine, Corzine, & Whitt, 1994).

Different patterns within subcategories of homicide and assault may also be useful in explaining the origins of the seasonal cycles. The current study has concentrated on the overall rates of both offenses, but these totals could hide unique seasonal patterns that operate within finer classifications. Using NCVS data, Carbone-Lopez and Lauritsen (2013) found that seasonality in assault victimization varies across locations and age groups. The same general types of differences might exist in the police-recorded incidents that the present study examines. Consistent with this possibility, McCleary and Chew (2002) showed that homicides involving infants follow an annual pattern that differs from that for other age groups. Block (1984) similarly reported that some forms of crime seasonality differ across urban and rural locations.

Homicide and assault may in fact have seasonal patterns that correspond or diverge across multiple dimensions. Attacks involving robbery and other instrumental vio-lence, for example, might follow different cycles than do expressive offenses that arise from arguments. Some categories of homicide and assault might also have opposing




seasonal fluctuations, reducing the apparent amount of seasonality in the offenses overall. Alternatively, only some of the categories might be seasonal, adding noise to the aggregates that would complicate efforts to detect their annual cycles.

The present study has considered only overall patterns, and the number of pos-sible categorizations makes a study of disaggregated seasonality beyond its scope. UCR data, although desirable for present purposes, also limit the range of topics that a disaggregated analysis might consider. The UCR lacks the detail necessary for extensive investigation of subcategories, and low monthly homicide frequen-cies in many cities would compromise the results of such analyses in any case. Future research might nevertheless examine assault and homicide subcategories using data from only the most populous cities or similarly large areal units. More directly, detailed characteristics of homicides and assaults are available in a variety of other major data systems. These include, for example, the National Incident-Based Reporting System, the National Violent Death Reporting System, and the Supplementary Homicide Reports.

As with comparisons to suicide, study of seasonal variations in categories of homi-cide and assault could provide evidence about how the aggregate patterns arise. Examining variations over many dimensions is accordingly a worthwhile strategy for better understanding seasonality, and it would be a useful topic for future study. Besides narrowing the analytic categories, future work might also broaden them. This could involve, for example, using a sample of nations to study seasonal patterns in an international context.

Conclusion

The nature of their sources aside, seasonal cycles in assault and homicide are simi-lar in their general form and operation. Criminologists often regard the two crimes as essentially identical in their basic characteristics (e.g., Block, 1987; Harries, 1989, 1997), and the results here are consistent with a weak version of that view. The seasonal patterns in homicide and assault are close to each other, and this sug-gests that many of the same conditions operate to generate both types of violence. The crimes nevertheless diverge too much in their seasonality to support the idea that the only difference is in their outcome. Additional side-by-side comparisons of the offenses should be helpful in gaining a better understanding of their individual characteristics.

At a broader level, a seasonal pattern is a source of variation that operates predict-ably over time. To the degree that seasonal fluctuations are a non-trivial component of the variation in a time series, the forces that underlie seasonality are a meaningful part of its causal structure. Seasonal patterns do not affect the overall volume of homicides and assaults within a year, but they do influence when they will occur. These differ-ences in the timing of offenses can be important for theoretical understandings of assaultive violence as well as for public policies designed to control crime. Despite many advances in understanding crime seasonality over the last century-and-a-half, the topic therefore remains a promising area for future criminological study.



20 Homicide Studies

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Notes

1. The Block analysis also considered slightly different time periods, but these differences were for the most part minor.

2. Cheatwood (1988) argues that a seasonal cycle must operate symmetrically over the cal-endar seasons, and he rejects the possibility of more general patterns. Other researchers often fail to as rigorously define seasonality, but they generally assume that it is any type of annually recurring cycle (e.g., Block, 1984; Ghysels & Osborn, 2001; Granger, 2001; Miron, 1996). The current article also uses this wider definition.

3. Also as an aid in presenting the results, the models in Table 1 include only trend variables for time (grand mean centered) and its second, third, and fourth powers. For both homi-cides and assaults, additional polynomials of time contributed significantly to the models, but polynomials higher than the third order did not affect the other estimates. To make the coefficients larger and easier to interpret, the time variables are all divided by 100.

4. The relative positions of the months differed from year to year, but this variation did not follow a clear pattern. To avoid a large amount of unnecessary detail, the figure therefore includes only the highest and lowest months.

5. The temperature data are from the U.S. Weather Service, and were current as of June 5, 2013. They were retrieved from http://www.eachtown.com. As before, the results do not include the coefficients for the cross-sectional fixed effects.

6. The before plots are identical to those in Figure 1.

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Author Biographies

David McDowall is a Distinguished Teaching Professor in the School of Criminal Justice at the University at Albany–SUNY, and codirector of the Violence Research Group. His current research interests include crime trends in the United States and the world, and the causes and consequences of interpersonal violence and criminal victimization.

Karise Curtis is a doctoral student in the School of Criminal Justice at the University at Albany-SUNY, where she also received her Master’s Degree. Her current research interests include criminally and politically violent crime trends cross-nationally, particularly in how violence is affected by temporal and spatial changes.



Documents

Seasonal Variation in Homicide and Assault Across Large U.S. Cities