Statistical Analysis Models

Cross-sectional time-series Poisson regression models were specified for this study. In particular, two DD models and one DDD model were constructed. Any geographical crime displacement/attraction effect were estimated based on the DDD

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Changes in police-patrol strategies at a district level typically result in changes in inputs of police- prevention activities and changes in numbers of arrests, influencing crime-incident outcomes. To address this police-patrol strategy change effect on crime outcomes, it might be ideal to use the police districts as the regional variables. However, there were frequent changes in sectioning police districts in Philadelphia during the period from 1998 and 2011. For consistency, the police division was used in the current paper instead.

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model, which is of the most interest in the current paper, since the DDD model is more rigorous approach than the DD model in examining whether crime patterns were altered when stores started to sell liquor on Sundays.

A DD method was employed to overcome problems associated with a single difference model. A simple pre-post comparison model that compares crime outcomes before and after a time of Sunday-open permission is the easiest model. However, such a simple model is unlikely to address potential threats to the internal validity. For example, a major historical event coinciding with the Sunday-sales-ban repeal by chance might be a true cause of changes in outcomes. A DD model yields more reliable estimates by including an additional difference ― when two comparable groups across which any confounding factor influences equally are compared, the equal effects by the confounding factor are differenced away.

The current paper suggested two sets of the DD model. The first DD model had the difference between Sundays and the other days of the week for the treatment group in addition to the pre-post comparison surrounding the repeal of the ban. In other words, the two differences were (1) whether changes in crime rates within 1/8 mile radius areas of the treatment group stores occurred on Sundays or on the other days of the week, and (2) whether the incident occurred before or after the time the treatment-group stores were allowed to sell on Sundays, February 9, 2003. Because only the treatment group stores were considered in this first DD model, the number of observations for this model was 30,678 (6 stores * 5,113 days). The effects of any confounding factor that influenced equally on Sundays and the other days of the week were expected to be differenced away.

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This DD modeling is especially useful in adjusting for local confounding factors at a micro-level, such as store-specific changes.

The first DD model can be specified in a regression form. Model 1 below

represents the first DD model, with the Sun*Postijt main interaction term variable, which

were assigned to one if a day falls on a Sunday after the repeal, and to zero otherwise. In the model, i stands for each of the six treatment W&S stores (i=1, 2..., 6), while j stands for a day of week (j=Sun, Mon, Tue…, Sat) and t for one of 5,113 dates from 1998 to 2011 (t=1/1/1998…, 12/31/2011). Yijt indicates a number of crime incidents occurring

within the relevant mile radius as a response variable. A dichotomous variable, Sunj, is

assigned to one if the day of the week is Sunday, and to zero otherwise. Postt is assigned

to one if the date falls during the pre-repeal period, and to zero if it falls during the post- repeal period. FE_Year/Month variables stand for the fixed effects of years and months. The five Police_Division variables are respectively assigned to one if a W&S store in located in the Northeast, Northwest, South, Southwest, or East police divisions, with the Central police division serving as the reference. Holidayt is assigned to one if a date falls

on any one of eight holidays or their eves but does not fall on Sunday, and to zero otherwise.

Yijt = α0 + α1·Sun*Postijt + α2·Sunij + α3·Postit + FE_Year + FE_Month +

Police_Division1 + ~ + Police_Division5 + Holidayt + εijt ……… (Model 1)

Another DD model exploited an additional difference of Sunday crime incidents between the treatment and control groups, looking instead at the difference in crime

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incidents for the treatment group only between Sundays and non-Sundays. This second model compared crime incidents occurring on Sundays only (1) between the treatment group and the control group among the entire sample, in addition to (2) the comparison between before and after the repeal of the Sunday liquor sales ban on February 9, 2003. Because this second DD model counted crime incidents occurring only on the 730 Sundays occurring between January 1, 1998, and December 31, 2011, the number of observation was 22,630 (31 stores * 730 days). Modeling with these two differences renders an advantage in that any effect of a confounding factor on crime outcome is differenced away, as long as the treatment and control groups are affected equally by that factor. This modeling is especially useful in addressing global confounding factors at a macro-level, including shifts in overall crime trends or the effect of the economic recession on crime.

The second DD model can also be specified in a regression form, as depicted in Model 2 below. Notations are in general identical to those in the first model but with a few differences. The main interaction term variable in this model, Treat*Postit, is

assigned to one if a W&S store is allowed to open on Sunday by the repeal and the date is on or after February 9, 2003, and to zero otherwise. Now i covers all the 31 W&S stores (i=1, 2..., 31), but t is limited to 730 Sundays occurring between January 1, 1998, and December 31, 2011. Also note that there is no j term in this model. Treat is assigned to one if a W&S store belongs to the treatment group, and to zero otherwise.

Yit = β0 + β1·Treat*Postit + β2·Treati + β3·Postt + FE_Year + FE_Month +

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The two DD models, however, have their own weaknesses. The strength of one can be regarded as the weakness of the other. The first model ― with the Sunday vs. non- Sunday and the pre- vs. post-repeal differences ― is easily influenced by macro-level changes, such as an effect of an economic boom or recession. On the other hand, the DD model ― including the treatment vs. control groups and the pre- vs. post-repeal

differences ― cannot appropriately respond to particular local variations.

Adding another (the third) difference to a DD model can address the limitations of the DD models. The DDD model has three differences: (1) whether a crime incident occurring within the 1/8 mile radius areas of the treatment group occurred on a Sunday or on the other days of the week, (2) whether a W&S store belonged to the treatment group or the control group, and (3) whether the incident occurred before or after February 9, 2003, when the six treatment-group stores were allowed to open and sell liquor on

Sundays. This DDD model is expected to yield the most reliable estimation results on the effects of the repeal on crimes occurring within the 1/8 mile radius areas of the W&S stores in Philadelphia.

The regression form of the DDD model is depicted in Model 3 below. Notations are in general identical to those in the previous DD models, except that now i stands for each of the 31 W&S stores (i=1, 2,.., 31) and t for each of 5,113 dates from 1998 to 2011 (t=1/1/1998…, 12/31/2011). Note also that the variable of main interest in this DDD model is Treat·Sun·Postijt, that is assigned to one only if a W&S store belongs to the

treatment group, the day falls on Sunday, and that a date is on or after the repeal; all other values are assigned to zero.

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Yijt = γ0 + γ1·Treat·Sun·Postijt + γ2·Treat·Sunij + γ3·Treat·Postit + γ4·Sun·Postjt +

γ5·Treati + γ6·Postt + γ7·Sunj + FE_Year + FE_Month + Police_Division1

+ ~ + Police_Division5 + Holidayt + εijt ……… (Model 3)