CONCEPTOS DE SOLUCIÓN INTEGRADOS

We used EViews 8 software for the regression analysis. The preparation of the data took considerable time – 238,208 rows of data by agent by month had to be aggregated into 21,117 rows of data by zip code by month. Lottery sales data were prepared in Excel and imported into EViews. The econometric software then automatically generated the zip-code and time-period fixed effects variables. Rather than using gross lottery sales for each period as our dependent variable, we run the models using the “change in sales” (Δ Sales) over the same month a year ago for each zip code. This way, factors that impact sales in both periods net out, and will help to better isolate the impact of the introduction of a new casino nearby. In addition, because there is likely to be seasonal variation in lottery sales (e.g., ticket sales may increase in December because many people buy lottery tickets for holiday gifts), the lottery sales variable is calculated as the change in sales over the same month in the prior year, rather than from the immediately preceding month.⁹

8 See Walker and Nesbit (2014, 28) for a discussion and application to intrastate casino competition in Missouri. The fact that 1/tt fits the data better here than 1/tt² does not conflict with the “gravity model”

analyses employed by Cummings and others. In those models, it is only the patronage of one casino versus another that is related to 1/tt², not casino patronage overall.

9 This procedure, however, does have the effect of reducing our 56-period dataset to one of just 44

Thus, our model explains the dollar change in sales for zip code i in period t, calculated:

Δ Sales$ i,t = Sales$ _i,t – Sales$_{(i, t-12)}.

The change in the gravity variable over the same month the prior year is our primary explanatory variable:

Δ Gravity i,t = Gravity i,t – Gravity(i, t-12),

where Gravity = 1 / travel time. Thus, a particular month’s change in sales over the same month the last year is explained in the model by the change in the gravity variable over the same month the year prior, plus an error term. For our basic model we have:

Δ Sales$ i,t = Constant + Δ Gravity + ε i,t

The constant term in this type of model represents the sum of the zip code and month fixed effects coefficients. We ran the model for each type of lottery game (refer back to Table 1 for definitions); the results are presented in Table 10.¹⁰

The positive constant term in each model shows the average change in sales over the same month last year, for all zip codes and all months in the sample. Given that the constant term represents the average of the time and zip code fixed effects, the coefficients are based on all the unique characteristics in the zip codes, as well as any factors that affect all zip codes during any particular month. The positive result suggests that all games would naturally see increased sales over the sample period. However, the negative sign on the Δ Gravity variables indicate that all game types, except “Other” games, see a decline in the Δ Sales as casinos get closer.

10 Results for fixed effects variables are usually not shown in econometric output; we omit the variables for brevity and because we are not going to be interpreting the coefficients. They may, however, ultimately be useful for some other type(s) of analyses the Maryland Lottery may wish to conduct so we

Table 10. Basic regression results.

(Dependent variable: Change in Sales over same month 1 year ago) Instant Monitor Pick Multi-State Other Constant^a

a The constant coefficients reported are the average of the fixed effects coefficients in the regression.

b The p-values in this and subsequent statistical tables provide a quick way to interpret whether the particular coefficient estimate is statistically significant. Generally, a p-value below 0.10 indicates statistical significance at standard levels in economics. Smaller p-values can be interpreted to mean increased statistical significance. The

“gravity” coefficients presented above are, for example, all significant at the “99%(+)” level.

The interpretation of the Δ Gravity coefficients in the table above is not straightforward because of the “change in” format of the variables. Recall that the Δ Gravity variable is the change in “1/travel time” over the same month the year before. If in May 2012 the closest casino to a particular zip is 100 minutes away, the value of the gravity variable would be 1/100, or 0.01.

If in May 2013 a new casino opens only 20 minutes away, the gravity variable increases to 1/20, or 0.05. So the Δ Gravity for May 2013 would be 0.04. Then a negative coefficient on the Δ Gravity variable indicates that an increase in the gravity variable results in a decline in sales (or negative Δ Sales$). Such is in fact the case we found for all game types except the “Other”

category. The magnitudes of the coefficients are not immediately obvious because they are in dollar terms, not percentages.

The somewhat low R²s in each case should not be of concern. Again, we are not trying to

“explain” all aspects of lottery sales – only to isolate the part that may be related to casinos. And in each case, that “part,” while only a small portion of “everything going on,” is highly

statistically significant. The results from Table 10 will be used later to calculate estimated overall impacts of casinos on Maryland lottery sales, as discussed below in Section 10.