State lotteries as a source of revenue: a re-examination.

I. Introduction

Most of the academic literature that has examined state lotteries has had a negative assessment of them. Studies of lotteries as a source of revenue have reported that even under the best circumstances, they generate only a tiny proportion of a state's revenues, and the lottery revenues are so volatile that states cannot and should not be dependent on them. In addition, the general consensus is that lotteries have high administrative costs, and that they may reduce other sources of state revenue.(1)

In contrast to this negative view prevalent in the academic literature, lotteries have been exceedingly popular with policymakers in state capitals and among the general public. Most of the literature which casts a negative light on lotteries as a revenue source looked at data through the mid-1980s. Since then, the number of states operating lotteries has grown from 17 in 1985 to 33 in 1992. In only one state (North Dakota) during the past 30 years has a lottery referendum failed to achieve a majority of votes.(2) Why are lotteries so popular? Is it because voters are ill-informed and unsophisticated? Do proponents mistakenly believe that they will help cure government revenue shortfalls? Or is it possible that lotteries really do have a significantly favorable fiscal impact, that lottery revenues are not as small and destabilizing as the academic literature maintains?

A reexamination of lotteries as a source of revenue is warranted for two reasons. First, previous studies that have examined this issue [13; 3] have used data only through the mid-1980s, when lottery sales consisted primarily of instant and numbers games and lotto was in its infancy. Several studies [4; 6] have found that lotto has become enormously popular, and that revenues generated by lotto are "new money," that is in addition to revenues from the other games. Also, lotto seems to attract a different clientele than the other games: instant and numbers sales are primarily to lower income, less-educated consumers, while participation in lotto is much more apt to include middle and high income groups. Thus, it is reasonable to surmise that as lotto sales have surged, both the magnitude and the stability of lottery revenues over time may have changed.

Another, more crucial reason lottery revenues merit further analysis is that previous studies have all examined the volatility of this revenue on a stand alone basis, rather than in terms of the contribution of lottery revenue to the stability of overall state revenue. As a consequence of this failure to examine lottery revenue in a portfolio context, the earlier studies imply that lottery revenues, being highly volatile, destabilize overall revenue. In fact, given the likely low correlation of lottery revenue with non-lottery revenues, it is entirely possible that lotteries provide states with an attractive diversification vehicle and, on balance, help to stabilize overall revenues. If so, this could at least partly account for their popularity.

The balance of the paper is organized as follows. Section II contains a review of the literature on the fiscal implications of lotteries. In section III, we discuss sources of data and the methodology used to examine the contribution of lotteries to the size and variability of overall state revenues. The results are presented in section IV, and section V concludes the paper.

II. Literature Review

Proponents of the lottery, according to Mikesell and Zorn [13], believe it is an outstanding state revenue source for a number of reasons. Participation is voluntary, unlike payment of taxes. It generates enough revenue to relieve pressure on the fiscal system, and provides an alternative to illegal gambling through legal games that have been virtually scandal-free.

Clotfelter and Cook [2] report that per capita sales in lottery states have grown from $23 in 1975 to an average of $88 in 1985, measured in 1985 dollars. This represents a per capita sales growth of 14 percent a year. By 1986, lotteries in the United States were selling $12 billion worth of tickets per year, and generated $4.7 billion of net income to states after prizes, commissions and administrative costs had been subtracted. Sales growth has slowed somewhat since then, but more states adopted lotteries and, by 1992, aggregate lottery net income was nearly $7.8 billion.

Opposition to lotteries in the academic literature is extensive. Critics assert that lotteries provide only a negligible proportion of state revenues, that lottery revenues are highly volatile, that lotteries have much higher administrative costs than alternative revenue streams, and that lotteries can reduce other sources of state revenue such as taxes on parimutuel wagering and sales taxes.

Previous studies find that despite their rapid growth, lotteries still represent only a small percentage of revenues in states that have them. Mikesell and Zorn [13] report that in the 17 states with lotteries operating in 1984, net revenues from lotteries averaged just 1.95 percent of own-source revenues (which they define as general revenue less intergovernmental revenue). In only two states, Pennsylvania and Maryland, did lotteries account for more than 4 percent of own-source revenue.

In addition to the low yield, Mikesell and Zorn [13] and Clotfelter and Cook [3] find that lottery revenues are highly volatile and unpredictable.(3) This volatility has been attributed to the introduction of new games, changes in consumer preferences, variations in the intensity and effectiveness of marketing efforts, the startup of competing lotteries in neighboring states, and rollovers in lotto jackpots. DeBoer [9] and Clotfelter and Cook [4] both find that lotto sales are highly sensitive to the size of the jackpot. When a state is "unlucky" and most lotto drawings produce a winner, lottery sales will be much lower than in periods when winners are infrequent and jackpots build to high levels.

A number of studies focus on the difficulty of making accurate revenue forecasts of lottery proceeds. Stover [14] concludes that for a state to make accurate forecasts on lottery revenues, the effects of substitution from competing games both inside and outside the state must be taken into account. Mikesell [11, 252] found that "States without lottery competition from any neighbor enjoy higher per capita sales than do those with such competition, other influences held constant." He also reported evidence that new states' lotteries significantly reduce the expected yields of existing lotteries. The age of the lottery was also found to affect sales. Over time, games lose their appeal to customers and new ones must be found by the lottery management. Summing up the consensus view that has emerged in the literature, Mikesell and Zorn [13] conclude that "Clearly, a state cannot rely on its lottery to be a stable, reliable source of net revenue."

The cost of administering lotteries is much greater than for other revenue sources, and a wide variation exists in administrative costs as a percentage of revenues across states. For example, Mikesell and Zorn calculate that in 1984 administrative costs ranged from 4.8% in Illinois to 76.8% in Maine [13]. They note that these costs are underreported, as commissions to vendors are not included in the figures. Deboer [8] finds that there are substantial economies of scale in the provision of lotteries, with smaller states having much higher administrative costs as a percentage of sales than large states.

The consensus in the literature is that the reduction in other revenues attributable to increases in lottery revenue is generally small. Borg, Mason and Shapiro [1] project receipts from all states' taxes for each of the 23 states that operated lotteries in 1987 and then perform time-series estimates of the impact of lotteries on sales and excise tax revenue in a sample of these states. The overwhelming majority of states were found to give up less than 15 cents per dollar of their lottery proceeds due to the impact of the lottery on other sources of state revenue. Similarly, Vasche [16] finds that even though California led the nation in 1989 in the amount of money wagered on both the lottery ($2.6 billion) and on horse racing ($2.7 billion), the state loses only about 4 cents in parimutuel revenue for each one dollar increase in lottery revenue.

There has been an evolution in the product mix offered by lotteries. In the early years of the modern lottery, the game consisted of periodic, usually weekly, drawings. Following this came "instant" games, where players scratch off a covering to instantly reveal whether their ticket is a winning one. The arrival of the computer age in the mid-1970s led to numbers games, consisting of daily drawings in many areas, allowing players the chance to choose their own numbers. Lotto is the most recent game, which requires players to pick six numbers from a larger set (usually between 30 and 49). The odds against winning at lotto are high and jackpots are allowed to accumulate when there is no winner, leading to huge jackpots (often exceeding $10 million).

Cook and Clotfelter [6] report that lotto sales represented 40% of total lottery sales in 1989, even though lotto was unknown before 1980. Indeed, Mikesell [12, Table 1] indicates most lottery states did not add lotto until 1983 or later. Moreover, Clotfelter and Cook [4, 115] present evidence that, contrary to their a priori expectations, lotto is not a substitute for other lottery products such as instant and numbers games. The growth rate in sales for these other games actually increased in the two years following the introduction of lotto in 9 of the 13 states they examined. In addition, an analysis of Massachusetts numbers sales data revealed that the size of the lotto jackpot, which had an enormous effect on lotto sales, had no discernible impact on sales of the numbers game. Thus, Clotfelter and Cook conclude that the additional betting on lotto was "new" money.

Recent findings showing that the introduction of lotto increases total lottery sales and that middle and higher income groups are major participants in lotto [2; 12] may have implications for the growth potential and stability of lottery revenues. Because virtually all states with longstanding lotteries had introduced lotto by 1985, we expect to find slower sales growth in the recent past than in the pre-1985 period examined in Mikesell and Zorn [13]. However, because lotto sales probably stabilize a few years after first availability and because higher income groups have greater access to credit (and can thus maintain steadier consumption patterns), we also expect greater stability in lottery revenues in the post-1985 period.

III. Data Sources and Methodology

In this paper, we examine the impact of lottery revenue on the level and variability of total state revenue for the years 1978-1992; this analysis extends the period examined in Mikesell and Zorn [13]. All of the data used in this study are from State Government Finances [15]. We define lottery revenue as "proceeds available from ticket sales" as reported in State Government Finances, [55] Table 35; these proceeds equal ticket sales (excluding commissions), less funds apportioned for prizes and administrative costs. Total revenue is defined as "general revenue"; general revenue includes all state government revenue (including intergovernmental revenue from the federal government) except insurance trust and liquor store revenue. Non-lottery revenue is defined as general revenue less lottery revenue.

To examine the impact of lotto on the growth of lottery revenue, we estimate the following regression with time series/cross section data for those 13 states operating lotteries since 1978:(4)

PL[R.sub.i,t] = [[Alpha].sub.i] + [[Beta].sub.i][TREND.sub.i,t] + [[Gamma].sub.i][LOTTTO.sub.i,t] + [[Epsilon].sub.i,t] (1)

Where PL[R.sub.i,t] is lottery revenue as a percent of total revenue in state i in year t, [TREND.sub.i,t] is a linear time trend, and [LOTTO.sub.i,t] is a 0/1 Dummy variable which equals one if state i adopted lotto in year t - 1 or earlier;(5) note that all three coefficients may vary across states.(6) We expect to find that the [[Gamma].sub.i] coefficients are positive and significant, especially for the large states which can achieve significant economies of scale in running lotto.

All prior studies concerning the desirability of lottery revenue from a stability and predictability perspective, including Mikesell and Zorn [13], have examined the lottery revenue on a stand-alone basis. According to Modern Portfolio Theory, first proposed by Markowitz [10], this type of analysis is flawed because what really matters is not the volatility of the lottery revenues, but the impact the addition of lottery revenue has on the variability of total state revenues.(7) This impact will depend not only on the variance of lottery versus non-lottery revenues, but also on the correlation between them. Indeed, if lottery sales fluctuate primarily because of non-systematic factors such as the introduction of new games and the frequency of rollovers in lotto jackpots, the correlation of lottery revenues with other revenues such as sales and income taxes is likely to be close to zero. The diversification benefits provided by the low correlation may more than offset the high variance of lottery revenue, resulting in total revenues actually being less volatile with the addition of the lottery. The effects of diversification in this context are illustrated more formally below.

One commonly-used measure of volatility is the standard deviation of year-to-year percent changes in a revenue stream:

[Mathematical Expression Omitted]

where [SD.sub.R] is the standard deviation of percent changes in stream R, T is the number of observations, [R.sub.t] is percent change in revenue in year t, and [Mathematical Expression Omitted] is the mean percent change in revenue. Given two component series of percent changes, lottery revenues (L) and non-lottery revenues (N), the percent change in total revenues can be calculated as:

[R.sub.t] = [W.sub.t-1][L.sub.t] + (1 - [W.sub.t-1])[N.sub.t] (3)

where [W.sub.t-1] is lottery revenue as a proportion of total revenue in year t - 1. If the weight of lottery revenue is relatively constant from year-to-year, the standard deviation of total revenues can be expressed as:

[Mathematical Expression Omitted].

Where [SD.sub.R], [SD.sub.L] and [SD.sub.N] are the standard deviations of percent changes in total, lottery and non-lottery revenues (respectively), W is the average weight of lottery revenue, and r is the correlation coefficient between percent changes in lottery and non-lottery revenues. If the revenue streams are perfectly positively correlated (r = 1), [SD.sub.R] will be a weighted average of [SD.sub.L] and [SD.sub.N]. In this case, provided [SD.sub.L] exceeds [SD.sub.N] (as we would expect), [SD.sub.R] must also exceed [SD.sub.N] and the lottery unambiguously increases the volatility of total revenue. However, if r [less than] 1, the lottery provides some measure of diversification, and the standard deviation of total revenue will be less than a weighted average of [SD.sub.L] and [SD.sub.N]. The magnitude of [SD.sub.R] relative to [SD.sub.N] will now be ambiguous, depending on the weight of lottery revenue, the size of [SD.sub.L] vis-a-vis [SD.sub.N] and the value of r.

It is entirely possible for the lottery to have a stabilizing effect on total revenue (i.e. for [SD.sub.R] to be less than [SD.sub.N]) and it is not necessary for r to be zero or negative for this result to occur. To determine the precise conditions under which the standard deviation of total revenue is less than or equal to the standard deviation of its non-lottery component, we proceed as follows: Let X denote the relative volatility of lottery revenue, that is, let X = [SD.sub.L]/[SD.sub.N]. Setting the left hand side of equation (4) equal to [SD.sub.N], substituting X[SD.sub.N] for [SD.sub.L] and squaring both sides yields

[Mathematical Expression Omitted].

Solving for the correlation coefficient r and simplifying, we obtain

r [approximately equals] (2 - W - W[X.sup.2])/2(1 - W)X. (6)

Thus, provided r is less than the right hand side of equation (6), the lottery stabilizes a state's total revenues in the sense that the standard deviation of total revenue ([SD.sub.R]) is less than the standard deviation of its non-lottery component ([SD.sub.N]).

The condition implied in equation (6) is illustrated graphically in Figure 1 for a hypothetical state with an average degree of reliance on lottery revenue (W = .02). It is clear from Figure 1 that the ratio [SD.sub.L]/[SD.sub.N] can be very high, yet the lottery can be stabilizing, so long as r is low. For example, the relative volatility of lottery revenue can be 2 if r = .45, 4 if r = .20, and as high as 10 if r = 0, and still the lottery would not increase the standard deviation of total revenue.(8) If our a priori expectation of near-zero correlation is borne out, lottery revenues can have a stabilizing effect in a portfolio context even if their volatility on a stand-alone basis is very large relative to other revenue sources. The empirical findings regarding these issues are reported in the next section.

IV. Results

Figures for lottery revenue as a percentage of total general revenue, by year for the period 1978-1992, for all states with lotteries operating in 1992, are provided in Table I. Panel A provides data for 13 states which have continuously operated lotteries since 1978, while Panel B contains data for states which have adopted lotteries more recently. Table II contains figures for annual percent changes in lottery revenue from the prior year, in a format similar to Table I.

For the average early-adopting state, lottery revenue represented 2.04% of general revenue in 1992; this figure is down from a peak of 2.31% in 1988 and about the same as in 1985. Of course, since many more states operated lotteries in 1992, total lottery revenues for all states have grown rapidly, but (possibly due to the increased competition) growth within the average state has slowed considerably. This same conclusion is also readily apparent from Table II: The average early-adopting state experienced about 30% annual growth in lottery revenue between 1981 and 1986, but growth then slowed to the 15% range in 1987 and 1988 and into the single digits in the post-1989 period.

In examining the time series nature of the data in Tables I and II, one major conclusion emerges: despite the slowdown in the growth of lottery revenue after 1986, the revenue potential of the lottery (as measured by the percentage of total revenue it contributes) is nonetheless considerably greater in the post-1985 period than in the late 1970s or early 1980s. The introduction and maturity of lotto seems a very plausible explanation for the patterns observed, e.g., the high growth rate of lottery revenues in the 1981-1986 period (when most states were initiating lotto) and the subsequent slowdown in revenue growth. To examine this issue formally, we estimate regression (1) using White's [17] heteroskedasticity-consistent technique.(9) The results in Table III show that the lotto variable is significantly positive for 9 of the 13 states. In three of the four states for which the coefficient on LOTTO is insignificant (Delaware, Maryland and Rhode Island), we would expect this result based on Cook and Clotfelter's [6] finding that small states have difficulty achieving necessary economies of scale in lotto operations. For the average state, we estimate that lotto is responsible for a one-time, permanent increase in lottery revenue of approximately [TABULAR DATA FOR TABLE I OMITTED] [TABULAR DATA FOR TABLE II OMITTED] .7% of total general revenue, and can thus account for most of the revenue growth that took place between 1980 and 1986.(10)

Table III. Impact of Lotto on Lottery Revenue

Model Estimated:(a) [PLR.sub.i,t] = [[Alpha].sub.i] +
[[Beta].sub.i][TREND.sub.i,t] + [[Gamma].sub.i][LOTTO.sub.i,t] +
[e.sub.i,t]

State               [[Alpha].sub.i]    Standard Error    t-stat.(b)

Connecticut             0.01663           0.00105        15.77(***)
Delaware                0.00543           0.00076         7.13(***)
Illinois                0.00736           0.00197         3.73(***)
Maine                   0.00009           0.00069         0.13
Maryland                0.03405           0.00223        15.30(***)
Massachusetts           0.00857           0.00207         4.14(***)
Michigan                0.02024           0.00094        21.42(***)
New Hampshire           0.00365           0.00131         2.79(***)
New Jersey              0.01780           0.00092        19.37(***)
New York                0.00369           0.00057         6.49(***)
Ohio                    0.00479           0.00103         4.65(***)
Pennsylvania            0.01501           0.00144        10.41(***)
Rhode Island            0.01256           0.00096        13.02(***)

Average(c)              0.01153           0.00037        31.16(***)

State               [[Beta].sub.i]     Standard Error    t-stat.(b)

Connecticut             0.00002           0.00031         0.06
Delaware                0.00048           0.00013         3.65(***)
Illinois                0.00022           0.00048         0.45
Maine                   0.00048           0.00016         3.04(***)
Maryland               -0.00038           0.00039        -0.97
Massachusetts           0.00146           0.00054         2.70(***)
Michigan               -0.00054           0.00014        -3.73(***)
New Hampshire           0.00045           0.00037         1.21
New Jersey              0.00022           0.00027         0.81
New York                0.00054           0.00018         2.99(***)
Ohio                    0.00171           0.00018         9.60(***)
Pennsylvania           -0.00050           0.00041        -1.21
Rhode island           -0.00009           0.00009        -1.03

Average(c)              0.00031           0.00009         3.58(***)

State               [[Gamma].sub.i]    Standard Error     t-stat.(b)

Connecticut             0.00703           0.00265         2.65(***)
Delaware                0.00027           0.00084         0.32
Illinois                0.01980           0.00495         4.00(***)
Maine                   0.00541           0.00146         3.70(***)
Maryland                0.00503           0.00333         1.51
Massachusetts           0.00148           0.00536         0.28
Michigan                0.00855           0.00117         7.33(***)
New Hampshire           0.00803           0.00375         2.14(**)
New Jersey              0.00791           0.00241         3.28(***)
New York                0.00527           0.00152         3.46(***)
Ohio                    0.00475           0.00179         2.65(***)
Pennsylvania            0.01863           0.00364         5.12(***)
Rhode Island           -0.00221           0.00140        -1.58

Average(c)              0.00692           0.00083         8.34(***)

a. [PLR.sub.i,t] = lottery revenue as a percent of general revenue
in state i, year t; [LOTTO.sub.i,t] = 1 if state i adopted lotto in
year t - 1 or before, and 0 otherwise. Regression [Mathematical
Expression Omitted]. Coefficient standard errors estimated using
White [17] heteroskedasticity-consistent covariance matrix.

b. ***, ** and *, respectively, denote significance at the 1%, 5%
and 10% levels.

c. The standard error reported is for an equally-weighted linear
combination of coefficients; the reported figure is less than a
weighted average of the individual standard errors due to low
correlation in the errors across states.

Consistent with the previous literature, the figures in Table II, with few exceptions, indicate that the variability of lottery revenues within each state is enormous. Of course, large percentage increases are to be expected with new lotteries as they grow and mature, but even for the established lotteries the volatility is large. For example, Illinois saw a 143% jump in revenue in 1981 but a 10% decline in 1988; Massachusetts experienced a 30% drop in 1981, then a 124% increase four years later.

Where the previous literature errs is in assuming from figures like those in Table II that the lottery must have destabilizing implications for total state revenue, without really examining the issue. We perform this analysis in Table IV, where we report the standard deviations of annual percent changes in lottery, non-lottery and total revenue, for each of the 13 states which continuously operated lotteries between 1978 and 1992. Consistent with the figures in Table II, the standard deviations of the lottery revenues are very large, nearly seven times as great as for non-lottery revenues in the average state during the whole 1979-1992 period. However, confirming our a priori expectations, the correlation of lottery revenues with non-lottery revenues is quite low in most cases: for the whole period, the correlation coefficient averages .18 across the states and in no case exceeds .5. These low correlations seem to offset the high variance of lottery revenues because, for the average state, the standard deviation of total revenues (for the 1979-1992 period) is approximately the same as the standard deviation of non-lottery revenues.

An even more relevant test of the impact of a lottery on revenue stability is provided by the 1986-1992 subperiod. This period, coming after the initiation of lotto in all of the 13 states, is not distorted by the growth spurt attributable to the startup of lotto and provides a better indication of what can be expected when all three major lottery games are operating simultaneously. In this latter subperiod, the standard deviation of lottery revenue in the average state is only four times as large as the standard deviation of non-lottery revenue, and the correlation of the two revenue streams averages only .10. As a consequence, consistent with equation (6), total revenues are actually less volatile than non-lottery revenues on average, and in 9 of the 13 individual states.(11) While the difference in volatility between non-lottery and total revenue is probably not large enough to be statistically significant, our results do show that there is no evidence whatsoever supporting the commonly-held belief that lotteries destabilize a state's total revenues. Thus, upon closer examination, one common criticism of lotteries as a revenue source is found to be invalid.

[TABULAR DATA FOR TABLE IV OMITTED]

V. Conclusion

This paper re-examines the revenue-generating potential of state lotteries using recent data, and extends the previous literature by modeling the volatility of lottery revenue in the context of a portfolio of different state revenue streams. We find that the introduction of lotto in the 1981-1985 period has had a strong positive impact on the net proceeds generated by state lotteries. In contrast to implications in earlier studies, we find that lottery revenues do not destabilize total state revenues, because the low correlation of lottery revenues with revenues from other sources offsets the high stand-alone risk of lottery funding. These results, at least in part, may help explain why so many states have adopted lotteries during the past decade. The lure of a mechanism which raises significant revenue, without coercion, and without destabilizing overall state revenues, is powerful and difficult to resist.

1. In addition, of course, there is substantial opposition to lotteries on essentially moral grounds. It is often argued that the lottery constitutes a highly regressive tax because lower income groups buy a disproportionate share of the tickets, that lotteries promote gambling, and that lotteries are deceptively advertised and marketed. While these issues are all important, they are beyond the scope of this paper.

2. It is clear from recent history that lotteries are popular with voters, but it is not necessarily true that their current structure is optimal. Clotfelter and Cook [5] note that, with few exceptions, states seek to maximize revenues, even if this requires deceptive advertising and imposes needless social costs.

3. Clotfelter and Cook [3] use several different methodologies to compute volatility, e.g., deviation from trend, number and frequency of absolute percentage declines, etc., and find that lottery revenues are less volatile using some of these measures. But all of the yardsticks they use are calculated on a stand-alone basis; to our knowledge, no study has examined lottery revenues in a portfolio context.

4. We tested for trend stationarity in the dependent variable of regression (1) by conducting Dickey-Fuller (DF) and Augmented Dickey-Fuller (ADF) tests as described in Davidson and MacKinnon [7, 700-15] models (20.05) and (20.16), with critical values in Table 20.1. In these tests, the constant and trend coefficients were allowed to vary across states, but the coefficient on the lagged dependent variable was restricted to be equal across states in order to provide a more powerful test. We obtained pseudo t-statistics on the lagged dependent variable of -4.12 using the DF specification, and -5.65 using the ADF with two lags. These both are below the critical value at the 1% level (-3.96), indicating that PL[R.sub.i,t] is trend stationary.

5. We set [LOTTO.sub.i,t] = 0 if the state adopted lotto in the current year due to the discrepency between fiscal and calendar years. The data in State Government Finances is for fiscal years, which run from July 1-June 30 in most states (e.g., fiscal 1986 is really July 1985-June 1986), whereas the only information available to us on the startup of lotto was the calendar year of first adoption. Thus, for example, if a state adopted lotto in 1985, the impact on lottery revenues in fiscal 1985 is likely to have been minimal.

6. We test the restrictions that the [[Alpha].sub.i]'s, [[Beta].sub.i]'s and [[Gamma].sub.i]'s are equal across states. These restrictions are all soundly rejected at any conventional significance level. Because (1) is essentially a time-series regression, we do not include other variables (e.g., size of state, number of years lottery has been in operation, existence of competing lotteries in border states) found to be significant in previous cross-section studies. We are confident that the effects of other variables are adequately captured in the intercept and the trend.

7. We implicitly assume that lottery revenue is fungible, i.e., that the legal dedication of lottery revenue to specific functions that exists in some states has no practical impact, because the state is free to add or subtract other funds to these functions if lottery revenue departs from predicted levels.

8. The curve in Figure 1 shifts down and to the left as W increases, implying that holding correlation constant, the greater the reliance on lottery revenue as a percent of total revenue, the lower the relative volatility of lottery revenue must be to avoid a destabilizing effect. However, within the range of W's observed in practice (e.g., .0027 to .0366 in 1992), these shifts are negligible.

9. Due to the large number of independent variables we could not apply the White [17] test for heteroskedasticity. We apply the White correction to the covariance matrix because there are a-priori grounds for believing that heteroskedasticity is present, and because the estimated standard errors for some of the variables were markedly different than in OLS estimates.

10. we also estimated an alternative model in which the [[Gamma].sub.i]'s were restricted to be equal across states but allowed to vary across time, to allow for the possibility that the impact of lotto may differ depending on how recently it was added. These results (not reported) showed that lotto had a significant positive effect on [PLR.sub.i,t] each year between 1 and 7 years after first adoption. As expected, the impact in the year of adoption was very close to zero because of the fiscal year/calendar year mismatch. The hypothesis that the impact was the same in each post-adoption year between 1 and 7 could not be rejected at the 10% level.

11. We measure the standard deviation of total revenue directly, rather than using equation (4), but our findings are consistent with equations (4) and (6). For the entire 1979-1992 period, based on observed combinations of r, [SD.sub.L]/[SD.sub.N], and average W, equation (6) correctly predicts the size of [SD.sub.R] vis-a-vis [SD.sub.N] for 11 of 13 states. For the more crucial 1986-1992 subperiod, equation (6) correctly predicts the size of [SD.sub.R] vis-a-vis [SD.sub.N] for all 13 states. The slight discrepancy for the whole period is caused by the fact that equations (4) and (6) hold exactly only if W is constant; in actuality, as Table I shows, the weight of lottery revenue fluctuates over time, and the fluctuations are larger in the early years.

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13. ----- and C. Kurt Zorn, "State Lotteries as Fiscal Savior or Fiscal Fraud: A Look at the Evidence." Public Administration Review, July/August 1986, 311-20.

14. Stover, Mark Edward, "Contiguous State Lotteries: Substitutes Or Complements?" Journal of Policy Analysis and Management, Fall 1990, 565-68.

15. U.S. Bureau of the Census. State Government Finances. Washington, D.C.: Government Printing Office, 1978-1992.

16. Vasche, Jim David, "The Net Revenue Effect of California's Lottery." Journal of Policy Analysis and Management, Fall 1990, 561-64.

17. White, Halbert, "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity." Econometrica, May 1980, 817-38.