Modelo de Acumulación y Derretimiento de Nieve

Finally, this section returns to the Working-Leser model considered above but relaxes the assumption that the budget share of lotto tickets is log linear in total expenditure. Instead,

a completely agnostic approach is taken towards the functional form by allowing the data to dictate the shape of the Engel curve using semi-parametric estimation. This is accomplished by using Robinson’s (1988) semi-parametric routine75 _{to estimate equation (4.3),}

𝑤₍ = 𝑋₍+_{𝛽 + 𝑓(ln 𝑌} () + 𝜀(.

Robinson’s estimator ensures consistent estimation of 𝛽 and estimates the unknown function,

𝑓, using kernel density estimation. Blundell et al (1998) and Blundell and Powell (2003) show that when the non-parametric component is endogenous, instrumental variables techniques can be employed and the residuals of this first-stage are to be included in the parametric component of equation (4.3). Therefore, the endogeneity of log total expenditure is controlled for by instrumenting with gross normal income and using the residuals from the estimates presented in Appendix Table A4.14. Following the standard notation for this procedure, these residuals,

𝜈₍ from the first stage are included in the semi-parametric regression as,

𝑤₍ = 𝑋₍+_{𝛽 + 𝑓(ln 𝑌}

() + 𝜌𝜈( + 𝜀( (4.12)

and 𝜌, the coefficient on these residuals, is estimated as part of the parametric component. Table 4.11 presents estimates of the parametric component of equation (4.12), 𝛽 and 𝜌. The effects of the control variables on the budget share of lotto using the Robinson’s semi- parametric estimator are considerably smaller than all parametric models estimated above.

Though the sex, education level, and some employment statuses of the HRP affect the budget share allocated to lotto in the same direction as previous models, these estimates suggest that having an unemployed or sick HRP increases the budget share for lotto, albeit by a small amount. The significant, though also small, coefficient on the residuals from the first-stage regression indicates that estimating equation (4.3) without instrumenting log total expenditure would yield biased estimates, thus the instrumental variables process and the inclusion of these residuals here is justified. The solid line in Figure 4.7 is the estimate of the non-parametric component, 𝑓, of equation (4.12) and the shaded area shows the 95% confidence interval. Due to the large dataset used here, this confidence interval is unsurprisingly small for much of the total expenditure range. The figure is trimmed to exclude the top and bottom 1% of households by total expenditure to allow the graph to be legible – though these observations were included

75 _{This routine is identical to that used in Chapter 0 and a more detailed description of the procedure is}

in the estimation process. The estimated budget share of lotto over the distribution is indicated on the left-hand y-axis. Superimposed in Figure 4.7 is a histogram of household weekly total expenditure which corresponds to the right-hand y-axis.

Table 4.11: Parametric component of semi-parametric estimation of the Working-Leser model for lotto demand

(1) Dependent Variable 𝑤₍ Female HRP -0.001*** (0.0001) Age HRP 0.000*** (0.0000) Self Employed -0.001*** (0.0001)

Unemp. (seeking work) 0.001***

(0.0003)

Unemp. (about to start work) 0.001

(0.0013) Sick 0.002*** (0.0003) Retired -0.001*** (0.0001) Unoccupied -0.000 (0.0001) Left Education 17-18 -0.003*** (0.0001) Left Education 18+ -0.004*** (0.0001) Constant 0.006*** (0.0006) 𝜌 -0.041*** (0.0027)

Month-Year Effects Yes

Observations 78,867

R-squared 0.044

Robust standard errors in parentheses. */**/*** denotes statistical significance at the 1%/5%/10% level, respectively. Omitted categories: Male HRP, employee, left education before 17, North East. 𝜌 is the coefficient on residuals from instrumenting log total expenditure with gross income as in Appendix Table A4.14.

Figure 4.7: Semi-parametric estimate of lotto budget share against log total expenditure with distribution of households by log total expenditure

Moving from left to right across the total expenditure distribution, the estimated non- parametric component, f, rises until a log weekly expenditure of around 5 (which translates to £148 per week). The peak in lotto budget share occurs at a level of total expenditure which is considerably lower than the average as shown by the superimposed distribution of total expenditure. The fitted relationship between lotto budget share and log total expenditure falls rapidly after this peak over the remainder of the distribution – furthering the conclusions drawn from the parametric estimates that the budget share of lotto declines with higher levels of income and the lotto taxes are regressive.

Hardle and Mammen (1993) provide a test of the fitted relationship between the budget share of lotto tickets and log total expenditure against a parametric fit of any degree polynomial of the latter. This test evaluates whether the fitted budget share of lotto tickets from the semi- parametric routine is statistically different from that of a given polynomial expansion, i.e. whether the parametric fit is a sufficiently close approximation of the underlying relationship. It is not surprising upon examination of the fitted function in Figure 4.7 that this test rejects the linear specification of the Working-Leser model, and even specifications which are quadratic and cubic in log total expenditure. Nonetheless, the Working-Leser model which forms the

basis of the parametric estimates presented in Section 4.6.2 has a more substantial theoretical underpinning than the linear expenditure model favoured by the existing lotto literature, indicating that the estimates contained in this chapter are still both an improvement and important.

4.7 Conclusion

Despite their notoriously low return rates, lotteries are one of the most popular forms of gambling enjoyed not just in the UK but around the world. Their usefulness as a means of raising public finance when it is otherwise difficult to do so has meant that many games are either operated directly, or have their operation licensed, by the state. However, unlike many other goods subjected to so-called “sin taxes”, there is little-to-no supporting evidence that playing lotto games generates any externality or is in some way otherwise harmful to the individual. Without a supporting paternalistic ‘moral high ground’ argument – and assuming one of the objectives of government is maintain a progressive tax structure – the question of whether, and to what extent, a high rate of tax imposed on such games is regressive becomes an important one.

This chapter has sought to answer this question by estimating a Working-Leser demand function for lottery tickets using household-level data. Since taxes on the UK lotto are constant, the question of whether taxes are regressive, proportional, or progressive can be answered by determining whether the income elasticity of demand for lotto tickets is less than, equal to, or greater than one, respectively. The preferred estimates of this chapter yield headline estimates of the income elasticity for lotto between -1.4 and -0.7. These estimates are significantly lower than those in the existing literature, and well below the critical value of +1 for proportionality in taxes. To the best of the author’s knowledge, this is the only estimate in the literature that indicates lotteries are actually inferior goods. Such low income elasticity estimates therefore suggest that lotto taxes are significantly more regressive than previously thought. This conclusion is supported by a Suits’ index estimate of -0.36, lower than any other study that also calculates an income elasticity and only exceeded by early estimates for lotteries in the US.

Using a Working-Leser demand specification provides an improvement upon the existing literature surrounding the income elasticity of lotto and the regressivity of lotto taxation. Previous studies exclusively use naïve models of demand in which the level of expenditure is some function – typically linear – of income. In the wider economics literature, this model has been criticised for its lack of foundation in microeconomic theory and is, at best,

merely an approximation of the true relationship between income and expenditure. Section 4.6.2 briefly replicates this approach and the resultant estimates are, unsurprisingly, very similar to those of the existing literature, estimating income elasticity to be between 0.22 and 0.46. Whilst they would still imply that lotto taxes are regressive, these naïve estimates understate the regressivity of such taxes and contrary to the Working-Leser estimates, class lotto as normal goods, rather than inferior.

Household-level expenditure data presents a number of econometric issues, primarily the presence of zeroes in lotto expenditure which would bias coefficient estimates from OLS. Moreover, the simplest approach to correcting for this bias, using a Tobit model, is undesirable because the independent variables are likely to have different effects on whether a given household participates in lotto and the level of play conditional on participation. The use of Heckman’s selection model and Cragg’s double-hurdle model overcomes this issue with Tobit by allowing the independent variables to affect the participation and consumption decisions separately, but their two-stage nature present identification issues of their own. This chapter deals with the identification of these models by exploiting differences in consumption behaviour which arises because of religious belief – namely, the identification strategy uses alcohol and pork consumption as an exclusion restriction. It is from these models which the headline estimates of income elasticity are obtained.

Such a large difference between the income elasticity estimates presented here and those in the existing literature highlights the importance of correctly specifying the model used when investigating the relationship between lottery expenditure and income. Moreover, despite its popularity in economics literature and theoretical foundations, the Working-Leser model favoured here still imposes a strict (log-linear) functional form on the relationship between income and the budget share of lotto tickets. Thus, as the final contribution of this chapter, Robinson’s semi-parametric estimator has been employed in which the budget share of lotto is fully flexible in its relationship to income – allowing the data to dictate the shape of this function. Thanks to the large dataset used here, this estimator is well determined and shows clearly that the effect of income on lotto consumption varies substantially over the income distribution. Testing against this fully flexible specification reveals that parametric fits of polynomial expansions up to cubic in log income are statistically different. Nonetheless, the Working-Leser model estimates presented here offer a novel finding for the literature and indicate that lotto taxes are even more regressive than previously thought. This is supported

with an estimate of Suits’ regressivity index of -0.36, far lower than any previous estimate for the UK lotto.

4.8 Appendix