Habit and long memory in UK lottery sales

Numerous time series studies have been conducted on the determinants of lottery (Lotto) sales in various countries. (See, for example, Beenstock and Haitovsky, 2001, Cook and Clotfelter, 1993, Walker and Young, 2001, and Forrest et al., 2000). One feature of the reported results is the persistence of lottery sales, as measured by the significance of a number of lags of the dependent variable, given the other determinants such as expected price, other higher moments or jackpot size. For example, Beenstock and Haitovsky (2001), employing Israeli data, report three significant lags which are all positive with declining weights, Forrest et al. (2000) report four positive lags employing UK data and Keeton-Stolk (2008) reports two significant lags employing South African data. The significance of the lags could be interpreted as evidence of habit in the purchasing decision of agents. Farrell et al. (1999) explicitly base the functional form they estimate on the “rational addiction” model of Becker and Murphy (1988). However, the lag structures reported in their empirical work do not have the form suggested by the Becker and Murphy model. Whilst the Becker and Murphy model illustrates that the structural demand functions of individuals can be autoregressive it does not necessarily follow that the aggregate demand curve will also exhibit the same autoregressive structure. Granger (1980) demonstrates how aggregation over heterogeneous agents, who exhibit different degrees of habit or autoregressive parameters, can generate a functionwhich exhibits the fractional property. A fractional process exhibits an autocorrelation function that exhibits hyperbolic rather than geometric decay and consequently the aggregate function has different properties to that of any individual in the aggregate. These results on aggregation are of more than theoretical interest since fractional processes can exhibit, like I(1) processes, the ‘spurious regressions’ problem with important implications for the appropriate methods for analyzing aggregate relationships (see, for example, Marmol, 1998, Davidson, 2002, and Sun, 2006). Our purpose in this letter is to determine whether two series of aggregate lottery sales in the UK exhibit the fractional property. The letter is structured as follows. In Section 2 we set briefly set out the models that can generate aggregate fractional behavior. Section 3 sets out our method of estimation, data and empirical results. The final section provides a brief conclusion.