Modelling heavy tails and skewness in film returns

Abstract

The average of box-office revenue is dominated by extreme outcomes, with most films earning little and most revenues flowing to a few blockbusters. In this paper the skewness and heavy tails of film returns are formally modelled using skew-Normal and skew-t distributions. Logarithmic skew-Normal and skew-t models of the distribution of box-office revenue are fitted conditional on star actors and directors, budget, release pattern, genre, rating, and year of release. The estimates show significantly more skewness and heavier tails than the log-Normal distribution. It is also found that a wide theatrical release has a much smaller impact on box-office revenue when heavy tails and skewness are explicitly modelled.

Acknowledgements

This research was supported by a grant from the Committee on Research and Conference Grants of the University of Hong Kong. The author would like to thank M. C. Auld for his most generous donation of the computing resources used in the production of this research, and Adelchi Azzalini for helpful comments on an earlier version of this paper.

Notes

¹ Walls (1997) estimates the unconditional distribution of movie revenues using Pareto and parabolic Pareto distributions; Hand (2001) and Maddison (2004) also perform similar calculations on different data sets. De Vany and Walls (2004) model the unconditional distribution of motion-picture profit using the stable Paretian model. Walls (2005) estimates the conditional distribution of movie revenues using a symmetric stable Paretian regression model that fixes the skewness parameter at zero. No prior work has modelled the conditional distribution of movie returns without imposing symmetry.

² For example, Harris and Kucukozmen (2001) use the exponential generalized beta and skew generalized t distributions, and Brannas and Nordman (2003) apply the log-generalized gamma and Pearson type IV specifications.

³ The development in this and the following subsection closely follows the simplified exposition of Azzalini and Kotz (2002).

⁴ Any good mathematical statistics book will have a thorough treatment of the log-Normal distribution. See, for example, the discussion in Hogg and Craig (1978).

⁵ A partial listing of papers would include Smith and Smith (1986), Prag and Cassavant (1994), Litman and Ahn (1998) and Ravid (1999).

⁶ In standard industry parlance ‘domestic’ refers to the USA and Canada.

⁷ The statistical models were estimated in the R language (Ihaka and Gentleman, 1996) using the ‘sn’ library developed by Adelchi Azzalini.

⁸ The differences between the skew-Normal and skew-t estimates of location and skewness are small relative to their estimated standard errors of 0.051 and 0.038, respectively.

⁹ The MAD estimator is also a maximum likelihood estimator when the disturbances follow a two-tailed exponential distribution. See Judge et al. (1985, pp. 836–37) and the references cited therein for further discussion.

¹⁰ They do not explicitly report elasticities in their paper. The elasticity has been calculated using their estimate regression coefficient on budget of 0.38254, their reported average budget of 31.38 million and average box-office revenue of 51.24.

¹¹ Again, Litman and Ahn (1998) do not explicitly report elasticity estimates in their paper. Point elasticity has been calculated using their estimated regression coefficient of 0.01982, and their reported average screens of 1669.9 and average box-office gross of 31.38 million.

¹² Litman and Ahn (1998) are more honest than most empirical researchers in stating that their regression results, ‘represent the final ‘best fit’ after initial screening of different groups of independent variables through correlation analysis' (p. 188).

¹³ Their sample of data includes only the films listed in Variety's Top-100 chart for the years 1993–1995 inclusive.

¹⁴ Readers are referred to the web page for The R Project for Statistical Computing at www.R-project.org.