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Black swan models for the entertainment industry with an application to the movie business

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Abstract

Success in popular entertainment is highly unpredictable. Yet it is the high-impact low-probability events—known as ‘black swans’—that drive entertainment industry profitability because success is highly concentrated on a small number of winners. In this research, we apply recently developed statistical tools to model motion-picture success; these tools explicitly account for rare extreme events while permitting valid statistical inferences to be made on how product attributes are associated with product success. The specific empirical application relates the attributes of a film and its theatrical release to the distribution of worldwide cumulative box-office revenue. A regression model with skew-stable random disturbances is applied to a large sample of motion pictures to quantify the correlates of film success while explicitly accounting for skewness and heavy tails. The skew-stable estimates are compared to estimates obtained from symmetric-stable regression, ordinary least-squares regression, and several alternative robust-to-outliers regression models. Failure to control explicitly for heavy tails and skewness generates misleading statistical inferences, particularly regarding the impact of production budget, opening week screens, and star power on a film’s success at the box-office.

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Fig. 1

Notes

  1. Several authors, including Walls (1997), Hand (2001), McKenzie (2008) and others fit models with Paretian tails to theatrical film returns data; Maddison (2004) performs a similar exercise for Broadway shows. Hadida (2009) discusses a number of conceptual and methodological issues relating to motion-picture performance.

  2. Even the encyclopedic Hennig-Thurau and Houston (2019) volume on entertainment science—applying data analytics and practical theory to the movie, gaming, book, and music industries—contains not a single mention of applied statistical models that account for skewness or heavy tails.

  3. It should be emphasized here that the stable distribution generalizes the normal distribution model; it does not rule it out. Because the normal model is nested within the general stable model, we can test the normal model against the stable model with heavy tails and asymmetry.

  4. In the Walls (2005a, b) paper, the logarithm of revenues is analyzed, which is not particularly skewed. But in general, it is strongly preferred to have a model that flexibly accommodates asymmetry. Walls (2005a, b) applies the skew-student-t regression model to film returns to accommodate skewness and heavy tails; although the model is ad hoc, it does accommodate skewness and heavy tails.

  5. See, for example, the discussion in Uchaikin and Zolotarev (1999).

  6. Ijiri and Simon (1977) investigated dynamic processes that possessed the momentum characteristic, where firm-level growth depends on firm size. They showed that these processes converged on distributions with Paretian tails and that this result is not sensitive to changes in initial conditions.

  7. Competing candidate distributions, such as the student-t, can be used to model heavy tails, and the skewed-student-t (Azzalini and Capitanio 2003) can accommodate skewness and heavy tails, but they are ad hoc. Other alternatives include nonparametric estimators such as Racine and Li (2004) and semi-parametric estimators such as those discussed in Hallin et al. (2008). Another flexible approach accounting for skewness and extreme outliers is the Tukey g-and-h family developed and fully operationalized by and Xu and Genton (2015, 2017) and applied to accessible linear modeling by Ricci et al. (2016).

  8. See, for example, the volume edited by Adler et al. (1998) for many papers that extend or apply the stable distribution model and Rachev and Mittnik (2000) and McCulloch (1996) for specific applications in finance. Nolan (2008) has compiled an exhaustive bibliography relating to nearly all aspects of stable modeling.

  9. See Nolan (1998a) for a useful discussion of the various parameterizations of the stable distribution.

  10. This property makes the stable model particularly appealing because it provides the connection between the behavioral model generating the data and the statistical model applied to data.

  11. The material in this section follows closely the exposition and notation of McCulloch (1998). See Judge et al. (1985, pp. 825–826) for an overview of econometric models with infinite variance.

  12. The parameter estimates do not have heavy tails. While the data may follow a stable distribution characterized by skewness and heavy tails, the parameter estimates are asymptotically normal as they would be for any maximum likelihood estimation.

  13. Due to the nature of the data set, the empirical results from this analysis will clearly be more appropriate for the mainstream commercial motion-picture industry seeking profits than for lower-budget limited-release ‘art’ films focusing solely on artistic integrity.

  14. The Ulmer Scale provides a metric for film industry talent classification: The Ulmer metric indexes on a 100-point scale that is based on the survey of dozens of film and television industry deal-makers. Complete details of the Ulmer Scale are available at http://www.ulmerscale.com. Multiple editions of the Hot List are available, and each edition covers more actors than previous editions—the most recent edition used in this research featured 1400 actors. Rankings change throughout time in movement with industry perceptions about the particular actor’s ‘bankability.’ As a guide, the cumulative distribution of rankings in the sixth edition were (A+) 0.5%, (A) 2.6%, (B+) 8.4% and (B) 17.1%. Given the relatively long time dimension of our study—and the fact that stars are ephemeral—we use various volumes of Ulmer’s Hot List to classify a star. In particular, we used volume 1 for the years 1997–1999, volume 3 for the years 2000–2002, volume 5 for the years 2003–2005, and volume 6 for 2006–2007.

  15. For a more complete listing and discussion of empirical studies on the economics of the film industry, see McKenzie (2012) and the references therein. See Walls (2013) for a survey of empirical methods used to quantify bestsellers and blockbusters in economics.

  16. The stable regression models were estimated using John Nolan’s R package ‘stable,’ version 5.1.10, serial number #231; see Nolan and Ojeda-Revah (2013) for details of the estimation procedure. Alternative approaches to estimation of stable regression models are provided by Lambert and Lindsey (1999) and Achcar and Lopes (2016).

  17. In the context of the stable model, the tail weight parameter falls into the region where the mean is defined and the variance is not defined.

  18. See, for example, the expositions in Rousseeuw and Leroy (1987), Huber (1981), Goodall (1983), and Berk (1990).

  19. The Tukey-M-regression estimates are presented for purposes of comparison only. The second-generation robust regression models presented as follows have superior Gaussian efficiency.

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  1. Department of Economics, University of Calgary, Calgary, Canada

    W. D. Walls

  2. Department of Economics, Macquarie University, Sydney, Australia

    Jordi McKenzie

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Correspondence to W. D. Walls.

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An earlier version of this paper was presented at the 20th Anniversary Carol and Bruce Mallen Scholars and Practitioners Conference in Filmed Entertainment Economics held at Film Universität Babelsberg, Germany, September 27–28, 2018, and at the Asian Meeting of the Econometric Society held at Xiamen University, Amoy, China, June 14-16, 2019. We are grateful to conference participants, and to two diligent referees, for comments that have helped to improve the content and clarity of the paper.

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Walls, W.D., McKenzie, J. Black swan models for the entertainment industry with an application to the movie business. Empir Econ 59, 3019–3032 (2020). https://doi.org/10.1007/s00181-019-01753-x

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