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Testing for complementarities among countable strategies

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Abstract

I study whether the pricing strategies of competing duopolists in the early US cellular telephone industry can be considered strategic complements or substitutes. In order to do so, I present a multivariate count data regression model that is suitable to test for the existence of strategic complementarities when firms make use of countable strategies. The estimator, which accommodates the underdispersion that characterizes the data, is shown to have better small-sample properties than common estimators based on the Gaussian copula. It also allows for correlations of any sign among counts independently of the dispersion parameters. Results show that in addition to screening consumers, competing firms imitated each other in the number of tariff options offered to their customers.

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Notes

  1. The need to estimate joint demands of countable products or services arises in many environments such as medical service, e.g., Munkin and Trivedi (1999) or Riphahn et al. (2003); job changes, Jung and Winkelmann (1993); types of food, Meghir and Robin (1992); and recreational trips, Hausman et al. (1995), Hellström (2006), or Terza and Wilson (1990).

  2. Windmeijer and Santos-Silva (1997) successfully deal with an endogenous count and an endogenous continuous variable while Hausman et al. (1995) address the case of an endogenous count and an endogenous dichotomous variable.

  3. It has long been recognized that the Poisson model is generally too restrictive when estimating univariate count data regressions. Implicit to the Poisson model is the assumption of equidispersion of the distribution of counts, which is customarily rejected by the data. Many models, such as the negative binomial regression, have been suggested to address the existence of unobserved heterogeneity in the data that could explain the commonly observed overdispersion but not the less frequent underdispersion of the distribution of counts. Hausman et al. (1984) even deal successfully with overdispersion in a univariate panel count data model.

  4. The multivariate Poisson-gamma mixture model of the random effects model of (Hausman et al. (1984), §3) is a restricted version of the model of Marshall and Olkin (1990), while Gurmu and Elder (2000) and Winkelmann (2000) suggest multivariate negative binomial models. In all these works, only overdispersion is allowed, but in the latter two cases, correlation is independent from dispersion, although still necessarily positive.

  5. The double Poisson introduced by Efron (1986) is one of the few discrete univariate distributions that can accommodate both over- and underdispersion. Winkelmann (1995) builds a similarly flexible unidimensional model but based on the continuous gamma distribution of latent waiting times that exploits the one-to-one relationship between the properties of the hazard rate of the distribution of waiting times and the over/underdispersion of the distribution of events that take place within an arbitrarily defined time interval.

  6. There are other models that share some but not all of these features. For instance, Hellström (2006), Munkin and Trivedi (1999), or Riphahn et al. (2003) build upon the bivariate Poisson-lognormal distribution of Aitchison and Ho (1989) to allow for any sign correlation between counts. But, these models can only address the case of overdispersed counts while estimation has to resort to simulation methods as the Poisson-lognormal mixture does not have a closed-form expression. Gurmu and Elder (2008) obtain a closed-form expression only after considering a first-order Laguerre polynomial approximation to the bivariate distribution of unobservables.

  7. See also Gurmu and Elder (2012). The Sarmanov regression model presented in this paper can address both over- and underdispersion separately from correlation.

  8. (Efron (1986), Table 2) explores the discrepancy between \(\tilde{f}_k(y_k|\mu _k,\theta _k)\) and \(f_k(y_k|\mu _k,\theta _k)\) as a function of parameters \(\theta _k\) and \(\mu _k\). For the application of Sect. 5, the approximate probability is just 1.17 % larger than the exact probability frequency function at the estimated value of the parameters. Notice, however, that all relations below make use of the exact double Poisson density.

  9. The use of Stirling’s formula is useful for practical purposes to ensure the stability of estimation for large counts (not an issue in the application of this paper). Moreover, using it twice for \(z=y_k\) and \(z=\theta _k y_k\) simplifies \({{y_k}^{y_k}}/{y_k !}\) in (1b) and helps showing the convergence of the infinite sums of the double Poisson–Sarmanov model in Sect. 3.2.

  10. In the literature on copulas, mixing functions \(\psi _k(y_k)\) are known as copula generators. See (Fisher and Klein (2007), §2) and Nelsen (2006).

  11. The GAUSS code used for the estimation of the bivariate model of Sect. 5 is available upon request.

  12. In the case of rescaled bootstrapping, we use a given number of bootstrap samples of size \(b\) where some of these samples might be repeated. This differentiates rescaled bootstrapping from subsampling where some or all \(n!/[b!(n-b)!]\) samples of size \(b\) (always without repetition) are employed in estimating each replication, e.g., see (Politis et al. (1999), §2.1).

  13. This is exactly the copula function employed by Heinen and Rengifo (2007) in a time series framework. Assuming double Poisson marginals has the advantage of reducing the risk of misspecification by wrongly imposing equidispersion of counts.

  14. (Heinen and Rengifo (2007), §2) make use of this continuization approach to estimate their Gaussian copula model. Alternatively, Bien et al. (2011) and Cameron et al. (2004) use first-order differencing operators to link the multivariate frequency function with the cumulative multivariate copula distribution function.

  15. van Ophem (1999) adopts this same approach to characterize the multivariate distribution of discrete variables based on the joint distribution of underlying continuous variables.

  16. For an institutional and historical account of the poorly designed awarding process of licenses in the early US cellular telephone industry, see Hausman (2002), Parker and Röller (1997), or Murray (2002).

  17. Other value-added services such as detailed billing, call waiting, no-answer transfer, call forwarding, three way calling, busy transfer, call restriction, and voice mail were priced independently and rarely bundled together with particular tariff options.

  18. Other regressors are available, but they are not significant in neither equation.

  19. Businesses with potential high cellular demand include service firms, health care, professional, and legal services, contract construction, transportation, finance, insurance, and real estate. The source of all these demographics is the 1989 Statistical Abstracts of the United States; US Department of Commerce, Bureau of the Census, using the Federal Communication Commission (FCC) Cellular Boundary Notices, 1982–1987, available in The Cellular Market Data Book, EMCI, Inc., as well as the 1990 US Decennial Census.

    Table 2 Descriptive statistics
    Full size table

  20. Busse (2000) addresses the relationship between multimarket contact on collusion so that the offering of certain tariff features allows firms to coordinate pricing.

  21. Shew (1994) claims that the possibility of having request approval of new tariffs in the future prompts firms in this industry to offer an “excessive” number of options when they enter the market, the only time when they do not have to seek such approval as cost data are not yet available.

  22. They are: Ameritech Mobile (AMERITECH), Bell Atlantic Mobile (BELLATL), Bell South Mobile (BELLSTH), Century Cellular (CENTEL), Contel Cellular (CONTEL), GTE Mobilnet (GTE), McCaw Communications (McCAW), Nynex Mobile (NYNEX), PacTel Mobile Access (PACTEL), SouthWest Bell (SWBELL), and US West Cellular (USWEST).

  23. It could be argued that the documented underdispersion was due to the fact that firms always offer at least one tariff option, and thus, the dependent variables do not include any zeros. Later in the regressions, the endogenous variables will be defined as the number of tariff options minus one. Notice that variances in the last line of Table 3 still exceed the means of the transformed endogenous variables, i.e., the means of that last line minus one.

  24. Notice that for the Sarmanov model, I have computed t statistics making use of rescaled bootstrapping. This criterion is more conservative than using standard bootstrapping and also the appropriate one to deal with the existence of constraints involving the parameter estimates. Every estimation of the scaled bootstrap is run on a sample size of one-third of the full sample and where the same observation may be present multiple times. Since in addition to the sample considered, the effective range of the correlation coefficient is partially determined by the regressors included in the exponential conditional mean function (3), I repeated the analysis without any regressors. While correlation becomes not significant, the simpler specification is rejected in favor of that of Table 5.

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Acknowledgments

Thanks are due to Jason Abrevaya, Stephen Donald, Jerry Hausman, and two referees for their comments. A previous version was presented at the Texas Camp Econometrics XIV in New Braunfels. It circulated as CEPR Discussion Paper No. 7463 “Multivariate Sarmanov Count Data Models.”

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  1. Department of Economics, The University of Texas at Austin, 1 University Station C3100, Austin, TX, 78712, USA

    Eugenio J. Miravete

  2. Centre for Competition Policy, UEA, Norwich, UK

    Eugenio J. Miravete

  3. CEPR, London, UK

    Eugenio J. Miravete

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Miravete, E.J. Testing for complementarities among countable strategies. Empir Econ 46, 1521–1544 (2014). https://doi.org/10.1007/s00181-013-0729-y

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