Econometric models of duration data in entrepreneurship with an application to start-ups' time-to-funding by venture capitalists (VCs)

Abstract

Because time is a key determinant of entrepreneurial decision making, time-to-event models are ubiquitous in entrepreneurship. Widespread econometric misconception, however, may cause complicated biases in existing studies. The reason is spurious duration dependency, a complicated form of endogeneity caused by unobserved heterogeneity, which is particularly pronounced in entrepreneurship data. This article discusses the endogeneity problem and methods to ‘debias’ time-to-event models in entrepreneurship. Simulations and empirical evidence indicate that only the frailty approach yields consistently unbiased parameter estimates. An application to start-up firms' time-to-funding shows that other methods lead to dramatic biases. Therefore, this article advocates a paradigm shift in the modeling of time variables in entrepreneurship.

Keywords:

• Entrepreneurship
• time-to-event model
• survival model
• frailty model
• entrepreneurial finance

JEL Classifications:

• C41
• M13
• M16

Acknowledgements

Thanks goes to Jie Chen (the editor), the associate editor, two anonymous reviewers, and Karl Wennberg for helpful comments, as well as to Sebastian Hohenberg for insightful discussions on an earlier version of this paper. I gratefully acknowledge the financial support for this research project from the Price Center for Entrepreneurship and Innovation at UCLA.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The statistic is based on published and ‘online first’ articles in leading entrepreneurship journals (i.e. Entrepreneurship Theory and Practice, Journal of Business Venturing, Strategic Entrepreneurship Journal, Research Policy, and Small Business Economics) between 01/2018 and 02/2020.

2 Mathematically, it is possible that Cox Proportional Hazards models with or without fixed effects and the frailty model hold simultaneously. But this requires positive stable distributions of the random effects (i.e. the distribution on the positive numbers is stable if the sum of independent random variables from a normalized distribution follow the same distribution [32]), which is unrealistic in most applied contexts.

3 For example, for non-censored time-to-event with gamma-distributed unobserved heterogeneity in a proportional-hazard Cox model, the bias is ${\beta }^{\mathrm{\prime }}=\frac{\beta }{1+{\sigma }^2}$ where β denotes the unbiased and ${\beta }^{\mathrm{\prime }}$ the biased coefficient, and ${\sigma }^2$ denotes the variance of the unobserved heterogeneity. But for other cases, no explicit solutions exist.

4 An important result from asymptotic theory due to [67] is that the asymptotic variance of $\hat{\beta }$ in frailty models is a weighted average of those in unadjusted and fixed effects Cox Proportional Hazards models. Specifically, the asymptotic variance is inflated in fixed effects models (upper bound), whereas it is deflated in unadjusted Cox Proportional Hazards models (lower bound).

5 In contrast, in unshared frailty models, $h_{ij}(t|X_{ij},u_i)=h_0(t)\times e^{{\beta }^{\prime }{X}_{ij}+{\omega }_j}$, it is assumed that random effects differ across subjects.

6 We thank an anonymous reviewer for pointing out this implicit assumption of our empirical research design.

7 We also checked the performance of an Accelerated Failure Time (AFT) model and found that it is outperformed by the frailty model; for this reason, we do not report it here in greater detail. AFT models model the change in the hazard as a horizontal shift, whereas Cox Proportional Hazard models model it as a vertical shift. With very few exceptions, AFT and Cox Proportional Hazard models are mutually exclusive with respect to representing underlying data. Thus, we conclude that our specific empirical context is best approached with the frailty model. It may therefore be interesting for future research to examine which entrepreneurial contexts are better model by the AFT vis–vis the frailty approach. We thank an anonymous reviewer for suggesting this check.

8 In Table 3, all coefficients are reported as hazard ratios. While this makes the interpretation of individual coefficients easy, it requires additional steps to interpret interaction effects. Specifically, to make interaction effects interpretable, it is required to take the logarithm of the hazard ratios, add them up, and then exponentiate them to arrive at the interaction hazard ratios.

9 Although it is beyond the scope of this paper to provide an extensive theoretical foundation for the identified effects in Table 3, the parameter estimates are largely consistent with the reasoning provide in related work (compare, for example, [39]).

Additional information

Funding

I gratefully acknowledge the financial support for this research project from the Price Center for Entrepreneurship and Innovation at UCLA.