Panel threshold model with covariate-dependent thresholds and unobserved individual-specific threshold effects

Abstract

This article introduces a panel threshold model with covariate-dependent and time-varying thresholds and unobserved individual-specific threshold effects (PTCDI). We develop methods for estimation and inference for threshold parameters in the proposed PTCDI model by employing the correlated random effects (CRE) device. We also suggest test statistics for linearity, threshold constancy, unobserved individual-specific threshold effects, and for determining the number of thresholds. We derive the asymptotic properties of the proposed estimator in the small-threshold-effect framework and establish the limiting distributions of the suggested test statistics. We also investigate the extension to dynamic panels and show that both the static and dynamic models can be handled uniformly in the CRE framework. Monte Carlo simulation results indicate that the estimation, inference, and testing procedures have the desired performance in finite samples. The model is illustrated with two empirical applications to the relationship between cash flow and investment and the nexus between inflation and economic growth.

Keywords:

• Cash flow/investment relationship
• inflation/economic growth nexus
• multiple covariate-dependent thresholds
• panel threshold model
• testing
• unobserved threshold effects

JEL Classification:

• C12
• C13
• C15
• C33

Acknowledgments

We would like to thank the Editor Prof. Esfandiar Maasoumi, an associate editor, and three anonymous referees for their very constructive comments and suggestions. We also thank Hujie Bai for research assistance. Any remaining errors are solely our responsibility.

Disclosure statement

The authors declare that they have no potential conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by the authors.

Notes

1. Yang et al. (2021b) extend the kink threshold model of Hansen (2017) to a panel framework with one covariate-dependent threshold.

2. The covariate-dependent threshold setting can also be treated as a normalization of the classical threshold model with a linear index (e.g., Lee et al., 2021; Seo and Linton, 2007).

3. We thank an anonymous referee for raising the issue of selecting the optimal set of covariates, which is important as the inappropriate choice of the covariates affecting the threshold can lead to biased estimates and distorted testing results. Future research can work on this issue.

4. We first focus on the model with one covariate-dependent and time-varying threshold and unobserved individual-specific threshold effects, and then discuss the extension to multiple covariate-dependent and time-varying thresholds.

5. In this article, we do not investigate how to select covariates (shaping the threshold) among many candidate variables. In applications, we can choose 𝐬_{1, it} based on economic intuition, as argued by Yu and Fan (2021). It is interesting to develop a systematic approach to choosing covariates that shape the threshold in the panel data framework with a covariate-dependent and time-varying threshold and unobserved individual-specific threshold effects.

6. If 𝐱_it does not contain q_it and 𝐬_it, ${\overline{\mathbf{x}}}_i$ should include ${\overline{q}}_i$ and ${\overline{\mathbf{s}}}_i$. This comment also applies to the dynamic setting where $q_{it}\ne y_{i,t-1}$.

7. It is worth noting that SSR(γ)∕NT is not asymptotically equivalent to $SSR(\gamma )/(NT-N-2p_{\mathbf{x}})$ under the framework with large N and fixed T, as $\frac{T}{T-1}\ne 1$. Thus, the limiting distributions of the proposed test statistics would differ if we replaced the denominator SSR(γ)∕NT with $SSR(\gamma )/(NT-N-2p_{\mathbf{x}})$. We thank an anonymous referee for raising this point with us.

8. In this article, we assume ${\mathit{\gamma }}_1^{\prime }{\mathbf{s}}_{it}<{\mathit{\gamma }}_2^{\prime }{\mathbf{s}}_{it}$ for all 𝐬_it’s, and exclude the case in which ${\mathit{\gamma }}_1^{\prime }{\mathbf{s}}_{it}<{\mathit{\gamma }}_2^{\prime }{\mathbf{s}}_{it}$ for some 𝐬_it’s while ${\mathit{\gamma }}_1^{\prime }{\mathbf{s}}_{it}\ge {\mathit{\gamma }}_2^{\prime }{\mathbf{s}}_{it}$ for other 𝐬_it’s. Future research can focus on the latter case.

9. We provide simulation evidence supporting the consistency of the estimator in the case of multiple covariate-dependent and time-varying thresholds. Future research can focus on providing a rigorous proof for this result, i.e., ${\hat{\mathit{\gamma }}}_1$ would be consistent for either 𝜸₁ or 𝜸₂ depending on which effect is stronger. We thank an anonymous referee for raising this point with us.

10. In the simulations, we set the magnitude of the threshold effect to be fixed, which implies α = 0 in Theorem 1.

11. https://www.ssc.wisc.edu/bhansen. We are grateful to Bruce Hansen for making his data and code publicly available.

12. We thank N.R. Ramírez-Rondán for sharing his dataset to us.

13. To save space, we do not report the simulation results for $({\psi }_{l1},{\psi }_{l2},{\psi }_{l3},,{\psi }_{l0})$ (l=1,2,3). The results show that the empirical mean of each parameter is also close to its true value for all combinations of T and N, and the standard deviations decrease as either T or N increases. These simulation results are available from the authors upon request.

Additional information

Funding

Lixiong Yang acknowledges the financial support from the National Natural Science Foundation of China (Grant No. 72273059).