Panel threshold model with covariate-dependent thresholds and its application to the cash flow/investment relationship

Lixiong Yang

doi:10.1515/snde-2022-0035

Abstract

This paper introduces a panel threshold model with covariate-dependent and time-varying thresholds (PTCT), which extends the classical panel threshold model of Hansen, B. E. 1999. “Threshold Effects in Non-dynamic Panels: Estimation, Testing, and Inference.” Journal of Econometrics 93: 345–68 to a framework with multiple covariate-dependent and time-varying thresholds. Based on the within-group transformation and Markov chain Monte Carlo (MCMC) technique, we develop methods for estimation and inference for threshold parameters in the proposed panel threshold model. We also suggest test statistics for threshold effect, threshold constancy, and for determining the number of thresholds. Monte Carlo simulations indicate that the estimation, inference and testing procedures work well in finite samples. Empirically, using the same data as in Hansen, B. E. 1999. “Threshold Effects in Non-dynamic Panels: Estimation, Testing, and Inference.” Journal of Econometrics 93: 345–68 we revisit the cash flow/investment relationship and find quite different results.

Keywords: cash flow/investment relationship; estimation; multiple covariate-dependent thresholds; panel threshold model; testing

Corresponding author: Lixiong Yang, School of Management, Lanzhou University, Lanzhou, China, E-mail: ylx@lzu.edu.cn

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 71803072

Award Identifier / Grant number: 72273059

Acknowledgement

The author would like to thank the editor and anonymous referees for very valuable comments and suggestions that result in a substantial improvement in this paper. Remaining errors and omissions are my own. The author acknowledges the financial support from the National Natural Science Foundation of China (Grant No. 72273059 and 71803072).

Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: This study was supported by the National Natural Science Foundation of China (Grant No. 72273059 and 71803072).
Conflict of interest statement: The author declares that he has no potential conflict of interest.
Ethical approval: This article does not contain any studies with human participants or animals performed by the author.

Appendix A: Simulation results of the model with multiple covariate-dependent thresholds

In this section, we conduct Monte Carlo simulations to examine the finite sample performances of the proposed estimation and testing procedures for the model with multiple covariate-dependent thresholds. To this end, we consider the following data generating process (DGP) with double thresholds:

(A1) y i t = β 1 x i t I ( q i t ≤ γ 1 , i t ) + β 2 x i t I ( γ 1 , i t < q i t ≤ γ 2 , i t ) + β 3 x i t I ( q i t > γ 2 , i t ) + β 4 z i t + α i + u i t ,

(A2) γ 1 , i t = γ 10 + γ 11 s i t ,

(A3) γ 2 , i t = γ 20 + γ 21 s i t ,

where x _it = 0.25α _i + u _x,it, z _it = 0.5α _i + u _z,it, α _i ∼ i.i.dN(0, 1), s _it ∼ i.i.dN(0, 1) and q _it ∼ i.i.dN(0, 1). The innovation processes u _it, u _x,it and u _z,it are independent of each other. u _it follows i.i.d.N(0, 0.5²). u _x,it and u _z,it follow i.i.d.N(0, 1). The number of replications is set as 1000.

In the first experiment, we examine the performance of the estimation procedure for the multiple threshold model, setting the true parameters as (β ₁, β ₂, β ₃, β ₄) = (1, 2, 3, 1) and (γ ₁₀, γ ₁₁, γ ₂₀, γ ₂₁) = (−0.5, 0.3, 0.2, 0.5). Table A1 reports the simulation results. These simulation results show that the proposed estimation procedure for multiple threshold case works well in finite samples.

Table A1:

Estimates of the covariate-dependent multiple threshold model using the MCMC-based algorithm in Section 2.

T	N	β ₁ = 1		β ₂ = 2		β ₃ = 3		β ₄ = 1		γ ₁₀ = −0.5		γ ₁₁ = 0.3		γ ₂₀ = 0.2		γ ₂₁ = 0.5
		Mean	Std. dev	Mean	Std. dev	Mean	Std. dev	Mean	Std. dev	Mean	Std. dev	Mean	Std. dev	Mean	Std. dev	Mean	Std. dev
5	50	1.009	0.064	1.994	0.081	3.002	0.054	1.003	0.037	−0.506	0.055	0.304	0.072	0.203	0.056	0.499	0.076
	100	1.007	0.045	2.003	0.058	2.992	0.042	0.999	0.026	−0.501	0.039	0.296	0.050	0.201	0.039	0.502	0.049
	200	1.011	0.035	2.001	0.045	2.990	0.029	0.999	0.019	−0.501	0.033	0.300	0.042	0.198	0.035	0.501	0.043
10	50	1.001	0.045	1.995	0.056	2.994	0.038	1.001	0.025	−0.503	0.042	0.303	0.052	0.200	0.036	0.494	0.043
	100	1.016	0.032	2.002	0.046	2.991	0.026	1.000	0.017	−0.500	0.037	0.304	0.044	0.202	0.032	0.500	0.042
	200	1.015	0.027	2.001	0.038	2.990	0.022	1.000	0.012	−0.501	0.030	0.300	0.038	0.202	0.031	0.501	0.037
20	50	1.011	0.033	2.000	0.046	2.990	0.028	1.000	0.017	−0.501	0.033	0.300	0.042	0.200	0.033	0.504	0.044
	100	1.014	0.027	2.001	0.038	2.990	0.020	1.000	0.012	−0.500	0.030	0.299	0.036	0.200	0.030	0.500	0.039
	200	1.012	0.021	2.002	0.034	2.999	0.017	1.000	0.009	−0.500	0.027	0.303	0.033	0.201	0.028	0.501	0.036

In the second experiment, we investigate the finite sample properties of the F-type test statistic F ₂ for the null hypothesis of one threshold against two thresholds (defined in (11)), and the test statistic F ₃ for the null hypothesis of two thresholds against three thresholds. To assess the size and power properties of F ₂, we compute the rejection frequencies under the null model given by (12) and (13) and the alternative model given by (A1)–(A3), respectively. To assess the size and power properties of F ₃, we compute the rejection frequencies under the null model given by (A1)–(A3) and the alternative model with three thresholds (the model is similar with (A1)–(A3), but with a third threshold 0 and slopes (β ₁, β ₂, β ₃, β ₄, β ₅) = (1, 2, 3, 4, 1)). As reported in Table A2, the test statistics have good size and power properties, especially when N × T ≥ 1000.

Table A2:

Finite-sample size and power of the F-type test statistics F ₂ and F ₃.

T	N	Test for one vs two (F ₂)		Test for two vs three (F ₃)
		Size	Power	Size	Power
5	50	0.026	0.958	0.046	0.459
	100	0.044	0.999	0.043	0.687
	200	0.046	1.000	0.045	0.816
10	50	0.043	0.998	0.048	0.713
	100	0.049	1.000	0.049	0.821
	200	0.045	1.000	0.048	0.886
20	50	0.053	1.000	0.050	0.818
	100	0.050	1.000	0.051	0.981
	200	0.049	1.000	0.050	0.997

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Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/snde-2022-0035).

Received: 2022-04-12

Accepted: 2023-02-07

Published Online: 2023-02-27

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