Bayesian Dynamic Panel-Ordered Probit Model and Its Application to Subjective Well-Being

Abstract

This article proposes a Bayesian method for estimating the dynamic panel-ordered probit model. We compare four alternative algorithms for the estimation of ordered probit models. Furthermore, this article presents the empirical results of the application of the dynamic ordered probit model to subjective well being based on the micro-level survey data extracted from the Japanese Panel Survey of Consumers. The results of the application reveal that income and savings have positive effects on life satisfaction, whereas our posterior results show that marriage and labor force participation have negative effects on the same.

Keywords:

• Generalized Gibbs sampler
• Life satisfaction
• Markov chain Monte Carlo (MCMC)
• Well-being function

Mathematics Subject Classification:

• 62F15
• 62P20

Acknowledgments

The author appreciates the comments of two anonymous referees, which improved the article greatly. He is also grateful to the Institute for Research on Household Economics for providing the micro data of the Japanese Panel Survey of Consumers. Further, this work was supported in part by a Grant-in-Aid for Scientific Research (No. 19530177) from the JSPS.

Notes

¹In this article, we use life satisfaction as the measurement of well being.

²This model is a dynamic version of the Hausman and Taylor (1981) model. See, for example, Baltagi (2001, p. 139).

³Further, to use the scale group defined in Appendix B, we have to specify the hyperparameters in the prior distributions  = 0,  = 0, ₀ = 0, ₀ = 0, and  = 0.

^a “Mean” and “SD” denote the sample mean and sample standard deviation. “Min,” “50%” and “Max” denote the minimum value, median and maximum value.

^a “Mean” and “SD” denote the posterior mean and posterior standard deviation. “95%CI” denotes the 95% credible interval. “CD” and “p-Val” denote the convergence diagnostic statistic of MCMC proposed by Geweke (1992) and its p value. ^b The values of parameters except for φ are transformed ones devided by σ.

⁴We obtained 12,000 samples by running 60,000 iterations and sampling one observation at every five iterations. After obtaining 12,000 samples, we discarded the first 2,000 samples.

⁵The results of parameters in Algorithm 4 except for φ are those of the transformed parameters divided by σ.

⁶CD can be defined as follows: for the given sequence {g(j)|j = 1,2,…, n _s }, if the sequence is stationary,

where

and and denote consistent spectral density estimates. In this numerical example, we set n _A  = 1,000 and n _B  = 5,000, and calculate and using Parzen windows with bandwidths of 100 and 500, respectively.

⁷The number of ordinal categories of life satisfaction is originally C = 5, including “strongly dissatisfied,” “dissatisfied,” “moderately satisfied,” “satisfied,” and “strongly satisfied.” However, since the frequency of “strongly dissatisfied” is small, this article combines two categories “strongly dissatisfied” and “dissatisfied” into a single category, “strongly dissatisfied and dissatisfied.”

⁸In the case of zero income or zero savings, we set loginc (x ₂) or logsav (x ₃) equal to 0. Further, we use the consumer price index as a deflator for constructing real variables.

^a “Mean” and “SD” denote the sample mean and sample standard deviation. “Min,” “50%” and “Max” denote the minimum value, median and maximum value. ^b The data of “age” and “educ” are obtained from Panel 6.

⁹We obtained 30,000 samples by running 150,000 iterations and sampling one observation at every five iterations. After obtaining 30,000 samples, we discarded the first 10,000 samples.

¹⁰The results of parameters in Algorithm 4 except for φ are those of the transformed parameters divided by σ.

^a “Mean” and “SD” denote the posterior mean and posterior standard deviation. 95%CI denotes the 95% credible interval. “CD” and “p-Val” denote the convergence diagnostic statistic of MCMC proposed by Geweke (1992) and its p value. ^b The values of parameters except for φ are transformed ones divided by σ.

¹¹Here, the term “significant” is used if the 95% credible interval for a parameter does not include zero. See also Koop (2003, p. 124).

¹²The details of partial effects of the continuous explanatory variables are provided in Appendix E.

^a “Mean” and “SD” denote the posterior mean and posterior standard deviation. ^b “CD” denotes the convergence diagnostic statistic of MCMC proposed by Geweke (1992).

¹³Using Damien and Walker's algorithm, we can sample φ from the following steps.

1.	Let , we have .
2.	Introduce the latent variable y which has joint density with x given by where .
3.	We have the following FCDs:
4.	Calculate .

¹⁴Since β has k, δ has p, β ₀ has k, δ ₀ has p, α has n, γ has C − 2, z has nT, z ₀ has n, μ has one g, the exponential part of g becomes (T + 2)n + 2k + 2p + C − 1.