A Gibbs Sampling Algorithm for a Changing Regression Model with Pooled Binary Response Data

Abstract

This article presents a Gibbs Sampling algorithm for a changing regression model with a pooled binary dependent variable. A Gibbs Sampling algorithm for a changing regression model with a continuous dependent variable is extended to binary response data using the chained data augmentation of Tanner (1996). The proposed algorithm is applied to numerical examples with a single change point and multiple change points. The results suggest that this algorithm robustly provides accurate estimates of change points, and that this algorithm also provides sharper estimates of regression parameters when it is applied to a pooled data with a larger cross-section sample size.

Keywords:

• Binary response
• Changing regression
• Data augmentation
• Gibbs sampling

Mathematics Subject Classification:

• Primary 65C05
• Secondary 62J02

Acknowledgments

The author gratefully acknowledges Professor Hiroki Tsurumi for his help in developing the first draft of this article. The author also thanks a referee for valuable comments and suggestions.

Notes

¹Chib's (1998) method is particularly useful when the data are dependent over time. This article focuses on the case where the data are independent over time.

²A formal presentation of this model is given in Sec. 2.

³Hinkley (1970) and Yao (1987) only deal with the single parameter case. Bhattacharya's (1987) approach is also limited to exponential families.

⁴If the change point takes a continuous value and the dependent variable is observable, the following equality constraint on the dependent variable is usually imposed (Stephens, 1994)

This article focuses on the cases when the change point takes discrete values. Then this equality constraint becomes irrelevant.

⁵Carlin et al. (1992) used a hyper prior to obtain a proper normal prior for the regression coefficient. This approach also can be used.

⁶Albert and Chib (1993) employed this chained data augmentation in their analysis of binary response data.

⁷ is the initial value of δ _t for a given (t∗₁, t∗₂).

⁸Our approach is somewhat different from Stephens (1994), which suggested to sample t∗₁ | t∗₂ and t∗₂ | t∗₁ using π(t∗₁ | t∗₂, D) and π(t∗₂ | t∗₁, D). The examples provided in Stephens (1994) are the cases when π(t∗₁ | t∗₂, D) and π(t∗₂ | t∗₁, D) are available. Specifically, the following integration can be done without much difficulty.

With binary response data, this integration is hard to obtain unless the number of independent variables is very small.

⁹The fluctuation test was originally introduced in Ploberger et al. (1989).

∗: True change point.

sd∗: posterior standard deviation.

AR(1)∗∗: autocorrelation coefficient of draws.

FT∗∗∗: P value of the fluctuation test statistic.

¹⁰The sum of squared distances between the posterior mean of the regression coefficient and the true value (∑ _ij ( − β _ij )², (: posterior mean)) was 0.0093 when the cross-section sample size was 15. It dropped to 0.0026 when the cross-section sample size was 30. In case of the difference of the regression coefficients, the sum of the squared distances between the posterior mean and the true value dropped from 0.0165 to 0.0066 as the cross-section sample size increased from 15 to 30.