A Dynamic Binary Probit Model with Time-Varying Parameters and Shrinkage Prior

Abstract

This article studies a time series binary probit model in which the underlying latent variable depends on its lag and exogenous regressors. The regression coefficients for the latent variable are allowed to vary over time to capture possible model instability. Bayesian shrinkage priors are applied to automatically differentiate fixed and truly time-varying coefficients and thus avoid unnecessary model complexity. I develop an MCMC algorithm for model estimation that exploits parameter blocking to boost sampling efficiency. An efficient Monte Carlo approximation based on the Kalman filter is developed to improve the numerical stability for computing the predictive likelihood of the binary outcome. Benefits of the proposed model are illustrated in a simulation study and an application to forecast economic recessions.

Keywords:

• Bayesian shrinkage
• Horseshoe
• Probit
• TVP

Supplementary Materials

Appendix: Appendices A–H for the article. (.pdf file)

Data: Data used in the simulation and empirical studies. (.xlsx file)

Code: Matlab programs to estimate the models for the simulation and empirical studies. (.m file)

Acknowledgments

I would like to thank Professor Atsushi Inoue (the editor), an AE and three referees for many invaluable comments that have greatly improved the paper. All remaining errors are my own. The views in this paper are solely the author’s responsibility and are not related to the company the author works in.

Disclosure Statement

The author reports that there are no competing interests to declare.

Notes

1 Allowing $\varphi$ to be time varying while keeping the stationarity condition can complicate the model estimation. In my experiments, a random walk process for $\varphi$ that ignores the stationarity condition actually produces estimates that are mostly within the $(-1,1)$ range and are largely flat over time, though the resulting out-of-sample forecast performance is outperformed by the constant $\varphi$ specification.

2 It is possible to specify AR processes for ${\beta }_t^{*}$ instead of random walks at the cost of more parameters to estimate. Moreover the AR coefficients of ${\beta }_t^{*}$ would become unidentified when $v^2$ approaches zero.

3 The model parameters $\delta$, $\varphi$ and $z_0$ are not fixed quantities, but are either estimated from the data or are integrated out in estimation.

4 See Cadonna, Fruhwirth-Schnatter, and Knaus (2020) for a discussion of the benefit from not restricting the sign of coefficients in such a reparameterization.

5 I experimented using a single-move Gibbs sampler that integrates out the regression coefficients and found that the mixing of the posterior draws tends to be unsatisfactory when the number of regressors increases.

6 The TVP probit model retains the RHS shrinkage priors for v and ${\beta }_0^{*}$ as in the TDP model.

7 An example is provided in the empirical application of Section 6.

8 When the binary outcome is observed with delays, the time of forecast n should be understood as the last time when the binary outcome is observed.

9 Generating 1000 posterior draws from the TDP model with shrinkage priors takes about 60 sec on a standard desktop computer with a 3.0 GHz Intel Core i5 CPU, running in MATLAB R2020b.

10 Appendix E estimates the TDP model for a “dense” DGP where most coefficients are time varying and reaches similar findings. Tests for $v_j=0$ via Bayes factor are discussed in Appendix G.

11 See the websites https://www.newyorkfed.org/research/capital_markets/ycfaq and https://www.clevelandfed.org/en/our-research/indicators-and-data/yield-curve-and-gdp-growth.

12 The running time of producing 1000 draws from the blocked Gibbs sampler is about 50 sec and is longer than the standard Gibbs sampler that takes about 20 sec. Considering the improvement in sampling efficiency, this computation cost is considered worthwhile.

13 Marginal likelihoods are computed by the method of Gelfand and Dey (1994) and are $-$228, $-$325, and $-$314 for the TDP, TDP-NS, and TVP probit models, respectively. The employed likelihood functions integrate out the time-varying coefficients analytically but condition on the latent variable $z_t$. See Chan and Grant (2015) for a demonstration of the pitfalls of marginal likelihoods where the likelihood function conditions on latent variables.

14 The forecasts are not strictly real time ones as in reality the recession indicator data are released with irregular delays.

15 The model amounts to () where $v_j$ = 0 and ${\beta }_{0,j}^{*}\sim N(0,10)$ for j = 1,…, K.

16 The forecast performance of the TDP model and its restricted version where $v_j=0$ for j = 1,…, K (i.e., a dynamic probit model under RHS shrinkage) is comparable, consistent with the modest Bayes factor supporting overall model instability discussed in Section 6.1