Selection of random coefficients in ordered response models: a framework to detect heterogeneity in household surveys

Abstract

This paper develops a Bayesian method to detect heterogeneity in the relationship between covariates and the outcome in models with ordered responses. To this end, we construct an efficient Markov chain Monte Carlo algorithm for a hierarchical Bayesian model that selects random coefficients in ordered models. This method extends an approach for selecting random coefficients in linear mixed models into the ordered setting by adding two enhancements that are relevant to the latter category of models. First, we construct steps to efficiently estimate cut-points by addressing identification and ordering constraints. Second, we develop a framework to evaluate marginal effects that combine the fixed and random effects of each covariate. The marginal effects additionally allow for model uncertainty by averaging across models visited by the selection algorithm. Simulation studies demonstrate that this method detects random effects when they are present, estimates parameters accurately and efficiently samples from the posterior with low autocorrelations across successive draws. On applying this method on data from the survey of consumer expectations, we find clear support for the presence of household-level heterogeneity in relationships between demographic variables, and current as well as expected financial conditions.

Keywords:

• Variable selection
• random coefficients
• ordered responses
• marginal effects
• Bayesian estimation
• household surveys

Acknowledgments

The author thanks the Editor, the Associate Editor and anonymous referees for their insightful and helpful comments that greatly improved the quality and presentation of the paper. The author also thanks Ivan Jeliazkov, Gary Richardson, Eric Swanson, David Brownstone and Nick Sly for their insightful comments. Thanks to Jacob Dice for excellent research assistance. The views expressed here are the opinions of the author only and do not necessarily represent those of the Federal Reserve Bank of Kansas City or the Federal Reserve System.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 The files are located at https://www.newyorkfed.org/medialibrary/interactives/sce/sce/downloads/data/frbny-sce-public-microdata-complete-17-19.xlsx

2 The data are available under ‘Unemployment’ and ‘Local Area Unemployment Statistics’ at https://www.bls.gov/data/

3 In Section 2 of the Supplemental Material, we discuss the rationale for the decomposition proposed by Chen and Dunson [8] over an alternate decomposition. The interpretation of the components of the covariance matrix are discussed in Section 3 of the Supplemental Material.

4 The specific initial values chosen in our simulation exercise, as well as alternative values considered in sensitivity exercises are reported in Table 4 within Section 6 of the Supplemental Material.

5 The results of a sensitivity exercise that considers alternative prior hyperparameters are provided in Section 8 of the Supplemental Material.

6 The rationale for the identifying restrictions are discussed further in Section 7 of the Supplemental Material.

7 Step 1(a) in the alternative algorithm only samples the cut-points marginally of the latent variable z. The coefficients β are sampled from $\beta |z,{\theta }_i,\gamma ,\lambda$ in a separate step.

8 The Equal Credit Opportunity Act prohibits discrimination on the basis of race, color, religion, national origin, sex, marital status, age, receipt of public assistance or good faith exercise of any rights under the Consumer Credit Protection Act (https://www.ftc.gov/enforcement/statutes/equal-credit-opportunity-act).

9 The full set of marginal effects are available upon request.