Price Dividend Ratio and Long-Run Stock Returns: A Score-Driven State Space Model

Abstract

Abstract–In this article, we develop a general framework to analyze state space models with time-varying system matrices, where time variation is driven by the score of the conditional likelihood. We derive a new filter that allows for the simultaneous estimation of the state vector and of the time-varying matrices. We use this method to study the time-varying relationship between the price dividend ratio, expected stock returns and expected dividend growth in the United States since 1880. We find a significant increase in the long-run equilibrium value of the price dividend ratio over time, associated with a fall in the long-run expected rate of return on stocks. The latter can be attributed mainly to a decrease in the natural rate of interest, as the long-run risk premium has only slightly fallen.

KEYWORDS:

• Equity premium
• Present-value models
• Score-driven models
• State space models
• Time-varying parameters

Supplementary Materials

The supplementary materials include an online appendix with the derivations of the proofs, examples, an extensive Monte Carlo study, extensions of the main algorithm and additional details on the application, as well the Matlab codes necessary to replicate the results in the article.

Acknowledgments

The views expressed in this article belong to the authors and are not necessarily shared by the Bank of Italy or by the European Central Bank. The authors would like to thank Juan Antolin-Diaz, Gino Cenedese, Laura Coroneo, Ana Galvao, Domenico Giannone, Arie Gozluklu, Andrew Harvey, Gary Koop, Siem Jan Koopman, André Lucas, Massimiliano Marcellino, Daniele Massacci, Pavol Povala, Barbara Rossi, Bernd Schwaab, Ron Smith, and Shaun Vahey for their useful suggestions.