### Discussion Paper

## Abstract

According to several empirical studies, the Present Value model fails to explain the behaviour of stock prices in the long-run. In this paper, the authors consider the possibility that a linear cointegrated regression model with multiple structural changes would provide a better empirical description of the Present Value model of U.S. stock prices. The methodology is based on instability tests recently proposed in Kejriwal and Perron (*The limit distribution of the estimates in cointegrated regression models with multiple structural changes*, 2008, and *Testing for multiple structural changes in cointegrated regression models*, 2010) as well as the cointegration tests developed in Arai and Kurozumi (*Testing for the null hypothesis of cointegration with a structural break*, 2007) and Kejriwal (*Cointegration with structural breaks: an application to the Feldstein-Horioka Puzzle*, 2008). The results obtained are consistent with the existence of linear cointegration between the log stock prices and the log dividends. However, the empirical results also show that the cointegrating relationship has changed over time. In particular, the Kejriwal-Perron tests for testing multiple structural breaks in cointegrated regression models suggest a model of three or two regimes.

## Comments and Questions

This paper by Esteve, Navarro and Prats is interesting and addresses a question of relevance for economic research. They use tests for cointegration under structural breaks to find that stock prices and dividends have been cointegrated for around a century with one or two breaks. They also find that the ...[more]

... slope of the cointegrating relationship increases with time, as the long-term price-dividend ratio is now larger than 80 years ago. I enjoyed reading the paper and I learned a lot from it.

My main comment about the paper is related to the results of the tests that identify the number of breaks. My understanding is that in Section 3.2 the authors say that the Kejriwal and Perron (2008, 2010) tests will be used to determine the number of regimes. When these tests are used in the following section, the results of the tests and the sequential procedure suggest zero breaks. Then the authors conclude that there are one or two breaks based on the LWZ and BIC, which are not explained in the paper. I would be interested in knowing why the authors believe that the LWZ-BIC results are more convincing than the results of the Kejriwal and Perron tests.

I also have a few suggestions on the tables used to display the results. In Table I, I assume that that the third column shows the critical values and the fourth column the values of the statistic used in one of the tests. It is not clear to me what the fifth and sixth columns show. In Table 2, Note d references the source of the critical values for MDF1. What about the critical values for MDF2? In Table 3, the values of the Cτ statistic are not shown, only the Cµ. And in Table 5 the critical values are not consistent with the * displayed.

A final suggestion would be to use the estimated long-run relationship between stock prices and dividends to obtain a measure of how much the US stock market is overvalued nowadays.

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