A Common-Feature Approach for Testing Present-Value Restrictions with Financial Data

It is well known that cointegration between the level of two variables (labeled Y_{t} and y_{t} inthis paper) is a necessary condition to assess the empirical validity of a present-value model (PVand PVM, respectively, hereafter) linking them. The work on cointegration has been so prevalentthat it is often overlooked that another necessary condition for the PVM to hold is that theforecast error entailed by the model is orthogonal to the past. The basis of this result is theuse of rational expectations in forecasting future values of variables in the PVM. If thiscondition fails, the present-value equation will not be valid, since it will contain an additionalterm capturing the (non-zero) conditional expected value of future error terms.Our article has a few novel contributions, but two stand out. First, in testing for PVMs, weadvise to split the restrictions implied by PV relationships into orthogonality conditions (orreduced rank restrictions) before additional tests on the value of parameters. We show that PVrelationships entail a weak-form common feature relationship as in Hecq, Palm, and Urbain (2006)and in Athanasopoulos, Guillén, Issler and Vahid (2011) and also a polynomial serial-correlationcommon feature relationship as in Cubadda and Hecq (2001), which represent restrictions on dynamicmodels which allow several tests for the existence of PV relationships to be used. Because theserelationships occur mostly with financial data, we propose tests based on generalized method ofmoment (GMM) estimates, where it is straightforward to propose robust tests in the presence ofheteroskedasticity. We also propose a robust Wald test developed to investigate the presence ofreduced rank models. Their performance is evaluated in a Monte-Carlo exercise.Second, in the context of asset pricing, we propose applying a permanent-transitory (PT)decomposition based on Beveridge and Nelson (1981), which focus on extracting the long-runcomponent of asset prices, a key concept in modern financial theory as discussed in Alvarez andJermann (2005), Hansen and Scheinkman (2009), and Nieuwerburgh, Lustig, Verdelhan (2010). Hereagain we can exploit the results developed in the common cycle literature to easily extractpermament and transitory components under both long and also short-run restrictions. Thetechniques discussed herein are applied to long span annual data on long- and short-term interestrates and on price and dividend for the U.S. economy. In both applications we do not reject theexistence of a common cyclical feature vector linking these two series. Extracting the long-runcomponent shows the usefulness of our approach and highlights the presence of asset-pricing bubbles.