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

Abstract

It is well known that cointegration between the level of two variables (labeled Yt and yt inthis paper) is a necessary condition to assess the empirical validity of a present-value model(PV and PVM, respectively, hereafter) linking them. The work on cointegration has been soprevalent that it is often overlooked that another necessary condition for the PVM to hold isthat the forecast error entailed by the model is orthogonal to the past. The basis of this resultis the use 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,we advise to split the restrictions implied by PV relationships into orthogonality conditions(or reduced rank restrictions) before additional tests on the value of parameters. We showthat PV relationships entail a weak-form common feature relationship as in Hecq, Palm, andUrbain (2006) and in Athanasopoulos, Guillén, Issler and Vahid (2011) and also a polynomialserial-correlation common feature relationship as in Cubadda and Hecq (2001), which representrestrictions on dynamic models which allow several tests for the existence of PV relationships tobe used. Because these relationships occur mostly with nancial data, we propose tests based ongeneralized method of moment (GMM) estimates, where it is straightforward to propose robusttests in the presence of heteroskedasticity. We also propose a robust Wald test developed toinvestigate the presence of reduced 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 nancial theory as discussed in Alvarez andJermann (2005), Hansen and Scheinkman (2009), and Nieuwerburgh, Lustig, Verdelhan (2010).Here again 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.The techniques discussed herein are applied to long span annual data on long- and short-term interest rates and on price and dividend for the U.S. economy. In both applications we donot reject the existence of a common cyclical feature vector linking these two series. Extractingthe long-run component shows the usefulness of our approach and highlights the presence ofasset-pricing bubbles.