Time Series under Present-Value-Model Short- and Long-run Co-movement Restrictions

This paper has two original contributions. First, we show that PV relationships entail a weak-form SCCF restriction, as in Hecq et al. (2006) and in Athanasopoulos et al. (2011), and implies a polynomial serial correlation common feature relationship (Cubadda and Hecq, 2001). These represent short-run restrictions on the dynamic multivariate systems, something that has not been discussed before. Our second contribution relates to forecasting multivariate time series that are subject to PVM restrictions, which has a wide application in macroeconomics and finance. We benefit from previous work showing the benefits for forecasting when the short-run dynamics of the system is constrained. The reason why appropriate common-cycle restrictions improve forecasting is because it finds linear combinations of the first differences of the data that cannot be forecast by past information. This embeds natural exclusion restrictions preventing the estimation of useless parameters, which would otherwise contribute to the increase of forecast variance with no expected reduction in bias. We applied the techniques discussed in this paper to data known to be subject to PV restrictions: the online series maintained and updated by Robert J. Shiller at https://www.econ.yale.edu/~shiller/data.htm. We focus on three different data sets. The first includes the levels of interest rates with long and short maturities, the second includes the level of real price and dividend for the S&P composite index, and the third includes the logarithmic transformation of prices and dividends. Our exhaustive investigation of six different multivariate models reveals that better forecasts can be achieved when restrictions are applied to them. Specifically, cointegration restrictions, and cointegration and weak-form SCCF rank restrictions, as well as all the set of theoretical restrictions embedded in the PVM.