Maximum-likelihood estimation of fractional cointegration with application to the short end of the yield curve

We estimate a multivariate autoregressive fractionally-integrated moving-average (ARFIMA) model to illustrate a cointegration testing methodology based on joint estimates of the fractional orders of integration of a cointegrating vector and its parent series. Although previous work has recognized that deviations from long-run relationships could exhibit long memory and go undetected in traditional 1(1)/i (0) cointegration analysis, previous tests for fractional cointegration relied on a two-step testing procedure and maintained the assumption in the second step that the parent series were known to have a unit root. In our example of fractional cointegration between 10-year government bond rates in the United States and Canada, we illustrate how uncertainty regarding the order of integration of the parent series can be even more important than uncertainty regarding the order of integration of the cointegrating vector when conducting a test for cointegration based on joint estimates. For this reason, the cointegration test based on joint estimates is less likely to reject the null of no cointegration and ought to have better size properties.