Stationary and Nonstationary Behaviour of the Term Structure: A Nonparametric Characterization

We provide simple nonparametric conditions for the order of integration of the term structure of zero-coupon yields. A principal benchmark model studied is one with a limiting yield and limiting term premium, and in which the logarithmic expectations theory (ET) holds. By considering a yield curve with a complete term structure of bond maturities, a linear vector autoregressive process is constructed that provides an arbitrarily accurate representation of the yield curve as its cross-sectional dimension <inline-formula> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ramf_a_666120_o_ilm0001.gif"/> </inline-formula> goes to infinity. We use this to provide parsimonious conditions for the integration order of interest rates in terms of the cross-sectional rate of convergence of the innovations to yields, <inline-formula> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ramf_a_666120_o_ilm0002.gif"/> </inline-formula> as <inline-formula> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ramf_a_666120_o_ilm0003.gif"/> </inline-formula>. The yield curve is stationary if and only if <inline-formula> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ramf_a_666120_o_ilm0004.gif"/> </inline-formula> converges <italic>a.s.</italic>, or equivalently the innovations (shocks) to the logarithm of the bond prices converge <italic>a.s.</italic> Otherwise yields are nonstationary and I(1) in the benchmark model, an integration order greater than 1 being ruled out by the <italic>a.s.</italic> convergence of <inline-formula> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ramf_a_666120_o_ilm0005.gif"/> </inline-formula> as <inline-formula> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ramf_a_666120_o_ilm0006.gif"/> </inline-formula>. A necessary but not sufficient condition for stationarity is that the limiting yield is constant over time. Our results therefore imply the need usually to adopt an I(1) framework when using the ET. We provide ET-consistent yield curve forecasts, new means to evaluate the ET and insight into connections between the dynamics and the long maturity end of the term structure.