Stationarity and the term structure of interest rates: a characterisation of stationary and unit root yield curves

The nature of yield curve dynamics and the determinants of the integration order of yields are investigated using a benchmark economy in which the logarithmic expectations theory holds and the regularity condition of a limiting yield and limiting term premium is satisfied. By considering a zero-coupon yield curve with a complete term structure of maturities, a linear vector autoregressive process is constructed that provides an arbitrarily accurate moving average representation of the complete yield curve as its cross-sectional dimension (n) goes to infinity. We use this to prove the following novel results. First, any I(2) component vanishes owing to the almost sure (a.s.) convergence of the innovations to yields, vt(n), as n. Second, the yield curve is stationary if and only if nvt(n) converges a.s., or equivalently the innovations to log discount bond prices converge a.s.; otherwise yields are I(1). A necessary condition for either stationarity or the absence of arbitrage is that the limiting yield is constant over time. Since the time-varying component of term premia is small in various fixed-income markets, these results provide insight into the critical determinants of the stationarity properties of the term structure.