Is integration I(d) applicable to observed economics and finance time series? Joseph L. McCauley a, ⁎, Kevin E. Bassler a,b , Gemunu H. Gunaratne a,c a Physics Department, University of Houston, Houston, Tx. 77204-5005, United States b Texas Center for Superconductivity, University of Houston, Houston, Texas, United States c Institute of Fundamental Studies, Kandy, Sri Lanka abstract article info Article history: Received 13 January 2009 Accepted 6 March 2009 Available online 2 April 2009 Keywords: Integration I(d) Cointegration Unit root Ergodicity Martingales White noise The method of cointegration in regression analysis is based on an assumption of stationary increments. Stationary increments with fixed time lag are called ‘integration I(d)’. A class of regression models where cointegration works was identified by Granger and yields the ergodic behavior required for equilibrium expectations in standard economics. Detrended finance market returns are martingales, and martingales do not satisfy regression equations. We ask if there exist detrended processes beyond standard regression models that satisfy integration I(d). We show that stationary increment martingales are confined to the Wiener process, and observe that martingales describing finance data admit neither the integration I(d) nor the ergodicity required for long time equilibrium relationships. In particular, the martingales derived from finance data do not admit the time (or ‘space’) translational invariance required for increment stationarity. Our analysis explains the lack of equilibrium observed earlier between FX rates and relative price levels. © 2009 Elsevier Inc. All rights reserved. 1. Definition of integration I(d) First, we establish our terminology and notation. Given a stochastic process x(t) or a time series realization of a process, economists call a point x(t) a level, and the increment/displacement x(t,−T)=x(t) −x (t −T) is called a level difference, or just difference. With logarithmic variables as in finance or discussions of the money supply, x(t,T)=ln(p (t +T)/ p(t)) is the log increment of a price, whereas the underlying stochastic process is defined by the random variable x(t)=ln(p(t)/ p c ) where p c is a reference price that can be identified as ‘value’ (McCauley, 2008a). There is exactly one, single time scale t in a random variable x(t). There are two time scales in an increment, the time t and a time lag T. In economics the time lag is typically (and unnecessarily) fixed at T =1 period (The Royal Swedish Academy of Sciences, 2003), e.g. T =1 period can mean 1 quarter. In economics, the fact that the lag time T is a time variable is completely ignored. This leads to confusion, because the time lag T may in principle increase without bound, and increments x(t,T) of a nonstationary process x(t) cannot be understood as ‘stationary processes’ as T is increased even if the increments are stationary, although by taking t fixed and letting T vary one can derive an Ito sde for the increment/difference x(t,T)(McCauley, 2009). That is, stationarity of increments is generally very different from stationarity of a process. In our recent FX data analysis (Bassler, McCauley, & Gunaratne, 2007) we studied increments of detrended returns with T =10 min, and in the accompanying theory it was made clear that T is a variable. Summarizing, in agreement with mathematics terminology and with our recent papers, we will denote as x(t) the (stochastic) process or point in a time series, and x(t,T) is an increment of the process. The central question for us is: what is the class of stochastic processes {x(t)} admitting stationary increments. By a stationary increment is meant that x(t,T)=x(0,T) ‘in distribution’, meaning only that the two increments have the same 1-point distribution. In particular, nothing at all is implied in this definition about correlations. When T is held fixed, e.g., at T =1, then we will make contact with economists' language and expectations, but keep in mind that x(t,T) with T fixed generally cannot lead to a well-defined stochastic process described by an Ito sde. The belief in econometrics (macroeconomic data analysis) is that the increments may be stationary, with fixed T, even if the underlying process x(t) is nonstationary (The Royal Swedish Academy of Sciences, 2003; Engle & Granger, 1987). Stated explicitly, if the process x(t) is nonstationary then form the first difference x(t,−T)= x(t) − x(t − T). If the first difference x(t,−T) is nonstationary with T fixed, then one is expected to study the second difference x(t −T ,−T)= x(t) −2x(t −T)+x(t −2T), and so on until either stationarity is found or else the chase is abandoned. If the process x(t) is already stationary then it's called I(0). If the process x(t) is nonstationary but the first International Review of Financial Analysis 18 (2009) 101–108 ⁎ Corresponding author. E-mail address: jmccauley@uh.edu (J.L. McCauley). 1057-5219/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.irfa.2009.03.004 Contents lists available at ScienceDirect International Review of Financial Analysis