Nonstationary Time Series, Cointegration, and the Principle of the Common Cause

Elliot Sober ([2001]) forcefully restates his well-known counterexample to Reichenbach's principle of the common cause: bread prices in Britain and sea levels in Venice both rise over time and are, therefore, correlated; yet they are ex hypothesi not causally connected, which violates the principle of the common cause. The counterexample employs nonstationary dataÐi.e., data with time-dependent population moments. Common measures of statistical association do not generally re ̄ect probabilistic dependence among nonstationary data. I demonstrate the inadequacy of the counterexample and of some previous responses to it, as well as illustrating more appropriate measures of probabilistic dependence in the nonstationary case. 1 A challenge to the principle of the common cause 2 Sober's argument and the attempts to rescue the principle 3 Probabilistic dependence 4 Nonstationary time series 5 Probabilistic dependence in nonstationary time series 6 Do Venetian sea levels and British bread prices violate the principle of the common cause? 1 A challenge to the principle of the common cause Hans Reichenbach's ([1956]) principle of the common cause and the causal Markov condition stand at the core of several modern accounts of causality (e.g., Spirtes, Glymour and Scheines [1993]; Hausman and Woodward [1999]). Reichenbach ([1956], p. 156) states the principle of the common cause: `If an improbable coincidence has occurred, there must exist a common cause' (emphasis in the original). He then goes on to elaborate the conditions for a common cause in terms of probabilities (pp. 156±67). Elliot Sober 1 The causal Markov condition states that any variable, V, in a causal graph, conditional on its parents, is independent of all other variables that are neither its parents nor its descendants (Spirtes, Glymour and Scheines [1993], p. 54; Hoover [2001], p. 157). # British Society for the Philosophy of Science 2003 (2001, p. 331) restates the principle as: (P) If events X and Y are correlated, then either X caused Y, Y caused X, or X and Y are joint effects of a common cause (one that renders X and Y conditionally probabilistically independent). Sober ([1994], pp. 161±2) had earlier challenged the principle of the common cause with a counterexample. In Sober's scenario, bread prices rise monotonically through time in Great Britain, and sea levels rise monotonically in Venice. Each process is causally independent of the other by assumption, yet, he asserts, the series are highly correlated. Sober regarded this as a violation of the principle of the common cause, and a demonstration that the principle fails in its key role in accounts of causality. Sober ([2001]) reiterates the point and offers an analysis aimed at showing that various attempts to rescue the principle of the common cause from his counterexample fail. The kind of time series that Sober employs in his counterexample is commonplace in macroeconomics. Prices, gross domestic product, consumption, investment, employment, and wagesÐto name just a few prominent economic time seriesÐtrend up in the manner of Sober's bread prices and sea levels. In the past quarter century, statisticians (most notably time-series econometricians) have developed special tools for the analysis of these `nonstationary' time series. One well-known exposition of some of the issues in time-series econometrics uses an example analogous to Sober's example, in which the level of the consumer price index plays the role of bread prices, and cumulative rainfall in the United Kingdom plays that of sea levels (Hendry [1980]). There are several reasons to believe that the principle of the common cause is not a successful foundation for causal analysis (see, e.g., Cartwright [1999], Ch. 5, or Hoover [2001], Ch. 4, x3). Nevertheless, the recent work in timeseries statistics suggests that Sober's counterexample to the principle of the common cause is defective. If the principle is to be rejected, it must be on some other grounds. 2 Sober's argument and the attempts to rescue the principle The principle of the common cause states that any correlation demonstrates a causal connection between the variables displaying the correlationÐeither direct or through a third cause. If the connection is through the third cause, then the third cause will screen off the correlation in the sense that the correlation of X and Y conditional on Z (the third or common cause) will be zero. Sober's counterexample is, then, simple: Venetian sea levels and British bread prices are truly correlated and are not causally connected by construction; therefore, neither causes the other and there can be no common cause. The assumption that sea levels and bread prices are truly correlated is central to 528 Kevin D. Hoover