Time Series, Nonsense Correlations and the PCC. Julian Reiss Complutense University, Madrid and Centre for Philosophy, LSE. Overview. The PCC in the context of time series A Well-Known Problem: British Bread Prices and Venetian Sea Levels Two Attempts to Fix it: Defusing the problem
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Time Series, Nonsense Correlations and the PCC
Julian Reiss
Complutense University, Madrid
and
Centre for Philosophy, LSE
PCC. If two variables X and Y are correlated then either
Consider the fact that the sea level in Venice and the cost of bread in Britain have both been on the rise in the past two centuries. Both, let us suppose, have monotonically increased. Imagine that we put this data in the form of a chronological list; for each date, we list the Venetian sea level and the going price of British bread. Because both quantities have increased steadily with time, it is true that higher than average sea levels tend to be associated with higher than average bread prices. The two quantities are very strongly positively correlated.
A time series is weakly (or covariance) stationary if, and only if, its mean and variance are both finite and independent of time, and the covariance between the values of the series at different times depends only on the temporal distance between them
Let d be the minimum integer such that {dXt} is weakly stationary. Then {Xt} is said to be integrated of order d, which is notated I(d). (By convention, a stationary time series is notated as I(0).)
Two time series {Xt} and {Yt} are cointegrated if, and only if, each is I(1) and a linear combination {Xt – 0 – 1Yt}, where 1 0, is I(0).
Yt = Yt-1 + Yt
Xt = .5Xt-1 + Xt
Yt = Yt-1 + Yt
Xt = .5Yt-1 + Xt
“[T]he above discussion illustrates how researchers interested in drawing conclusions from statistical data can design their investigation so that counter-examples like Sober’s are not a concern. For instance, if the series is non-stationary but transformable into a stationary one via differentiating with respect to time, then differentiate. Then PCC can be invoked without concern for the difficulty illustrated by the Venice-Britain example.”
“Applying the program [that incorporates the PCC] to real data requires a lot of adaptation to particular circumstances: […] data must be differenced to remove auto-correlation…” (Clark Glymour, philosopher)
“A Simple Message to Autocorrelation Correctors: Don’t.” (Grayham Mizon, econometrician)
PCC*: A correlation between two variables X and Y is explanation-seeking. If all kinds of non-causal (e.g., statistical, logical, mathematical, conceptual, nomological) explanation can be ruled out, then either X causes Y, Y causes X or X and Y are the joint effects of a common cause Z, which screens off X and Y