Spurious Regression

Abstract

If a theory suggests that there is a linear relationship between a pair of random variables X and Y, then an obvious way to test the theory is to estimate a regression equation of form

$$$$

Estimation could be by least-squares and the standard diagnostic statistics would be a t-statistic on β, the R² value and possibly the Durbin-Watson statistic d. With such a procedure there is always the possibility of a type ii error, that is accepting the relationship as significant when, in fact, X and Y are uncorrelated. This possibility increases if the error term e is autocorrelated, as first pointed out by Yule (1926). As the autocorrelation structure of e is the same as that for Y, when the true β = 0, this problem of ‘nonsense correlations’ or ‘spurious regressions’ is most likely to occur when testing relationships between highly autocorrelated series.