1 Introduction, History, and Definitions

If Xt, Yt are a pair of time series, independent of each other and one runs the simple ordinary least squares regression

Yt = a + bXt + e t, (26.1)

then one should expect to find no evidence of a relationship, so that the estimate of b is near zero and its associated {-statistics is insignificant. However, when the individual series have strong autocorrelations, it had been realized by the early 1970s by time series analysis that the situation may not be so simple; that appa­rent relationships may often be observed using standard interpretations of such regressions. Because a relationship appears to be found between independent series, they have been called "spurious". To appreciate part of the problem, note that if b = 0, then e t must have the same time series properties as Yt, that is will be strongly autocorrelated, and so the assumptions of the classical OLS regression will not be obeyed, as discussed in virtually any statistics or econometrics text­book. The possibility of getting incorrect results from regressions was originally pointed out by Yule (1926) in a much cited but insufficiently read paper that discussed "nonsense correlations." Kendall (1954) also pointed out if Xt, Yt both obeyed the same autoregressive model of order one (AR (1))

Yt = a2Yt-1 + Єyt J

with a1 = a2 = a, where ext, eyt are a pair of zero-mean, white noise (zero autocorrelated) series independent of each other at all pairs of times, then the sample correlation (R) between Xt, Yt has

var(R) = n 2(1 + a2)/(1 – a2),

where n is the sample size. Remember that R, being a correlation must be between -1 and 1, but if a is near one and n not very large, then var(R) will be quite big, which can only be achieved if the distribution of R values has large weights near the extreme values of -1 and 1, which will correspond to "signifi­cant" b values in (26.1).

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