8.4 Serial Correlation, and Why It Breaks the Test

Mann–Kendall assumes the observations are independent. Series measured through time frequently are not: a wet year tends to follow a wet year, and a warm month a warm month. Positive serial correlation means consecutive observations carry overlapping information, so the series contains less independent evidence than its length suggests. The true variance of \(S\) is then larger than the formula above, and using the uncorrected formula rejects \(H_0\) too often — it reports trends that are not there.

Hamed and Rao (1998) give the standard remedy: rescale \(\operatorname {var}(S)\) by a factor computed from the autocorrelation of the data.

(i).

Estimate the slope by Theil–Sen and detrend: \(y_i = x_i - \widehat {\beta }\,(t_i - t_1)\). Detrending matters — the trend itself induces autocorrelation, and without removing it first the correction would count the signal as if it were noise.

(ii).

Rank the detrended series and compute the lag-\(\ell \) autocorrelations \(\rho _\ell \) of those ranks.

(iii).

Keep only the autocorrelations that are significant, setting \(\rho _\ell = 0\) whenever \(\left |\rho _\ell \right | \leq \dfrac {1.96}{\sqrt {n}}\).

(iv).

Form the variance inflation factor \[\dfrac {n}{n^{*}} = 1 + \dfrac {2}{n(n-1)(n-2)} \sum _{\ell } (n - \ell )(n - \ell - 1)(n - \ell - 2)\,\rho _\ell \] and use \(\operatorname {var}^{*}(S) = \dfrac {n}{n^{*}}\operatorname {var}(S)\) in place of \(\operatorname {var}(S)\).

The quantity \(n^{*}\) is an effective sample size: the number of independent observations the correlated series is really worth.

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