This graph shows three trend lines fitted to a standard sine function over one full period (x axis units are pi/16 radians). A sine function is used because it gives a particularly tidy example of highly autocorrelated data having strong internal variation but no actual trend. Non-periodic, but still highly autocorrelated, data will frequently show a similar general shape to this (see the last graph of the previous post on this topic, for example), in the sense that positive values tend to cluster together, as do the negatives.
Clearly, a standard OLS trend line fit to such a series gives a seriously wrong trend estimate. Tukey’s “robust” method (function “line” in R) is even worse. An average of the slopes at lag 1, by contrast, gives a perfect estimate. Why? Because the standard OLS method optimizes an “objective” function that minimizes the sum of the squared deviations from the trend line, solved analytically. The problem is that that’s the wrong objective function when data are highly autocorrelated. One should instead minimize the sum of the departures of some set of slope lines (= lag-x differences), taken from the data, from a central tendency thereof. Which is to say, take the mean or median thereof.
When the autocorrelation in the data is due mainly to the existence of a real trend, the latter method will return an estimate not greatly different from that of OLS methods. But when the autocorrelation is instead due to series that are approaching a random walk status (AR(1) -> 1.0), but with no real trend, it will be far less biased than OLS (or Tukey’s method). That fact might have some importance.
Updated comment. To clarify a little on the cause of the OLS mis-estimates, it’s not actually the autocorrelation itself that causes the OLS-derived mis-estimates. It just increases the frequency of them (in proportion to the ac level). When noise is purely white, OLS still gets it wrong p percent of the time for any given p value. When noise is red, OLS gets it wrong more often, and more severely.
Comment #2. Note that the “mean slope” estimate is always superior to the OLS estimate, regardless of the phase of the sine wave, with the exception of phases starting at 0.5*pi and 1.5*pi, where they are equal.