I’ve been posting up on some fundamental articles on spurious regression, involving autocorrelated processes. Here are some illustrations of what different examples look like, with specific comment on a realclimate article.
First, from realclimate here , we have an illustration of a trend with i.i.d. errors (i.e. independent identically distributed errors – I’ll
use "independent" errors to mean i.i.d. errors here). In this case they look like normal errors.
Original Caption: Fig. 1. An example showing two different cases, one which is statistically stable (upper) and one that is undergoing a change with a high occurrence of new record-events. Green symbols mark a new record-event. (courtesy William M Connolley)
I ran across an article recently with very similar illustration, but in a different context. See the right second row example.
Figure 2: Figure 1 from G. Mizon, Empirical Analysis of Time Series: Illustrations with Simulated Data, in A.J. de Zeeuw, Advanced Lectures in Quantitative Economics II
Mizon was trying to show the difference in look between series generated by different processes. Let’s now return to our example of the tropospheric temperatures as measured by satellite, shown below.
Figure 3. Tropospheric Temperatures Measured by Satellite
If you compare this series with the examples in Mizon, it’s self-evident that this dataset is not a simple trend with normal errors (which is what the Durbin-Watson statistic tells us as well.) The econometric literature on trend estimation is vast and I do not pretend to have mastered it. The look of the series is obviously much more like a random walk with or without drift, or an ARMA (1,1) or more complicated process with or without drift, and any effort to estimate trend should take this into account. For Benestad at realclimate to trot out a graph with independent errors and (leaving out the first part of the phrase) tell us it’s raining is not only statistics for dummies , it’s statistics by….. realclimate.
Another issue which I’ll illustrate, while I’m at it is how "finite-sample" properties of high-AR1 series approach random walk. "Asymptotically" ARMA(1,0) processes with an autoregression coefficient >0.9 do not yield biased estimates of slope, but for finite samples, it can be a problem. Phillips [Econometrica 1988] cited simulations by Evans and Savin [1981, 1984], showing that the coefficient estimator and the t-test in a stationary AR1 process with a root near unity had statistical properties in moderately large (T=50,100) that "are closer to the asymptotic theory for a random walk than they seem to be to the classical asymptotic theory that applied for stationary time series." The problems for paleoclimate where you have short (N=79 in MBH98) calibration and shorter (N-48) verification periods should be obvious. Here are some images from a teaching course which illustrate how high AR1 series behave like random walks in finite samples of moderate length, showing realizations for T=100, T-1000 and T=10000,
The above diagrams are AR1. I pointed out that some features of temperature series were remarkably well modeled by ARMA(1,1) processes with both an autoregressive and moving average term. I previously mentioned some statistics by Vogelsang  about problems with ARMA (1,1) statistics. Here is a table of the percentage of times that OLS statistics incorrectly record a "significant" trend (rejecting the null hypothesis of no trend). AS you see, the inter-relationship between the AR1 and MA1 coefficients in yielding spurious trends is highly non-linear. The results do not change much between T=250 and T=500, so this is a reasonable guide for T=320 (the length of the tropospheric record.) The ARMA(1,1) coefficients that I estimated for the tropospheric temperature series were AR1=0.9215 and MA1= -0.32. The table doesn’t precisely cover this combination; but my guess at interpolating would be that the effect would be no lower than 0.3 – very much in the red zone.
Table 1. Excerpt from Vogelsang , Table 1. Proportion of Trends Incorrectly Identified as "Significant" by AR and MA Coefficients (T=250.)