Independent and Autocorrelated: Some Examples

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 [1998] 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.


AR Coefficient

MA Coefficient 0.8 0.9 0.95 1.0
-1.0 0.0 0.0 0.0 0.062
-0.8 0.066 0.247 0.446 0.819
-0.4 0.211 0.308 0.409 0.717
0.0 0.082 0.095 0.122 0.436
0.4 0.016 0.027 0.044 0.275
0.8 0.007 0.012 0.029 0.223

Table 1. Excerpt from Vogelsang [1998], Table 1. Proportion of Trends Incorrectly Identified as "Significant" by AR and MA Coefficients (T=250.)


  1. TCO
    Posted Aug 27, 2005 at 8:45 PM | Permalink

    (being undisciplined) Well…you tried. I guess this is supposed to be a tutorial to the basic concepts at play in some of your stats blogs. But you still have a lot of places, where you assume the reader already knows what you are teaching him. Like the reference to AR and MA, without explanation (maybe ok in your other posts…but not in one that is supposed to be explanatory.

    On content:
    1. What is white noise. What is red noise?
    2. I’m still thinking about run order which is a variable in DOE. This would seem to equate to time in these climate studies. Do you think that we handle run order appropriately in manufacturing based studies? I wonder if there is some way to help equate what is done with run order in DOE with the concerns about trend estimation.

  2. fFreddy
    Posted Aug 28, 2005 at 5:31 AM | Permalink

    Steve, if you have time (hollow laugh) it might be a good idea to put together a Statistics FAQ with brief definitions and links, either to posts here where you expand on a particular subject, or else external sources that you think useful. If it all went through a single FAQ, it would be relatively easy to link to it each time you use a term that you recall is defined in your FAQ. Or just have the FAQ near the top of the side bar.

    TCO, apologies if I’m being thick, but what is DOE ?

  3. Michael Ballantine
    Posted Aug 28, 2005 at 5:56 AM | Permalink

    1. What is white noise. What is red noise? A reasonably clear explanation can be found at this link. It refers to music but can apply to anything with a “frequency”.

  4. Michael Ballantine
    Posted Aug 28, 2005 at 5:56 AM | Permalink

    Let’s try that link again.

  5. Chas
    Posted Aug 30, 2005 at 3:35 AM | Permalink

    Re #1
    I sort of agree with TCO, it would be nice to have a single location where simple autocorrelation tutorial could live.
    To make matters a bit worse than they are, presently (‘gambling runs’ is falling off the bottom of the page):
    To make an AR1 series (value 0.9, say)in Excel
    Create a column of normally distributed random numbers (tools=> data analysis) with mean zero and SD 1
    Place a value of zero at the top of the column adjacent to it.
    Below this zero make the cell’s value equal to 0.9 times the value above it (in this case zero) and add the adajcent random number.
    Drag this formula down.

    This is a bit dull; to liven things up a bit download the 30 day free demo of Resampling Stats for Excel add-in
    The program has a volatile random number generator (RXL normal)function so when one changes any cell on the page the random series is regenerated afresh (one can also stick a trend lines through the series and use its ‘resample and score’ to do a few thousand runs and plot these out with it’s autobin histogram )

    Im not too clear on the MA1 part but I beleive it refers to random series itself: A series with an MA1 of 0.3 adds the error of the adjacent random cell plus 0.3 times the value of the random no. above it
    -Please correct me if I have this wrong.

  6. Chas
    Posted Aug 30, 2005 at 7:11 PM | Permalink

    Luckily the recipe gives identical trend-line results to Steve’s R script [using AR1 =0.92 MA= -0.32 and SD for the random numbers of 0.11] ,so the MA1 part seems to be OK.

  7. Chas
    Posted Sep 3, 2005 at 2:38 PM | Permalink

    Further to #5 There are Freeware Excel add-ins for autocorrelation and ARMA at
    They seem to integrate extremely well into Excel. [They do have a quick flash screen explaining that the author is looking for a job].
    Use the ‘standard error of regression’ from the ARMA output worksheet for the standard deviation of the random number series, in a reconstruction.
    Incidently to create a volatile normally-distributed random number series using Excel’s inbuilt functions: Drag down RAND (which is volatile) and then drag down NORMINV setting its ‘probability’ value to the value given by the RAND function in the same row.

    -As clear as mud, I suspect!

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