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Autocorrelation: Durbin-Watson tests

Hi x-perts,

I am using the attached sheet models to run Durbin-Watson tests. The above example is for 2 regressors.

How can I use the same model to test autocorrelation of time series to itself?

I guess, I can replace the 2nd series (MSCI) to a simple incrementing series like 1,2,3,4,....n

Is it a correct way of running DW test for a single series?

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I'm not familiar with Durbin-Watson, but apparently, it is used specifically to correct problems with normal autocorrelation estimators when dealing with regressions on time series in econometrics. See
http://en.wikipedia.org/wiki/Autocorrelation section "regression analysis"

For a single series, just apply the normal definition (see attached sheet). the assumption is that a monthly return is independent from the previous month's.




Thank you, but I would rather keep DW general model for two series and transform one series to get autocorrelation to itself.

actually, it finds DW by building regression model (Y-dependent and X-independent series). If we replace the independent series by numbers 1,2,3,4,...,n, it should give autocorrelation to itself. Am I right?


Or maybe I should use lagged series as the 2nd one?
I'm sorry, but the DW model simply does not apply to one series. If you create an artificial constant series (all zeroes) or your 1,2,3,... suggestion, the errors will be so large that they will mask any residual autocorrelation. The same happens if you use the "lagged" series.

Notice that your current two sets of data overlap very well. They have a very strong correlation. The DW test is specifically designed to catch autocorrelation not of the data itself for of the regression of the data, which is very small in this case.

I read the formulas again, and I'm quite confident that there is no way to cheat or massage a single series into any meaningful DW-based test. Comparing raw data and comparing residual errors just isn't the same thing.

What is wrong with simple correlation?

I suspect the theoretical background is missing.

Say you have two time tracks, of gold prices and oil prices. You notice that they are correlated and attempt to predict the price of gold based on the price of oil and perform a regression. If it's reasonably good, you could then ask if, perhaps, there is a time delay, a response lag. That would mean that the two curves are slightly offset from one another. In such a case, the residual errors will not be random: they will show a positive autocorrelation. This would be an argument for direct or indirect causation. If you look only at the returns, the same thing would happen if and only if a good month for oil is likely to be followed by a good month for gold.

When you have only one track, there is no residual error. If you use the price of gold to predict the price of gold, well, you don't guess. There can be no systematic lag in the effect of the price of gold on the price of gold.

Does that help?


What I need is to find a robust metric to estimate autocorrelation for a given time series. There is only one time series. Obviously, I could measure correlation between the series and its 1st lag, but I am trying to find something better.

My assumption was simple:

1) DW test takes regression residuals between two series
2) If I replace one series by a straight line, it will measure residuals to that line, i.e. correlation to itself.

Does it make sense?
No, I'm afraid it doesn't. The straight line cannot lag or anticipate, since it's straight. Also, since there is no correlation, there cannot be any autocorrelation of the regression. Finally, if if it did make sense, the residual errors will be so large that the tiny variations between two tracks measured by DW will be totally drowned.

In econometrics, you tent to look for autocorrelations over the base period (the month in your data) or some meaningful time frame (yearly variations in crop and weather related values). There are also several techniques to test for randomness of absolute returns (are too many positive or negative returns following each other).

DW gives the impression that it doesn't require a time frame in the hypothesis. You want a measure of autocorrelation without choosing a time frame, so you try to bend DW to your needs.

I don't know what I can add, really. You can make any computation you like, but it will not be a test, it will be exploratory statistics, or, if you are really lucky, descriptive statistics.