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For 100 companies, I have collected (i) tweets and (ii) corporate website pageviews for 148 days. The tweetvolume and pageviews per day are two independent variables corpaired against the stock trading volume for each company, resulting in 100 x 148 = 14,800 observations. My data is structured like this:

company  date  tweetVol  pageviewVol  tradingVol
------------------------------------------------
1        1     200        150          2423325
1        2     194        152          2455343
1        3     214        199          3100429
.        .      .          .              .
.        .      .          .              .
1       148    205        233          2563463
2        1     752        932          7434124
2        2     932       2423          7464354
2        3     600       1435          5324323
.        .      .          .              .
.        .      .          .              .
.        .      .          .              .
100      148     3         155           32324

Because there is much difference in company-size (some companies only receive 2 tweets per day, where others like Apple get over 10,000 per day), all variables are logged to smoothen distribution. (This is in line with previous research - this is for my thesis).

I just performed a linear regression on this data, including both independend variables. R-Squared is .411 but Durbin-Watson only .141 (!) Without looking for the exact bounderies, I know this directly means my residuals are non-linear, eg. auto-correlated, right?

My question is: how can I solve this? When I think about it, this data should not be autocorrelated, so I don't really understand. Is it due to this actually being a timeseries analysis? I wouldn't think that either, since for instance trading volume today is independent of yesterdays trading volume. Can somebody explain this to me?

P.S. At my university, we use SPSS/PASW without additional modules, so I am unable to perform a timeseries analysis on this like you could in STATA or R.

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    $\begingroup$ Perhaps I could be of more help . Please post data for 148 days for say 5 of your companies and I will try and be specific about my recommended analysis. $\endgroup$
    – IrishStat
    Aug 7, 2012 at 14:33
  • $\begingroup$ @IrishStat Thanks a bunch. See the data here: bit.ly/ML20Ez. It is in xls file format. In the second tab, the variables are explained. A regression for stockvol0Log with variables wikiLog, 'svi' and 'hbVolLog' returns a Durbin-Watson of 0.276, see here for SPSS output: i.imgur.com/Nq0YI.png. SPSS does not support calculating p-value for the Durbin-Watson. I found an Excel-plugin that could (bit.ly/OMZZZ0) but with my list of residuals, it froze. I applied for a student license for SHAZAM (econometrics.com), a statistical package that does calculate p-value. $\endgroup$
    – Pr0no
    Aug 7, 2012 at 18:13
  • $\begingroup$ @IrishStat BTW, return0Pct and volatility0 are the other two dependent variables in my research. 'hbBull' and 'hbAgree' are independent variables, who can be added to the regression. I'm planning on performing a regression of all 5 independent variables (wikilog, svi, hbVolLog, hbBull, and hbAgree) on all three dependent variables (return0Pct, stockvol0Log, and volatility0). In reality, I have more independent variables, but these are the most important ones. $\endgroup$
    – Pr0no
    Aug 7, 2012 at 18:31
  • $\begingroup$ Oh my word! I have had a similar issue with DW being very low. I have financial data that I am applying panel regression to. I reordered the data - using the ln of the DV and dividing by an IV. Then ordered the cases in increasing order. DW is now perfect. Thank you!! $\endgroup$
    – Gill
    Oct 31 at 12:44

5 Answers 5

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The Durbin-Watson test may suggest the need for an ARIMA model to render the error term free of structure IFF there are no outliers/inliers/pulses AND no unspecified evel/step shifts AND no unspecified Seasonal Pulses AND no unspecified Local Time Trends AND the models' parameters are constant/homogeneous over time AND the error variance is constant/homogeneous over time AND the error variance is not related to the level/expected value AND the error variance can't be modelled as a random variable via GARCH.

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    $\begingroup$ My guess is that the correlation is weak and the Durbin-Watson test does not reject lack of correlation. But yes if there is structure in the residuals then of course all the time series structural checks you recommend should be looked at. $\endgroup$
    – Michael R. Chernick
    Aug 6, 2012 at 12:11
  • $\begingroup$ I'm really trying here to follow but it goes over my head. To simplify, according to the Durbin-Watson test the residuals autocorrelate. Could you describe me briefly why this is the case, if I have data from three different sources? The way I understood, autocorrelation is the outcome of biased data, for instance interviewing children of the same school every year instead of a different school (if your goal is to research schools in general). Such is not the case here? $\endgroup$
    – Pr0no
    Aug 6, 2012 at 12:26
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Autocorrelation has nothing to do with nonlinearity. The Durbin-Watson test is used to determine if the residuals from your model have significant autocorrelation. So you look at the p-value for the test and conclude that there is autocorrelation if the p-value is small (usually taken as less than 0.05). Is 0.141 the p-value for the test or the value of the test statistic? If it is the p-value it is not low enough to conclude that there is significant autocorrelation. If it is the value of the test statistic you need to find out what the corresponding p-value is. If you do have a p-value less than 0.05, a way to account for this would be to construct a model that includes residual correlation structure such as an autorgerssive model for the residuals.

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  • $\begingroup$ Thanks for your reply. NO, 0.141 is the value of the test statistic. I'm looking on how to get the p-value...it is not given in the (standard) SPSS output. Do you have a suggestion? $\endgroup$
    – Pr0no
    Aug 6, 2012 at 12:19
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    $\begingroup$ It is very probably off the chart. It is not always easy to convert test statistics to a p value. $\endgroup$
    – IrishStat
    Aug 6, 2012 at 12:26
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    $\begingroup$ Yes checking the residuals locally ( i.e. for each company ) would be more appropriate ALTHOUGH I strongly disagree with you using 8858 observations. What you have is called pooled cross-sectional time series .... 86 groups 103 observations within each group. Form a suitable model for each group separately, decide on a common model then estimate that common model globally ensuring that the prediction for the 104 observation does not depend on the 103rd etc..Compare the error sum of squares globally with the sum across the groups to perform an F test of the hypothesized similar coefficients. $\endgroup$
    – IrishStat
    Aug 6, 2012 at 13:17
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    $\begingroup$ @Pr0no Did you determine the p-value for the Durbin-Watson test? Seems like pooling companies may not be the right way to analyze your data. How do the Durbin-Watson statistics look for separate groups? $\endgroup$
    – Michael R. Chernick
    Aug 6, 2012 at 14:07
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    $\begingroup$ @Pr0no How can you tell that the number .258 listed below Durbin-Watson is the value of the test statistic and not the p-value? It doesn't look obvious to me. The p-value would seem more likely to me as it would be interpretable by the user whereas the value of the test statistic would not. $\endgroup$
    – Michael R. Chernick
    Aug 6, 2012 at 15:10
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I know I'm late to the party on this one but I stumbled across this post when having a similar issue myself. The good news is, I figured out why I also had such a low Durbin Watson statistic. I followed a textbook procedure for linear regression in SPSS (probably like you), and when looking at residuals for those +/- 3 SDs, I ordered my data from largest to smallest. All of a sudden my regressions were coming out with very low DW values... I then read this on Laerd Statistics: 'If there is a reason that your observations could be related, SPSS Statistics assumes that you have entered your data in the order you expect the autocorrelation to occur. For example, if you tested your participants over time, you might enter your participants in the order you tested them (i.e., the first participant you tested would be on the first row, the second participant you tested would be on the second row, etc.).' So all I did was go back to data view and order my data in ID order (that's my first column for participant number in my study) and the DW value is back to normal! Hope this helps. P.S. If there is no reason that your observations could be related, you do not need to interpret the Durbin-Watson test.

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I am of the idea that the value of .141 that is mentioned ambiguous. If it is Durbin-Watson test statistic then it means the auto correlation is very low.

Following is the definition of Durbin-Watson statistic:- A number that tests for autocorrelation in the residuals from a statistical regression analysis. The Durbin-Watson statistic is always between 0 and 4. A value of 2 means that there is no autocorrelation in the sample. Values approaching 0 indicate positive autocorrelation and values toward 4 indicate negative autocorrelation.

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if you work with a transverse sample and do not have variables that imply time or space (in your case you have: company), BE SURE TO SORT THE DATA BY CASE NUMBER BEFORE STARTING THE REGRESSION ANALYSIS. If the data is sorted in some way, for example because you have previously used the commands select cases or sort by X variabkle, the Durbin-Watson value will not be correct. I had the problem, an unusually low DW, and there was no logical explanation for autocorrelation, since there were no variables of time or space.

Trying to understand what was happening, I repeated the analysis several times, and in also, I ordered the database according to my variables, according to my typified residues, according to Mahalanobis D values, etc. A lot of different sorts of variables! And I realized that every time I got a different DW value without changing the variables of the equation! It was very weird, so I checked the formula and realized that the input order of the data is taken into account. I ordered the base by number of cases, a random variable, without any pattern or order, and changed DW again, leaving me a fairly normal value that indicated no autocorrelation. I do not know if this is what is happening to you, but you could try it. Note: if you have the database ordered by conpany or country, etc., (for example cases 1 to 30 are from company A, from 31 to 60 are from company B,etc.) It is best that you do not use a case number to sort the base. Instead, you can create a new variable case number and sort the database randomly and to verify what happens to the DW value. In this way, you would be breaking the order by company, country, etc., which is influencing DW.

If the variables company, country (or any other variable of time or space, are relevants for your research), you will have to consider it in your model, as said by other colleagues above. I hope I've helped.

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