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Our problem is to determine if there is a relationship between the return on equity of firms (ROE) and the presence (number of indicators used) (Y) of a specific type of indicator showing up in the annual report. Initially we had 118 observations and after some adjustments we had 104 firms. In order to verify if there is a linear relationship between Y (Poisson distributed) and the predictor ROE we built 8 clusters based on ROE’s distribution and we computed for each class the mean of ROE (ROEm), the mean of the number of indicators used (ym) and, finally, the total amount of indicators used by each class. Then we plotted in a graph the log_ym and ROEm, their relationship was not linear but quadratic.

scatterplot

Thus we decided to apply a Poisson regression using as the dependent variable Y and as predictor ROE and ROE^2. This model was not so good: pseudo R^2 was very small (about 0,02) and the hypothesis of over-dispersion was accepted.

enter image description here As we read in some papers, we concluded that there is a tendency for observations to cluster. At this point we decided to apply the Poisson regression on clusterized data using as dependent variable the total amount of indicators used by each class (ym is not discrete) and as predictor ROEm and ROEm^2. As the offset variable we used the number of cases per each cluster. The output of the model is summarized below:

enter image description here

  • Is our approach correct?
  • Are there simpler methods?
  • Are 8 clusters few?
  • Results get worst if we use for example 14 clusters instead of 8; does that mean that the model doesn’t fit well?
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    $\begingroup$ Instead of putting in screenshots of the code output, it would be preferable to just copy/paste the output... as it is I cannnot read it. $\endgroup$ – bdeonovic Apr 16 '14 at 20:04
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    $\begingroup$ Your impression that it's quadratic is driven almost entirely by a single point (cover that point, the relationship looks linear). You're very reliant on that one point being representative of firms with high ROE (what if there's something that makes it atypical?). Check its influence (e.g. see how responsive the fit at its ROE is to a small change in its value). Could your replace the images in your post with your new images you linked in comments? $\endgroup$ – Glen_b Apr 16 '14 at 22:53
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Partially answered in comments:

Your impression that it's quadratic is driven almost entirely by a single point (cover that point, the relationship looks linear). You're very reliant on that one point being representative of firms with high ROE (what if there's something that makes it atypical?). Check its influence (e.g. see how responsive the fit at its ROE is to a small change in its value). – Glen_b

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