# Aggregate vs. firm-level regressions - How can the regression coefficients differ that much?

I recently came across a study that finds the following two results:

• On the firm-level, the independent variable $X_{it}$ has a positive impact on the dependent variable $Y_{it}$. Concretely, the study shows that the time-series regression coefficient of realized returns on earnings changes (deflated by stock prices) is positive for almost all firms $i$. That is, they run a time-series regression for each firm and look at the distribution of the betas. Both the mean and the median are positive, only 10.7% have a negative beta.
• On the aggregate level, they find the inverse relation, i.e. aggregate earnings changes (either value- or equally weighted) have a negative beta on aggregated stock returns.

I am rather surprised by this finding. I tried to search for it online and it first seemed my understanding problem could be solved with the ecological fallacy (see Wikipedia). However, I don't think so anymore.

If I am correct, the mechanism behind the ecological fallacy is a different one. For instance, take the literacy-immigration example from Wikipedia: within each group/state, illiteracy is higher for immigrants, but since immigrants settle in states with higher literacy, on aggregate, there is a negative effect between percentage of immigration and illiteracy. So the effect, if I understand it correctly, occurs because the groups, in this case the states, are different to start with and the immigrants can choose the states. Let's assume that immigrants are randomly sampled to a state. Then this effect wouldn't work, right?

However, in my example, there is no sample selection, at least non that I am aware of since the groups are the time periods. That is, each time period all firms in a sample are aggregated and firms can't really choose the time period.

Of course, it could be that firms are bankrupt in bad states of the economy, so the sample size varies between different periods. But let's ignore that for a second and assume that the number of firms stays constant through the whole sample: How can it be that the regression coefficient is so different on the aggregate in comparison to the distribution of the regression coefficients on the firm level? Both a formal answer and an intuitive one (maybe a small example) would be great.

• Sounds like Simpson's paradox, to me. Wikipedia has a nice explanation of this – Peter Flom - Reinstate Monica Jul 31 '13 at 13:32
• Definitely Simpson's paradox. – user25658 Jul 31 '13 at 14:02
• Thanks for the comments. Although I know Simpson's paradox and I think that this could be the solution, I can't transfer it to this case. What are the groups in my case? How does one aggregate? I just have a hard time making a simple example in which (almost) every firm has a positive correlation between earnings changes and stock returns, yet I don't see it on the aggregate. For me, Simpson's paradox is more about differences in groups. For instance, the Berkeley case between men and women. What is the equivalent in this case? – user28673 Jul 31 '13 at 15:21
• Adding to my former comment, the Berkeley case can be summarized like this (simplified): "For each faculty, more women are accepted than men. However, men choose faculties with higher acceptance rates. Ergo, in total more men are accepted than women." I can't translate that logic to my case: "For each firm, earnings changes are higher in periods with high returns. However, positive earnings changes occur in periods of low returns and negative earnings changes occur in periods of high returns. Ergo, in total there is a negative relation." – user28673 Jul 31 '13 at 17:15
• This logic clearly does not add up: How can on the aggregate positive earnings changes occur in periods of low returns, if every firm has positive earnings changes when its returns are high? – user28673 Jul 31 '13 at 17:16

As pointed out by the commenters, this seems to be an instance of Simpson's paradox. The difference between the relationships at the aggregate and within-group level can really be as large as you want it to be (no correlation at one level, strong positive or negative correlation at the other, etc.)

I am not sure about the mechanism that could account for it in your case but I think this is easiest to understand by looking at some plots. I don't have time to create a fictional example tailored to your situation and I obviously don't have your data at hand but here is one I created some time ago: As I said, this plot was generated in another context but let's imagine that points of the same color/shape represent measures from the same firm and the two variables are some sort of interesting quantitative characteristics. Within each firm, there is basically no relationship. At the aggregate level there is a perfect positive correlation. Pooling all the data and ignoring the structure of the data set, there seems to be a high positive correlation, driven by the aggregate level correlation and dampened by the (smaller) within-firm variance. Roughly, one interpretation for this example would be that the evolution of both variables are not related but that different firms have a different, stable, baseline level on each and that those are related.

Since one of your variables seems to be a difference and the other a ratio, providing an intuitive interpretation is more difficult but graphically and mathematically pretty much everything is possible, the correlation and regression coefficients at both level are just two different things. I think that this is the key insight behind Simpson's paradox and the examples with dichotomous variables and the interpretations that go with it are just special cases.

• Thanks, this plot helps a lot. Together with my thoughts, I think I understand it now. – user28673 Aug 1 '13 at 10:15

OK, I think I found a possible mechanism that works. I'm not sure if this is Simpson's paradox though, some confirmation would be great.

Assume that there are two type of firms:

• Firms HIGH with a high earnings growth/return relation, i.e. a high beta.
• Firms LOW with a low, but still positive, earnings growth/return relation, i.e. a low beta.

That is, in this economy every single firm has a positive earnings growth/return relation, as found by the cited study.

Now, assume for simplicitiy that there are only two states:

1. State: HIGH have a moderate positive earnings growth, but very large returns (due to the large beta), LOW have zero earnings growth and therefore, zero returns.
2. State: LOW have a large positive earnings growth, but only moderate positive returns (due to the small beta), HIGH have zero earnings growth and therefore, zero returns.

Now aggregating the two states, one gets:

1. State: Moderate mean earnings growth, large mean stock returns.
2. State: Large mean earnings growth, moderate mean stock returns.

Therefore, regression realized returns on earnings growth yields a negative beta. I think this explanation is better to understand when drawing each case in a diagram.

This is certainly an extreme example and I'm still surprised that they find it in their study empirically (because my explanation basically needs a huge negative correlation between the two types of firms: precisely in those periods in which one type does great, the other has to do bad, and vice versa...not sure how realistic that is), but at least that is a simple mechanism that works. There is also a study by Kothari/Lewellen/Warner (2006, p. 539): They write:

It is useful to note that a negative reaction to aggregate earnings is entirely consistent with a positive reaction to ﬁrm earnings (a result conﬁrmed in our data). The economic story is simple. Aggregate earnings ﬂuctuate with discount rates because both are tied to macroeconomic conditions, while ﬁrm earnings primarily reﬂect idiosyncratic cash-ﬂow news. As a result, the confounding effects of discount-rate changes show up only in aggregate returns. Put differently, cash-ﬂow news is largely idiosyncratic while discount-rate changes are common across ﬁrms. By a simple diversiﬁcation argument, discount-rate effects play a larger role at the aggregate level. In short, our evidence suggests that common variation in discount rates explains an important fraction of aggregate stock market movements.

Maybe some of you find this explanation helpful.

• Should be helpful to many, but full literature references please. – Nick Cox Aug 1 '13 at 9:38