# How would you test the hypothesis “There are no idiosyncratic returns available in the market”?

A commentary attributed to Matt Rothman had recently (in the past six months) been making the rounds of the internet echo chamber claimed "There are no idiosyncratic returns available in the market". How would one turn this into a proper statistical hypothesis test?

As part of the commentary, a plot of 'implied correlation' was given: This does not tell me much, but it suggest perhaps a test based on the two leading eigenvalues of the correlation matrix, but I am not sure how to pose the problem.

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Take the CAPM regression (it's not exactly correct, but it's instructional) $$(R_i - R_f) = \alpha_i + \beta_i (R_{mkt} - R_f) + \epsilon_i$$ The author is saying that these days the $(R_{mkt} - R_f)$ term is driving all returns and that the $\alpha_i$ and $\epsilon_i$ terms are not significantly different than zero because all returns are correlated.

Another way of looking at it is that before this period of high correlation you could find the stocks that were mis-priced by looking at fundamentals. If you bought the stocks that were under-priced, then the market would soon enough find out that you were correct, buy the stock and return it to the correct price, and you would capture this return in excess of predicted by $\beta_i$. He is saying that even if these stocks are mis-priced, because everything is correlated, there are no price corrections and you're only seeing returns based on $\beta_i$.

That the correlation isn't 1.0 shows that he's not absolutely correct, but it's a good point. Because everything is "correlated" you can't get an excess return without buying a high $\beta_i$ stock and accepting market risk (i.e., this is a "risk on" period when all stocks are correlated).

I can think of two things to test:

1. Fama-MacBeth regressions -- calculate $\beta_i$ for all the stocks, then do cross-sectional regression of the excess returns on $\beta_i$ and take the time series average. If he's correct, then the average coefficient on $\beta_i$ will be positive and significant.
2. Standard CAPM regressions (although to get a decent estimate of $\alpha_i$ you will need more risk factors) -- if he's correct, then the intercepts should not be significantly different than zero and the idiosyncratic error $\epsilon_i$ should be smaller than previous periods.
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It seems for either of the two proposed tests I will have to correct for the correlation of stocks returns to each other (beyond just via the market) and possibly make some correction for multiple hypothesis tests. I was hoping for a more consolidated approach that looked at all stocks returns simultaneously. The graph presented with the commentary is supposed to be illustrative, I guess, but I am not sure what it tells me. – shabbychef Feb 20 '11 at 22:23
@shabbychef -- You're right. These tests assume that the model is correct -- this was Roll's big critique from the 70s. But the author is saying that returns are correlated, whether it is through $R_{mkt} - R_f$ or through some other risk factor. Even if that risk factor is sun spots, that factor is driving all returns and makes it impossible for a stock-picker to exploit mis-pricing. Even if he were correct, the market has everything moving in unison according to $\beta$, whether it's a market $\beta$ or a sun spot $\beta$. – Richard Herron Feb 20 '11 at 23:34

One way of thinking about the problem is with a statistical factor model. Consider the two cases:

• You have more assets than time points

In this case if you accept enough factors, then there is no idiosyncratic risk. But there will be idiosyncratic risk if you restrict the number of factors.

• You have more time points than assets

In this case even if you accept as many factors as there are assets, you will in general have idiosyncratic risk as well. In order for your hypothesis to be true (over the time period of the data), there would need to be no idiosyncratic risk visible in this case.

I don't see a good way of testing the hypothesis, but it doesn't seem like a realistic possibility to me. I think a more reasonable hypothesis is that the distribution of idiosyncratic risk is [smaller, more skewed, ...] in time frame X relative to time frame Y.

Such hypotheses can be tested reasonably by estimating factor models in the two periods. Use the same number of factors in each period and then plot the densities of idiosyncratic risks. And probably try a few different choices of the number of factors.

You could bootstrap to get a sense of how variable the idiosyncratic distributions are.

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A proper analysis of whether idiosyncratic returns exist requires a "joint hypothesis" test. In other words, an equilbirum risk factor model must be assumed to test that the intercept across stocks is not statistically different from zero. You can use the GRS statistic to test the joint hypothesis that the idiosyncratic returns are zero.

The CAPM test as described in @richardh's answer above is the right approach go about estimating intercepts (especially since, visually, it seems that the market factor is dominant), or another model such as the Fama-French 3 factor model.

The chart illustrates the pairwaise correlation of stocks is increasing (i.e. we are in a "macro market" as opposed to a stock-pickers market). This means that there are likely a few common factors driving returns rather than a high variety of residual dispersion.

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