# Correlations

If I had to compress everything that ails us today into one word it would be correlations.  Basically, everything bad that has happened recently from the financial crisis to political gridlock is due to undesired correlations.  That is not to say that all correlations are bad. Obviously, a system without any correlations is simply noise.  You would certainly want the activity on an assembly line in a factory to be correlated. Useful correlations are usually serial in nature like an invention leads to a new company.  Bad correlations are mostly parallel like all the members in Congress voting exclusively along party lines, which reduces an assembly with hundreds of people into just two. A recession is caused when everyone in the economy suddenly decides to decrease spending all at once.  In a healthy economy, people would be uncorrelated so some would spend more when others spend less and the aggregate demand would be about constant. When people’s spending habits are tightly correlated and everyone decides to save more at the same time then there would be less demand for goods and services in the economy so companies must lay people off resulting in even less demand leading to a vicious cycle.

The financial crisis that triggered the recession was due to the collapse of the housing bubble, another unwanted correlated event.  This was exacerbated by collateralized debt obligations (CDOs), which  are financial instruments that were doomed by unwanted correlations.  In case you haven’t followed the crisis, here’s a simple explanation. Say you have a set of loans where you think the default rate is 50%. Hence, given a hundred mortgages, you know fifty will fail but you don’t know which. The way to make a triple A bond out of these risky mortgages is to lump them together and divide the lump into tranches that have different seniority (i.e. get paid off sequentially).  So the most senior tranche will be paid off first and have the highest bond rating.  If fifty of the hundred loans go bad, the senior tranche will still get paid. This is great as long as the mortgages are only weakly correlated and you know what that correlation is. However, if the mortgages fail together then all the tranches will be bad.  This is what happened when the bubble collapsed. Correlations in how people responded to the collapse made it even worse.  When some CDOs started to fail, people panicked collectively and didn’t trust any CDOs even though some of them were still okay. The market for CDOs became frozen so people who had them and wanted to sell them couldn’t even at a discount. This is why the federal government stepped in.  The bail out was deemed necessary because of bad correlations.  Just between you and me, I would have let all the banks just fail.

We can quantify the effect of correlations in a simple example, which will also show the difference between sample mean and population mean. Let’s say you have some variable $x$ that estimates some quantity. The expectation value (population mean) is $\langle x \rangle = \mu$.  The variance of $x$, $\langle x^2 \rangle - \langle x \rangle^2=\sigma^2$ gives an estimate of the square of the error. If you wanted to decrease the error of the estimate then you can take more measurements. So let’s consider a sample of $n$ measurements.  The sample mean is $(1/n)\sum_i^n x_i$. The expectation value of the sample mean is  $(1/n)\sum_i \langle x_i \rangle = (n/n)\langle x \rangle = \mu$. The variance of the sample mean is

$\langle [(1/n)\sum_i x_i]^2 \rangle - \langle x \rangle ^2 = (1/n^2)\sum_i \langle x_i^2\rangle + (1/n^2) \sum_{j\ne k} \langle x_j x_k \rangle - \langle x \rangle^2$

Let $C=\langle (x_j-\mu)(x_k-\mu)\rangle$ be the correlation between two measurements. Hence, $\langle x_j x_k \rangle = C +\mu^2$. The variance of the sample mean is thus $\frac{1}{n} \sigma^2 + \frac{n-1}{n} C$.  If the measurements are uncorrelated ($C=0$) then the variance is $\sigma^2/n$, i.e. the standard deviation or error is decreased by the square root of the number of samples.  However, if there are nonzero correlations then the error can only be reduced to the amount of correlations $C$.  Thus, correlations give a lower bound in the error on any estimate.  Another way to think about this is that correlations reduce entropy and entropy reduces information.  One way to cure our current problems is to destroy parallel correlations.

## 7 thoughts on “Correlations”

1. Rick Gerkin says:

Good thought. I think this is somewhat related to what economists call pro-cyclical and anti-cyclical forces. Correlation tends to make things more pro-cyclical than anti-cyclical.

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2. josic says:

Interesting – but do we need the technical term “correlations”. Could we make a similar argument as a consequence of dependencies between parts in a complex system – dependencies that are very difficult to predict and are hence frequently just ignored? I am not sure whether you meant something stronger when you used “correlations” as a specific measure of such dependencies.

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3. Kreso: I just meant run of the mill second moment correlations. However, I claim that whatever complicated nonlinear dependencies you may have will at minimum induce linear correlations that we can quantify.

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4. Thank goodness some bloggers can still write. Thank you for this read

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