Financial Mathematics Text

Monday, January 6, 2014

Correlation as a Substitute for Critical Thinking in Finance?

I frequently come across graphs like these on various blogs and articles. I'm going to single out one, not because there is anything particularly wrong about it (they're all wrong), but only to use it for illustrative purposes. I could really pick out any of these and offer the same criticisms.

This one I found in the article The Declining Inflation Expectations Chart That Should Have Stock Investors Very Concerned. The original chart apparently come from Dan Greenhaus via twitter.

So here's a quick reproduction of the chart:

So what's the deal?

Well, there is a decent correlation from 2009 to the end of 2012 (r=0.834). And then the two graphs diverge quite a bit.

But they ought to not diverge. So something has to give. Either inflation expectations have to go up or stocks have to come down. . . . or so the story goes. But why should that be?

In my blog on the problem of induction, I outlined two things that can be done to improve our knowledge:

1) We can test the hypothesis on a different data set.
2) We can develop theories that explains why we're observing the data we are observing (which may give us guidance on those which will work in the future).

These correlation type graphs typically fail on both grounds as I shall illustrate with this example.

First, we can actually see if this relationship did well in the past. FRED offers us data on TIPS that go back to 2003. In fact, it did not:

While the two actually appear to be decently correlated between 2007-2012, the correlation breaks down prior to 2007.

Notice also that from mid-2013 (when the article above was written) and the end of the year, the two diverged even further.

So clearly this one fails the first test. What about the second?

Now there is good reason to think stock prices might be influenced by inflation. After all, if inflation is higher then we might expect earnings to go up with inflation (more or less) and therefore stock prices ought to follow with it.

That's all fine and good (as a simplistic analysis at least; stock prices are definitely more complicated than that.) But why should it be a linear relationship between inflation expectations and price? Because that's what's being proposed here.

And to this, I offer the following answer: There should not be any!

Consider the following scenario. Suppose that earnings do go up with inflation and suppose that we can perfectly predict inflation will be exactly 2% a year. Then inflation expectations will be a nice 2% a year so earnings will go up, say, 4% a year (2% of real growth along with 2% inflation). So you might expect the graph to look something like this (assuming a well behaved stock market and constant growth rates):

OK, so that looks kind of boring. But the point is that if we accept that there is a relationship inflation and stock prices, that doesn't mean it's going to be the linear one that was found via an historical correlation which will not likely continue into the future.

Now of course, there is likely to be a more complex relationship between inflation and stock prices than the which I've proposed. But I think whatever your more complex theory will end up being (and we'd have to test the theory), it seems to me unlikely that it will predict that those two variables (inflation expectations and stock prices) will continue to be linearly correlated well into the future.

So this fails test two.


So the next time you see one of these silly correlation charts, ask yourself if it will pass the above two tests. If it can't pass both, it only exhibits the author's ability to scale charts appropriately. And that is not critical thinking. And it has no predictive value.

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