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Thursday, 14 May 2009

Correlation is not Causality (and is often Spurious)

Portfolio Theory

Correlation’s a powerful tool in stockmarket investment, because it allows investors to identify assets whose price tends not to move in the same direction at the same time. The branch of the Efficient Market Hypothesis labelled “Portfolio Theory” shows that by selecting groups of uncorrelated stocks it’s possible to create portfolios with lower levels of volatility than the individual stocks themselves.

The same is true of different asset classes – you can create a portfolio of these to provide a defence against any particular asset suffering a nasty fall. This has usually proven to be really useful, right up to the point it was actually needed.

Causal Connections

Correlation is a statistical measure of how things vary together over time. If every time stock A goes up 10% stock B also goes up 10% the two stocks are perfectly positively correlated. If stock B goes down 10% when A goes up 10% then they’re negatively correlated. If they move completely randomly to each other they're uncorrelated. Correlation (or lack of it) applies between asset classes as well – so, for instance, often (but not always) government bonds fall as equities rise and vice versa.

Correlation between assets is assessed on the basis of historical data. Which is fair enough, history is all we’ve got, but it’s only a guide to future performance. As soon as we assume it guarantees the shape of things to come we can be fairly sure we’re living on borrowed time.

The danger of correlation is that in order to assess it properly you need to understand the causal connections between the variables being assessed. If, say, luxury goods companies have tended to outperform in a recession, it’s important to ask why this is, rather than simply taking the statistics at face value.

Fire Engines, Lice and Syphilis

There’s an old example of this that shows the absurdities of the use of correlation as a mindless tool. Plot the number of fires in any city against two variables – the cost of the damage in dollars and the number of fire engines attending. You’ll find a clear statistical correlation between the cost of the damage and the number of fire engines attending the blaze. Should we conclude that the fire engines cause the damage?

This is clearly absurd, but many so called “obvious” links between variables are no less ridiculous when analysed dispassionately because of hidden third variables. All this shows is that correlation is not causality – just because we can see that two things vary together it doesn’t mean we can safely infer that one is causing the other.

Consider the people of the New Hebrides who, after many generations of careful observation, came to the slightly surprising conclusion that lice make people healthy and it was therefore important to make sure you had your fair share. Observationally they were correct – sick people had no lice. Unsurprisingly really, since lice don’t like hot bodies and sick people tend to have fevers.

Or what about the crazy idea that people with syphilis are cured by malaria?

Only that one’s correct as it happens: Julius Wagner-Jauregg got a Nobel Prize in 1927 for discovering it. Which shows that psychiatrists occasionally do have their uses.

The Hidden Third Variable

Many correlations are both correct and useless because of the hidden third variable. So breast fed children have better academic records than bottle fed children? Well, maybe mothers who breastfeed can afford to stay at home longer, spend more time with the children, read to them more and generally give them a better start in life. Maybe mother’s milk doesn’t directly link to brainpower. Maybe.

The hidden variable is the critical thing when assessing correlation. Correlation on its own proves nothing – even if you can establish that correlation exists you still have to figure out the direction of causality. Consider the relationship between obesity and illness. Fat people get ill more often, obviously.

Well, maybe, but maybe ill people get fat. Getting the direction of causality wrong can play havoc with correlation related decisions. In the world of stocks and assets it’s not easy to tease apart the issues of correlation and causality and it’s also extremely dangerous to build an investment approach on the basis that assets that historically aren’t correlated can never be so.

Luxury Goods, Premium Investments?

So why do luxury goods companies do better in a recession?

We can hypothesise that when people feel poorer, because their earnings are declining and they’re more likely to lose their jobs, they like to go out and treat themselves to an expensive new handbag or a splendid new suit. Only, on balance, that doesn’t sound very likely. More probable is that luxury goods companies’ target customers are High Net Worth (HNW) individuals whose spending power isn’t much diminished in a recession. That sounds more likely: David and Victoria Beckham’s consumption habits are unlikely to change much due to an overall contraction in the global economy.

So, should we all go out and buy luxury brand owners? Well, maybe. A closer look at the luxury goods companies suggests some of them have stretched their brands since the last recession. They’ve been targeting aspirational wannabees, rather than genuine HNW individuals. That’s been good for earnings in the meantime but is not so good now money’s tight. Posh and Becks are still spending but the chavs down the street are wondering how to pay off the credit card bill.

Macro and Micro Correlations

Investors have no choice but to rely on historical data to assess correlation between assets. History’s useful, but there’s a terrible danger that individuals live in their own bubble of space and time, failing to recognise that the correlations they’re used to are simply artefacts of their particular age, not unwritten laws of investment. Nonetheless, the past is the best guide we’ll get so using its data at the macro level is a reasonable approach, while being exceptionally careful at the level of individual investments.

Even at the macro level investors need to worry about the hidden third variable – a lot. The past few years have given us ample evidence of this. Asset classes which historically have only been loosely correlated suddenly started to move together with uncanny synchronism – generally in a downwards direction. Equities, property, commodities, tulips – you name it, down they went.

What had been missed was that the underlying connectedness between assets was growing due to the habit of investors taking on large amounts of leverage to buy them. When it suddenly became necessary to deleverage then all asset classes fell together. Liquidity was the hidden third variable. When the dash for cash became irresistible everyone sold everything in unison, and previously uncorrelated asset classes fell together. That’s the trouble with humans, they just keep on doing unexpected things.

Equities, Bonds and Time

In fact the polarisation of asset classes revealed a simple underlying truth – the best hedge against equity investments are high quality government bonds. At the margin, at the points when the market embarks on one of its fully fledged panics, the final recourse is a flight to safety when investors are prepared to pay virtually any price for safety. The rest of the time, a balanced portfolio offers the ability to generate risk weighted returns.

For most stockmarket investors the real hidden variable is time. Give markets long enough and they’ll trend upwards taking investor wealth with them due to general economic growth. In between times volatility can be terrifying. The irony is that short-term investors, who take the biggest risks, are the least likely to look for portfolio insurance while long-term investors will usually seek safety in correlation.

So just remember the next time some special interest group shows you the link between two things – correlation is not causality.

Related Posts: Darwin's Stockmarkets, Of Black Swans, Tsunamis and Cardiac Arrests


Frank Curmudgeon said...

Good post. I don't think that Treasuries and stocks are really inversely correlated, although it sure looked that way in the fall. I believe that all investments are positively correlated with each other to some extent. After all, they all compete with each other for investment dollars.

I also think that the widely held belief that inflation is bad for stock returns is a good example of the false correlations that you discuss. There's really no theoretical justifiation for it, as stocks are a real asset, but as it happens periods of high inflation have tended to also be periods of poor economic performance and low confidence, so the numbers seem to suggest a (false) link.

timarr said...

Hi Frank

Agree with both points but think there are a couple of exceptional circumstances to be considered. Mostly asset correlation is useful in hitting the efficient frontier (ok, I know ...) but in extreme cases of fear I think this breaks down. We don’t get many examples of this in a lifetime but in both the ‘90’s Asian crisis and last year there was a flight to “quality” and damn the correlation. The real surprise is the lack of a flight to gold, but perhaps that says something about the stability of the world we think we live in?

On inflation and equities I agree but think the argument can be extended a bit – those companies that can pass on inflationary costs in increased prices are very good defences against inflation, those that can’t will get hurt. I’ve some numbers somewhere on the effect of hyper inflation on stocks in Germany in the 1920’s – a clear case of where holding ‘safe’ fixed income bonds was a one-way ticket to penury.

P.S. To any readers I recommend clicking through to Frank's blog, it's a refreshing dose of reality :)

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Commentor said...

"If every time stock A goes up 10% stock B also goes up 10% the two stocks are completely correlated." Well, actually, no; but perhaps this caricature is excusable in the context. "If stock B goes down 10% when A goes up 10% then they’re uncorrelated." If we accept the first caricature, then this caricature is one of negative correlation---not non-correlation.

timarr said...

"If every time stock A goes up 10% stock B also goes up 10% the two stocks are completely correlated." Well, actually, no; but perhaps this caricature is excusable in the context. "If stock B goes down 10% when A goes up 10% then they’re uncorrelated." If we accept the first caricature, then this caricature is one of negative correlation---not non-correlation.

It's been so long since I wrote this I can't remember the context any more, but the latter point is completely correct, mea culpa. Modified. The former, well, I suppose it is a caricature, but if the value of two assets moves in the same direction by the same amount at the same time they're perfectly positively correlated. No?