Curious Cases Of Correlation

Have a good look at the below graphs and try to explain them before reading on:

The first one, taken from one of my International Economics lectures, shows a strong correlation between US Rock Music quality and US Oil Production. What was your explanation behind this? How on earth can US Oil production improve rock music quality?? Perhaps music producers have large stakes in oil firms and higher oil profits flow through to more investment in musical talent. Maybe it’s the other way around! Perhaps listening to some really good rock music generates some of the great production innovations in the oil industry. This makes sense. Who needs fracking when all you need is just some good rock music to ensure self-reliance on oil in the US?

Clearly I am being sarcastic. I am guessing if you had a proper look at the graph, you also probably concluded that something odd is going on here. If you tried to link the two movements in the top graph, I have faith that at least you didn’t attribute higher usage of Internet Explorer to decreasing murder rates in the US in the second graph!

These are obvious cases of correlation without causation.  There are some fantastic and hilarious examples of this in the real world. Just have a look at the Buzzfeed link (http://www.buzzfeed.com/kjh2110/the-10-most-bizarre-correlations). I put a few more examples below just to drive home the point:

Clearly, in contrast to what these graphs might suggest, organic food sales do not cause autism and credit card debt does not cause obesity.

In reality something else, a factor outside the view of these graphs is affecting both the relationships. For example, increasing world GDP can have the effect of increasing access to Internet explorer and reducing crime rates in the US. All too often, the human eye sees a relationship and tries to attribute reasoning behind the relationship. In the hands of deceiving politicians, media and policymakers, graphs can be a destructive tool. Take a look at this subtler graph below:

This is a graph from Google showing the US unemployment rate in red and the US real minimum wage in black from 1990 to just before the crisis. It is sensible to associate the unemployment rate with the minimum wage. When two variables seem likely to be co-related, it is only natural to attribute causation. This graph tells us something very interesting – increasing the minimum wage decreases unemployment. Interest groups, media and politicians to a certain extent leave the story here. Armed with this graph and a convincing dialect, this is a dangerous weapon. If you have ever studied Economics, this will puzzle you. Increasing the minimum wage should increase unemployment. There should be a strong positive correlation here. From here a lot of economists and policymakers will try to explain the causation underlying the correlation. Perhaps increasing the minimum wage creates a multiplier effect driving up incomes and hence jobs. Once satisfied with the reason, a politician may decide to increase the minimum wage. After all, it has increased employment in the past. Wrong.

All too often, people forget that correlation is not causation, especially when the variables are supposed to be affected by each other.  There is a simple explanation for this graph. Improving economic conditions tend to reduce unemployment and also encourage policymakers to increase the minimum wage. Why do you think George Osborne wishes to raise the UK’s minimum wage now? It is because of a recovery that he believes it is a good time to make the political move. Take a look at what happens after the dotcom bubble. Unemployment naturally shoots up. At the same time, real minimum wages fall as politicians are less inclined to hike the minimum wage and damage employment further.

In fact, what is more likely is that the unemployment rate, in the absence of an increasing minimum wage, would have fallen even further. 

Econometrics

I have never really been a huge fan of econometrics. All this inverting matrices and proving consistency all seems a bit pointless (and still does!) But it has only been this year that I have truly realised the entire point of Economics. The role of Economics is to understand where there is and is not causation. If all graphs showed causal relationships, all economists may as well pack up their stuff and start throwing their constrained optimisations into other disciplines (perhaps physics where it belongs J). It is the role of the economist to break down these graphs and explain what is truly going on, first by creating theoretical hypotheses and secondly by robustly testing them econometrically.

I am fascinated by how often the media or a policymaker will show you a graph to convince the public. Statistics in the wrong hands can be wildly deceiving. Graphs are merely starting points.

Having said this, I want to look at some examples of where economists and the media all too often attribute causation where perhaps it is inappropriate.

Debt

The top graphs show Greek government debt alongside its sovereign ten-year yields for similar time periods. The bottom two are for the US, where the yield graph starts in 1982 just as the US debt explodes on the left hand graph.

The Greek debt crisis seems to provide concrete evidence that higher public debt increases the yield on government bonds (thanks to a higher risk of default). But if you actually look at the graphs, bond yields only began to rise at the start of the crisis in 2008. Government debt hit 106% in GDP by the end of 2006 yet markets seemed extremely calm. In 2001, debt was already above the worrying level of 100%.

I am not saying that higher debt does not contribute to higher default risk and larger yields but I want to make two distinct points:

1)  Even from the Greek graphs you cannot conclude that higher debt causes higher yields.  If the relationship were purely causal, yields would have increased well before the crisis. Of course, at the onset of the financial crisis, investors reassessed the risk of Greek debt and priced it accordingly. But there were other factors driving both graphs in their respective directions. Worryingly low competitiveness in Greece drove investors to safe havens at the start of the recession simultaneously increasing yields on Greek bonds and kicking into place automatic stabilisers driving up public debt further. There is also some reverse causality here. As the interest on government debt increases (and GDP falls) so does its level as a percentage of GDP. It might be more appropriate to argue that flight to safety was the cause of capital outflows in Greece and that rising yields was a consequence leading to a subsequent sovereign debt crisis.

2) Even if the relationship holds for Greece, it would be wrong to assume the relationship is causal globally. The US is an obvious example. It is cheating to a certain extent given that the dollar has the benefit of being the globe’s reserve currency, but it is useful for this point. Yields have fallen dramatically in spite of an enormous explosion of debt to GDP in the US. In fact, during the talk of a technical default, investors could not buy Treasuries quick enough. What holds for one country may not hold for another.

Forward Guidance

A lot of people have had their dig at forward guidance (myself included) for its pitiful effect in the markets. The introduction of forward guidance was intended to reduce long-term interest rates and flatten the yield curve. However, the introduction of forward guidance was correlated with increasing yields. There is the obvious argument that forward guidance brought forward interest rate rise expectations. However, this can be seen as a case of correlation without causation.

Forward guidance is an insurance policy against rising rates in an economy that was not improving. The idea was to prevent yields from increasing in a stagnant economy. It was brought out due to the fear that rates were starting to increase damaging the recovery. But rates were increasing due to the fact that the economy was improving. It is a vicious cycle – an improving economy pushed up rates and jolted the central banks into action to prevent steep rate rises in the potential scenario where the recovery was not as good as expected. Forward guidance did not necessarily push up yields but was an insurance policy against a scenario that did not happen. If anything, it probably reduced the rate at which yields were rising.

Austerity

“Our austerity plan is working,” is something you will have heard a lot recently. This statement is usually accompanied by recent growth and unemployment figures. All too often in politics and the media, policies are linked directly to macroeconomic outcomes. If you really wanted to prove causation here you would need some counterfactual (i.e. growth in the absence of austerity) to compare with.  Macroeconomic outcomes are a function of a huge number of variables and it is extremely ambitious to claim any single policy was the cause of an uptick in growth. In reality, the benefits of austerity are strongly contested amongst leading economists. In my opinion, economies should follow a cyclical fiscal strategy, fixing the roof while the sun is shining and allowing automatic stabilisers to kick into place in busts. This is especially applicable in economies like the UK and US that are deemed creditworthy even at high levels of national debt. I believe austerity should only take place in the bad times when it is forced upon you by huge capital flight leading to high borrowing costs and untenable debt. Whilst trying to avoid a Keynesian fiscal policy debate, my point is that a lot of well-established economists would argue that austerity would only have damaged UK growth. In this case, growth may have returned in spite of austerity and not because of it.

So…. Never believe anything?

So what’s the conclusion? Graphs are useless? No. Graphs are interesting. They show us correlations. The point of this blog was to take the insights that politicians and the media draw from statistics or graphs with a pinch of salt. Usually people will use data to emphasise their point where in fact the data is caused by another reason entirely.

Austerity may very well have lead to growth. Internet explorer may in fact reduce murder rates. The point is that the graph is just a starting point and anyone claiming it shows causation is undeniably wrong. If you can think of another reason that causes both variables to increase together then you may have proved them wrong entirely.

The idea is to be cynical. Is their conclusion sensible? What else can cause the co-movements? What does common sense tell me? Next time somebody throws you a statistic in an argument, throw this back at them (or ask to see their econometric analysis!)  Correlation does not mean causation. Remember that when reading the paper tomorrow.

The Game Theory Behind Golden Balls

Golden Balls is a simple, fascinating and sometimes hilarious game show in the UK that must have been created by an economist. It is a game show where prizes should theoretically never be won. Yet to the dismay of the economist who orchestrated the game something very interesting happens in reality.

The game involves players accumulating prize money in two rounds of pre-play. In the final round players have the opportunity to share or steal the accumulated prize money. This round is a one-stage game where two players can communicate briefly before simultaneously choosing whether to “Split” or “Steal”. If both players Split, the prize money is distributed evenly. If one Splits and the other Steals, the latter wins all the money. Finally, both Splitting means no prize money is won at all. The game can be written in normal form below, where M represents the accumulated prize money, green relates to the first player and orange to the second:

P1\P2	Split	Steal
Split	M/2 , M/2	0 , M
Steal	M , 0	0 , 0

This game is a “weak prisoner’s dilemma.” For the economists reading, it has three pure-strategy Nash equilibiria in (Split, Steal), (Steal, Split) & (Steal, Steal), as well as an infinite number of mixed-strategy equilibiria where one person Steals with certainty and the other randomises. For those that have not studied game theory, the analysis of the game is simple. If your opponent chooses to Split, you should choose to Steal since winning all the money is more than half of it. If your opponent chooses to Steal, you are indifferent between Splitting and Stealing. Given that Nash Equilibrium relies on knowing what the other player is going to play, it seems most rational to always play Steal. In other words, if there is a non-zero probability of your opponent ever playing Split, then you should always play Steal. This is because Steal is a weakly dominant strategy (you are never worse off by playing it). Only if you correctly believe your opponent is choosing to Steal with certainty can a (Split, Steal) equilibrium be possible.

It is called a prisoner’s dilemma because the most likely equilibrium of mutual Stealing is Pareto dominated by mutual Splitting. In other words, both players can be made strictly better off if they deviate together to Split. However, in a stage game without repetition, this equilibrium should never occur. If you believe your opponent will Split, you should always Steal. A working paper at the University of Zuirch* found some very interesting results. By studying outcomes of the game it finds that there is a 33% mutual cooperation rate of both Splitting. The paper then goes on to investigate what increases the likelihood of both players Splitting such as handshakes, racial bias and so on. However I am more interested in why Splitting could ever possibly result in equilibrium in a one-stage game such as this. I came up with two potential explanations, one of which I dislike but felt the need to mention and the other which, with its pitfalls, does seem to explain the data.

1) Miscoordination

It is important to realise, given the payoff structure, that both Splitting is not a matter of mutual cooperation but instead a situation of miscoordination. The reason being is that without repetition of the game, there is no incentive to coordinate. Once one player Splits, the other must Steal. Hence the cheap talk before choosing your action is meaningless. If you have convinced your opponent to irrationally Split then you should rationally choose to Steal. As a side note, I find it ironic that players try to convince each other to Split with the lure of Splitting as well. Theoretically the best strategy to ensure you win the entire prize is to convince her you will Steal. Only in this situation will she be indifferent between her two actions. When she believes you will Split, she will be better off by Stealing.

The reason both Splitting can be a result of miscoordination is when both players choose different Nash equilibrium. If one believes that the equilibrium will be (Split, Steal) and the other (Steal, Split) then both Splitting may result. In other words, if both players believe their opponent is Stealing with certainty, an equilibrium where they both Split is possible. The problem with this explanation is that Nash equilibrium relies on correctly knowing your other player’s strategy. As I mentioned above, if there is any non-zero probability of a player choosing to Split then the opponent should always choose Steal. Hence, you would only really choose to Split if you are 100% sure your opponent is Stealing and hence you are indifferent. 

Furthermore, your payoff from mutual Stealing may actually be higher than when you Split and your opponent Steals. This is because in the former you “fail to win” but in the latter you have “lost.” Or put more simply, if you know your opponent is going to Steal, are you really indifferent between Stealing and Splitting? Perhaps I am filled with too much animosity but I would rather my opponent got nothing than stole everything. Hence in this sense, stealing is a strictly dominant strategy and miscoordination cannot be a good explanation.

2) Payoff Re-model

The results of the Zurich study are interesting. Stake size and communication do actually have significant effects on the outcome. There is a negative correlation between cooperation and stake size. Actions such as mutual promising to Split and a handshake increase cooperation. This suggests that the payoff of players do not wholly depend on the monetary gain. I believe we can crudely remodel the game with an additive term to the Split payoff of k(x), where k is increasing in x and x is a set of characteristics such a reputational concerns. They key point is that x can be endogenously given, in the sense that the cheap talk and affection towards your opponent can affect it. For example, if you do not want to be seen stealing from an old lady who you really admire on national television then x may be very high. The normal form becomes:

P1\P2	Split	Steal
Split	M/2 + k(x) , M/2 + k(x)	0 + k(x) , M
Steal	M , 0 + k(x)	0 , 0

Now Splitting is a dominant strategy for a particular player if their value of k(x) is larger than M/2, meaning that regardless of what the other player does, this player should Split. If this is the case for both players then (Split, Split) is a dominant strategy equilibrium. For example, a handshake and a mutual promise will greatly increase reputational concerns or guilt from not Splitting and hence lead to a large k(x) for both players. Note we could have modelled the game by subtracting k(x) from the Steal payoffs as well.

The larger the monetary prize for a given level of x, the less likely k(x) will exceed M/2, explaining the negative correlation between prize money and cooperation. In other words, at some prize level, the potential gain from Stealing all the money exceeds the costs of a damaged reputation or guilt.

Where one player's k(x) exceeds M/2 but the other player's does not, the only Nash equilibrium is (Split, Steal). Where neither player's k(x) exceeds M/2, both (Split, Steal) and (Steal, Split) are Nash equilibria. These explain the unilateral cooperation rate of 55% by players.

The problem with this revised model is that now (Steal,Steal) isn't ever a Nash equilibrium if k(x) is some non-zero value. In light of this, it might be more appropriate to remove k(x) when the other play Steals meaning that players only avoid repetitional damage by Splitting when the opponent is Splitting as well.

Another problem with this argument is that it simply restates the payoffs rather than provide rational explanations of cooperation in the original game.  However, if players are only concerned with their monetary payoff, then how can you explain a situation where both Split?

* http://www.econ.uzh.ch/faculty/graetz/publications/wp1006.pdf