## Monday, March 14, 2016

### Brexit: Getting the Probabilities Right

Brexit is the next big known unknown macro event this year. And probably you have already heard enough about it from a multitude of different sources.

And if you have, one thing you must have noticed. While the opinion polls are neck to neck mostly, the betting market pricing is quite different. For example the latest from Telegraph poll tracker puts it at 51% for "stay" and 49% for "exit". The best current offer for "stay" from the bookies is at $4/11$ - which prices a 73% probability of stay.

This has (and continues to) caused quite a bit of confusions among economists and strategists - cursory glances at the estimates and research notes doing the rounds will give you the idea. It seems the opinion polls and the betting market are not consistent with each other. And it seems most sell-sides (and some buy sides too) are siding with the tight calls from opinion poll.

This confusion is utterly wrong and in all likelihood, both results are right and support each other.

No fault of the economists and the strategists of course. There are many cases where we humans have a good intrinsic sense of chance - like sensing the movement in our peripheral vision to determine the probabilities if it is friend or foe, and converting that to a swift "stay calm" or a "fight-or-flight" decision. Unfortunately, we are not naturally evolved to understand how the probability works in opinion polls!

The opinion polls and the betting market present two connected, yet different, measures. The poll figures shows how many will, if the referendum is held right now, choose "stay" - for example. The betting market odds indirectly gives a probability of "stay". The connection between these two measures is subtle. To understand that, assume an extreme case where we have no undecided voters in the opinion polls. Also the voters are absolutely certain and will not change their mind come what may. If we have 51 to 49 in favor of "stay", what is the probability of a "stay" outcome?

It is 100%! An absolute certainty. We will have 51% in favor votes and 49% against. Since this outcome is guaranteed and $51 > 49$, the resulting win for the "stay" choice is guaranteed as well. This hold true for a 50.5 to 49.5 split. We can go even further. This apparent 50-50 results are actually far from 50-50.

Of course, in real life, there are three deviations from this scenario. Firstly, we are not polling the entire population, hence that 51-49 split is just an estimate from a smaller sample. Secondly, opinions can change on the actual day of voting. Thirdly we do have undecided voters who will eventually vote one way or the other, and decided voters who will end up missing it.

We have statistics as our tool to do our best with the first observation. The entire thing called opinion poll is asking some $n$ number of people about a binary choice ("stay" or "exit"). This is much like a bi-nomial trial. Let's assume the true fraction in the entire population that supports "stay" is $\Pi$. What we are trying to estimate is $Pr(\Pi > 0.5| \pi )$. Here $\pi = X/n$ is our estimate of the fraction that support the "stay" outcome, $X$ being the number votes in favor.

The rest is straightforward, although a bit mathematically involved. We assume prior probability distribution of $\Pi$ as $f_\Pi(\pi')$, carry out the opinion polls, and from the results compute the posterior probability distribution $f_\Pi(\pi)$, using Bayes theorem. Doing just that, we plot the probability of a "stay" outcome against the percentage point difference in the opinion polls.

As you can see, the probability of "stay" quickly converges to $0$ or $1$ as the lead diverges from $0$ on either side. Around $0$, it is highly non-linear. In fact for our extreme case above, it would just jump from impossible (probability = $0$), just below lead $0$ to probability of $1$ (complete certainty) slightly above that - a step function.

The current spread between "stay" and "exit" vote (51 to 49) reflects a probability of "stay" at $0.80$. Inversely, the betting market probability of $0.73$ indicates a lead of $+1.55$ priced in. So obviously both the betting markets and the opinion poll results are not very different. Also another very good crowd-based event forecasting source with expert-beating results projects the odds at 75%.

All of these are quite comparable. And is nowhere near the 40% to 50% odds thrown in most research notes. Take a note before you put your position.

What still can change from here? Well we still have the observations two and three from above - a change in mind for the voters, and swing of the undecided. They add to the uncertainties. On the other hand, we have assumed an unbiased prior - but most likely the people of the United Kingdom has a certain bias to stay to start with. That will add support to a "stay" outcome.

Technical details: Here we have taken the YouGov January survey results - with a sample size of $n=1735$. Also we have assumed the prior distribution as conjugate beta (which leads to posterior beta distribution as well), with unbiased (non-informative) prior - beta distribution shape parameters as $\alpha=\beta=0.5$. The upper and lower bound is computed based on the standard error $\pm\frac{1}{\sqrt{n}}$. For more technical details, you can see here.