Our next government? (updated x2)

Eric Crampton has an interesting piece on the iPredict blog discussing Our next government? Eric writes

The markets today are saying 70% National, 30% Labour as next PM. Let's have a look at the underlying numbers.

At high bid for each party, assuming that Peters does not return to Parliament, Hide takes Epsom, Dunne and Anderton return to Parliament and the Maori Party take 6 electorates, we're looking at a 123 seat Parliament arrayed as follows:

GRN: 9
PRG: 1
LAB: 45
MAO: 6
UF: 1
NAT: 57
ACT: 4

62 seats are needed for a majority. National + Act + UF are at 62; Labour + Green + Progressive + Maori are at 61.

He goes on to look at bit deeper,

The markets also are saying there's a 17% chance that National's true vote share will be above 50%. If National's price is normally distributed around the true mean, though, that means there's also a 17% chance that National's true vote share will be below 43.3%. Again, we can take the combination of the prices in VOTE.NATIONAL and MAJORITY.NAT to work out the standard deviation of VOTE.NATIONAL: it's currently at 3.5. Let's assume that the standard deviation of VOTE.LABOUR is the same. We'll benchmark minor party vote share standard deviation by that of NZ First, which we can work out from the combination of VOTE.NZFIRST, TAURANGA.PETERS and MP.PETERS. At current prices, that suggests that the minor party standard deviation is 1.15.

Ok. National's coalition's expected vote share is 50.65%, getting it 61 seats plus Dunne. While the vote share point estimate is 50.65, the standard deviation will be the sum of the standard deviations of the underlying components: 3.5 + 1.15 = 4.65. National's seat share is then expected to be 62, with standard deviation 5.58 seats (as 1% of the vote gets you 1.2 seats).

Labour's coalition's expected vote share is 49.37%, getting it 59 seats plus 1 Maori overhang plus 1 Anderton. The standard deviation will be (as a rough cut) the same as that for National's coalition because we don't need to worry as much about the variance of the Maori party vote if they're set anyway for overhang, so we're really only worried about variance in Green and Labour. So, 4.65. Labour's seat share is then expected to be 61, with standard deviation 5.58 seats.

Now, what's the probability that the realized seat share for the National coalition really lies above the realized seat share for the Labour coalition? A cheap way of checking this is just to check the probability that National lies at 61 seats or less. Our standard normal table tells us the chances of this are 42.86% as only 7.14% of the distribution lies between 62 and 61 seats.

So, National's chances of getting more seats than Labour look more like 57% than like 70%. And those chances get much worse in the 26% probability case that Peters returns to Parliament. I'm not going to bother working them out at this point though.

Don't you just love economists and statisitics?! But what this says is that either Eric has gotten his calculation very wrong or the iPredict market has gotten something wrong. If it is the latter then there is money to be made. Go to it!

Update: Eric has added the following post to the iPredict blog,

Our next government redux

One of the advantages of being at Canterbury in October is that Sir Clive [Granger]comes to visit. Sir Clive won the Nobel a couple of years back for his work in economic forecasting and economic statistics. So I asked him if I'd screwed up anything major in the post below. He said the overall technique should be reasonable for what we're trying to do here, but that I'd made a minor error: the correct probability of National forming the next government, at the current prices in the vote share market, is 60.6%, not 57%. I'd made the rather silly error of adding up the standard deviations rather than adding up the variances.

So we have four options.
1. The prices in the PM markets are wrong
2. The prices in the VS markets are wrong
3. The underlying distributions are highly skewed rather than symmetrical.
4. The Maori party is more likely to go with National than NZ First is likely to make it into Parliament.

We've eliminated 3a, which was the "Eric's completely screwed this up" option as I now have the Sir Clive seal of approval. And you're not going to do better than that in New Zealand currently.

I'm complementing my positions in the PM markets with the appropriate inverse positions in the VS markets as a quasi-arbitrage.

Update 2: Eric comments on And so is solved the mystery of the inconsistent prices