Opinion polls are increasingly common; UK Polling Report lists 125 polls for both the 1974 elections combined, between 1970 and 1974, while the same website lists 1868 for the 2015 election, with still 67 days to go. However opinion polls only report the vote share implied by the surveying of the pollster; while undoubtedly vote share will influence election outcomes, anomalies are still possible. For example, in February 1974 the Conservatives won more votes yet fewer seats than Labour, an outcome it is reported the Tories are preparing for this time around.

Seats matter more, and trends vary across the country for the different parties; many Tory heartlands are supposedly at threat from Ukip, while Labour’s Scottish seats appear lost to the SNP. Constituency polling, led it seems by Lord Ashcroft since 2010, is viewed as the way forward. Despite this, traditional nationwide polls continue to attract attention – not least the current neck-and-neck nature of Labour and the Tories. Can we glean anything from such polling?

Is there any kind of relationship between how much a party is polling, and how many seats they can expect to get? There naturally is, but the more pertinent question is how strong and robust is that relationship over the years? We focus on polls since 1970, hence 10 elections and 8,253 polls. We include information on the time horizon until the election (number of days), the political party (in case of any biases), and consider any kind of incumbency effect, and we interact all these variables together in a linear regression model in order to see whether the resulting model had any explanatory power. Surprisingly enough, it manages to account for almost 90% of variation in seats won, and I’m happy to provide any interested party with the regression output (I’m working on tidying up a more general set of code for this).

A few plots to help:

Firstly this is a cross plot between actual opinion poll shares and election outcomes in terms of seats – the black circles signify such points. They show the Lib-Dem cluster below 100, and the Labour/Tory cluster spread between 200 and 400. It suggests something of a non-linear relationship in the Labour/Tory cluster, since polling shares in the 50s are consistent with seat totals of 300 and 400. The red apparent scribbles are the fitted values, or predicted values, from our linear regression model. They show that within the sample, to some extent, the two clusters are captured. Clearly improvements are possible, but it’s a reasonable model to begin with.

How did the model fare in 2010? Here’s the implied seat forecasts from each poll against the actual seat totals:

Note this is the same model estimated over all polls prior to the 2005 election – any poll for the 2010 election is excluded in order that this is actually a forecasting exercise rather than an in-sample fitting exercise. The resulting forecasts for seats are quite surprising in that Labour were forecast as the election neared to get more seats. This appears to be the incumbency effect, which is strongly significant in the regression model. On average, forecast errors were about 11 seats for this election, but clearly biased up for Labour, and down for the Tories and Lib Dems. The same was true, but on a smaller scale, in 2005.

What does all of this mean for 2015? It’s harder to tell whether there’s an incumbency effect since there is no comparable full coalition term; the model automatically attributes this to the Conservatives. The forecasts look like:

Hence unlike much of the current media narrative (e.g here, here), this very simple model appears to point towards the Tories not just being the largest party in a hung parliament, but potentially winning a majority; the most recent polls indicate based on historical data that the Tories will win somewhere between 280 and 300 seats, but Labour only 220-240. Could the incumbency effect be too strong here, as it appears to have been in forecasts in 2010?

There are no Ukip forecasts since there is no recent historical precedent for Ukip, hence no data upon which to base a forecast. We could apply the model, estimated over Lib-Dem historical performance, to Ukip, but there’s only so far one should take a very basic statistical model such as this. On the most important question of most seats between Labour and the Tories, it has already provided a thought provoking forecast.