A Naive 2022 Senate Forecast

I do not study elections. I am not a forecaster. So take what follows accordingly.

There are a number of people who have built sophisticated and probably quite fine election forecasting models. I wanted to see something a little different. How close can I get with a very naive forecasting model?

So first, what am I trying to forecast? I’m gonna go for the Democratic two party voteshare, or

$$ \delta \triangleq \dfrac{\text{Number of Democratic Votes}}{\text{Number of Democratic Votes} + \text{Number of Republican Votes}}. $$

The Model

Now like I said, for kicks I’m going to try to forecast δ_i for each state i where there’s a Senate race in a very unsophisticated way: Specifically, I’m only going to use information voters have given me about whether they prefer to vote for the Democrat or the Republican. This information comes in two flavors: the information I had beforehand and what I’ve learned through the campaign. This leads very naturally to a very simple Bayesian model, where we place a Beta prior on δ_i, whose parameters are shaped by election returns from past Senate races in state i, and update our belief about δ_i using polls of the voters in state i in the current election cycle. (Specifics about how the prior parameters are calculated from past election returns are given for the interested reader after the forecasting results are discussed).

Model “Validation”

Since we have historical polls and election returns for the past couple cycles of Senate elections, we can go ahead and see how this naive model would have fared in 2018 and 2020 before seeing what its predictions for tomorrow will be. (I omitted some cases to avoid dealing with wrinkles for this simplified naive model; this is discussed at the end).

Surprisingly, the results are not that bad?? The dashed line here represents if the estimated two party voteshare was exactly what we actually observed, so one thing to notice is the model does overestimate Democrats slightly since most of the points are just a bit on the upper-left side of that line, but the points do track the line pretty well actually.

Here are how the races would have been classified vs the actual results:

##           Prediction
## Result     Dem lost Dem won
##   Dem lost       22       5
##   Dem won         0      33

This super simple model predicted 91.7% of 2018 and 2020 US Senate elections correctly LMAO. BUT, notice that all the errors were overestimating Democrats, so that’s something to be wary of. For reference, here’s how far off the predictions were in those classification misses:

Year	State	Actual voteshare	Estimated voteshare
2018	FLORIDA	0.4993874	0.5063907
2018	INDIANA	0.4691855	0.5037296
2020	IOWA	0.4660038	0.5031730
2020	MAINE	0.4539646	0.5113481
2020	NORTH CAROLINA	0.4908670	0.5261374

The 2022 Senate Forecast

Okay, so if we were to apply this method to the 2022 races, what would we get?

(The completely solid red or blue states were states that could not be estimated by this model due to data issues such as lack of polling, but there is also a very strong consensus on the outcome in those states so I treated them as a particular party winning them with probability 1).

We’d end up with 51 Democrats in the Senate and 49 Republicans in the Senate… at least just looking at the point estimates alone and using 0.5 as the prediction cutoff. However, just like with all the other models of this election, this naive model gives lots of close races; here are the 10 closest according to this model:

State	Estimated Dem Voteshare
Nevada	0.50
Georgia	0.51
Wisconsin	0.49
Ohio	0.49
North Carolina	0.49
New Hampshire	0.52
Pennsylvania	0.52
Indiana	0.47
Arizona	0.53
Florida	0.47

So, we can do the usual thing that these modellers do and simulate a whole bunch of election outcomes from the model and then summarize them for you to get a sense of the uncertainty and the likely range of outcomes according to the model:

Dem-held seats	N Simulations
49	24
50	9017
51	30749
52	210

Unsurprisingly, there’s less uncertainty in this model than some others given its almost total lack of sophistication. The modal outcome of this model, by far, is the Dems keeping the Senate, and even gaining a seat, up to 51-49. But Dems ending up with more than that is quite a rare outcome of this model.

…And the model overestimated Dems by just a little in the last two cycles… so it’s gonna be a nail-biter 😬️

Appendices

Setting the Prior Parameters

A Beta distribution is defined by two parameters, a and b. The mean and variance of a Beta distribution are given by

$$ \mu = \dfrac{a}{a + b} $$

and

$$ \sigma^2 = \dfrac{ab}{(a + b)^2 (a + b + 1)} $$

respectively.

So we can use the method of moments to set a and b for our Beta prior over δ_i by taking the election returns from the last N (say 3) Senate elections in state i and setting μ equal to their mean and σ² equal to their variance, then solving for a and b using the equations above.

Omitted cases

To avoid dealing with wrinkles in just what Democratic two party voteshare means in elections where there’s multiple Democrats, I omit races in California, Louisiana, and the 2020 Georgia special.