How the model works
The model follows the state-space approach of Jackman (2005), Pooling the polls over an election campaign, as adapted for New Zealand by Peter Ellis (2017, 2020) and the NZ Herald (2023). Latent vote intention moves week by week; polls are noisy, biased measurements of it; official results measure it exactly. The main changes from the 2023 model are listed at the end.
Notation
| Symbol | Meaning |
|---|---|
| k = 0, \dots, K-1 | tracked parties; k=0 is National, k = K-1 is “Other” |
| t = 0, \dots, T-1 | weeks (starting Sunday); t=0 is the week of the anchor election (2011) |
| \pi_t | vote shares in week t, a point on the simplex |
| \theta_t \in \mathbb{R}^{K-1} | additive log-ratios, \theta_{t,k} = \log(\pi_{t,k} / \pi_{t,0}) |
| i | a poll, with week t_i, pollster j_i, house segment h_i, term c_i and sample size n_i |
\pi_t = \operatorname{softmax}([0, \theta_t]), so shares are always positive and sum to one.
Latent dynamics
The state is a random walk with correlated innovations:
\theta_t = \theta_{t-1} + \eta_t, \qquad \eta_t \sim \mathcal{N}(0,\ v_t\, Q), \qquad Q = \operatorname{diag}(\sigma)\, \Omega\, \operatorname{diag}(\sigma).
- \sigma_k \sim \text{HalfNormal}(0.06) is the weekly volatility of each log-ratio.
- \Omega is a correlation matrix with an LKJ(2) prior, so parties can move together.
- v_t = \kappa^2 in the 8 weeks up to and including each election week, and 1 otherwise, with \kappa \sim \text{LogNormal}(0, 0.3). Support can move faster during campaigns; past terms estimate how much.
- In the
heavyvariant, v_t is also multiplied by \frac{\nu - 2}{\nu}\, w_t^{-1} with w_t \sim \text{Gamma}(\nu/2, \nu/2) and \nu = 4. This makes the weekly shocks multivariate Student-t with the same average variance: most weeks are quiet, and a news week can move every party at once.
Pinning the state to election results
At the anchor election and every completed election e in the window, \theta_{t_e} = \operatorname{alr}(\text{result}_e) exactly. Parties absent from a result, such as TOP in 2011, are floored at 0.2% before taking logs.
Sampling a random walk that must pass through fixed points is stiff: the 2023 Stan model needed a maximum tree depth of 20. Here the path is sampled directly from its conditional distribution. With non-centred innovations W_t = \sum_{s \le t} \sqrt{v_s}\, L z_s, where L L^\top = Q and z_s \sim \mathcal{N}(0, I), and cumulative variance V_t = \sum_{s \le t} v_s, the path between consecutive anchors a < b is the Brownian bridge
\theta_t = \theta_a + \frac{V_t - V_a}{V_b - V_a}(\theta_b - \theta_a) + (W_t - W_a) - \frac{V_t - V_a}{V_b - V_a}(W_b - W_a), \qquad a \le t \le b.
After the last anchor the walk runs freely: \theta_t = \theta_{\text{last}} + W_t - W_{\text{last}}.
Conditioning on the anchors also requires their probability, which the bridge alone omits. The likelihood therefore includes the transition density of every interval,
\sum_{\text{intervals } (a, b)} \log \mathcal{N}\big(\theta_b - \theta_a;\ 0,\ (V_b - V_a)\, Q\big),
so large swings between elections, such as 2017 to 2020 to 2023, inform \sigma, \Omega and \kappa.
Observation model
Offsets
Each poll measures the state with an offset \delta_i \in \mathbb{R}^{K} on the logit scale, with one value per party so that no party is privileged:
\delta_i = \beta_{h_i} + \gamma_{j_i, c_i} + (1 - f_i)\, s_{c_i} + f_i\, e_{c_i}.
| Term | Meaning | Prior |
|---|---|---|
| \beta_h | persistent house effect of a pollster, or of a pollster’s method segment (Reid Research after its 2017 change) | \mathcal{N}(0, 0.12^2) per party |
| \gamma_{j,c} | how pollster j deviates from its persistent effect in term c | \mathcal{N}(0, \tau_\gamma^2), \tau_\gamma \sim \text{HalfNormal}(0.05) |
| s_c, e_c | industry-wide error shared by all pollsters at the start and at the end of term c | t_4(0, (m_k \tau_s)^2), t_4(0, (m_k \tau_e)^2), \tau_s, \tau_e \sim \text{HalfNormal}(0.15) |
f_i \in [0, 1] is how far through its term the poll was taken, from 0 at the previous election to 1 at the next. The first polls of a term identify s_c against the previous result, and each result identifies e_c. For the term being forecast, e_c has no result yet, so its posterior is its prior: a fresh election-day polling error with spread \tau_e learned from past elections.
Two things about that error are deliberate, because four completed elections cannot settle either.
The error is Student-t, not normal (priors.industry_df), rescaled so \tau still means its standard deviation. Four elections can estimate how big a typical polling miss is; they cannot bound how big a rare one could be, and polling industries do fail occasionally, as in Britain in 2015. Under this prior an error of three standard deviations is about four times as likely as under a normal.
Small and untested parties get a wider error: m_k multiplies the scale by priors.minor_party_factor for a party polling under priors.minor_party_share at the cutoff, or one that has never crossed the threshold at a completed election, and is 1 otherwise. The reason is measured, not assumed. Over 2011–2023, parties polling under 8% in the final fortnight missed their result by about 21% of their own support, against about 8% for larger parties; a single error scale on the log-ratio scale gives every party roughly the same relative error, so it understates the small ones by a factor of about two and a half. For 2026 the wider error applies to Te Pāti Māori and TOP. How much of a party’s stated support turns into votes has never been tested for a party that has not been elected, which is the same argument in a different guise.
The offsets enter on the log-ratio scale as \Delta_{i,k} = \delta_{i,k} - \delta_{i,0}. The expected poll shares are
p_i = \operatorname{softmax}([0,\ \theta_{t_i} + \Delta_i]).
A poll that did not report a tracked party is treated as having folded it into “Other”: that party is unobserved, and p_{i,\text{Other}} absorbs its expected share. It is never treated as 0%.
Likelihood
Each pollster has a design effect d_j = 1 + \epsilon_j with \epsilon_j \sim \text{LogNormal}(0, 0.4), a prior median of 2. This is the ratio of the pollster’s true variance to simple random sampling, learned from the data instead of the 2023 model’s fixed \sqrt{2} inflation of the standard error.
- Dirichlet-multinomial (
base): the reported counts y_i = n_i \times \text{share}_i follow \text{DirMult}(n_i, \phi_i p_i) over the reported categories, with \phi_i = (n_i - d_j)/(d_j - 1) so that the variance is d_j times the multinomial variance. - Gaussian (
gauss,heavy,fund): each reported share, including Other, follows \mathcal{N}\big(p_{i,k},\ d_j\, p_{i,k}(1 - p_{i,k})/n_i + u_i^2/12\big). Here u_i is the rounding unit of the published figures: 1 point when every figure is a whole number, 0.5 when they are halves, otherwise 0.1. Rounding matters for small parties, where “1%” can mean anything from 0.5% to 1.5%.
The real sample size is used when Wikipedia reports one; otherwise the pollster’s default from config/pollsters.yml.
Fundamentals prior (fund only)
Ellis (2020) added a prior on the swing of the Prime Minister’s party:
\pi_{T, \text{PM}} - \pi_{\text{prev}, \text{PM}} \sim \text{Student-}t_4(m, s),
where m is the mean swing at elections since 1996 before the target, and s = \sqrt{\text{sd}^2 + \text{sd}^2 / n}. For 2026 this gives a mean swing m of −2.0 points and a scale s of 9.6 points, from 10 elections. The prior is estimated without look-ahead in backtests. It is not in the published ensemble because it made the backtests worse (see How it is tested).
Forecasts
- Election day is the posterior of \pi at the election week. For the term being forecast this includes the random-walk movement still to come, the campaign volatility, and the fresh election-day industry error.
- If held now is the posterior of \pi at the last week with polls.
The published forecast mixes variants by their stacking weights; see How it is tested.
Seats
For each of 20,000 simulated elections:
- Electorates. Each electorate in
config/electorates.ymlis won by the named party, an independent, or anyone else. The three outcomes are mutually exclusive and match the configured probabilities when the party’s vote equalsat_share. The party’s chance moves on the logit scale byslopeper percentage point of simulated party vote, and the independent keeps the same share of the remaining probability. These probabilities are editorial judgements, and such judgements tend to be wrong for all of a party’s electorates at once: Te Pāti Māori held no Māori electorate in 2017 and six in 2023. Each simulation therefore shifts the logit of every electorate of a party by one shared draw with standard deviationelectorate_group_sd, so the seats move together. The intercept is calibrated to preserve the configured probabilities atat_shareafter averaging over that shared shift. With the current assumptions that leaves Te Pāti Māori a 1% chance of holding none of its configured electorates, against <1% if each seat were an independent coin. - Eligibility. A party qualifies for list seats with at least 5% of the party vote or one electorate. “Other” never qualifies.
- Allocation. 120 seats, less one for each electorate won by an independent, are divided among qualifying parties by the Sainte-Laguë method (divisors 1, 3, 5, …).
- Overhang. A party keeps every electorate it wins. If that exceeds its proportional entitlement, the House grows.
- Majorities. A coalition has a majority when its seats reach half the simulated House size, rounded down, plus one.
- Balance of power. For each bloc in
balance_of_power(National + ACT, and Labour + Green + Te Pāti Māori), each simulation is classed as a majority alone; short, but a majority with one pivot party on its own (NZ First or TOP); a majority only with two or more pivots together; or short even with every pivot. With two pivots, the third class means needing both.
The allocation reproduces the official 2011, 2014, 2017, 2020 and 2023 outcomes; see tests/test_seats.py.
Inference
All variants are written in NumPyro and sampled with NUTS on CPU through JAX: 4 chains of 1,000 warmup and 1,000 draws for the live forecast, and 2 chains of 500 and 500 for backtests. Each fit records the number of divergences, the largest split R-hat and the smallest effective sample size over the hyperparameters, from NumPyro’s diagnostics. These appear in summary.json and at the foot of the forecast page.
Kalman variant (kal, experimental)
With a Gaussian likelihood on log-ratios, the model given its hyperparameters is linear and Gaussian, so the weekly state can be integrated out exactly. Each poll is observed as the log-ratios of its published shares against National, with covariance from the delta method: R_i = \tfrac{d_j}{n_i}\big(\operatorname{diag}(1/s_{i,k}) + \tfrac{1}{s_{i,0}} \mathbf{1}\mathbf{1}^\top\big) over the reported parties. An information-form Kalman filter combines all polls in a week in one update, NUTS samples only the hyperparameters, and a Rauch-Tung-Striebel smoother recovers the states afterwards. tests/test_kalman.py checks the likelihood and smoothed moments against brute-force Gaussian integration.
This is a different likelihood from gauss, not a faster route to the same posterior. It is not used in the ensemble because each gradient of the sequential filter is slower on CPU than the full-state model. It is the exact counterpart of the ensemble Kalman filters used for very high-dimensional states: with a seven-dimensional state, the exact filter costs little and has no Monte Carlo error.
2023 model replica (legacy)
For backtests, src/pollofpolls/model/legacy.py re-implements the 2023 NZ Herald model (legacy-r/stan/model2023.stan):
- a random walk on raw proportions with \sigma \sim \mathcal{N}(0.002, 0.001) truncated at zero and LKJ(1);
- constant house effects \mathcal{N}(0, 0.03) per pollster and party, with the Reid Research method change;
- n = 1000 for every poll, a fixed standard error per pollster and party from its average share, multiplied by \sqrt{2};
- unreported parties recorded as 0%;
- election results as exact observations, including their transition density.
Two deliberate differences keep the comparison about the model: the standard errors are matched to the right party columns, which the R code misaligned; and the replica sees the same pollsters and anchor election as the new model.
Changes from the 2023 model
| 2023 model | This model |
|---|---|
| Random walk on raw proportions, which can go negative | Log-ratio random walk; shares stay on the simplex |
| n = 1000 for every poll, standard error \times \sqrt{2} | Real sample sizes, learned design effect per pollster, rounding error |
| Constant house effects plus one method-change dummy | Persistent plus per-term house effects, method segments |
| No industry-wide polling error | Start- and end-of-term industry error; fresh election-day error for 2026, Student-t and wider for small or untested parties |
| Electorate outcomes fixed by hand | Editorial electorate probabilities, with a shared error across a party’s seats |
| Constant volatility | Campaign multiplier; optional heavy-tailed weekly shocks |
| Unreported parties recorded as 0% | Unreported parties treated as unobserved |
| Stan, maximum tree depth 20 | NumPyro, bridge sampling, maximum tree depth 10 |
| Four hard-coded pollster slots | Any number of pollsters from config/pollsters.yml |
| No evaluation | Rolling-origin backtests, proper scoring rules, stacking |