📈 economics

Drivers of Success: A Bayesian State-Space Model to Disentangling Latent Driver and Constructor Abilities in Formula One

This paper introduces a Bayesian state-space model that successfully disentangles the dynamic, latent abilities of Formula One drivers and constructors by analyzing qualifying lap times and race rankings from 2014 to 2021, revealing that while driver performance remains relatively stable, constructor capabilities exhibit greater variability and often play a more dominant role in race outcomes.

Original authors: Tim Lindner, Rui Jorge Almeida, Nalan Baştürk, Stephan Smeekes

Published 2026-08-06

📖 3 min read☕ Coffee break read

CC BY 4.0

Original authors: Tim Lindner, Rui Jorge Almeida, Nalan Baştürk, Stephan Smeekes

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a high-stakes cooking competition. You see the final dish on the plate, but you can't see the ingredients or the chef's technique. Was the meal amazing because the chef is a genius, or because the kitchen was stocked with the world's best ingredients? In the world of sports statistics, this is a classic puzzle: how do you separate the skill of the individual athlete from the quality of the team or equipment they represent? This question sits at the intersection of statistics and sports science, a field that uses math to untangle hidden patterns in performance. To solve this, researchers often use "latent variable models," which are fancy math tools that guess at invisible qualities (like "talent" or "machine quality") based on what we can actually see (like race times or scores). They also use "state-space models," which are like time machines for data, allowing these invisible qualities to change and evolve as the season goes on. Understanding who is really driving the success in Formula One matters because it settles the age-old debate: is the driver the hero, or is the car the real star?

In this paper, Tim Lindner and his colleagues tackle this exact mystery using a new mathematical recipe called a Bayesian state-space model. They looked at Formula One racing data from the "hybrid era" (2014 to 2021), a time when the cars used complex hybrid engines. Instead of just looking at race results, they fed two different types of data into their model: the fastest lap times during the qualifying session (which is like a solo sprint) and the final race rankings (which is a group race). The model treats the driver and the car manufacturer (called a "constructor") as two separate, invisible forces that are constantly shifting and changing over time.

The researchers found that while both the driver and the car matter, they behave very differently. The model suggests that a driver's ability is like a steady river; it flows relatively smoothly and stays stable over time. A driver might get slightly better or worse, but their core talent doesn't swing wildly from one race to the next. The constructor's ability, however, is more like a stormy sea. The performance of the car teams fluctuates much more dramatically, with big jumps and drops in capability as they develop new parts or fix problems.

Perhaps the most interesting discovery is that for many driver-and-car combinations, the car actually contributes more to the final result than the driver does. The model shows that the "constructor ability" often has a bigger impact on the shared performance score than the "driver ability." However, the authors are careful to note that their model isn't perfect. When they tested it by simulating race results, it did a great job at predicting the general flow of qualifying times. But when it came to race rankings, the model struggled to fully capture the sheer dominance of the very best teams. It suggested that while the math works well for the middle of the pack, the "super-storms" of the top teams are a bit harder to predict with this specific tool. Ultimately, the study provides a way to mathematically split the credit, showing us that in Formula One, the engine under the hood is often just as important as the hands on the wheel.

Technical Summary: Drivers of Success in Formula One

Problem Statement
Formula One performance is a joint product of driver skill and constructor (team) capability, yet these contributions are unobserved and evolve dynamically over time. Separating these effects is methodologically challenging due to three factors: the time-varying nature of abilities, the frequent changes in driver–constructor assignments, and the heterogeneity of observed outcomes. Specifically, the data consists of two distinct outcome types: continuous fastest qualifying lap times and ordinal race finishing positions (partial rankings). Existing literature often addresses these issues in isolation—using static models, single-outcome analyses, or dynamic ranking models that do not jointly incorporate continuous performance metrics. Furthermore, standard two-way worker-firm models in econometrics typically analyze a single continuous outcome and lack frameworks where both individual and group effects follow separate stochastic evolution equations within a joint state-space structure.

Methodology
The authors propose a Bayesian state-space model that jointly analyzes standardized qualifying lap times and race rankings to disentangle latent driver and constructor abilities.

Observation Equations: The model treats the two outcomes using outcome-specific sampling distributions linked by shared latent states.
- Qualifying: Modeled as a continuous outcome using a Normal distribution. The observed time is standardized within each Grand Prix to account for circuit and weather variations.
- Race: Modeled as a partial ranking outcome. The race finishing position is induced by a latent race performance variable following a Gumbel distribution, which analytically yields a Plackett-Luce likelihood. The race equation includes the starting grid position as a covariate, acknowledging that grid position is determined by qualifying performance.
Latent State Dynamics: Both driver abilities ( $a_{d,t}$ ) and constructor abilities ( $c_{k,t}$ ) are modeled as separate random-walk processes evolving at the Grand Prix level. These states are decomposed additively into the performance predictors for both outcomes. The observed contractual assignment ( $z_{d,t}$ ) determines which constructor state enters a specific driver's predictor.
Identification Strategy: Identification of the dynamic decomposition relies on three pillars:
1. Joint Likelihood: The continuous qualifying equation and the ranking likelihood provide complementary information, restricting common shifts in latent components that a ranking model alone cannot identify.
2. Sum-to-Zero Constraints: To fix the location and scale of the latent states, the model imposes constraints that center driver and constructor abilities at zero for every Grand Prix. This is implemented via specific prior specifications that induce negative correlations in the innovation terms of the state equations.
3. Assignment Variation: The variation in driver–constructor assignments over time (e.g., drivers switching teams) allows the model to distinguish persistent driver ability from constructor ability, analogous to worker mobility in two-way fixed effects models.
Inference: Bayesian inference is performed using the No-U-Turn Sampler (NUTS), a Hamiltonian Monte Carlo algorithm, under weakly informative priors that treat driver and constructor abilities symmetrically. This approach is chosen over standard Kalman filters due to the non-Gaussian nature of the ranking likelihood.

Key Contributions

Joint Modeling of Heterogeneous Outcomes: The paper integrates a continuous outcome and a partial ranking outcome within a single dynamic framework, extending mixed-outcome latent-variable models to the state-space domain.
Dynamic Two-Way Decomposition: Unlike static worker-firm models, this framework allows both the individual (driver) and group (constructor) effects to evolve stochastically over time while being identified through observed assignment structures.
Methodological Synthesis: The model bridges the literature on mixed-outcome models, dynamic ranking models (e.g., dynamic Plackett-Luce), and econometric two-way effects, providing a unified approach for decomposing shared performance predictors.

Empirical Results
Applied to the Formula One hybrid era (2014–2021), the model yields the following findings:

Heterogeneity: There is substantial heterogeneity in both driver and constructor abilities across the sample.
Stability vs. Volatility: Driver abilities are generally more stable over time, exhibiting gradual evolution. In contrast, constructor abilities display greater variation, with more pronounced and sometimes abrupt changes within and across seasons.
Relative Contributions: For several driver–constructor combinations, the constructor component contributes more strongly to the shared performance predictor than the driver component. However, the relative contribution varies significantly depending on the specific pairing and time period.
Predictive Performance: The model reproduces continuous qualifying outcomes reasonably well. However, the race-ranking component shows greater difficulty in reproducing the sustained dominance of top-tier pairings (e.g., Verstappen/Red Bull in 2021), tending to underestimate the frequency of top-two finishes compared to observed data.
Uncertainty: Posterior uncertainty is generally higher for constructor abilities than for driver abilities, reflecting the greater volatility in constructor performance and the specific structure of the assignment data.

Significance and Claims
The paper claims to provide a rigorous statistical framework for decomposing joint performance in team-based sports where individual and group contributions are unobserved and dynamic. By leveraging the specific structure of Formula One data—where drivers switch teams and outcomes are measured on different scales—the model successfully identifies separate latent trajectories for drivers and constructors. The authors note that while the model is tailored to Formula One, the underlying structure is applicable to other settings involving heterogeneous longitudinal outcomes that depend on separately evolving individual and group effects, such as organizational outcomes or other team-based competitions. The study modestly concludes that while the decomposition is successful, the race-ranking model retains limitations in capturing the extreme tails of performance dominance.

Technical Summary: Drivers of Success in Formula One

More like this