Behavioral Science Dictionary

Fixed effects

Methods & Evidence

A modeling trick that controls away everything stable about each unit to isolate real change.

What it means

In panel or grouped data, fixed effects gives each unit - a person, firm or period - its own intercept and estimates relationships from within-unit variation alone. This nets out time-invariant confounders that shift a unit's level by a fixed amount, whether measured or not, but it leaves time-varying confounding untouched, and a stable characteristic whose influence grows or fades over time survives too. Because it relies solely on within-unit variation, it cannot estimate effects of variables that never change within units, and sets aside between-unit information. In econometrics it is usually contrasted with random effects, but the phrase also carries a second, incompatible meaning: in mixed-model and multilevel usage it denotes the coefficients held constant across groups, estimated with complete pooling, which coexist with random effects rather than compete with them. It matters because much of what separates units is unmeasured and probably unmeasurable, and this design does not require measuring it, though a causal reading rests on conditions the intercepts do not supply.

How it works

The arithmetic is mundane. Units - people, stores, schools, countries - are observed repeatedly, each gets its own intercept, and the slope of interest is estimated from movement around those intercepts. Two implementations are algebraically identical: adding one indicator variable per unit, or subtracting each unit's own average from every variable and regressing on the deviations. The second, the within transformation, is what software uses at scale. The estimate therefore answers a within-unit question: is a unit's outcome higher in the periods when its predictor sits above that unit's own usual level? Anything that does not move across the window - a person's upbringing, a store's catchment area - is subtracted away with the mean, including characteristics nobody thought to measure. Most panel work adds a second set of intercepts, one per period, absorbing whatever is common to all units at a given moment: a recession, a holiday, a platform-wide change. Standard errors are a separate matter, usually clustered by unit, because repeated observations from one unit stay correlated in ways the intercepts do not fix.

What the evidence shows

The core result here is algebraic, not empirical: in a linear additive model the intercepts absorb unit-constant terms exactly. Yair Mundlak's 1978 paper made the sharper point that the fixed-versus-random dichotomy is a matter of specification, since the within estimate can be recovered from a random-effects model that adds each unit's average of the time-varying predictors. Jerry Hausman's test, published the same year, formalized the comparison. The empirical record concerns how the number misleads. Stephen Nickell showed in 1981 that when a lagged outcome sits among the predictors the within estimator is biased, the bias of order one over the number of periods - serious in short panels, negligible in long ones. Marianne Bertrand, Esther Duflo and Sendhil Mullainathan showed in 2004 that conventional standard errors are badly understated when outcomes are serially correlated: inserting randomly generated placebo policies into state-level wage data, they found spurious effects significant at the five percent level in up to forty-five percent of runs. The most consequential revision is recent. When treatment starts at different times for different units, the single two-way coefficient is a weighted average of many two-group, two-period comparisons, some using already-treated units as controls. Clement de Chaisemartin and Xavier D'Haultfoeuille showed the implied weights can be negative, so the pooled estimate can carry the opposite sign to every underlying effect, and Andrew Goodman-Bacon's 2021 decomposition made the source visible. The 2023 synthesis by Jonathan Roth and colleagues treats this as established, while noting that how far corrections move a given answer depends on the design. A counterweight has since appeared: Dmitry Arkhangelsky and colleagues showed in 2024 that weighting units by a model of how treatment was assigned gives an estimator consistent if either that model or the regression is correct, placing the trouble in the pairing of design and estimator rather than in the intercepts as such.

Where it shows up

Behavioral research reaches for unit intercepts whenever the same participants, teams or sites are observed more than once. In experience sampling and diary studies, where one person supplies dozens of observations, person intercepts convert a question about which sort of person is happier into a question about when a given person is happier. The two answers can differ in magnitude and even in sign, a distinction Patrick Curran and Daniel Bauer laid out for psychologists in a 2011 review. Field work uses the device structurally rather than psychologically. A pricing or messaging change rolled out to some locations before others is usually evaluated with site and period intercepts, so each site serves partly as its own comparison. The attraction is that much of what separates units is unmeasured and probably unmeasurable, and this design does not require measuring it. The cost is that stable differences between units become unanswerable inside the model.

Where it breaks down

What the intercepts remove is routinely overstated. They net out unit-constant influences that shift a unit's level by a fixed amount, and nothing more. Confounders that move over time survive, and so does a stable characteristic whose influence grows or fades across the window: a trait that mattered little early and a great deal later leaves a residue that subtracting unit means cannot reach. Kosuke Imai and In Song Kim showed in 2019 that a causal reading also needs conditions rarely stated aloud: past treatments must not affect current outcomes, and past outcomes must not affect current treatment. Where behavior responds to a policy that then responds to behavior, both fail. Measurement error bites harder here than in cross-sections. Subtracting a unit's mean strips out much of the genuine variation in a predictor while leaving the noise, so a poorly measured variable is attenuated more severely after the transformation than before. Predictors that barely move within units leave little identifying variation, so the estimate turns imprecise even where it is unbiased. Nonlinear models bring separate trouble. One intercept per unit in a logit or probit adds a parameter per unit, and the estimates are inconsistent in short panels - the incidental parameters problem. Two escapes are standard, not one. Conditional logit removes the unit parameters by conditioning on how often each unit's outcome was positive, discarding units whose outcome never changed. Fixed-effects Poisson, introduced by Jerry Hausman, Bryan Hall and Zvi Griliches in 1984 and shown by Jeffrey Wooldridge in 1999 to stay consistent even when the Poisson distribution is wrong, is the workhorse for counts and for outcomes that would otherwise be logged. Beyond those, no general fix exists.

Related but distinct

One confusion must be cleared before any comparison. The phrase carries at least two incompatible meanings. In the econometric sense used throughout this entry, fixed effects are unit-specific intercepts, estimated with no pooling across units. In the mixed-model sense familiar from the block headed "Fixed effects" in multilevel software output, they are the coefficients held constant across groups, estimated with complete pooling, and they sit alongside random effects in one model rather than as an alternative to them. Andrew Gelman and Jennifer Hill catalogued five definitions in circulation and judged the terminology beyond repair. A report that a study "used fixed effects" is uninformative until the sense is settled. Random effects, the usual foil in the econometric sense, also gives units their own intercepts but treats them as draws from a distribution assumed uncorrelated with the predictors. Where that holds it uses both within- and between-unit variation and is more precise; where it fails, it is biased. The Hausman test was built to adjudicate, though simulations by Tom Clark and Drew Linzer show a test result is a poor substitute for deciding which quantity is wanted: when a predictor moves little within units, the within estimate can be imprecise enough that a slightly biased alternative carries less error. The Mundlak device - a random-effects model augmented with unit means of the time-varying predictors, or correlated random effects - sits between the two, reproducing the within estimate while still permitting coefficients on time-invariant variables; multilevel work calls it the within-between formulation. With two periods, first differencing removes unit-constant terms just as subtracting means does; with more, the two diverge. Clustering standard errors is different again, adjusting uncertainty for correlated observations rather than changing the estimate.

Examples

Studying whether minimum-wage changes affect employment, a fixed-effects model compares each state with itself over time, netting out fixed differences like local culture or industry mix.

A subscription business wants to know whether raising the price of a plan reduced usage. Comparing heavy and light users side by side confounds the price tier with the kind of customer who chose it, whereas account-level intercepts restrict the comparison to how each account's own usage moved before and after its own price change. What the intercepts cannot do is separate the price change from anything else that moved with time inside the same account - usage typically decays with tenure whatever the price - so period intercepts, and staggered rather than simultaneous price changes, are doing as much work here as the account intercepts.

In a two-week diary study, person-level intercepts ask whether an individual reports sharper focus on the days they slept longer than their personal average. That is a different question from whether well-rested people are more focused overall, and the two estimates can point in opposite directions.

A retail chain introduces new shelf labels across its stores in waves and evaluates the change with store and week intercepts. Because later-adopting stores are compared partly against stores that were already treated, a benefit that fades after a few weeks can shrink the headline coefficient or reverse its sign.

An analyst asks whether a stable personality trait predicts absence from work. A specification with employee intercepts cannot answer it, because the trait never varies within an employee and is removed along with everything else that stays put.

Key references

  1. Mundlak, Y. (1978). On the pooling of time series and cross section data. Econometrica, 46(1), 69-85. doi.org/10.2307/1913646
  2. Hausman, J. A. (1978). Specification tests in econometrics. Econometrica, 46(6), 1251-1271. doi.org/10.2307/1913827
  3. Nickell, S. (1981). Biases in dynamic models with fixed effects. Econometrica, 49(6), 1417-1426. doi.org/10.2307/1911408
  4. Hausman, J., Hall, B. H., & Griliches, Z. (1984). Econometric models for count data with an application to the patents-R&D relationship. Econometrica, 52(4), 909-938. doi.org/10.2307/1911191
  5. Wooldridge, J. M. (1999). Distribution-free estimation of some nonlinear panel data models. Journal of Econometrics, 90(1), 77-97. doi.org/10.1016/S0304-4076(98)00033-5
  6. Bertrand, M., Duflo, E., & Mullainathan, S. (2004). How much should we trust differences-in-differences estimates? The Quarterly Journal of Economics, 119(1), 249-275. doi.org/10.1162/003355304772839588
  7. Gelman, A., & Hill, J. (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press. doi.org/10.1017/CBO9780511790942
  8. Curran, P. J., & Bauer, D. J. (2011). The disaggregation of within-person and between-person effects in longitudinal models of change. Annual Review of Psychology, 62, 583-619. doi.org/10.1146/annurev.psych.093008.100356
  9. Clark, T. S., & Linzer, D. A. (2015). Should I use fixed or random effects? Political Science Research and Methods, 3(2), 399-408. doi.org/10.1017/psrm.2014.32
  10. Imai, K., & Kim, I. S. (2019). When should we use unit fixed effects regression models for causal inference with longitudinal data? American Journal of Political Science, 63(2), 467-490. doi.org/10.1111/ajps.12417
  11. de Chaisemartin, C., & D'Haultfoeuille, X. (2020). Two-way fixed effects estimators with heterogeneous treatment effects. American Economic Review, 110(9), 2964-2996. doi.org/10.1257/aer.20181169
  12. Goodman-Bacon, A. (2021). Difference-in-differences with variation in treatment timing. Journal of Econometrics, 225(2), 254-277. doi.org/10.1016/j.jeconom.2021.03.014
  13. Roth, J., Sant'Anna, P. H. C., Bilinski, A., & Poe, J. (2023). What's trending in difference-in-differences? A synthesis of the recent econometrics literature. Journal of Econometrics, 235(2), 2218-2244. doi.org/10.1016/j.jeconom.2023.03.008
  14. Arkhangelsky, D., Imbens, G. W., Lei, L., & Luo, X. (2024). Design-robust two-way-fixed-effects regression for panel data. Quantitative Economics, 15(4), 999-1034. doi.org/10.3982/QE1962

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