Fixed Effects Models

Using Margins.jl with FixedEffectModels.jl

Overview

Margins.jl supports FixedEffectModels.jl via a package extension. When you load both packages, marginal effects and predictions become available for models with high-dimensional absorbed fixed effects.

using Margins, FixedEffectModels, DataFrames

model = reg(df, @formula(y ~ x1 + x2 + fe(state) + fe(year)))

# Average marginal effects
population_margins(model, df; type=:effects)

# Predictions (requires save=:fe at fit time)
model = reg(df, @formula(y ~ x1 + x2 + fe(state) + fe(year)); save=:fe)
population_margins(model, df; type=:predictions)

The extension loads automatically when both Margins and FixedEffectModels are imported. No additional setup is required.

Marginal Effects

Why Effects Work Without FE Estimates

For linear models with absorbed fixed effects, the model is:

\[y_i = X_i \beta + \alpha_{g(i)} + \varepsilon_i\]

where $\alpha_{g(i)}$ are the absorbed fixed effects. The marginal effect of a continuous variable $x_k$ is:

\[\frac{\partial y}{\partial x_k} = \beta_k + \sum_j \beta_{kj} x_j\]

This depends only on the non-FE coefficients $\beta$, not on the absorbed $\alpha$. Similarly, delta-method standard errors use only $\text{Var}(\hat{\beta})$, which is what vcov(model) returns. This means marginal effects are valid without saving the fixed-effect estimates.

Average Marginal Effects (AME)

model = reg(df, @formula(y ~ x1 + x2 + fe(state)))

# All variables
result = population_margins(model, df; type=:effects)
DataFrame(result)

# Specific variables
result = population_margins(model, df; type=:effects, vars=[:x1])

Marginal Effects at Means (MEM)

grid = means_grid(df)
result = profile_margins(model, df, grid; type=:effects, vars=[:x1, :x2])
DataFrame(result)

Interaction Models

Marginal effects for interaction models are computed correctly. When the model includes $x_1 \times x_2$, the marginal effect of $x_1$ varies with $x_2$:

model = reg(df, @formula(y ~ x1 * x2 + fe(state)))

# AME: averaged over the sample distribution of x2
result = population_margins(model, df; type=:effects, vars=[:x1])

# Counterfactual: effect of x1 when x2 is set to specific values
result = population_margins(model, df; type=:effects, vars=[:x1],
                            scenarios=(x2=[0.0, 1.0],))

Backend Selection

Both computation backends are supported:

# Automatic differentiation (default)
population_margins(model, df; type=:effects, backend=:ad)

# Finite differences (zero allocation)
population_margins(model, df; type=:effects, backend=:fd)

Predictions

Predictions require the absorbed FE estimates, since the full predicted value is $\hat{y}_i = X_i \hat{\beta} + \hat{\alpha}_{g(i)}$. Pass save=:fe when fitting the model:

model = reg(df, @formula(y ~ x1 + x2 + fe(state) + fe(year)); save=:fe)

Without save=:fe, prediction requests produce an informative error.

Population Predictions (AAP)

Average adjusted predictions incorporate FE estimates:

result = population_margins(model, df; type=:predictions)
DataFrame(result)

The predicted value for each observation is $X_i \hat{\beta} + \sum_k \hat{\alpha}_{k,g_k(i)}$, averaged across the sample. Standard errors reflect uncertainty in $\hat{\beta}$ only, not in the FE estimates — this is standard practice.

Profile Predictions (APM)

Profile predictions evaluate the model at specific covariate combinations. Fixed-effect variables can be included in the reference grid to select specific FE levels:

# Predictions at specific covariate values (FE averaged across sample)
grid = cartesian_grid(x1=[0.0, 1.0, 2.0], x2=[0.0])
result = profile_margins(model, df, grid; type=:predictions)

# Predictions at specific FE levels
grid = DataFrame(x1=[0.0, 0.0], x2=[0.0, 0.0], state=["CA", "NY"])
result = profile_margins(model, df, grid; type=:predictions)

When FE variables are in the reference grid, the corresponding FE estimate for that level is used. When FE variables are not in the reference grid, their population-average FE value is used.

Multiple Fixed Effects

Models with multiple absorbed FE sets work as expected:

model = reg(df, @formula(y ~ x1 + x2 + fe(state) + fe(year)); save=:fe)

# Average all FEs
grid = cartesian_grid(x1=[0.0, 1.0])
result = profile_margins(model, df, grid; type=:predictions)

# Specify state, average year
grid = DataFrame(x1=[0.0], x2=[0.0], state=["CA"])
result = profile_margins(model, df, grid; type=:predictions)

# Specify both
grid = DataFrame(x1=[0.0], x2=[0.0], state=["CA"], year=[2020])
result = profile_margins(model, df, grid; type=:predictions)

Instrumental Variables

IV models estimated with the (endogenous ~ instrument) syntax are supported. Margins are computed using the structural (second-stage) coefficients and covariance matrix, consistent with Stata's margins after ivregress and R's marginaleffects package.

model = reg(df, @formula(y ~ x2 + (x1 ~ z) + fe(state)))
result = population_margins(model, df; type=:effects)

An informational note is logged once per session when an IV model is detected. Effects for all variables (both endogenous and exogenous) are computed — the package does not distinguish between them, following the convention of existing implementations.

Restrictions

Absorbed FE Variables Cannot Be in vars

The coefficients of absorbed fixed effects are not part of coef(model), so marginal effects for these variables are not computable:

# This will error:
population_margins(model, df; type=:effects, vars=[:state])

# This works:
population_margins(model, df; type=:effects, vars=[:x1, :x2])

Counterfactual Scenarios on FE Variables

For predictions, counterfactual scenarios on absorbed FE variables are blocked. Setting everyone to a specific FE group conflates causal effects with unobserved heterogeneity:

# This will error for type=:predictions:
population_margins(model, df; type=:predictions, scenarios=(state=["CA"],))

Use the profile approach instead to examine specific FE levels.

Standard Error Coverage

Standard errors for predictions reflect uncertainty in $\hat{\beta}$ only. They do not incorporate estimation uncertainty in the absorbed FE estimates $\hat{\alpha}$, which are treated as fixed. This is the standard approach in the fixed-effects literature and matches the behavior of Stata and R.

Stata Translation