reghdfe

reghdfe is a Stata package designed to estimate linear regressions with multiple levels of fixed effects. It serves as a high-performance generalization of Stata's built-in areg, xtreg,fe, and xtivreg,fe commands.

Key capabilities include:

• Efficient Fixed Effect Absorption: Uses a robust algorithm to handle multiple fixed effects, performing well even in 'hard cases' where other algorithms fail to converge.
• High Speed: Significantly faster than alternatives like reg2hdfe, a2reg, felsdvreg, and res2fe, especially when dealing with multiple fixed effects.
• Advanced Clustering: Supports two-way and multi-way clustering of standard errors.
• Robust Estimators: Integrates with the avar package for an extensive list of robust variance estimators.
• IV and GMM Support: Works with instrumental-variable and GMM estimators (e.g., two-step-GMM, LIML) via the ivreg2 routine.
• Flexible Modeling: Supports multiple heterogeneous slopes, frequency/probability/analytic weights, and time-series/factor variables.
• Convenience Features: Allows expressing fixed effects and clusters as factor interactions (e.g., state#year) and supports postestimation commands like predict and test.
• Optimization: Includes a cache() option to precompute results for faster subsequent regressions.

The constant term in reghdfe can be problematic and often provides little useful information. Keep the following in mind:

• Interpretation: The constant is the expected value when all independent variables are zero, not the average of the dependent variable.
• Singleton Groups: Using the dropsingleton option will change the estimated value of the constant.
• Conflicts: The constant can conflict with the over() option and may mess up Variance-Covariance Matrix (VCV) calculations, specifically when using ivreg2 or avar with clustering.
• Dropping the constant: To disable the automatic addition of the constant term after partialling-out fixed effects, use the noconstant option. Note that by construction, all reghdfe regressions include a constant through the absorb() terms.
• ivreg2 behavior: When using ivreg2, the constant is always omitted to prevent errors regarding orthogonality conditions (where S is not of full rank) when using clustered standard errors.

The Kaczmarz algorithm is a sketch for iteratively solving for fixed effects (alphas) in a linear system $Ax=b$.

Given a system where $A$ is $N imes M$, the process starts with an initial guess $p=0$ (an $M imes 1$ vector). The algorithm iterates through every observation $i$ until convergence. For each observation, it calculates a residual $s$ by subtracting the current estimated effects (intercepts and slopes) from the target value $b[i]$, then updates the effects using a denominator term to ensure convergence.

Key components:

• Residual calculation ($s$): For each observation $i$, $s$ is initialized to $b[i]$ and adjusted by subtracting the current alphas for each fixed effect group $g$ where the observation belongs to a specific level.
• Denominator (denom[i]): This term scales the update. If only intercepts are used, denom[i] = G (the number of groups). If slopes are included, denom[i] = num_intercepts + cvar[i]'cvar[i] (where cvar represents the covariance of the slopes).

s = b[i]
for g=1:G
	level = F[g].levels[i]
	if has_intercept[g]: s -= F[g].alphas[level, 1]
	// if has_slope: iterate for every slope; don't forget to add back stdev
next
s = s / denom[i]
for g=1:G
	F[g].alphas[level, 1] += s
	// also update slopes
next

A singleton is an individual that appears in only one observation. In regressions with fixed effects, singletons do not contribute useful information for estimating regressors because the fixed effect perfectly explains the dependent variable for that individual.

Excluding singletons is recommended for three reasons:

1. Correcting Standard Errors: When using clustered standard errors with fixed effects (especially in High-Dimensional Fixed Effects/HDFE settings), including singletons can lead to underestimated standard errors and overestimated P-values. This is because singletons can artificially inflate the sample size $N$ without increasing the effective number of clusters $M$ or providing within-cluster variability.
2. Identifying Effective Cluster Size: Removing singletons reveals the effective number of clusters, which is critical for determining if the cluster count is large enough for asymptotic theory to apply.
3. Computational Efficiency: Excluding singletons speeds up HDFE computations by reducing the number of ancillary parameters that need to be estimated.

Singletons do not affect coefficient estimates, but they significantly bias standard error estimates.

In a regression with clustered standard errors, the variance-covariance matrix $V$ is adjusted by a finite-sample factor $q$:

$$q = \frac{M}{M-1} \times \frac{N-1}{N-K-L}$$

Where:

• $M$ is the number of clusters.
• $N$ is the number of observations.
• $K$ is the number of regressors (excluding the constant).
• $L$ is the number of fixed effects.

When fixed effects are nested within clusters, singletons cause the 'meat' and 'bread' of the variance (the residuals and score matrix) to remain unchanged, but they change the $q$ factor (the 'wooden stick'). Because $L$ is often excluded from the denominator of $q$ when nested, the denominator does not adjust for the loss of information, leading to biased results.

Mitigation: A common solution is to remove singleton clusters from $M$ (using $M_S$ as the number of singleton clusters) to calculate a corrected $q^*$:

$$q^* = \frac{M-M_S}{M-M_S-1} \times \frac{N-M_S-1}{N-M_S-K-L^*}$$

Alternatively, excluding singletons or using bootstrapped standard errors can ameliorate this bias.

When performing regressions where fixed effects are nested within clusters (e.g., firm fixed effects nested within industry clusters), keeping singleton groups (groups with only one observation) can lead to incorrect statistical inference.

The Problem

If singletons are not dropped, the finite-sample adjustment factor ($q$) converges toward 1. This causes:

1. Underestimated standard errors: Statistical significance will be overstated.
2. Overstated cluster counts: The reported number of clusters may be higher than the actual number of informative clusters.
3. Slower estimation: More ancillary parameters are estimated than necessary.

Note that coefficient estimates and the asymptotic part of the variance estimator remain unchanged; the issue is strictly with the finite-sample adjustment used for robust and cluster-robust standard errors.

The Solution

You can mitigate this by either:

• Removing singleton groups from the dataset before estimation.
• Excluding singleton counts from the number of clusters ($M$) and observations ($N$) used in the variance calculation.

Note: Future versions of reghdfe will automatically drop singletons to prevent this issue.