Clustered standard errors and hypothesis tests in fixed effects models

I’ve recently been working with my colleague Beth Tipton on methods for cluster-robust variance estimation in the context of some common econometric models, focusing in particular on fixed effects models for panel data—or what statisticians would call “longitudinal data” or “repeated measures.” We have a new working paper, which you can find here.

The importance of using CRVE (i.e., “clustered standard errors”) in panel models is now widely recognized. Less widely recognized, perhaps, is the fact that standard methods for constructing hypothesis tests and confidence intervals based on CRVE can perform quite poorly in when you have only a limited number of independent clusters. What’s worse, it can be hard to determine what counts as a large-enough sample to trust standard CRVE methods, because the finite-sample behavior of the variance estimators and test statistics depends on the configuration of the covariates, not just the total sample size. For example, suppose you have state-level panel data from 50 states across 15 years and are trying to estimate the effect of some policy using difference-in-differences. If only 5 or 6 states have variation in the policy variable over time, then you’re almost certainly in small-sample territory. And the sample size issues can be subtler than this, too, as I’ll show below.

One solution to this problem is to use bias-reduced linearization (BRL), which was proposed by Bell and McCaffrey (2002) and has recently begun to receive attention from econometricians (e.g., Cameron & Miller, 2015; Imbens & Kolesar, 2015). The idea of BRL is to correct the bias of standard CRVE based on a working model, and then to use a degrees-of-freedom correction for Wald tests based on the bias-reduced CRVE. That may seem silly (after all, the whole point of CRVE is to avoid making distributional assumptions about the errors in your model), but it turns out that the correction can help quite a bit, even when the working model is wrong. The degrees-of-freedom correction is based on a standard Satterthwaite-type approximation, and also relies on the working model. There’s now quite a bit of evidence (which we review in the working paper) that BRL performs well even in samples with a small number of clusters.

In the working paper, we make two contributions to all this:

  1. One problem with Bell and McCaffrey’s original formulation of BRL is that it does not work in some very common models for panel data, such as state-by-year panels that include fixed effects for each state and each year (Angrist and Pischke, 2009, point out this issue in their chapter on “non-standard standard error issues”). We propose a generalization of BRL that works even in models with arbitrary sets of fixed effects. We also address how to calculate the correction when the regression is fit using the “within” estimator, after absorbing the fixed effects.
  2. We propose a method for testing hypotheses that involve multiple parameter constraints (which, in classical linear regression, you would test with an F statistic). The method involves approximating the distribution of the cluster-robust Wald statistic using Hotelling’s T-squared distribution (a multiple of an F distribution), where the denominator degrees of freedom are estimated based on the working model. For one-parameter constraints, the test reduces to a t-test with Satterthwaite degrees of freedom, and so it is a natural extension of the existing BRL methods.

The paper explains all this in greater detail, and also reports a fairly extensive simulation study that we designed to emuluate the types of covariates and study designs encountered in micro-economic applications. We’ve also got an R package that implements our methods (plus some other variants of CRVE, which I’ll explain some other time) in a fairly streamlined way. Here’s an example of how to use the package to do inference for a fixed effects panel data model.

References

  • Angrist, J. D., & Pischke, J.-S. (2009). Mostly harmless econometrics: An empiricist’s companion. Princeton, NJ: Princeton University Press.
  • Angrist, J. D. and Pischke, J.-S. (2014). Mastering ’metrics: The Path from Cause to Effect. Princeton, NJ: Princeton University Press.
  • Bell, R. M., & McCaffrey, D. F. (2002). Bias reduction in standard errors for linear regression with multi-stage samples. Survey Methodology, 28(2), 169-181.
  • Cameron, A. C., & Miller, D. L. (2015). A practitioner’s guide to cluster-robust inference. URL: http://cameron.econ.ucdavis.edu/research/Cameron_Miller_JHR_2015_February.pdf
  • Carpenter, C., & Dobkin, C. (2011). The minimum legal drinking age and public health. Journal of Economic Perspectives, 25(2), 133-156. doi:10.1257/jep.25.2.133
  • Imbens, G. W., & Kolesar, M. (2015). Robust standard errors in small samples: Some practical advice. URL: https://www.princeton.edu/~mkolesar/papers/small-robust.pdf

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