Questions tagged [sandwich]

Sandwich variance estimation is a method of variance estimation in which parameters are defined as the roots to estimating equations. In likelihood based inference, estimating equations are called score equations (given by the derivative-log likelihood); in econometrics, moment conditions.

One can show by way of multivariable calculus that parameter estimation using estimating equations is consistent and has a variance of the form of a sandwich in which the "meat" of the sandwich is the empirical variance estimates of the score equations, with the Jacobian and its transpose as two slices of "bread". All of this is set up in Huber (1967) paper, and the various version of robust standard errors (GEE, White heteroskedasticity-corrected standard errors in cross-sectional econometrics, Newey-West variance estimates in time series, linearized variance estimates in survey statistics, Satorra-Bentler standard errors in structural equation modeling, etc.) are special forms.

Inference with sandwich estimates is semi-parametric in nature. The way the model contributes to them is (1) by providing the estimating/score equations, and (2) by providing the "bread" of the sandwich. Robustness properties of sandwich estimates stem from the second moments being taken with respect to the data, not with respect to the assumed model, although in most situations, the statistician needs to make a judgement call as to which units can be assumed (approximately) independent and can enter the relevant summations (e.g., what the clusters are for the clustered standard errors, or at what lag length autocorrelations drops to zero for Newey-West standard errors).

If all of the model assumptions hold, the model based estimates of standard errors (e.g., those based on inverse Hessian in the likelihood context) are efficient than sandwich based estimates, but these assumptions may be untestable, and/or could easily be found to be in violation for very large samples.

Sandwich variance estimation is, by default, the variance estimate of parameters estimated from Generalized Estimating Equations (GEE).

For most sandwich variance estimators, there is an asymptotically equivalent bootstrap method available.

Implementations:

• in R, sandwich package (https://cran.r-project.org/web/packages/sandwich/index.html)
• in Stata, many estimation commands support robust option for data that are independent but not necessarily identically distributed. See https://stats.idre.ucla.edu/stata/dae/alternative-methods-of-estimating-standard-errors/

Related topics:

• robust-standard-error
• gee
• clustered-standard-errors
• neweywest