Sandwich estimators for standard errors
- Where is the model fitting information stored in MLwiN?
- Should the comparative SD output when I calculate the residuals be different for each row?
- MLwiN is giving the standard errors of parameter estimates as 0, but I know from comparison with other software packages that the standard errors should not be 0
Sandwich estimators for standard errors are often useful, eg when model based estimators are very complex and difficult to compute and robust alternatives are required.
Consider the fixed part parameter estimates
The covariance matrix is given by
If we replace the central covariance term by the usual (Normal) model based value, V, we obtain the usual formula
with sample estimates being substituted. The sandwich estimator is formed by replacing the estimate of the central covariance term, , by an empirical estimator based on the (block diagonal structure) cross product matrix, namely
For residuals the estimated set of residuals for the j-th block at level h, using a similar notation to Goldstein (1995, App. 2.2) omitting the sub/superscript h, is given by
To obtain consistent estimators of the covariance matrix of these residuals (ignoring variation in the fixed parameter estimates) we can choose comparative or diagnostic estimators.
The diagnostic estimator is given by
If the model based estimator is used this reduces to the expression given by Goldstein (1995, Appendix 2.2), otherwise the cross product matrix estimator is used.
For the comparative estimator we have
which reduces to the expression in Goldstein (1995, Appendix 2.2) when the model based estimator is used.
In MLwiN 1.1 access to the sandwich estimators is via the FSDE and RSDE commands
For residuals, sandwich estimators will automatically be used when weighted residuals are specified in the residuals section on weighting for details of residuals produced from weighted models.