## Tuesday, February 14, 2012

### "Asymptotic" Properties of Estimators and Tests

We're so familiar with "large-sample" asymptotics as a way of characterizing the behaviour of our estimators and tests in econometrics, that we tend to forget that there are other, very interesting ways of evaluating their behaviour, and approximating small-sample behaviour.

I touched on this in an earlier earlier post when I discussed "small-sigma" (or "small error") asymptotics. However, that's by no means the end of the story.

There are at least two other types of "asymptotics" that have proven themselves to be very useful in econometric analysis. The first of these dates back at least as far as the work of  Kunitomo (1980), Morimune (1983), and others, during the hey-day of work on simultaneous equations models.

It's what is nowadays called "many instruments" asymptotics. In this set-up, the number of instruments and the sample size are allowed to grow at the same rate. (Of course, the number of regressors, and hence the number of parameters to be estimated, is constant). This type of asymptotic behaviour has been shown to be especially useful in the construction of asymptotic covariance matrices, and hence asymptotic standard errors, for a number of problems. For example, see Hansen et al. (2008).

The other type of "asymptotics" is the so-called "small bandwidth" asymptotics, as exemplified by the work of Cattaneo et al. (2010, 2011). In this case, a non-parametric approach is taken, and the asymptotics are based on a sequence of bandwidths, rather than a sequence of sample sizes (as with the usual large-n asymptotics), or a sequence of values for the disturbance variance (as with "small-sigma" asymptotics).

Very recently, Cattaneo (2012) showed that actually there is a very close connection between "small bandwidth" asymptotics and "many instruments" asymptotics.

One of the many things that is interesting about these other types of asymptotic behaviour is that they enable us to construct asymptotically valid covariance matrix estimators in situations where estimators such as Hal White's "heteroskedasticity-consistent" estimator fails.

A great example of this is in the context of a linear panel-data model with fixed effects. If you thought that White's covariance matrix estimator is consistent in this context, then think again. Better yet, take a look at Stock and Watson (2007). It is inconsistent if T (> 2) is fixed, and the number of cross-section units increases without limit. Stock and Watson supply an alternative covariance matrix estimator that is (nT)1/2 consistent.

In short, the "asymptotic behaviour" of our estimators and tests can depend on what we mean by "asymptotics". There's more than one way to think about this type behaviour.

Note: The links to the following references will be helpful only if your computer's IP address gives you access to the electronic versions of the publications in question. That's why a written References section is provided.

References

Cattaneo, M. D., R. K. Crump, and M. Jansson  2010. Robust data-driven inference for density-weighted average derivatives. Journal of the American Statistical Association, 105, 1070-1083.

Cattaneo, M. D., R. K. Crump, and M. Jansson 2011. Small bandwidth asymptotics for density-weighted average derivatives. Forthcoming in Econometric Theory.

Cattaneo, M. D., M. Jansson and W. K. Newey, 2012. Alternative asymptotics and the partially linear model with many regressors. CREATES Research Paper 2012-02, Department of Economics and Business, Aarhus University.

Hansen, C., J. Hausman, and W. K. Newey, 2008. Estimation with many instrumental variables. Journal of
Business and Economic Statistics, 26, 398-422.

Kunitomo, N., 1980. Asymptotic expansions of the distributions of estimators in a linear functional relationship and simultaneous equations. Journal of the American Statistical Association, 75, 693-700.

Morimune, K., 1983. Approximate distributions of k-class estimators when the degree of overidentifiability is large compared with the sample size. Econometrica, 51, 821-841.

Stock, J. H., and M. W. Watson, 2007. Heteroskedasticity-robust standard errors for fixed effects panel data regression. Econometrica, 76, 155-174.