Estimation of SEs for heteroscedastic and cross-sectionally correlated data

Abstract

This study develops SE estimators for heteroscedastic and cross-sectionally correlated data. The new estimators are a cross-sectional version of the White and Domowitz (1984) and Newey and West (1987) estimators, and therefore, consistent in the presence of heteroscedasticity and cross correlation of unknown form. Unlike the estimators in the literature, these estimators can control for cross correlation even for single-period cross-sectional data.

Acknowledgements

This work was supported by the Hankuk University of Foreign Studies Research Fund.

Notes

¹If X is stochastic, the assumptions about the disturbances U hold conditionally on X.

²For notational simplicity, a subscript of k (or −k) is not added to and without causing any confusion.

³In fact, the two sample residual-vectors are identical: because WX _{− k} = 0.

⁴The regressor cross correlation are defined for stochastic regressors. For cases of nonstochastic regressors, it is understood in terms of calculated sample quantities, as in Bernard (1987).

⁵Petersen (2006) does not explicitly state about the co-variability. However, since his model includes firm and/or time effects in both the error term and the regressor, the residual cross correlation always co-varies with the regressor cross correlation.

⁶From this point of view, it can be understood that the autocorrelation of residuals in time-series data affect the SEs of coefficient estimates only when regressors are also autocorrelated. Thus, implicitly assuming that the regressors and the error term are correlated, respectively between observations which are near in time, White and Domowitz (1984) and Newey and West (1987) control for autocorrelation by including only cross-product terms within a specified time lag in a time-series version of Equation 12. In proving the consistency of their estimators, they assume that the autocovariances vanish fast with the time lag.

⁷According to the second term in Equation 12, the cross products matter for a possible bias of SE estimation. As the cross-sectional correlation of residuals is positive in most cases, i.e. the ordering should be based on the signed value of a regressor. It is to make sure that the product is positive for observations near in order and can maintain the sign of . Otherwise, for example, ordering by the absolute or squared values will cause a cancellation due to positive and negative values of .

⁸The consistency also holds when the truncation bound grows at a rate of slower than N ^1/2, under the condition that is an infinite order moving average with absolutely summable coefficients and i.i.d. innovations, where the innovations have finite fourth moments (Newey and West, 1987, p. 705). I have also simulated using the truncation bound of N ^1/2instead of N ^1/4. The NW estimators produced qualitatively same results. However, the WD estimators did not work because the variance estimates were negative in several cases.

⁹The correct SE is obtained by calculating the SD of the OLS estimates from 500 iterations. Since the OLS coefficient estimator is unbiased even in the presence of cross correlation, the correct SEs are calculated in the same way for the tables in what follow.

¹⁰As the original variable x ₁ was used to order observations in generating data, ordering by x ₁ produced more accurate estimates for SEs than ordering by the orthogonalized variable x ₁, although both produced qualitatively same results. In this article, we report the results obtained from using x ₁.

¹¹The asymptotic results hold when the number of observations increases at a higher rate than the cluster size does, i.e. the ratio of the sample size to the cluster size increases to infinity.

¹²In all cases, the WD estimator produces more accurate SE estimates than the NW estimator does. It is because cross-correlated components were added without weights in generating the datasets, but the NW estimator applies a weight function to guarantee positive semi-definite estimates in finite samples.