Assessing Differences Between Grouped and Individual-Level Regression Coefficients

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Abstract
Several techniques for assessing differences between least-squares estimators of regression coefficients from group and individual-level data are summarized. The structural equations approach (Hannan and Burstein, 1974) and the X-rule approach (Firebaugh, 1978) generate estimates of predicted bias when individual-level data are available, and it is important to determine the consequences ofgrouping according to a known procedure. While the X-rule provides slightly more accurate forecasts, both approaches identify the same methods of grouping as resulting in either large or small bias. An approach based on the statistical significance of differences between estimators at two levels (Feige and Watts, 1972) is also described along with an alternative test based on a different interpretation of the standard test for significant differences among regression models. The empirical results indicate that both the Feige- Watts test and the proposed alternative significance test behave well for cases with small differences between group and individual-level estimates. However, the Feige- Watts test was less satisfactory when group and individual-level estimates diverged greatly. The utility and suitability of all four approaches, when (a) data are grouped by a nominal characteristic, (b) there are multiple regressors, or (c) individual-level data cannot be reconstructed, are discussed.