statistics calculated only over the balanced panel of households. The estimated effects are even smaller, though statistically significant for the 1995-96 and 1997-98 endpoints. It is the combination of these two changes in specification–excluding the household-level covariates and using only the balanced panel sample to calculate initial village conditions–that drives the drop in the estimated effect of inequality from the household to the village-level regressions: It has nothing to do with aggregation from one level to the other _per se_ .

This is seen most clearly in column (9), where we estimate the household-level Equation (A1.4) by instrumental variables, using as instruments the village mean values of Mean Log Income, the Mean Log Deviation, and Mean Education.<sup>25</sup> This exercise yields the same results as column (8). We thus obtain the same coefficient estimates three ways: Estimation at the household level with either the villagelevel regressors (column 8) or village-means as instruments (column 9); and the village-level results in column (11).<sup>26</sup> Finally, in column (10) we implement the Deaton (and Devereux) corrections for small sample bias. This entails only a small variation on the specification in column (9): We use the jackknifed village means instead of the straight village means as instruments for the household variables in Equation A1.4. For this specification, the estimated effects of inequality are slightly smaller, and significant only for the “peak” year of 1995-96. Because it accounts for the small-sample measurement error bias, this is the preferred “bottom-line” village-level result: inequality has a negative but generally insignificant effect that fades quickly after 1995-96. However, our overall preferred specification is column (5), which uses the same jackknifed inequality measure, but with the inclusion of the full suite of household covariates, and with village statistics calculated using all available observations from 1987-88. As explained in Appendix 1, column (5) is a richer “reduced form” for the Deaton and Devereux specification presented in column (9).

# _4.3 Regression Results: Heterogeneous Responses_

Who is hurt most by high inequality? In Table 3 we present estimates of two variants of Equation (1), adding interaction effects between inequality and key dimensions of household covariates: education and cohort in one variant, and own-income in the other. For each time period, in column (1), we estimate interaction effects for education and cohort. For the cohort interaction, while we maintain the more flexible base effects for household age, we define the interaction in terms of “young” and “old”, with age 40 as the cut-off. In column (2) we report interactions between inequality and a household’s position in the income distribution. We assign households to their quartile within their village, where the quartiles

> 25 Identically, we can use Village Dummies as instruments, creating a “Visual Instrumental Variables” estimate (intimated by Figure 1).

> 26 As usual, the standard errors are slightly higher at the village level, as the Cluster Correction is not a perfect substitute for estimating at the village-level.

_Inequality and Growth in Rural China_ , Page 16