bounds are useful, in the sense of delineating a reasonably narrow range of values for ���� , is an empirical question to which we turn in the remainder of the paper.

# **4.  Comparing Actual and Approximate Mobility in the PSID**

In this section, we verify that our methodology for placing bounds on the mobility parameter ���� provides reasonably informative bounds on true mobility, using data from the US Panel Study of Income Dynamics (PSID). We first estimate bounds on ���� by applying the approach described in the previous sections to the time-series of the cross-sectional mean and standard deviation of log income in the PSID. We then take advantage of the panel structure of the PSID micro data to estimate the mobility parameter ���� directly, and compare these micro estimates with the bounds obtained using only aggregate data.

We work with annual rounds of the PSID between 1977 and 1997.<sup>9</sup> The unit of observation in the PSID is the household, and we take nominal family income per capita deflated by the national consumer price index as our measure of real income per capita. Since measures of inequality are sensitive to extreme observations, we discard a small number of household-year observations corresponding to implausibly low and implausibly high per capita incomes.<sup>10</sup> We then compute the mean and variance of log per capita income, using the sampling weights provided in the PSID, and apply our methodology for estimating � and obtaining bounds on ���� to the resulting two time-series of aggregate moments.

Table 1 summarizes our results. Panel A reports the estimates and standard errors of the autoregressive parameter � . We first report estimates based on the dynamics of the mean of log income (Equation (8)) and the variance of log income (Equation (9)). The three columns correspond to the OLS estimates, the small sample bias-corrected OLS estimates, and the MSE-minimizing linear combination of the two. Comparing the first two columns suggests that the bias correction is important – the bias corrected estimates of � are substantially larger than the OLS estimates, increasing from 0.65 to 1 (based � in the dynamics of �� ), and increasing from 0.76 to 0.96 (based on the dynamics of �� ). However, the

> 9 After 1997, the PSID switches to biannual frequency.  We also considered working with a biannual version of the PSID from 1977 to the present.  However, over this longer time series, there is clear evidence of a structural break in the time series for the standard deviation of log income. Accommodating this trend break in a series of only 17 biannual observations led to estimates of � based on Equations (8) and (9) that were highly imprecise.  For this reason, we work with the shorter 1977-1997 time period with annual data.  In the following subsection, we allow for structural breaks when estimating Equation (8) and (9) using longer time series on aggregate moments available in the World Wealth and Income Database.

> 10 Specifically, we drop households log per capita income below $400 or above $440,000, corresponding to the bottom 0.8 percent and top 0.04 percent of the household-year observations in the raw data.

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