Job Stability in Developing and Developed Countries - Evidence from Colombia and the United States (Appendix)

Appendix 1 Interpolated Quantiles When describing current job tenure distributions, I use interpolated (Farber, 1995) rather than simple quantiles, because tenure in years is reported in integer values, and thus contains many observations with exactly the same reported value. Simple quantiles can be quite insensitive to shifts in such distributions. For example, in each of the following two samples the sample median is 2, though there is a sense in which the sample median is "just 2" in the first and "almost 3" in the second. Tenure observations in Sample A: 1111222233 Tenure observations in Sample B: 1122223333 If true job tenures are continuous but are rounded to integer reports in years, then one might guess that the true mean is closer to 1.5 in the population corresponding to Sample A and closer to 2.5 in the case of Sample B. The median appears closer to 3 in sample B because a larger fraction of the observations with the value 2 (the integer quantile) must be included with the observations with value 1 (all values lower than the integer quantile) to achieve 50 percent of the observations. That is, the 50 percent line is farther to the right in the list of observations with the value 2. The interpolated quantile, is related to the simple quantile 7, (ie. the integer value of tenure such that a population-weighted fraction q of the observations have values less than or equal to that value), in the following way (for T₁>0): where q is a fraction defining the quantile of interest', w, is the normalized weight associated with individual i, normalized so that the weights sum across all observations to one, N is sample size, t is the integer tenure value for the ij-th observation, and I() is the indicator function taking the value In the tables I presem 10%, 25, 50, 75 and 90 quantiles, which correspond to q's of 10, 25, 50, 75 and 90. 1 if the expression in parentheses is true and zero otherwise. (When T, equals zero, the formula is adapted to cause the quantiles to fall in the range of zero to 5.) If one conditions on T, (as suggested by Farber, 1995), one may derive standard errors by applying the delta method, as discussed in Appendix 2. If the sample were self-weighting, so that w-1 for all i, the formula for the interpolated quantile could be reexpressed in a somewhat more intuitive way, as Appendix 2 Weighting, Standard Errors and Testing Development of standard errors and test statistics is complicated somewhat by the need to employ weights for describing population distributions. Weights are employed because tenure distributions are suspected to differ systematically across groups whose proportions in the sample differ from their proportions in the population. In deriving test statistics and standard errors, I must acknowledge explicitly these potential differences in group-specific distributions. Both the ENH and the CPS employ stratified random samples of Primary Sampling Units (PSUs), which are small neighborhoods containing on the order of 10 to 20 households, all of which are typically interviewed. In principle the survey is self-weighting, in the sense that each stratum is sampled in proportion to its share in the population. In practice, however, both DANE and the CPS provide weights that adjust for three potential departures from a self-weighting sample: (1) When interviewers discovered more than 20 households in a PSU, they followed rules for selecting a random subsample of the PSU's households. The weights for those households are inflated accordingly. (2) Some households refused to respond. (3) Errors in either the sampling frame or the sampling process led to divergences between sample-based population estimates and independent population estimates within demographic groups. The weights that adjust for all three potential problems vary across demographic groups (defined by gender, age, and, in the case of the U.S., race) and PSUs. I made two adjustments to these weights. First, because I am interested in comparing population tenure distributions across countries and because I suspect that tenure distributions differ significantly by age, education and job type (private sector wage employment, public sector wage employment, self employment), I adjusted the weights for nonresponse to the job tenure question, multiplying them by the inverse of the response rate among workers within age-education-job type categories. Tenure nonresponse rates were quite low (on the order of five percent or less) for the Colombian datasets and for the U.S. 1987 and 1991 datasets, but were much higher (on the order of 20 percent) for the U.S. 1988 dataset. After I make this adjustment, the weights vary by education level and job category, as well as demographic groups and PSUs. Second, because I am willing to assume that tenure distributions are the same across PSUs