1. Introduction - Econometrics at Illinois - University of Illinois at ...

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24 Inference on the Quantile Regression Process Indicator for whether the claimant was registered in one of the low unemploymentshort unemployment duration districts: Coatesville, Reading, or Lancaster.In Figure 6.1 we present a concise visual representation of the results from theestimation of this model. Each of the panels of the Figure illustrate one coordinate ofthe vector-valued function, ^(), viewed as a function of 2 [ 0 ; 1 ]. Here we choose 0 = :20 and 1 = :80, eectively neglecting the proportion of the sample that areimmediately reemployed in week one and those whose unemployment spell exceedsthat insured limit of 26 weeks. The lightly shaded region in each panel of the gurerepresents a 90 percent condence band. We omit the plots for the 5 quarter of entryindicators, and for the for the low unemployment district variable to conserve space.Before turning to interpretation of specic coecients, we will try to oer somebrief general remarks on how tointerpret these gures. The simplest case is the purelocation shift model for which wewould have the classical accelerated failure time(AFT) model,log T i = x > i + u iwith fu i g's iid from some F . For F of the form F (u) =1, exp(, exp(u)), this isthe well known Cox proportional hazard model with Weibull baseline hazard. In thiscase we would expect to see slope coecients ^ j () that oscillate around a constantvalue indicating that the shift in the response due to a change in the covariate isconstant over the entire estimated range of the distribution. The conditional meaneect estimated by least squares is asymptotically equivalent in this case to integratingthe estimated coecients over the unit interval.Another conventional model with linear quantile functions is the linear locationscalemodel,log T i = x > i +(x > i )u iwhere again, u i is taken to be iid. Nowthecovariates are allowed to inuence the scaleas well as the location of the conditional distribution of durations. In this case theplots of the \slope" coecients ^ j () should look just like the \intercept" coecientup to a location and scale shift. The intercept coecient estimates a normalizedversion of the quantile function of the u i 's and all the other coecients are simplylocation and scale shifts of this function.6.2. Interpretation of the Estimated Eects. No treatment eect is observed ineither tail implying that the treatment had no eect in changing the probability ofimmediate reemployment (inweek one), or in eecting the probability of durationsbeyond the 26 week maximum. The high bonus and long qualication period treatment,yielded roughly a 15% reduction in median duration. This eect is considerablystronger statistical signicance than that seen in the other treatments.The randomization of the experiment was quite eective in rendering the potentiallyconfounding eects of other covariates orthogonal to the treatment indicator.
Roger Koenker and Zhijie Xiao 25Figure 6.1. Quantile Regression Process for Log Duration ModelIntercept0.5 1.5 2.5Treatment-0.25 -0.10 0.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0Female0.0 0.1 0.2Black-1.2 -0.8 -0.4 0.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0Hispanic-1.0 -0.6 -0.2N-Dependents0.0 0.05 0.150.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0Recall Effect0.0 0.5 1.0Young Effect-0.4 -0.2 0.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0Old Effect0.0 0.1 0.2 0.3 0.4Durable Effect0.0 0.1 0.2 0.30.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0
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Page 21 and 22: and the transformed process isRoger
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