Notice that this decomposition sums to the total observed change \u0394 Q X \u03b8 \u0394 Q b

# Notice that this decomposition sums to the total

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Notice that this decomposition sums to the total observed change: Δ Q X θ Q b θ Q w θ = Δ Q θ . This is an important advantage over the JMP procedure, in which the ’residual price and quantity component’ must be estimated as a remainder term after the other two com- ponents are calculated. 5.6.2 Quantile implementation of DFL Interestingly, the notation above makes it apparent that the DFL procedure is simply the first step of the JMP decomposition above. In particular, Δ Q DF L θ = Q θ f g τ ( X ) , β t b , β t w Q θ f g t ( X ) , β t b , β t w = Δ Q θ X . 24 ( ( )) ( )) ( ( )) ( )) ( ( )) ( )) ( ( )) ( ( )) ( ( (
Why does DFL only do Step 1? Because the DFL procedure simply reweights the function n mapping X n s to w s (given by β t b , β t w in our QR model) using the change in the density of X n s between t and τ ( g t ( X ) to g τ ( X )). Since DFL do not explicitly model ’prices’ ( β t b , β t w in the quantile model), they do not take Steps 2 and 3 where these prices are varied. Similarly, the Lemieux (2004) procedure for reweighting residual densities can be written as Δ Q L θ = Q θ f g τ ( X ) , β t b = 0 , β t w Q θ f g t ( X ) , β t b = 0 , β t w = Δ Q θ X . Unlike the quantile approach, Lemieux estimates β t w using OLS. But this difference is unlikely to be important. 5.6.3 Advantages and disadvantages of the quantile decomposition relative to other approaches An advantage of the QR approach is that it nests JMP, DFL, and all extensions to DFL that have been recently proposed. I would also argue that it handles each of these cases somewhat more transparently than the competing techniques. A second virtue of QR is that procedure explicitly models the separate roles of quantities, and between- and within-group prices to overall inequality. That is, DFL and extensions never explicitly estimate prices, although these prices are implicit in the tool. By contrast, JMP do estimate prices (both observed and unobserved). But in practice, their residual pricing function does not quite work as advertised unless one conditions the residual distribution ( F t ( θ | X it )) very finely on all combinations of X n s . A third virtue of QR is that it satisfies the adding-up property. That is, if the QR model fits the data well, the sum of the components of the decomposition will add up to the total. (Of course, it is still a sequential decomposition; the order of operations matters.) Finally, unlike JMP and extensions, QR provides a consistent treatment of between- and within-group prices (there is no ’hybridization’ of OLS and logit/probit models). The QR decomposition has two notable disadvantages. First, it is parametric. The precision of the simulation will depend on the fit of the QR model which in turn depends on the characteristics of the data and the richness of the QR model. By contrast, the DFL procedure and its extensions never actually parameterize the conditional distribution of wages, F ( w | X ). Hence, the treatment of F ( w | X ) in DFL is fully non-parametric. Notably, DFL must parameterize the reweighting function (through the probit/logit). I’ve never seen any work documenting the

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• Spring '15
• Prof. David Autor
• Economics, Normal Distribution, Probability theory, probability density function, Cumulative distribution function, Wage Distribution