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Different episodes of economic growth display widely varying distributional characteristics, both across countries and over time. Growth is sometimes accompanied by rising and sometimes by falling inequality. Applied economists have come to rely on the Growth Incidence Curve, which gives the quantile-specific rate of income growth over a certain period, to describe and analyze the incidence of economic growth. This paper discusses the identification conditions, and develops estimation and inference procedures for both actual and counterfactual growth incidence curves, based on general functions of the quantile potential outcome process over the space of quantiles. The paper establishes the limiting 0 distribution of the test statistics of interest for those general functions, and proposes resampling methods to implement inference in practice. The proposed methods are illustrated by a comparison of the growth processes in the United States and Brazil during 1995-2007. Although growth in the average real wage was disappointing in both countries, the distribution of that growth was markedly different. In the United States, wage growth was mediocre for the bottom 80 percent of the sample, but much more rapid for the top 20 percent. In Brazil, conversely, wage growth was rapid below the median, and negative at the top. As a result, inequality rose in the United States and fell markedly in Brazil.

Poverty Reduction

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Gearing, Reduction. --- Reduction gearing --- Gearing

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Wastewater --- processing --- Phosphorus --- Reduction. --- Reduction --- Purification --- Belgium

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FATTY ACIDS --- OXIDATION-REDUCTION --- REDUCTION --- METABOLISM --- CONGRESSES

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Data are not only ubiquitous in society, but are increasingly complex both in size and dimensionality. Dimension reduction offers researchers and scholars the ability to make such complex, high dimensional data spaces simpler and more manageable. This Element offers readers a suite of modern unsupervised dimension reduction techniques along with hundreds of lines of R code, to efficiently represent the original high dimensional data space in a simplified, lower dimensional subspace. Launching from the earliest dimension reduction technique principal components analysis and using real social science data, I introduce and walk readers through application of the following techniques: locally linear embedding, t-distributed stochastic neighbor embedding (t-SNE), uniform manifold approximation and projection, self-organizing maps, and deep autoencoders. The result is a well-stocked toolbox of unsupervised algorithms for tackling the complexities of high dimensional data so common in modern society. All code is publicly accessible on Github.

Dimension reduction (Statistics) --- Dimensionality reduction (Statistics) --- Reduction, Dimension (Statistics) --- Reduction, Dimensionality (Statistics) --- Statistics