After constructing factors for all securities in the investable universe, each factor is analyzed individually. Presenting the time-series and cross-sectional averages of the mean, standard deviations, and key percentiles of the distribution provide useful information for understanding the behavior of the chosen factors.

Although we often rely on techniques that assume the underlying data generating process is normally distributed, or at least approximately, most financial data is not. The underlying data generating processes that embody aggregate investor behavior and characterize the financial markets are unknown and exhibit significant uncertainty. Investor behavior is uncertain because not all investors make rational decisions or have the same goals. Analyzing the properties of data may help us better understand how uncertainty affects our choice and calibration of a model.

Below we provide some examples of the cross-sectional characteristics of various factors. For ease of exposition we use histograms to evaluate the data rather than formal statistical tests. We let particular patterns or properties of the histograms guide us in the choice of the appropriate technique to model the factor. We recommend that an intuitive exploration should be followed by a more formal statistical testing procedure. Our approach here is to analyze the entire sample, all positive values, all negative values, and zero values. Although omitted here, a thorough analysis should ...

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