Three essays on quantile factor analysis
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In the first chapter of this dissertation, I develop a method that extends quantile regressions to high dimensional factor analysis. In this context, the conditional quantile function of a panel of variables is endowed with a factor structure. Thus, both factors and factor loadings are allowed to be quantile-specific. I provide a set of conditions under which these objects are identied, and I propose a simple two-step iterative procedure called Quantile Principal Components (QPC) to estimate them. Uniform consistency of the estimators is established under general assumptions when both the cross-section and time dimensions (N and T, respectively) become large jointly. In the second chapter, I propose a novel measure to quantify systemic risk from the information contained in asset returns. In the context of the external habits formation model of Campbell and Cochrane (1999) and heteroskedastic stock returns, I show that the equilibrium risk premium has a factor structure where factors are a monotonic transformation of the systemic risk variable in the structural model, and one of the factors affects the variance of excess returns. I estimate the factor model using the QPC estimation procedure. Simulations of the model calibrated to the US show a good performance of the proposed metric computed at quantiles different than the median. When estimated using post-war data, the proposed measure displays signicant hikes that coincide with both several US recessions and financial market turbulence periods; and it can forecast extreme tight and loose financial conditions, and sharp shifts in both economic activity and industrial production up to one year ahead. The third chapter provides limiting distributions of the QPC estimators proposed in the first chapter. Under certain additional assumptions related to the density of the observations about the quantile of interest, and the relationship between N and T, I show that the QPC estimators are asymptotically normal with convergence rates similar to the ones derived in the traditional factor analysis literature. Monte Carlo simulations suggest that the proposed theory provides a good approximation to the finite sample distribution of the QPC estimates.