Identification of Joint Distributions in Dependent Factor Models

Abstract

This paper studies linear factor models that have arbitrarily dependent factors. Assuming that the coefficients are known and that their matrix representation satisfies rank conditions, we identify the nonparametric joint distribution of the unobserved factors using first and then second-order partial derivatives of the log characteristic function of the observed variables. In conjunction with these identification strategies the mean and variance of the vector of factors are identified. The main result provides necessary and sufficient conditions for identification of the joint distribution of the factors. In an illustrative example, we show identification of an earnings dynamics model with a subset of arbitrarily dependent income shocks. Closed-form formulas lead to estimators that converge uniformly and despite being based on inverse Fourier transforms have tight confidence bands around their theoretical counterparts in Monte Carlo simulations.

Dynamic deconvolution and identification of independent autoregressive sources

“…a) at most one source distribution is Gaussian (see e.g. Comon, 1994; Hyvarinen et al ., 2001; Eriksson and Koivunen, 2004 Th 5. iii; Chan et al ., 2006; Ben‐Moshe, 2018b) the moments exist up to order three, four or six (see, e.g., Comon and De Lathauwer, 2010; Erickson et al ., 2014c) the characteristic functions of sources have no exponential factor (Eriksson and Koivunen, 2004, Th 5, ii), iv), d) restrictions on the support or exclusion restrictions are imposed (Williams, 2020; Ben‐Moshe, 2021). Moreover, the number of sources can be assumed upper bounded (De Lathauwer, 2008; Bonhomme and Robin, 2010; Zinde‐Walsh, 2013, Section 4.2.2.3), or Bayesian approaches can be used(Klepper and Leamer, 1984; Leonard, 2011).…”

“…One may proceed further to the estimation of the density function itself in Step 4. Step 4: If is a continuous variable, its density can be obtained through a kernel regularized Fourier transform, to have a well‐posed inverse problem (Ben‐Moshe, 2018, eq.22 and p150):…”

Recovering Latent Variables by Matching

“…Lastly, note that the methods introduced in this paper naturally deliver counterparts to the Mallows algorithm for other models beyond deconvolution, such as general linear independent factor models. 8 Strictly speaking, Mallows (2007) 1 for all i = 1, ..., N at the end of step s, and then applies the random permutation σ (s+1) to the new X (s+1) values. This difference with the algorithm outlined here turns out to be immaterial, since the composition of σ (s+1) and σ (s) is also a random permutation of {1, ..., N }.…”