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This note considers a three-step non-Gaussian quasi-maximum likelihood estimation (TS-NGQMLE) of the double autoregressive model with its asymptotics, which improves efficiency of the GQMLE and circumvents inconsistency of the NGQMLE when the innovation is heavy-tailed. Under mild conditions, the estimator not only can achieve consistency and asymptotic normality regardless of density misspecification of the innovation, but also outperforms the existing estimators, such as the GQMLE and the (weighted) least absolute deviation estimator, when the innovation is indeed heavy-tailed.

This note reconsiders the marginal density of a threshold moving average process and proposes a simple yet effective numerical algorithm to implement that by solving an associated integral equation. This algorithm can also be applied to calculate stationary probability density or distribution functions of a few other types of nonlinear stationary stochastic processes numerically.

CONDITIONAL QUANTILE VARIATIONAL AUTOENCODER" S1. Additional empirical studies on US equity data. S1.1. Characteristics importance. The usefulness of all 94 characteristics so far has been demonstrated. However, we are not clear which characteristic makes a major contribution to the prediction of returns. To identify those influential characteristics, we compute the variable importance of a given characteristic (Gu, Kelly and Xiu, 2020), which is defined as the reduction in out-of-sample total R 2 by setting all values of this given characteristic to zero while holding the remaining model estimates fixed. After getting the values of variable importance for all 94 characteristics, we normalize these values to sum to one for a given model.

The supplementary contains five parts. Section S.1 discusses information criteria for order selection and regime specification, and provides an example of regime coalescence. Section S.2 gives the extended weighted Nadaraya-Watson (WNW) and modified resampling algorithms with all the parameters estimated. In order to compare with the diminishingthreshold-effect framework in Chen et al. ( 2012), Section S.3 discusses on confidence interval construction for thresholds, provides theoretical support and gives simulation results for different distributions. Section S.4 and S.5 contain proofs of the main results and some auxiliary lemmas.