Abstract
We introduce and investigate some properties of a class of nonlinear time series models based on the moving sample quantiles in the autoregressive data generating process. We derive a test fit to detect this type of nonlinearity. Using the daily realized volatility data of Standard & Poor’s 500 (S&P 500) and several other indices, we obtained good performance using these models in an out-of-sample forecasting exercise compared with the forecasts obtained based on the usual linear heterogeneous autoregressive and other models of realized volatility.
Highlights
We propose a nonlinear autoregressive time series model where the nonlinearity stems from the presence of moving order statistics-linked quantities in the data generating process (DGP)
Before turning to the empirical part, which aims at illustrating the application of the model, we introduce a minimal set of tools that are needed to evaluate the statistical properties of processes with moving quantiles, define some properties of a simple estimator of parameters and propose a statistical test for evaluating the significance of moving sample quantiles (MQ) terms
We show that coupling the MQs with the constrained linear autoregression, where we use the exponential Almon polynomial restriction that is employed extensively in realized volatility forecasting models based on the mixed frequency data sampling (MIDAS) literature, robustly outperforms the standard benchmark, i.e., the heterogeneous autoregression model of realized volatility (HAR-RV) proposed by [30], as well as other constrained and unconstrained models that do not contain the MQ terms
Summary
We propose a nonlinear autoregressive time series model where the nonlinearity stems from the presence of moving order statistics-linked quantities in the data generating process (DGP). In the models with moving order statistics and moving sample quantiles (MQ) as their generalization, the order statistics-linked effects directly reveal the importance of the different sizes of realizations observed within a sample These models do not contain any exogenously given threshold levels, because they evolve endogenously depending on the realizations of the recent values of the process. They convey an infinite number of potential regimes, which are represented by the specific local distributions that are characterized by the order statistics/quantiles in the moving samples. The proofs of the propositions are given in Appendix A (unless explicitly stated otherwise), while Appendices B to E provide additional information about some properties of the process and the empirical applications
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