Modeling Autoregressive Processes with Moving-Quantiles-Implied Nonlinearity

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.

Order statistics from overlapping samples: bivariate densities and regression properties

“…Here we only consider the simplest case of os's from overlapping samples (X 1 , X 2 ) and (X 2 , X 3 ), that is the case of r = 1, m = n = 2. Then (15) gives…”

“…That is, both regressions we are interested in are represented through rather complicated expressions (15) and (16). Thus characterizations or identifiability questions for parent distributions through the form of E(X i:m |X (r) j:n ) or E(X (r) j:n |X i:m ) seems to be a difficult task in such a general framework.…”

“…There are two special cases which until now have been studied in the literature: (1) moving os's, when the subsequent samples are of the same size and have the same size of the overlap -see e.g. Inagaki (1980), David and Rogers (1983), Ishida and Kvedaras (2015) or Balakrishnan and Tan (2016); (let us mention that moving samples have a long history in quality control and time series analysis -in particular, the moving median is a simple robust estimator of location and the moving range is a current measure of dispersion complementing the moving average); (2) special cases of os's from the original and extended sample which except the original sample contains a number of additional observations -see e.g. Siddiqui (1970), Tryfos and Blackmore (1985), Ahsanullah and Nevzorov (2000) or López-Blázquez and Salamanca-Miño (2014).…”