Quantiles, expectiles and splines

Abstract

A time-varying quantile can be fitted by formulating a time series model for the corresponding population quantile and iteratively applying a suitably modified state space signal extraction algorithm. It is shown that such quantiles satisfy the defining property of fixed quantiles in having the appropriate number of observations above and below. Like quantiles, time-varying expectiles can be estimated by a state space signal extraction algorithm and they satisfy properties that generalize the moment conditions associated with fixed expectiles. Because the state space form can handle irregularly spaced observations, the proposed algorithms can be adapted to provide a viable means of computing spline-based non-parametric quantile and expectile regressions

Change Point and Trend Analyses of Annual Expectile Curves of Tropical Storms

“…The underlying concept of expectiles was first discussed by Newey and Powell (1987) and further analyzed in several directions, e.g. Efron (1991) or Rossi and Harwey (2009) focused on time-varying expectiles. Most relevant to our setting is the paper by Schnabel and Eilers (2009), which extended the work of Eilers and Marx (1996).…”

Bayesian Analysis of Time-Varying Quantiles Using a Smoothing Spline

“…Under these additional assumptions (i)(ii), if λ m = λ/(2σ 2 η ), then the mode of the distribution of (h(1), h(2), ..., h(n)|y 1 , ..., y n ) converges to the solution of the smoothing spline problem as κ → ∞ (De Rossi and Harvey (2009)). Noting that equation (8) can be represented in the following state space form (see, e.g., Wecker and Ansley (1983), Kohn and Ansley (1987)),…”

“…In this paper, we use the smoothing spline for that purpose as discussed in De Rossi and Harvey (2009) and propose an efficient Bayesian estimation using the MCMC method in which we exploit a state space representation to apply a simulation smoother (de Jong and Shephard (1995), Durbin and Koopman (2002)) to generate the latent time-varying quantiles from the posterior distributions. The model is further extended to incorporate a correlation between the dependent variable and its one-step-ahead quantile.…”

Dependence Uncertainty Bounds for the Expectile of a Portfolio

“…Expectiles are well known in regression analysis [14][15][16]; they are used for forecasting financial time series [17] and estimating VaR and ES [18]. A penalized least squares approach in portfolio optimization was suggested by [19]; the expectile is a special case when a quadratic downside penalty is used.…”