Nonparametric Conditional Quantile Estimation for Locally Stationary Functional Time Series: Applications in Financial and Economic Modeling

Conditional distribution estimation (CDE) is central in nonparametric forecasting and risk analysis. While considerable progress has been made for finite-dimensional and stationary settings, functional data and nonstationary settings pose new challenges. We propose a Nadaraya-Watson (NW) conditional quantile estimator for regularly mixing locally stationary functional time series (LSFTS). It incorporates three kernel functions: one for time rescaling, another for the functional covariates, and an integrated kernel to act as a cumulative distribution function (CDF) of the response variable. A theoretical framework and the uniform convergence of the estimator were provided. To demonstrate the consistency of the estimator, a numerical experiment was conducted. Finally, we apply the method to financial data, specifically the Nikkei 225.