Tests for conditional heteroscedasticity with functional data and goodness-of-fit tests for FGARCH models

Functional data objects that are derived from high-frequency financial data often exhibit volatility clustering characteristic of conditionally heteroscedastic time series. Versions of functional generalized autoregressive conditionally heteroscedastic (FGARCH) models have recently been proposed to describe such data, but so far basic diagnostic tests for these models are not available. We propose two portmanteau type tests to measure conditional heteroscedasticity in the squares of financial asset return curves. A complete asymptotic theory is provided for each test, and we further show how they can be applied to model residuals in order to evaluate the adequacy, and aid in order selection of FGARCH models. Simulation results show that both tests have good size and power to detect conditional heteroscedasticity and model mis-specification in finite samples. In an application, the proposed tests reveal that intra-day asset return curves exhibit conditional heteroscedasticity. Additionally, we found that this conditional heteroscedasticity cannot be explained by the magnitude of inter-daily returns alone, but that it can be adequately modeled by an FGARCH(1,1) model.