Learning CHARME models with neural networks
This work provides incremental theoretical foundations for applying neural networks to time series modeling in econometrics or finance.
The authors established theoretical properties for the CHARME time series model under weaker conditions than previous literature, proving stationarity, ergodicity, and weak dependence, and developed a neural network-based learning theory with guarantees of strong consistency and asymptotic normality for parameter estimation.
In this paper, we consider a model called CHARME (Conditional Heteroscedastic Autoregressive Mixture of Experts), a class of generalized mixture of nonlinear nonparametric AR-ARCH time series. Under certain Lipschitz-type conditions on the autoregressive and volatility functions, we prove that this model is stationary, ergodic and $τ$-weakly dependent. These conditions are much weaker than those presented in the literature that treats this model. Moreover, this result forms the theoretical basis for deriving an asymptotic theory of the underlying (non)parametric estimation, which we present for this model. As an application, from the universal approximation property of neural networks (NN), we develop a learning theory for the NN-based autoregressive functions of the model, where the strong consistency and asymptotic normality of the considered estimator of the NN weights and biases are guaranteed under weak conditions.