A long range dependent model with nonlinear innovations for simulating daily river flows

We present the analysis aimed at the estimation of flood risks of Tisza River in Hungary on the basis of daily river discharge data registered in the last 100 years. The deseasonalised series has skewed and leptokurtic distribution and various methods suggest that it possesses substantial long memory. This motivates the attempt to fit a fractional ARIMA model with non-Gaussian innovations as a first step. Synthetic streamflow series can then be generated from the bootstrapped innovations. However, there remains a significant difference between the empirical and the synthetic density functions as well as the quantiles. This brings attention to the fact that the innovations are not independent, both their squares and absolute values are autocorrelated. Furthermore, the innovations display non-seasonal periods of high and low variances. This behaviour is characteristic to generalised autoregressive conditional heteroscedastic (GARCH) models. However, when innovations are simulated as GARCH processes, the quantiles and extremes of the discharge series are heavily overestimated. Therefore we suggest to fit a smooth transition GARCH-process to the innovations. In a standard GARCH model the dependence of the variance on the lagged innovation is quadratic whereas in our proposed model it is a bounded function. While preserving long memory and eliminating the correlation from both the generating noise and from its square, the new model is superior to the previously mentioned ones in approximating the probability density, the high quantiles and the extremal behaviour of the empirical river flows.