I have developed a recurrent neural network (RNN) model for time-series forecasting. I now want to test its performance against more standard statistical/econometric models such as ARIMA or VAR. The model outperforms ARIMA in a few of the typical datasets used for model testing (shampoo sales, minimum temperatures...).
However, what I am really interested in is finding out how the model performs on macroeconomic and financial time series (both univariate and multivariate). Here is where the problems start. When I apply the model to macroeconomic indicators (GDP, unemployment...) or stock price indices (opening prices, realized volatility), forecasts appear to be very good at first (high R-squared, low RMSE). However, once I compare them with a baseline random walk model (i.e. y_t=y_t-1), I discover that this always gives the best forecast possible. In other words, both ARIMA and RNN models approximate the random walk forecast but always remain below it (this becomes especially clear after parameter tuning where the best estimates are always selected with a lag never greater than one). This has led me to believe that the series I am considering all more or less exhibit a random walk behaviour.
Therefore, I would like to know if anyone could point me to any macroeconomic or financial indicator which has been shown to not exhibit this kind of behaviour (perhaps one with strong seasonal components which can be learned by both ARIMA and RNN models). Ideally this would have daily or monthly observations in order to guarantee as much data as possible, but even quarterly or annually series will do if the time span is long enough.