• Kernel estimation;
  • long memory property;
  • non-parametric ARCH;
  • non-parametric cointegrating regression;
  • non-stationarity;
  • volatility persistence

Summary  We investigate a new non-stationary non-parametric volatility model, in which the conditional variance of time series is modelled as a non-parametric function of an integrated or near-integrated covariate. Importantly, the model can generate the long memory property in volatility and allow the unconditional variance of time series to be time-varying. These properties cannot be derived from most existing non-parametric or semi-parametric volatility models. We show that the kernel estimate of the model is consistent and its asymptotic distribution is mixed normal. For an empirical application of the model, we study the daily S&P 500 index return volatility using the VIX index as the covariate. It is shown that our model performs reasonably well both in within-sample and out-of-sample forecasts.