ARCH/GARCH representations of financial series usually attempt to model the serial correlation structure of squared returns. While it is undoubtedly true that squared returns are correlated, there is increasing empirical evidence of stronger correlation in the absolute returns than in squared returns (Granger, Spear and Ding 2000). Rather than assuming an explicit form for volatility, we adopt an approximation approach; we approximate the γ-th power of volatility by an asymmetric GARCH function with the power index γ chosen so that the approximation is optimum. Asymptotic normality is established for both the quasi-maximum likelihood estimator (qMLE) and the least absolute deviations estimator (LADE) estimators in our approximation setting. A consequence of our approach is a relaxation of the usual stationarity condition for GARCH models. In an application to real financial data sets, the estimated values for γ are found to be close to one, consistent with the stylised fact that strongest autocorrelation is found in the absolute returns. A simulation study illustrates that the qMLE is inefficient for models with heavy-tailed errors, while the LADE estimation is more robust.

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We put forward a new method to construct jump‐robust estimators of integrated volatility, namely realized information variation (RIV) and realized information power variation (RIPV). The ‘information’ here refers to the difference between two‐grid of ranges in high‐frequency intervals, which preserves continuous variation and eliminates jump variation asymptotically. We show that such kind of estimators have several superior statistical properties, i.e., the estimators are generally more efficient with sufficiently using the opening, high, low, closing (OHLC) data in high‐frequency intervals, and have faster jump convergence rate due to a new type of construction. For example, the RIV is much more efficient than the estimators that only use closing prices or ranges, and the RIPV has faster jump convergence rate at Op(1/n), while the other (multi)power‐based estimators are usually Opfalse(1false/nfalse). We also extend our results to integrated quarticity and higher‐order variation estimation, and then propose the corresponding jump testing method. Simulation studies provide extensive evidence on the finite sample properties of our estimators and tests, comparing with alternative prevalent methods. Empirical results further demonstrate the practical relevance and advantages of our method.