Bias in the estimation of non-linear transformations of the integrated variance of returns

Volatility models such as GARCH, although misspecified with respect to the data-generating process, may well generate volatility forecasts that are unconditionally unbiased. In other words, they generate variance forecasts that, on average, are equal to the integrated variance. However, many applications in finance require a measure of return volatility that is a non-linear function of the variance of returns, rather than of the variance itself. Even if a volatility model generates forecasts of the integrated variance that are unbiased, non-linear transformations of these forecasts will be biased estimators of the same non-linear transformations of the integrated variance because of Jensen's inequality. In this paper, we derive an analytical approximation for the unconditional bias of estimators of non-linear transformations of the integrated variance. This bias is a function of the volatility of the forecast variance and the volatility of the integrated variance, and depends on the concavity of the non-linear transformation. In order to estimate the volatility of the unobserved integrated variance, we employ recent results from the realized volatility literature. As an illustration, we estimate the unconditional bias for both in-sample and out-of-sample forecasts of three non-linear transformations of the integrated standard deviation of returns for three exchange rate return series, where a GARCH(1, 1) model is used to forecast the integrated variance. Our estimation results suggest that, in practice, the bias can be substantial.