The Fixed Volatility Bootstrap for a Class of Arch(q) Models

The ‘fixed regressor’ – or ‘fixed design’ – bootstrap is usually considered in the context of classic regression, or conditional mean (autoregressive) models, see for example, Gonçalves and Kilian, 2004). We consider here inference for a general class of (non)linear ARCH models of order q, based on a ‘Fixed Volatility’ bootstrap. In the Fixed Volatility bootstrap, the lagged variables in the conditional variance equation are kept fixed at their values in the original series, while the bootstrap innovations are, as is standard, resampled with replacement from the estimated residuals based on quasi maximum likelihood estimation. We derive a full asymptotic theory to establish validity for the Fixed Volatility bootstrap applied to Wald statistics for general restrictions on the parameters. A key feature of the Fixed Volatility bootstrap is that the bootstrap sample, conditional on the original data, is an independent sequence. Inspection of the proof of bootstrap validity reveals that such conditional independence simplifies the asymptotic analysis considerably. In contrast to other bootstrap methods, one does not have to take into account the conditional dependence structure of the bootstrap process itself. We also investigate the finite sample performance of the Fixed Volatility bootstrap by means of a small scale Monte Carlo experiment. We find evidence that for small sample sizes, the Fixed Volatility bootstrap test is superior to the asymptotic test, and to the recursive bootstrap-based test. For large samples, both bootstrap schemes and the asymptotic test share properties, as expected from the asymptotic theory. Its appealing theoretical properties, together with its good finite sample performance, suggest that the proposed Fixed Volatility bootstrap may be an important tool for the analysis of the bootstrap in more general volatility models.