Marc Henry

Long and short memory conditional heteroskedasticity in estimating the memory parameter of levels

Semiparametric estimates of long memory seem useful in the analysis of long financial time series because they are consistent under much broader conditions than parametric estimates. However, recent large sample theory for semiparametric estimates forbids conditional heteroscedasticity. We show that a leading semiparametric estimate, the Gaussian or local Whittle one, can be consistent and have the same limiting distribution under conditional heteroscedasticity as under conditional homoscedasticity assumed by Robinson (1995a). Indeed, noting that long memory has been observed in the squares of financial time series, we allow, under regularity conditions, for conditional heteroscedasticity of the general form introduced by Robinson (1991) which may include long memory behaviour for the squares, such as the fractional noise and autoregressive fractionally integrated moving average form, as well as standard short memory ARCH and GARCH specifications.

Bandwidth Choice in Gaussian Semiparametric Estimation of Long Range Dependence

We consider covariance stationary processes with spectral density which behaves according to a power law around zero frequency, where it can be infinite (long range dependence), finite and positive (short range dependence), or zero (antipersistence). This behaviour is governed by a self-similarity parameter which can be estimated semiparametrically by one of several methods, all of which require a choice of bandwidth. We consider a Gaussian estimate which seems likely to have good efficiency, and whose asymptotic distributional properties have already been determined. The minimum mean squared error optimal bandwidth is heuristically derived and feasible approximations to it are proposed, these being assessed in Monte Carlo experiments and applied to financial and Nile river data.

Robust Automatic Bandwidth for Long Memory

The choice of bandwidth, or number of harmonic frequencies, is crucial to semiparametric estimation of long memory in a covariance stationary time series as it determines the rate of convergence of the estimate, and a suitable choice can insure robustness to some non-standard error specifications, such as (possibly long-memory) conditional heteroscedasticity. This paper considers mean squared error minimizing bandwidths proposed in the literature for the local Whittle, the averaged periodogram and the log periodogram estimates of long memory. Robustness of these optimal bandwidth formulae to conditional heteroscedasticity of general form in the errors is considered. Feasible approximations to the optimal bandwidths are assessed in an extensive Monte Carlo study that provides a good basis for comparison of the above-mentioned estimates with automatic bandwidth selection.

Averaged Periodogram Spectral Estimation with Long Memory Conditional Heteroscedasticity

Semiparametric spectral methods seem particularly appropriate for the analysis of long financial time series, providing they are robust to a variety of forms of conditional heteroscedasticity, which is generally recognized as a dominant feature of financial returns. This paper analyses the averaged periodogram statistic in the framework of a generalized linear process with (possibly long memory) conditional heteroscedasticity in the innovations. It is shown that the averaged periodogram statistic is appropriate for asymptotically normal estimation of the spectrum of a weakly dependent process at frequency zero and for consistent estimation of long memory and stationary cointegration in strongly dependent processes. The asymptotic results are coupled with extensive small sample investigations of the performance of the estimates considered.

Bandwidth Choice, Optimal Rates and Adaptivity in Semiparametric Estimation of Long Memory

Semiparametric estimation of long memory refers to periodogram based estimation of the shape of the spectral density f(λ) at low frequencies, where all but the lowest harmonics of the periodogram are discarded, so as to forego specification of the short range dynamic structure of the time series, and avoid bias incurred when the latter is misspecified. Such a procedure entails an order of magnitude loss of efficiency with respect to parametric estimation, but may be warranted when long series (earth scientific or financial) can be obtained. This paper presents strategies proposed for the choice of bandwidth, i.e. the number of periodogram harmonics used in estimation, with the aim of minimizing this loss of efficiency. Such strategies are assessed with respect to minimax rates of convergence, that depend on the smoothness of |λ|−2d f(λ) (where d is the long memory parameter) in the neighbourhood of frequency zero. The plug-in strategy is discussed in the case where the degree of local smoothness is known a priori, and adaptive estimation of d is discussed for the case where the degree of local smoothness is unknown.

Nonparametric specification analysis of dynamic parametric models

Time series parametric models generally cater to a particular objective, such as forecasting, and it is therefore desirable to judge such models solely on the basis of their performance in the full llment of that objective. We propose a speci cation testing procedure which concentrates power on the parametric models ability to estimate a set of characteristics of the nite dimensional distributions of the process. It is based on the comparison between a nonparametric estimate of the said characteristic and its parametric bootstrap analogue. Applications of this principle are proposed for the assessment of recursive dynamic models in the estimation of conditional means and conditional quantiles for mixing processes and for the estimation of dependence in long memory processes.