This paper deals with the factor modeling for high-dimensional time series based on a dimension-reduction viewpoint. Under stationary settings, the inference is simple in the sense that both the number of factors and the factor loadings are estimated in terms of an eigenanalysis for a nonnegative definite matrix, and is therefore applicable when the dimension of time series is on the order of a few thousands. Asymptotic properties of the proposed method are investigated under two settings: (i) the sample size goes to infinity while the dimension of time series is fixed; and (ii) both the sample size and the dimension of time series go to infinity together. In particular, our estimators for zero-eigenvalues enjoy faster convergence (or slower divergence) rates, hence making the estimation for the number of factors easier. In particular, when the sample size and the dimension of time series go to infinity together, the estimators for the eigenvalues are no longer consistent. However, our estimator for the number of the factors, which is based on the ratios of the estimated eigenvalues, still works fine. Furthermore, this estimation shows the so-called "blessing of dimensionality" property in the sense that the performance of the estimation may improve when the dimension of time series increases. A two-step procedure is investigated when the factors are of different degrees of strength. Numerical illustration with both simulated and real data is also reported.

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This document is the author's final manuscript accepted version of the journal article, incorporating any revisions agreed during the peer review process. Some differences between this version and the published version may remain. You are advised to consult the publisher's version if you wish to cite from it. Estimation of Latent Factors for SummaryThis paper deals with the dimension reduction of high-dimensional time series based on common factors. In particular we allow the dimension of time series p to be as large as, or even larger than, the sample size n. The estimation of the factor loading matrix and the factor process itself is carried out via an eigenanalysis of a p × p non-negative definite matrix. We show that when all the factors are strong in the sense that the norm of each column in the factor loading matrix is of the order p 1/2 , the estimator of the factor loading matrix is weakly consistent in L 2 -norm with the convergence rate independent of p. This result exhibits clearly that the 'curse' is canceled out by the 'blessing' of dimensionality. We also establish the asymptotic properties of the estimation when factors are not strong. The proposed method together with their asymptotic properties are further illustrated in a simulation study. An application to an implied volatility data set, together with a trading strategy derived from the fitted factor model, is also reported. Short Title: Estimation of Large Latent Time Series Factors.Some key words: Convergence in L 2 -norm; Curse and blessing of dimensionality; Dimension reduction; Eigenanalysis; Factor model.

ARCH and GARCH models directly address the dependency of conditional second moments, and have proved particularly valuable in modelling processes where a relatively large degree of fluctuation is present. These include financial time series, which can be particularly heavy tailed. However, little is known about properties of ARCH or GARCH models in the heavy–tailed setting, and no methods are available for approximating the distributions of parameter estimators there. In this paper we show that, for heavy–tailed errors, the asymptotic distributions of quasi–maximum likelihood parameter estimators in ARCH and GARCH models are nonnormal, and are particularly difficult to estimate directly using standard parametric methods. Standard bootstrap methods also fail to produce consistent estimators. To overcome these problems we develop percentile–t, subsample bootstrap approximations to estimator distributions. Studentizing is employed to approximate scale, and the subsample bootstrap is used to estimate shape. The good performance of this approach is demonstrated both theoretically and numerically.

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