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This paper establishes consistency and asymptotic normality of the generalized quasi-maximum likelihood estimate (GQMLE) for a general class of periodic conditionally heteroskedastic time series models (PCH). In this class of models, the volatility is expressed as a measurable function of the infinite past of the observed process with periodically time-varying parameters, while the innovation is an independent and periodically distributed sequence. In contrast with the aperiodic case, the proposed GQMLE is rather based on S instrumental density functions where S is the period of the model while the corresponding asymptotic variance is in a “sandwich” form. Application to the periodic asymmetric power GARCH model is given. Moreover, we also discuss how to apply the GQMLE to the prediction of power problem in a one-step framework and to PCH models with complex periodic patterns such as high frequency seasonality and non-integer seasonality. [ABSTRACT FROM AUTHOR]

This paper proposes a stochastic volatility model (PAR-SV) in which the log-volatility follows a first-order periodic autoregression. This model aims at representing time series with volatility displaying a stochastic periodic dynamic structure, and may then be seen as an alternative to the familiar periodic GARCH process. The probabilistic structure of the proposed PAR-SV model such as periodic stationarity and autocovariance structure are first studied. Then, parameter estimation is examined through the quasi-maximum likelihood (QML) method where the likelihood is evaluated using the prediction error decomposition approach and Kalman filtering. In addition, a Bayesian MCMC method is also considered, where the posteriors are given from conjugate priors using the Gibbs sampler in which the augmented volatilities are sampled from the Griddy Gibbs technique in a single-move way. As a-by-product, period selection for the PAR-SV is carried out using the (conditional) deviance information criterion (DIC). A simulation study is undertaken to assess the performances of the QML and Bayesian Griddy Gibbs estimates in finite samples while applications of Bayesian PAR-SV modeling to daily, quarterly and monthly S&P 500 returns are considered. [ABSTRACT FROM AUTHOR]

This paper proposes a class of generalized random coefficient autoregressions ( $$GRCA$$ ) in which the autoregressive coefficient is a linear regression of the innovation, so that the corresponding volatility is linear in parameters while having a quadratic expression. The proposed model allows a flexible representation, including level and volatility asymmetries, while being fairly simple to implement. We study the dynamic structure of the model and we propose a four-stage weighted least squares estimate ( $$4SWLSE$$ ) for which we establish consistency and asymptotic normality in both stationary and nonstationary cases. The proposed $$4SWLSE$$ , with a closed form, has the same asymptotic distribution as the quasi-maximum likelihood estimate under the same mild assumptions. Application to real data is given. [ABSTRACT FROM AUTHOR]

For an ARCH model, we propose a multistage weighted least squares (WLS) estimate which consists of repeated WLS procedures until the corresponding asymptotic variance equals that of the quasi-maximum likelihood estimate (QMLE). At every stage, the current estimate is of a WLS type weighted by the squared conditional variance evaluated at the estimate of the previous stage. Initially, the weighting parameter is any fixed and known value in the parameter space. The procedure provides, without any moment requirement, an asymptotically Gaussian estimate having the same asymptotic distribution as the QMLE even in the unstable case. Apart from the initialization stage, two additional stages are required in the stable case to obtain the same asymptotic distribution as the QMLE, while in the unstable case only one stage is enough. So in all, the proposed procedure involves three stages WLS in the stable case and two stages WLS in the unstable case. [ABSTRACT FROM AUTHOR]

This paper studies asymptotic properties of the quasi maximum likelihood and weighted least squares estimates (QMLE and WLSE) of the conditional variance slope parameters of a strictly unstable ARCH model with periodically time varying coefficients (PARCH in short). The model is strictly unstable in the sense that its parameters lie outside the strict periodic stationarity domain and its boundary. Obtained from the regression form of the PARCH, the WLSE is a variant of the least squares method weighted by the square of the conditional variance evaluated at any fixed value in the parameter space. In calculating the QMLE and WLSE, the conditional variance intercepts are set to any arbitrary values not necessarily the true ones. The theoretical finding is that the QMLE and WLSE are consistent and asymptotically Gaussian with the same asymptotic variance irrespective of the fixed conditional variance intercepts and the weighting parameters. So because of its numerical complexity, the QMLE may be dropped in favor of the WLSE which enjoys closed form. [ABSTRACT FROM AUTHOR]

This paper studies the covariance structure and the asymptotic properties of Yule–Walker (YW) type estimators for a bilinear time series model with periodically time-varying coefficients. We give necessary and sufficient conditions ensuring the existence of moments up to eighth order. Expressions of second and third order joint moments, as well as the limiting covariance matrix of the sample moments are given. Strong consistency and asymptotic normality of the YW estimator as well as hypotheses testing via Wald’s procedure are derived. We use a residual bootstrap version to construct bootstrap estimators of the YW estimates. Some simulation results will demonstrate the large sample behavior of the bootstrap procedure. [ABSTRACT FROM AUTHOR]