Squared-residual autocorrelations have been found useful in detecting nonlinear types of statistical dependence in the residuals of fitted autoregressive-moving average (ARMA) models (Granger and Andersen, 1978; Miller, 1979). In this note it is shown that the normalized squared-residual autocorrelations are asymptotically unit multivariate normal. The results of a simulation experiment confirming the small-sample validity of the proposed tests is reported.

We generalize the Gaussian mixture transition distribution (GMTD) model introduced by Le and co-workers to the mixture autoregressive (MAR) model for the modelling of non-linear time series. The models consist of a mixture of K stationary or non-stationary AR components. The advantages of the MAR model over the GMTD model include a more full range of shape changing predictive distributions and the ability to handle cycles and conditional heteroscedasticity in the time series. The stationarity conditions and autocorrelation function are derived. The estimation is easily done via a simple EM algorithm and the model selection problem is addressed. The shape changing feature of the conditional distributions makes these models capable of modelling time series with multimodal conditional distributions and with heteroscedasticity. The models are applied to two real data sets and compared with other competing models. The MAR models appear to capture features of the data better than other competing models do.

Time series with a changing conditional variance have been found useful in many applications. Residual autocorrelations from traditional autoregressive moving-average models have been found useful in model diagnostic checking. By analogy, squared residual autocorrelations from fitted conditional heteroskedastic time series models would be useful in checking the adequacy of such models. In this paper, a general class of squared residual autocorrelations is defined and their asymptotic distribution is obtained. The result leads to some useful diagnostic tools for statisticians using conditional heteroskedastic time series models. Some simulation results and an illustrative example are also reported.

This article introduces a new model to capture simultaneously the mean and variance asymmetries in time series. Threshold non-linearity is incorporated into the mean and variance specifications of a stochastic volatility model. Bayesian methods are adopted for parameter estimation. Forecasts of volatility and Value-at-Risk can also be obtained by sampling from suitable predictive distributions. Simulations demonstrate that the apparent variance asymmetry documented in the literature can be due to the neglect of mean asymmetry. Strong evidence of the mean and variance asymmetries was detected in US and Hong Kong data. Asymmetry in the variance persistence was also discovered in the Hong Kong stock market.