Traditional white noise testing, for example the Ljung-Box test, studies only the autocorrelation function. If its values are sufficiently close to zero, we fail to reject the white noise hypothesis. This is often sufficient for classic theory of linear time series models: autoregressions and moving averages. However, condition of being independent identically distributed random variables with fourth finite moment is stronger than white noise. Sometimes, it is assumed in the models that the sequence is IID but it is tested only for white noise. For example, heteroscedasticity in financial time series means periods of high variance (financial crises) can alternate with periods of low variance (calm times). Time series can be heteroscedastic and therefore not IID but still white noise. Examples include stochastic volatility or GARCH models. Indeed, in this case, absolute values or squares of time series terms are not white noise. Classic white noise tests thus can fail to identify that the sequence is not IID In this article we propose a new construction method for tests which can capture heteroscedasticity. We create a flexible framework generalizing Box-Pierce and Ljung-Box omnibus tests. We apply tests to simulated data, both autoregressive and heteroscedastic.
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