One remarkable feature of wavelet decomposition is that the waveletcoefficients are localized, and any singularity in the input signals can only affect the waveletcoefficients at the point near the singularity. The localized property of the wavelet coefficientsallows us to identify the singularities in the input signals by studying the wavelet coefficients atdifferent resolution levels. This paper considers wavelet-based approaches for the detection ofoutliers in time series. Outliers are high-frequency phenomena which are associated with the waveletcoefficients with large absolute values at different resolution levels. On the basis of thefirst-level wavelet coefficients, this paper presents a diagnostic to identify outliers in a timeseries. Under the null hypothesis that there is no outlier, the proposed diagnostic is distributedas a χ_1~2. Empirical examples are presented to demonstrate the application of the proposeddiagnostic.
The concept of cointegration describes an equilibrium relationship among a set of time-varying variables, and the cointegrated relationship can be represented through an error-correction model (ECM). The error-correction variable, which represents the short-run discrepancy from the equilibrium state in a cointegrated system, plays an important role in the ECM. It is natural to ask how the error-correction mechanism works, or equivalently, how the short-run discrepancy affects the development of the cointegrated system? This paper examines the effect or local influence on the error-correction variable in an error-correction model. Following the argument of the second-order approach to local influence suggested by reference [5], we develop a diagnostic statistic to examine the local influence on the estimation of the parameter associated with the error-correction variable in an ECM. An empirical example is presented to illustrate the application of the proposed diagnostic. We find that the short-run discre pancy may have strong influence on the estimation of the parameter associated with the error-correction model. It is the error-correction variable that the short-run discrepancies can be incorporated through the error-correction mechanism.
