The Bounded Influence Estimator Explained
ARCH (autoregressive conditional heteroskedasticity) models recognize the presence of successive periods of relative volatility and stability. The error variance, conditional on past information, evolves over time as a function of past errors. The model was introduced by Engle [1982]. Bollerslev [1986] proposed the GARCH (generalized ARCH) conditional variance specification that allows for a parsimonious parameterization of the lag structure. Considerable interest has been in applications of ARCH/GARCH models to high frequency financial time series.
Among all assumptions of ARCH (autoregressive conditional heteroskedasticity) models, a very strong one is the conditional normal distribution of the disturbance. However, numerous studies have shown that the distribution of the changes in exchange rates is, unconditionally as well as conditionally, far from being normal. In fact, leptokurtosis and skewness are frequently present. Hence, the normality assumption seems to be inadequate and often leads to spurious or inefficient inferences. This is mainly due to the fact that exchange rates are contaminated by some outliers or extreme values so that the conditional distribution looks heavy-tailed. To account for heavy tails of the conditional distribution, student-t, among other alternatives, is often adopted instead of normal.
However, estimated residuals from GARCH (generalized ARCH with conditional variance specification that allows for a parsimonious parameterization of the lag structure) models are still frequently observed with excess kurtosis. An alternative approach is to study the implied standard deviation (ISDs) derived from currency and exchange rate options. Previous studies indicated that ISDs were either biased forecasts for future volatility, or less efficient in predicting than historical time-series (Lamoureux and Lastrapes, 1993; Canina and Figlewski, 1993; Jorion, 1995). Andersen et al. (2001) constructed model-free estimates of daily exchange rate volatility. when conditional student-t errors are allowed (Franses and Ghijsels, 1999).
Alternatively, Hsieh (1989b) and Nelson (1991) used the generalized error distribution (GED), which encompasses the normal, exponential, and uniform distributions. Nelson (1991), nevertheless, noted that the GED has only one parameter to control the shape of the conditional distribution, which may not be flexible enough in the presence of many outliers. This paper proposes a BIE that is robust against departure from normality (of the conditional distribution) to describe the behavior of exchange rate changes. BIE limits the influence of any small subset of the data and is asymptotically normal.
By construction, BIE provides a mechanism to detect over-influential observations and limit their impact on the parameter estimation. In this paper, the BIE is used with GARCH to identify additive outliers (AO) and other outliers caused by abnormal information arrivals that may be triggered by changes in domestic policies and international shocks. The identification of outliers allows us to analyze major economic and political factors that are not described by the statistical model but directly contribute to dramatic changes in exchange rates. Balke and Fomby (1994) and Dijk et al. (1999) found that neglecting AOs can erroneously suggest misspecification or inadequate descriptive models for financial (particularly ARCH) modeling.
Franses and Ghijsels (1999) documented that neglected AOs substantially dampen the forecasting properties of GARCH models. The proposed BIE brings robustness to GARCH models against outliers including AOs. Previous studies attempted to detect and remove an outlier, in order to obtain “true” estimators of GARCH models. However, one concern arises that no observation can be regarded as an outlier with 100% assurance. Hence, an observation may be deleted by error. In addition, it is widely contended that an outlier may contain important information indeed, though its influence should not be assigned as high as its nominal magnitude suggests. The BIE identifies abnormal deviations from normality and downweights the influence of these observations accordingly. Thus, it achieves efficiency and robustness simultaneously. In this paper, the performance of the BIE will be compared with the maximum likelihood estimate (MLE), and a semiparametric estimator (SP) (Engle and Gonzalez-Rivera, 1991). Issues related to the assumption of the distribution such as non-normality, leptokurtosis, and outlying observations will also be addressed.