Additive Outlier Detection and Estimation for the Logarithmic Autoregressive Conditional Duration Model

Abstract

This study investigates the influences of additive outliers on financial durations. An outlier test statistic and an outlier detection procedure are proposed to detect and estimate outlier effects for the logarithmic Autoregressive Conditional Duration (Log-ACD) model. The proposed test statistic has an exact sampling distribution and performs very well, in terms of size and power, in a series of Monte Carlo simulations. Furthermore, the test statistic is robust to several alternative distribution assumptions. An empirical application shows that parameter estimates without considering outliers tend to be biased.

Keywords:

• Additive outlier
• Duration
• Log-ACD
• Lognormal distribution

Mathematics Subject Classification:

• Primary 62M10
• Secondary 62J20

Notes

The empirical results of Lunde (1999) and Grammig and Maurer (2000) indicate that the inverted U-shaped form of hazard function is strongly supported by observed duration data.

The AO property in the original duration series still keeps in the logarithmically transformed duration series since the logarithmic transformation is monotonic in duration series.

The omit-one method used here is to eliminate the impact of the potential outlier; see the proof in the Appendix.

Multiple duration outliers could be detected using the sequential detection method in which one single outlier is detected at each iterative step. Nevertheless, Chen and Liu (1993) indicated that the sequential detection is prone to producing biased estimates, especially for adjacent outliers.

Note: α and β are the parameters of the conditional expected duration equation of the lognormal Log-ACD model. σ² is the shape parameter of the lognormal distribution. Power is calculated as the proportion of the number of samples of correct outlier identification over 10,000 samples. Outlier size denotes the magnitude deviated from , where and s _i represent the sample mean and standard deviation of logarithmic durations of the ith sample. L denotes the time location of the additive outlier.

Another experiment with sample size of 2,000 is conducted both in size and power simulation studies and has the similar results.

Power is calculated as the proportion of number of samples of correct outlier identification over 10,000 samples. The critical value at 5% significance level is equal to 3.9 herein as the sample size is 1,000.

The power increases with the value of the parameter α when the outlier sizes are small (e.g., smaller than 4.00s _i ). This is because that the variance of the DGP is larger for the models with larger α, and the outlier size is just defined as a proportion of standard deviation of each sample.

The results are available from the authors upon request.

Note: T denotes the sample size.

We divide the whole sample into several subsamples to facilitate the additive outlier detection procedures because the proposed method is time-consuming with regard to the large sample size.

The details are available from the authors upon request.

Note: The Weibull density function is , where γ > 0, ξ = Γ(1 + (1/γ)), Γ(·) is the gamma function, and γ is the shape parameter. The Burr density function is , where 0 < φ < κ are two shape parameters. The generalized gamma density function is , where a > 0, δ > 0 are two shape parameters. Original represents the original duration data. Adjusted represents the additive outlier adjusted duration data. The two datasets are all diurnally adjusted using the cubic spline method. Numbers in parentheses denote the standard errors of estimated parameters. Statistically significant estimates at the 5% significance level are shown in boldface.

The duration data used in parameter estimation is the original dataset without diurnally adjusted.