Outlier model

In time series analysis, an outlier model is understood to be a univariate time series model, in which conspicuous observation values ​​are to be discovered and modeled. In contrast to the intervention model , the time t at which the conspicuous observation value occurs is not known. The conspicuous observation value or outlier can appear in two forms:

• as an additive outlier or ${\displaystyle Y_{t}={\begin{cases}X_{t},\qquad \quad \mathrm {f{\ddot {u}}r} \quad t\not =\tau \\X_{t}+\omega ,\quad \mathrm {f{\ddot {u}}r} \quad t=\tau \end{cases}}}$
• as an innovative outlier . ${\displaystyle Y_{t}=X_{t}+{\frac {\Theta (L)}{\Phi (L)}}\omega \varepsilon _{t}^{(\tau )}}$

The additive outlier is only effective in the period in which it occurs. The innovative outlier continues to have an effect in the following periods. This forms the memory of the process. ${\displaystyle \tau }$ ${\displaystyle {\frac {\Theta (L)}{\Phi (L)}}}$

For the development of an outlier model, the process equation is transformed in such a way that one obtains the outlier-contaminated shock variable . You can then proceed as follows: ${\displaystyle e_{t}}$

• Step 1: The outlier-contaminated shock variable is estimated using least squares estimation . ${\displaystyle e_{t}}$
• Step 2: The hypotheses : "Time series has additive outliers" and "Time series has innovative outliers" versus : "Time series has neither an additive nor an innovative outlier" are tested. ${\displaystyle H_{1}}$ ${\displaystyle H_{2}}$ ${\displaystyle H_{0}}$
• 3rd step: This procedure is repeated until no more outliers are detected in the last run.