Equity index replication with standard and robust regression estimators

Approximate equity index replication, based on a linear regression setting, is critically reviewed. It is shown that tracking a performance index like the German DAX necessarily leads to violations of basic assumptions of the classical regression model. Violations occur even if the model is formulated in terms of stock price levels. When the model is based on discretely or continuously compounded returns the situation is more critical. Due to these violations, the optimality properties of the regression estimators are generally weak. In the time series context, outliers in financial time series may additionally affect the standard least squares estimator. Despite these critical points, it is argued that regression techniques may still provide a useful tool for index replication. With respect to outliers, robust estimators can potentially provide an alternative to least squares. Hence, apart from least squares, a non-redescending and a redescending robust estimator is fitted. The empirical results for the DAX are obtained with a subset portfolio containing the most heavily weighted index members. Compared to a naive weighting scheme, the results document that least squares estimation highly improves out-of-sample replication performance. Typically, the use of robust estimators does not show replication improvements. However, when substantial market movements are present in the sample, superior replication can be obtained.