A multivariate time series analysis based on frequency domain decomposition and Hilbert space projection in the presence of missing data

The objective of this study is to develop a performance measurement tool that companies can use to assess their historical performance in terms of decision-making by their management as compared to other companies in their sector. Using the daily close value of the stock price as the only available information, we need to extract the fundamental performance of the companies. By applying the Kolmogorov-Zurbenko (KZ) filter in conjunction with frequency domain techniques, we filter out the seasonal and daily fluctuations introduced by various market forces and retain the long term component. To validate our results we also ensure that the transfer function of the filter is not altered by the presence of missing observations (weekend effect). Specifically, we show theoretically and by simulation that the difference between the actual and theoretical transfer function asymptotically follows $c22$ and it vanishes when the missing rate approaches zero.; After the successful decomposition, we model and compare each component separately (Iasonos and Zurbenko, 2001). The long term is captured by a linear trend, the seasonal is estimated by an annual empirical estimate, and the short term component is modeled by a first-order autoregressive model. For the scope of our analysis, we focus on the long term component of the data since by definition, it represents the company's underlying performance. Subsequently, we remove the market factor by subtracting the L₂ projection of the company to the market leaving out orthogonal residuals. These time-dependent vectors provide a performance evaluation tool since they show at any specific point in time how the company was performing relative to the overall market and relative to their competitors. Furthermore, lagged correlation coefficient analysis can identify the leaders and the delayed companies within a sector. In addition, the above methodology is followed to model and compare two NYSE indexes with the NASDAQ index. The results verify that the model we introduce is robust and can be used to model any bond or stock market index or well established and dot-com companies.; Finally, we conclude by comparing our approach and methodology with existing financial models, such as the CAPM and the GARCH models.