Statistics for structural break detection and their application to forecasting and statistical process control

Most econometrics and forecasting models rely on the key assumptions that there is no structural change in the model, which means model parameters are constant over time. Model stability is thus necessary for prediction and econometric inference. Any changes in the relationship among the variables will cause difficulties in interpreting and analyzing the regression results for making decisions and policies. The need for pursuing stable models has received broad attention over the years and this has resulted in a rich literature about developing tests of model stability. Centered CUSUMS (Inclan and Tiao 1994), which has been developed as one of the most well-recognised structural break detection methods, has been one of the focuses in our research on improving structural break detection. In this thesis, we propose a modified statistic called "Modified Centered CUSUMS" and a new statistic called "SUMSRM" for structural break detection. In addition, we further extend both our modified and new statistics into two application areas: forecasting and Statistical Process Control (SPC). The modified statistic and the new statistic have been developed based on our detailed analysis of Centered CUSUMS. Our operating condition analysis reveals that the increase of the pre-break data size is able to enhance the structural break detection performance of Centered CUSUMS. On the contrary, increasing the post-break data size weakens its detection performance. The empirical evidence shows that it is quite difficult for Centered CUSUMS to detect the structural change if the pre-break data size is relatively small while its post-break data size is relatively large. We propose Modified Centered CUSUMS to overcome this weakness through searching for an appropriate post-break data size. Then, we further analyze the bias of the estimated break location of Centered CUSUMS. The analysis shows that the bias of the estimated break location will increase greatly when there is a large change in the regression coefficients. However, there is no direct relationship between the bias of the estimated break location and the change of the variance. In order to minimize the bias of the estimated break location when there is a large change in the regression coefficients, we propose the new SUMSRM Statistic based on the sliding window prediction residual and the square deviation about the median. While the sliding window prediction residual can reduce the bias of the estimated break location, the square deviation about the median increases the statistic's sensitivity to the structural change. The empirical evidence shows that SUMSRM can effectively minimize the bias of the estimated break location when there is a change of the regression coefficients. In the research, our Modified Centered CUSUMS and SUMRUM are applied to time series prediction to improve the forecasting performance. We analyze several common existing forecasting approaches such as Rolling Window, Exponential Weight Regression and Reversed CUSUMS for the structural change situation. These approaches ignore either the structural break location or some relevant important data before the most recent break, or even both. All these limitations weaken the forecasting performance. In order to overcome these problems, we propose a data selection method named "Accumulated Segment Method", which consists of two processes: Splitting and Combining. We first use our proposed structural break detection methods to detect all possible break points, and split the time series into multiple segments. Then, we group all segments together if their structure is the same as the most recent segment. We call this group of segments "Accumulated Segment". The data within the Accumulated Segment is supposed to have the same structure, and they will be used to train the forecasting model. The empirical evidence shows that our Accumulated Segment Method outperforms Rolling Window, Exponential Weight Regression and Reverse CUSUMS. Another application area of concern in our research is Statistical Process Control (SPC). We extend our Modified Centered CUSUMS and SUMSRM; and propose the Special Version of Modified Centered CUSUMS control chart (SVMC) and Special Version of SUMSRM control chart (SVS) for phase I. Control chart is one of the important tools in Statistical Process Control. The aim of the control chart is to detect the anomaly, which is the shift of the in-process parameters such as process mean and process variance. There is a clear distinction between Phase I control chart and Phase II control chart. Phase I control chart, which is our focus in SPC, mainly analyzes and detects the parameter shift for historical data. If no shift is detected in Phase I, in-process parameters will be estimated and used in phase II for checking the anomaly on line. Our research contributes to the detection performance of the process parameter shift in Phase I which is one of the determining factors for the success of the control chart in Phase II. In our research, we compare the process shift detection performance of SVS and SVMC with the well-recognized and very effective Phase I Shewhart X control chart and MR control chart. Empirical evidence shows that the SVMC and SVS outperform Phase I Shew hart X control chart and MR control chart.