## Graduate Theses and Dissertations

12-2016

Thesis

#### Degree Name

Master of Science in Statistics and Analytics (MS)

#### Department

Mathematical Sciences

Giovanni Petris

Qingyang Zhang

#### Second Committee Member

Avishek Chakraborty

#### Keywords

Pure sciences, Break-points, time series

#### Abstract

A time series is a set of random values collected at equal time intervals; this randomness makes these types of series not easy to predict because the structure of the series may change at any time. As discussed in previous research, the structure of time series may change at any time due to the change in mean and/or variance of the series. Consequently, based on this structure, it is wise not to assume that these series are stationary. This paper, discusses, a method of analyzing time series by considering the entire series non-stationary, assuming there is random change in unconditional mean and variance of the series. Specifically, this paper emphasizes financial time series. The main goal in this process is to break the series into small locally stationary time series on which stationary assumption applies. The most interesting part of this procedure is locating the break-points, where the unconditional mean and/or variance of the series change. After having found what the break-points are, we divide the series into smaller series according to the break points; the number of break-points determines how many small stationary time series we have. The analysis by this method considers each interval on which the series is stationary as an independent time series with its specific parameters. Hence, the overall time series that is naturally nonstationary is broken into small stationary time series that are easier to analyze. Afterwards, by using Bayesian Information Criterion (BIC) we are comparing the local stationary model to the model considering the entire series stationary. In a simulation study with known sample size, unconditional means and variances, for each small stationary series, the result shows that we can locate the exact true break-points when the sample size is greater than 500. After our simulation study, this method is also applied to the real data, S&P 500 series of returns, which is a financial time series. The results obtained by using Maximum Likelihood Estimation (MLE) show that BIC is smaller for the locally stationary model.

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