TIME SERIES

Financial Time Series

Financial Time Series

1. Summary

Introduction

A variable measured in distinct point of time is called “Time Series”. Time Series is a series of successive order of numerical data points that occur at uniform intervals. Values that are variable (monthly overheads, weekly orders and daily sales) and plotted as chronologically ordered data points. Example of time series: 1) Blood pressure in biomedical science 2) Vibration in mechanical engineering 3) Risk evaluation 4) Option pricing 5) Portfolio construction Time series comprises of four components: 1) Trend Variations: The variations that move up and down in a predictable series or pattern. 2) Seasonal variations: those variations that repeat over a period of time such as season,week, month etc 3) Cyclical variations: Those variations that follow on an economic or business cycles or follow their own random cycles. 4) Random variations: It is the variability of a process occurred by erratic and irregular factors that cannot be identified or eliminated. Main goals of time series: 1) It identifies the nature of the phenomenon that is by a sequence of observations. 2) It forecast the future values of the time series variable Objective of Time series: Time series analysis is useful in extracting information from a time series. The objectives of the time series are as follows: 1) Testing of Hypothesis: it is helpful in obtaining the statistical measure of a time series. 2) Forecasting and control: it is helpful in forecasting values and to determine corrective methods to optimize settings and to keep process operating correctly. 3) Prediction: it helps in predicting the future values of a time series. 4) Description of data: it explains the variation of a related time series, which can help in better understanding the relationship. Time series is denoted by Yt,t=1, ...,T. Time series forecasting is performed to optimize the performance of functions such as production, staffing levels, inventory levels etc. The are two main structural variables for forecasting time series: 1) the aggregation level. They include the common periods such as week, month and days in supply chain (inventory optimization) 2) The number of periods that need to be forecasted known as the Horizon.

1.2 Fourier Series Model:

The Fourier series was introduced by Joseph Fourier. It may be defined as an infinite series having constant terms multiplied by cosine and sine functions that can approximate a wide variety of functions. The Fourier Series Model involves a periodic function into sums of oscillating functions that are sine and cosine. A function of real variables x denoted by f(x).this function usually taken for a 2p-periodic function. 2p-periodic function f(x) is integrable on [-p, p], the numbers. These numbers are termed as Fourier coefficients of f. The partial sums for f are Trigonometric polynomials. The functions SNƒ are approximate the function ƒ, and the approximation improves as N becomes infinite. This infinite sum is called the Fourier Series. Applications of Fourier series: It simplifies the analysis of real and periodic valued ...