What is Economic time series b. Discuss the four components of an economic time series, stationary time series, use of filter in time series analysis

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                  STELLA MARIS MTWARA UNIVESITY COLLEGE
                                                  (STEMMUCO)
                (A Constituent College of St Augustine University of Tanzania)
                                                    Stella_Maris_Mtwara_University_College_Logo
FACULTY OF EDUCATION
ATTEMPTED BY:                            LAXFORD   HAULE      
REG ; NUMBER  :                            STE/BSCMS/0001
DEPARTIMENT:                             MATHEMATICS
COURSE TITTLE:                          TIME  SERIES ANALYSIS
COURSE CODE:                              ST 202
COURSE INSTRUCTOR:              DR. BATHO
NATURE OF ASSIGNMENT:      INDIVIDUAL ASSINGMENT
DATE OF SUBMISSION:                26thNOV, 2019
TASK :
                Question one
What is a Time Series Analysis? Provide suitable examples
Question Two
a.       What is Economic time series
b.      Discuss the four components of an  economic time series, stationary time series, use of filter in time series analysis
What   is time analysis?
A time series is a sequential set of data points, measured typically over successive times. It
is mathematically defined as a set of vectors x(t),t = 0,1,2,... where t represents the time
elapsed [21, 23, 31]. The variable x(t) is treated as a random variable
or
A time series is a set of statistics, usually collected at regular intervals. Time series data occur naturally in many application areas.
        • Economics - e.g., monthly data for unemployment, hospital admissions, etc.
         • Finance - e.g., daily exchange rate, a share price, etc.
         • Environmental - e.g., daily rainfall, air quality readings.
         • Medicine - e.g., ECG brain wave activity every 2−8 secs.

Example of the time series

ü     Exchange rate , interest rate ,inflammation rate , national DDP
ü     Retail sales
ü     Number of accident fatalities 

Economic time series
Are the statistical records  of the evolution   of economic processes      through time , generally  compiled for consecutive periods such as   month   , quarters   or year 

Components of a time series , Any time series can contain some or all of the following components:
1.        Trend (T) 2.
2.         Cyclical (C)
3.         Seasonal (S)
4.         Irregular These components may be combined in different ways. It is usually assumed that they are multiplied or added, i.e., yt = T × C × S × I yt = T + C + S + I To correct for the trend in the first case one divides the first expression by the trend (T). In the second case it is subtracted.

 Trend component
The trend is the long term pattern of a time series. A trend can be positive or negative depending on whether the time series exhibits an increasing long term pattern or a decreasing long term pattern. If a time series does not show an increasing or decreasing pattern then the series is stationary in the mean.

Cyclical component
 Any pattern showing an up and down movement around a given trend is identified as a cyclical pattern. The duration of a cycle depends on the type of business or industry being analyzed.

Seasonal component
 Seasonality occurs when the time series exhibits regular fluctuations during the same month (or months) every year, or during the same quarter every year. For instance, retail sales peak during the month of December.

 Irregular component This component is unpredictable. Every time series has some unpredictable component that makes it a random variable. In prediction, the objective is to “model” all the components to the point that the only component that remains unexplained is the random component.
A stationary time series is one whose statistical properties such as mean, variance, autocorrelation, etc. are all constant over time.
Most statistical forecasting methods are based on the assumption that the time series can be rendered approximately stationary i.e. "stationeries" through the use of mathematical transformations.
A stationeries series is relatively easy to predict: you simply predict that its statistical properties will be the same in the future as they have been in the past!   (Recall our famous forecasting quotes.) 
The predictions for the stationeries series can then be "untransformed," by reversing whatever mathematical transformations were previously used, to obtain predictions for the original series. (The details are normally taken care of by your software.) Thus, finding the sequence of transformations needed to stationeries a time series often provides important clues in the search for an appropriate forecasting model
Another reason for trying to stationeries a time series is to be able to obtain meaningful sample statistics such as means, variances, and correlations with other variables. Such statistics are useful as descriptors of future behavior only if the series is stationary. For example, if the series is consistently increasing over time, the sample mean and variance will grow with the size of the sample, and they will always underestimate the mean and variance in future periods. And if the mean and variance of a series are not well-defined, then neither are its correlations with other variables. For this reason you should be cautious about trying to extrapolate regression models fitted to non stationary data.
      Uses of filter in time series analysis
There are several time-series filters commonly used in macroeconomic and financial research to separate the behavior of a time series into trend vs. cyclical and irregular components. These
techniques can usually be expressed in terms of linear algebra, and reliable code exists in other matrix languages for their  implementation.
I briefly describe the concept of time-series filtering, and then present several new implementations of time-series filters for Stata users written in Mata. These routines avoid matrix size constraints and are much faster than previous versions translated from Fortran
written in the ado-file language




REFERENCES
Eubank, R.L., (1988), Spline Smoothing and Nonparametric Regression, Marcel
Dekker Inc. New York.
Hamming, R.W., (1989), Digital Filters: Third Edition, Prentice{Hall Inc.,
Englewood Cli®s, N.J.
Ratkowsky, D.L., (1985), Nonlinear Regression Modelling: A Uni¯ed Approach,
Marcel Dekker Inc. New York.
Reinsch, C.H., (1967), \Smoothing by Spline Functions", Numerische Mathematik





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