Time Series AnalysisTIME-SERIES ANALYSIS
When data is collected, observed or recorded at successive intervals of time, such data are referred to as “Time Series” i.e a Time Series consists of statistical data in chronological order (in accordance with time).
In time series analysis, we analyze the past behavior of a variable in order to predict its future behavior.
When we observe numerical data at different points of time, the set of observations are known as “Time Series”.
Time series analysis is one quantitative method which is used to determine patterns in the data collected over regular intervals of time. We project these patterns to arrive at an estimate to cope up with uncertainty about the future.
Ex. Data of production, sales, imports etc. at different points of time.
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Reduced inventory costs.
Educated guess.
Expert opinions.
Past history of data values, known as a time series.
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COMPONENTS OF
TIME SERIES
Secular Trend
The general movement persisting over a long period of time represented by the diagonal line drawn through the irregular curve is called Secular Trend.
The general tendency of the data to grow or decline over a long period of time.
Sudden, Erratic and short term movements in either direction have nothing to do with trend.
The steady increase in cost of living recorded by CPI over time.
Example: GDP growth, population growth, prices, literacy rate etc.
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Seasonal Variations
SV are the fluctuations which completes the whole sequence of change within the span of a year and tend to be repeated year after year.
When a repetitive pattern is observed over some time horizon, the series is said to have seasonal behavior.
Seasonal effects are usually associated with calendar or climatic changes.
It includes any kind of variation which is of periodic natures & whose repeating cycles are of relatively short durations.
SV can be because of:-
- Climate and weather conditions
- Customs, traditions & habits.
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3. Cyclical Variations
These refers to the recurrent variations in time series that usually last longer than a year and are regular.
Cyclical fluctuations are long term movements that represent consistently recurring rises and declines in activity.
Most common Example: Business cycles. The time between hitting peaks or falling to low points is at least one year or could be more.
Irregular Variations
Refers to variations in business activities which do not repeat in a definite pattern.
These are variations caused by unpredictable factors like sudden political instability, earthquakes, strikes, wars etc.
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The Trend Component
decline, or constant.
-- greater supply of products and services
Technology -- impacts on efficiency, supply, and demand
Innovation -- impacts efficiency as well as supply and demand
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time each year.
Weather -- both outdoor and indoor activities can impact
demand because of the number of people involved
-- supplies of products and services may depend on the
weather
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The Cyclical Component
Similar to seasonal variations except that there is likely not a
relationship to the time of the year.
Examples of cyclical influences include:
Inflation/deflation -- energy costs, wages and salaries, and
government spending
Consequences of unique events -- severe weather, law suits
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These are short-term effects, usually. We treat them as
independent from one time period to the next. The length of
the duration of these effects would then be shorter than one
time period, that is, one month for monthly data, one year for
annual data.
USES OF A TIME SERIES
It enables us to study the past behavior of the phenomenon under consideration. Ex: Effectiveness of recruiting prog. Organised by University
It helps to study the components which are of paramount importance to a businessman in the planning of future operations and in the formulation of executive and policy decisions.
It helps to compare the actual current performance or accomplishments with the expected ones and analyze the causes of such variations.
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Least Squares Methods.
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BY METHOD OF LEAST SQUARES
Let Yc=a+bX represents equation of a straight line, where:--
Yc : represents calculated values of Y.
a : designates the Y-intercept.
b : represents the slope of the line, i.e., rate of change of ‘Y’ per unit change in ‘X’.
X : The ‘X’ variable in time series analysis represents time.
In order to determine the values of constants ‘a’ and ‘b’, the following two normal equations are to be solved:-
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Ques 1. Determine the trend line which best fits the following data and also find the trend values for the given years.
Year
Deviations from middle year: (X), i.e. 2002
X2
XY
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Ques 2. Given below are the figures of production (in lakh kg.) of a sugar factory. Fit a straight line trend by the least square method and tabulate the trend. Also estimate the trend for the year 2006.
Year
Production
1999
40
2000
45
2001
46
2002
42
2003
47
2004
50
2005
46
X2
XY
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Ques 3. Fit a straight line trend by the method of least squares to the following data. Assuming that the same rate of change continues what would be the predicted earnings for the year 1980?
Year
X2
XY
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Qs 4. From the data given below fit a straight line trend by the method of least squares and find the trend values. Calculate the estimated milk consumption for the year 1997, assuming same trend continues.
Year
Deviations from middle year: i.e. 1991.5
Multiplying deviations by 2 (X)
X2
XY
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Qs 5. The following data show the experience of machine operators and their performance ratings as given by the number of good parts turned out per 100 pieces.
Develop a linear trend for this data and estimate the probable performance if an operator has 10 years experience.
Operator experience
Performance Rating