Business Statistics Notes: Time Series Analysis

Unit – 4: Time Series analysis:
Q.N.1. Define time series. What are its utilities? What the components of time series?
Ans: - Definition: - A time series is a set of statistical observations arranged is chronological order. Time series may be defined as collection of magnitudes of some variables belonging to different time periods. It is commonly used for forecasting.
Utility of Time Series Analysis
The analysis of Time Series is of great significance not only to the economist and businessman but also to the scientist, geologist, biologist, research worker, etc., for the reasons given below:
a)      It helps in understanding past behaviors: By observing data over a period of time one can easily understanding what changes have taken place in the past, Such analysis will be extremely helpful in producing future behavior.
b)      It helps in planning future operations: Plans for the future cannot be made without forecasting events and relationship they will have. Statistical techniques have been evolved which enable time series to be analyzed in such a way that the influences which have determined the form of that series to be analyzed in such a way that the influences which have determined the form of that series may be ascertained.
c)       It helps in evaluating current accomplishments: The performance can be compared with the expected performance and the cause of variation analyzed. For example, if expected sale for 1995 was 10,000 refrigerators and the actual sale was only 9,000, one can investigate the cause for the shortfall in achievement. Time series analysis will enable us to apply the scientific procedure for such analysis.
d)      It facilitates comparison: Different time series are often compared and important conclusions drawn there from. However, one should not be led to believe that by time series analysis one can foretell with 100percnet accuracy the course of future events.
The four components of time series are:
1. Secular trend
2. Seasonal variation
3. Cyclical variation
4. Irregular variation
Secular trend: A time series data may show upward trend or downward trend for a period of years and this may be due to factors like increase in population, change in technological progress, large scale shift in consumer’s demands etc. For example, population increases over a period of time, price increases over a period of years, production of goods on the capital market of the country increases over a period of years. These are the examples of upward trend. The sales of a commodity may decrease over a period of time because of better products coming to the market. This is an example of declining trend or downward trend. The increase or decrease in the movements of a time series is called Secular trend.  
Seasonal variation: Seasonal variation are short-term fluctuation in a time series which occur periodically in a year. This continues to repeat year after year. The major factors that are responsible for the repetitive pattern of seasonal variations are weather conditions and customs of people. More woolen clothes are sold in winter than in the season of summer .Regardless of the trend we can observe that in each year more ice creams are sold in summer and very little in Winter season. The sales in the departmental stores are more during festive seasons that in the normal days.
Cyclical variations: Cyclical variations are recurrent upward or downward movements in a time series but the period of cycle is greater than a year. Also these variations are not regular as seasonal variation. There are different types of cycles of varying in length and size. The ups and downs in business activities are the effects of cyclical variation. A business cycle showing these oscillatory movements has to pass through four phases-prosperity, recession, depression and recovery. In a business, these four phases are completed by passing one to another in this order. It has four important characteristics: i) Prosperity ii) Decline iii) Depression iv) Improvement
Irregular variation: Irregular variations are fluctuations in time series that are short in duration, erratic in nature and follow no regularity in the occurrence pattern. These variations are also referred to as residual variations since by definition they represent what is left out in a time series after trend, cyclical and seasonal variations. Irregular fluctuations results due to the occurrence of unforeseen events like floods, earthquakes, wars, famines, etc.
Q.N.2. What are the different models of time series?
Ans: - In Traditional time series analysis, it is ordinarily assumed that there is a multiplicative relationship between the components of time series.        Symbolically, Y=T X S X C X I
                                Where T= Trend
                                S= Seasonal component
                                C= Cyclical component
                                I= Irregular component
                                Y= Result of four components.
Another approach is to treat each observation of a time series as the sum of these four components Symbolically, Y=T + S+ C + I
Q.N.3. Mention the different methods of measuring trend and seasonal variation.
Ans: - The following four methods are commonly used for measuring trends:-
i)        Graphic method
ii)       Semi-average method
iii)     Moving average method
iv)     Method of least squares.
Again, The following methods are commonly used for measuring seasonal variation:-
i)        Method of simple averages
ii)       Ratio to trend method
iii)     Ratio to moving average
iv)     Link relative method.
i) Graphic method: - This is the simplest method of studying trend. The procedure of obtaining a straight line trend is:
a) Plot the time series on a Graph.
b) Examine the direction of the trend based on the plotted information.
c) Draw a straight line which shows the direction of the trend.
The trend line thus obtained can be extended to predict future values.
i) This method is simplest method of measuring trend.
ii) This method is very flexible. I can be used regardless of whether the trend is a straight line or curve.
i)  This method is highly subjective because it depends on the personal judgement of the investigator.
ii)  Since this method is subjective in nature it cannot be used for predictions.
ii) Semi-average method: - Under this method, the given data is divided into two parts. After that an average of each part is obtained which gives two points. Each point is plotted at the mid-point of the class interval covered by the respective part and then the two points are joined by a straight line which gives the required trend line.
i) This method is simple to understand as compared to the moving average method and the method of least square.
ii) This is an objective method of measuring trend as everyone who applies this method gets the same result.
i) It is affected by extreme values.
ii) This method assumes straight relationship between the plotted points whether this exist or not.
iii) Method of moving average: - Under this method the average value for a certain time span is secured and this average is taken as the trend value for the unit of time falling at the middle of the period covered in the calculation of the average. While using this method it is necessary to select a period for moving average.
i) This method is simple to understand and apply.
ii) It is particularly effective if the trend of a series is very irregular.
iii) It is a flexible method of measuring trend because all figures are not changed if a few figures are added to the data.
i) Trend values cannot be computed for all years.
ii) No there is no hard and fast rule for selecting the period of moving average.
iii) this method is not appropriate if the trend situation is not linear.
iv) Method of Least Square: - This method is most commonly used method of measuring trend. It is a mathematical method and a trend line is fitted to the data in such a manner that the following two conditions are satisfied:-
i) the sum of deviation of the actual values from their respective mean is zero.
ii) the sum of square of the deviations of the actual and compute values is least from this line. That is why this method is called method of least square.
The straight line trend is represented by the equation: Y = a + bx
Where, y = denotes the trend values
                a = represents the intercept on y axis.
                b= represents slope of the trend line.
i) This is a mathematical method of measuring trend.
ii) Trend values can be obtained for all the given time periods in the series.
i) This method is more tedious and time consuming.
ii) This method cannot be used to fit the growth curves.
Q.N.4. What is meant by shifting of trend origin and deseasonalised data?
Ans: - Shifting:-Shifting of trend origin means replacing the origin with new base. Shifting can be done by using the following formula: Y = a + b (X + k)
Where k is the number of time units shifted. If the origin is shifted forward in time, k is positive and if shifted backward in time, k is negative.
Deseasonalised Value: - The value which show how things would have been or would be if there were no seasonal fluctuations is called Deseasonalised data. In order to obtain Deseasonalised data, the effect of seasonal variations have to be removed. For this purpose, the actual data is divided by the appropriate seasonal indices.    

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