Time Series

 

                                                                                Unit... 1.1

Time series

There is nothing constant in the universe

All ebb and flow, and every shape that’s born

Bears in its womb the seeds of change.

                                   -Ovid

introduction:©

                                The world is dynamic. There is nothing stable or constant in world. Changes are taking place at every moment of time. It is rightly said that the importance of time is great.  Changes are taken place continuously in statistical time series also. It is necessary to study the changes and to classify these changes and  to compare with similar changes in the other series. If we are able to identify the reasons behind these changes, we may predict the future changes by knowing the direction of past changes. Really, is it not the significant advantage? So, in modern world especially in economic and business world the analysis of time series becomes important day by day.

 1.1  Meaning of time series

       Forecasting is often necessary in the field of economics, commerce and management. We know that many variables vary with time. For example if we look at the figures of population, agricultural production, industrial production, exports, sales, employment, prices, electricity consumption etc. we find that the figures change with time.

 Definition:

A set of figures relating to a variable arranged according to time is called a time series.

Consider the following series of production (in thousands of tones) of a company.

 

Year	1964	1965	1966	1967	1968	1969	1970	1971
Production	35	39	44	33	45	54	57	56

 

This is a time series. It is used for forecasting. It is also called as historical series since it describes the history of the variable. Some other definitions of time series are:

According to Morris Hamburg, Time series is a series of statistical observations arranged in a chronological order. The observations in the chronological order indicate the order of occurrence, taken at regular successive intervals of time. The time intervals may be years, months, days, minutes and in some cases seconds also.

Mathematically time series is defined by the values Y₁,  Y₂  , ----- , Y_n, ----of the variable Y at times t₁,  t₂  , ----- , t_n, ----. The time points t₁,  t₂  , ----- , t_n, --- are equidistant. This definition is due to Spigel. Here time series is a function of time i.e. Y = F(t). Thus, in time series time is an independent variable and Y(t) is dependent variable. We denote time series by Y(t) or Y_t. In the form of function the time series is represented as follows:

 

t	t1	t2	t3	----	tn
Yt	Y1	Y2	Y3	----	Yn

 

The examples of time series are:

1.  Daily price of gold

2.  Weekly sales of departmental store

3.  Monthly deposits in a certain bank

4.  Yearly production of food grains of a country

5. Daily record of maximum or minimum temperature of a city

6. Population of a country at census years.

If we carefully observe the figures in the above given example of time series of production, we see that the production is increasing, although for some years it has decreased. The graph of this time series is: 

 

       

Figure 1.1

 

The graph of time series is called as Historigram and its nature is usually zig-zag or haphazard. It is obtained by plotting the data on a graph paper taking the independent variable time (t) along X - axis and dependent variable y along Y -axis.

 

1.2  Components of time series

The factors responsible for the changes in the time series are called as components of time series. There are four components of time series.

1.   The secular trend or trend:  An overall tendency of the series to rise or to fall.

2.   Seasonal variations: A regular up and down due to seasons.

3.   Cyclical variations: Changes due to booms and depressions.

4.   Irregular variations: Changes due to unpredictable causes.

                                                     

                       

1.3  Utility (Importance) of analysis of time series:

The analysis of time series is of great importance to economists and businessmen, because it helps them to understand past, to control the present (comparison) and to plan for the future (forecasting).

1. It helps to understand past

By analysis of time series, we can understand the past behavior of the variable. It is assumed that it will behave in future in the same manner as it did in the past.

2. It helps to plan for the future

By analysis of time series, we are able to predict the future requirements and to plan our activities accordingly.

3. It helps to control present performance 

By analysis of time series, we compare the actual performance with estimated performance and take the necessary steps. We also compare the two related time series by analysis of time series. For example; prices of gold and prices of shares.

1.4  Trend or Secular trend

                            The term secular trend or simply trend refers to the general tendency of the variable to increase or decrease. For ex; the population, agricultural production, prices, literacy etc. are increasing while death rate, illiteracy, travel by bullock cart, yearly birth rates, cost of electronic goods decreasing. The rise or fall may be steep or gradual. When we say that there is an increasing trend, it does not mean that the variable is always increasing. There may be some places where the variable decreases.

There are many factors which cause secular trend, the most important of which is the rise in population. The rise in prices, in population, in sales, in travel etc. is primarily due to the rise in population. Besides these there are other factors which cause secular trend. They are progress in science, improvements in technology, changes in culture, habits and tastes of people.  

Remarks:

1. The word secular is defined from Latin word saeculum which means generation or age.

2. Trend is a long period movement. Period can not be precisely defined ex. gold prices.

3. Trend is mostly monotonic, although original series is not.

4. Apart from the long term growth component, there are some short term periodic   rhythmic variations. These variations disturb the smoothness and monotony.

5. Trend is useful for two reasons: i) For comparison of two series

                                                       ii) It helps to extrapolate.  

1.5  Seasonal variations

                      This kind of variation is basically due to the seasons of the year. But it also includes the variations of any kind which are periodic in nature and whose period is less than a year. The ups and downs due to bazaar days or on weeks of month or on months of a year are seasonal variations. The factors causing seasonal variations are:

1. Seasons of a year

                      The most important factor which causes seasonal variations is the seasons of a year. There will be an increased demand for umbrellas, raincoats etc. in rainy season; for hats, sunglasses, cold drinks etc. in summer and for warm clothing in winter.

2. Festivals and Customs

                    The factors such as festivals, customs or traditions cause seasonal variations. For ex the demand for clothe, sweet, crackers etc. increases during festivals like Diwali, Id, Christmas. Also there is greater demand for gold, costly cloth, presentation articles etc. during marriage seasons.   

3. Practical needs

                      There are certain practical needs which cause variation in the time series. There are large withdrawals on the 1^st of every month for payment of salaries. The demand of electricity is very high during certain hours of the day. There is heavy rush for local transport during certain hours of the day.   

 1.6  Cyclical variations

                    The third component in time series is the cycle which is different from trend or seasonal variations. Cyclic variations are very common in economic and business activities. They show oscillating movements upward and downward forming a wave i.e. a cycle. There are four phases of a business cycle

i) Prosperity ii) Decline ( Recession) ii) Depression iv) Improvement ( Recovery)

The interval of time from one prosperity to next is called the period of cycle and it is anywhere between 3 to 10 years. Business cycle is the main cause behind the cyclic variations. Imbalances in economy, import and export facilities & policies, availability of loans, inflation, over development, decreasing efficiency of personnel (automation) and business competitions are some reasons for the business cycle.

 

Difference between seasonal variations and cyclic variations:

i) Reason: Seasonal variations are due to seasons, festivals, customs and needs of people    while cyclic variations are due to business cycle.

ii) Period: Seasonal variations are up and down swings of period less than a year. Cyclic variations are of period 3 to 10 years (or higher).

iii) Intensity: The intensity of seasonal variations is less than the cyclic variations.

 

 

 

1.7  Irregular variations

                   These variations are also called as random variations. They include all those variations which are not covered by the above three components and which are caused by factors other than those discussed earlier.

                  The causes of irregular variations are accidental like wars, earthquakes, floods, famines, fires, strikes etc. These factors are unpredictable, irregular variations are uncontrollable.

1.8 Analysis of time series

The study of time series i.e. the study of change in the figures of time series is called as analysis of time series. It’s purpose is two fold.

i) Identifying the four components which cause variations

ii) Isolating, studying and measuring each of them independently.

 

1.9  Mathematical Models of Time Series:

In analysis, it is required to know how the components interact and give the joint effect.  This can be done with the help of models. There are two models commonly used for the decomposition of time series in to its components.

i) Additive Model

                               Under additive model, the time series can be expressed as,

                                    Y_t = T_t + S_t + C_t + I_t

 where Y_t is the time series value at time t and T_t, S_t, C_t and It represents the trend, seasonal, cyclical and random variations at time t. In this model St, Ct and It are absolute values and can have positive and negative. The model assumes that all the four components of time series are independent of each other. In this model, the decomposition of time series is done on the assumption that the effect of various components are additive in nature.

ii) Multiplicative Model

                          Under multiplicative model, the time series can be expressed as

                                          Y_t = T_t × S_t × C_t × I_t

According to this model the decomposition of time series is done on the assumption that the effects of the four components are not necessarily independent. In this model S, C, and I are not absolute amounts as in case of additive model. They are relative variation and are expressed as rate or indices below or above unity. This model assumes that these four components are mutually independent.

 

1.10  Measurement of Trend

 

The following are the four methods which are generally used for the measurement of the trend in a time series.

i) Inspection (or Free hand curve fitting) method

ii) Moving averages method

iii) Progressive averages method

iv)Principle of least squares method.

 

1.11  Moving average method

                               This method consists of determining the arithmetic means for given number of years, months or days etc. This average value is supposed to be the proper or trend value for the middle period. By taking such averages the effects of the other variations is reduced. The period of moving averages is taken between 3 years and 10 years. Period depends upon the cycle of data.

If the period of moving averages is an odd number i.e. 3,5,7, --  years, the average calculated is written at the middle year. If it is an even number ( 2,4,6,8,--years ), after calculating the averages for that period, again the averages of each pair of adjacent moving averages are calculated. They are called the centered moving averages.

Merits:

1.   It is simple than method of least squares

2. If period of cycle is equal to period of moving averages, cyclic variations are     removed. 

Demerits:

1.   Irregular variations are not removed. So can’t used for prediction only for estimation.

2. Trend values can not be calculated for all the years.

1.12   Progressive average method

                 In the early years of a firm, data over a long period are not available. So the method of moving average is not used. Progressive average method is used.  Progressive averages are the cumulative averages. To calculate the progressive averages, we calculate the cumulative sums and divide the sums by 1, 2, 3, ---- and so on. A column is also prepared for the difference between the actual values and the corresponding progressive averages. Thus, if the time series values are y₁, y₂, y₃, ------ the progressive averages are

 Merit:

            It is highly useful to study the trend during the childhood of a concern.

Demerits:

          Progressive averages are not useful as the industry is grown up.

1.13   Least square method

Fitting of Linear Trend:

                           The idea of least squares was developed by Gauss in 1975 and is most widely used method of fitting a mathematical function to a given set of data.

Σ y = n a + b Σ t    .... (4)

Σ t y = a Σ t + b Σ t²  .... (5), where n is the number of time series  pairs (t, y).

It is seen that equation (4) is obtained by taking sum of both sides in equation (1) and equation (5) is obtained on multiplying equation(1) by t and then summing both sides over the given values of the series. Solving (4) and (5) for a and b and substituting these values in(1), we finally get the equation of the straight line trend.

 

Merits:

1.   It is more accurate than method of moving averages and progressive averages least squares

2. Trend line can be used for prediction as well as estimation. 

3. It is objective method free from personal bias. 

Demerits:

1.   It is quite tedious and time consuming , so difficult to nonmathematical person. 

2.  Addition of one value would require making calculations freshly, which is not so in other methods.  

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Multiple Choice Questions

1. The long term regular movement in a time series is called as ………

a) seasonal variation b) cyclical variations c) secular trend d) irregular variations

2. Time series analysis helps to ………

a) make predictions b) compare two or more series c) know behavior of business d) all of these

3. Given that;

Statement I: Seasonal variation has period of less than one year.

Statement II: Cyclical variation has period of more than one year.

a) Statement I is false b) Statement II is false c) Both are false d) Both are true

4. Periodic change in values of time series is ………

a) seasonal variation b) cyclical variations c) both a and b d) irregular variations

5. If all four components of time series operate independently then we use………

a) additive model b) multiplicative model c) exponential model d) none of these

6. A time series is a set of data recorded ………

a) periodically b) at time or space interval c) both a and b d) neither a nor b

7. Trend in a time series means ………

a) long-term regular movement b) short-term regular movement

c) at successive points of time d) all of these

8. Moving average method suffer from ………

a) the loss of information b) the element of subjectivity

c) the decision about the number of years in group d) all of these

9. Which of the following is not a method of measuring trend?...………

a) Moving averages b) Simple averages c) Least squares  d) Progressive averages

10. The variation in the production due to strike in a company is ....…

a) seasonal variation b) cyclical variations c) secular trend d) irregular variations

11.

Year	2005	2006	2007	2008	2009
Sale in lakhs	4	7	10	13	16
X

 

From the above information of the time series, X is ………

a) 4 yearly moving averages b) 2 yearly moving averages

c) 3 yearly moving averages d) none of these

12. Time series analysis helps to ………

a) understand the behavior of a variable in the past b) plan future observation

c) predict the future behavior of a variable d) all the above

13. To which component of time series the term recession is attached?…

a) trend b) seasonal c) cyclical d) random variation

14. Moving average method of fitting trend in a time series data removes the effect of …

a) long term effect b) cyclical variation c) short term movements d) none of these

15. An additive model of time series with components T, S, C and I is…

a) Y = T + S + C × I b) Y = T + S × C × I c) Y = T + S + C + I d) Y = T + S × C + I

Short answer questions

1. What is time series? State four components of time series.

2. Describe moving average method for determining trend.

3. State different components of time series and explain any one of them.

4. What is secular trend? Describe the method of least squares to determine secular trend.

5. State utility of time series.

Big answer questions

1. Explain the following terms with suitable illustration;

i) secular trend and ii) seasonal variation

2. What is secular trend? What are the methods for measuring trend? Describe any one of them.

3. What is time series? State four components of time series. Describe any one of them

4. What are the different models for time series? Describe them.

5. Discuss the simple average method for measuring seasonal variation.

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