Time Series Analysis : Models, Components, Methods – Secular, Trend, Cyclical, Seasonal & Irregular

Time Series Analysis: Introduction, Objectives, models of time series, Components & Methods of Time Series Analysis: Secular, Trend Cyclical, Seasonal, and irregular variations.

Time series analysis:

In the context of economic and business research, we may often obtain data relating to some time period concerning a given phenomenon. Such data is labeled as ‘Time Series’. More clearly it can be stated that a series of successive observations of the given phenomenon over a period of time is referred to as time Such series are usually the result of the effects of one or more of the following factors:

• Secular trend or long-term trend that shows the direction of the series over a long period of The effect of the trend (whether it happens to be a growth factor or a decline factor) is gradual, but extends more or less consistently throughout the entire period of time under consideration. Sometimes, secular trend is simply stated as trend (or T).
• Short-time oscillations e., changes taking place in a short period of time only and such changes can be the effect of the following factors:
  • • Cyclical fluctuations (or C) are the fluctuations as a result of business cycles and are generally referred to as long-term movements that represent consistently recurring rises and declines in an
    • Seasonal fluctuations (or S) variations are of short duration occurring in a regular sequence at specific intervals Such fluctuations result from changing seasons. Usually, these fluctuations involve patterns of change within a year that tend to be repeated from year to year. Cyclical fluctuations and seasonal fluctuations taken together constitute short-period regular fluctuations.
• Irregular fluctuations (or I), also known as Random fluctuations or variations, are variations that take place in a completely unpredictable

For example,

• Sales (y) of a product are influenced by (i) advertisement expenditure, (ii) the price of the product, (iii) the income of the people, (iv) other competitive products in the market, (v) tastes, fashions, habits, and customs of the people and so on.
• Similarly, the price of a particular product depends on its demand, various competitive products in the market, raw materials, transportation expenses, investment, and so on.

All these factors stated above are termed as components of time series and when we try to analyze time series, we try to isolate and measure the effects of various types of these factors on a series. To study the effect of one type of factor, the other type of factor is eliminated from the series. The given series is, thus, left with the effects of one type of factor only.

Objectives of the Time Series Analysis

The time series analysis is of great importance not only to businessmen or an economist but also to people working in various disciplines in natural, social, and physical sciences. Some of its uses are enumerated below :

• It enables us to study the past behavior of the phenomenon under consideration, i.e., to determine the type and nature of the variations in the data.
• The segregation and study of the various components is 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 (on the basis of past performances) and analyze the causes of such variations, if any.
• It enables us to predict or estimate or forecast the behavior of the phenomenon in the future which is very essential for business planning.
• It helps us to compare the changes in the values of different phenomena at different times or places, etc.

Mathematical Models of Time Series Analysis

For analyzing time series, we usually have two models of time series analysis; (1) multiplicative model; and (2) additive model.

(1) Multiplicative model of Time Series Analysis :

The multiplicative model of time series Analysis assumes that the various components interact in a multiplicative manner to produce the given values of the overall time series and can be stated as under:

Y = T × C × S × I

where

Y = observed values of time series,

T = Trend,

C = Cyclical fluctuations,

S = Seasonal fluctuations / variations

I = Irregular fluctuations.

(2) Additive model of Time Series Analysis

Additive model considers the total of various components resulting in the given values of the overall time series and can be stated as:

Y = T + C + S + I

There are various methods of isolating trends from the given series viz., the free hand method, semi-average method, method of moving averages, method of least squares, and similarly there are methods of measuring cyclical and seasonal variations, and whatever variations are left over are considered as random or irregular fluctuations.

Implication and Limitation of Additive Model

• This implies that the trend, however, fast or slow, it may be, has no effect on the seasonal and cyclical components ; nor do seasonal swings have any impact on cyclical variations and conversely.
• However, this assumption is not true in most of the economic and business time series where the four components of the time series are not independent of each other.
• For instance, the seasonal or cyclical variations may virtually be wiped off by very sharp rising or declining trend.
• Similarly, strong and powerful seasonal swings may intensify or even precipitate a change in the cyclical fluctuations.

(3) Mixed Models of Time Series Analysis

In addition to the additive and multiplicative models discussed above, the components in a time series may be combined in a large number of other ways. The different models, defined under different assumptions, will yield different results. Some of the mixed models resulting from different combinations of additive and multiplicative models are given below :

Y = TCS + I

Y = TC + SI

Y = T + SCI

Y = T + S + CI

The analysis of time series is done to understand the dynamic conditions for achieving the short-term and long-term goals of the business firm(s). The past trends can be used to evaluate the success or failure of management policy or policies practiced hitherto. On the basis of past trends, future patterns can be predicted and policies or policies may accordingly be formulated. We can as well study properly the effects of factors causing changes in a short period of time only, once we have eliminated the effects of trend. By studying cyclical variations, we can keep in view the impact of cyclical changes while formulating various policies to make them as realistic as possible. The knowledge of seasonal variations will be of great help to us in making decisions regarding inventory, production, purchases, and sales policies so as to optimize working results. Thus, analysis of time series is important in the context of long-term as well as short-term forecasting and is considered a very powerful tool in the hands of business analysts and researchers.

METHODS FOR TIME SERIES ANALYSIS

In business forecasting, it is important to analyze the characteristic movements of variations in the given time series. The following methods of time serries Analysis serve as a tool for this analysis:

1. Methods for Measurement of Secular Trend Analysis

i. Freehand curve Method (Graphical Method)

ii. Method of selected points for secular trend analysis

iii. Method of semi-averages for secular trend analysis

iv. Method of moving averages for secular trend analysis

v. Method of Least Squares for secular trend analysis

2. Methods for Measurement of Seasonal Variations

i. Method of Simple Average for seasonal variations

ii. Ratio to Trend Method for seasonal variations

iii. Ratio to Moving Average Method for seasonal variations

iv. Method of Link Relatives for seasonal variations

3. Methods for Measurement for Cyclical Variations

4.Methods for Measurement for Irregular Variations

Methods for Measurement of Secular Trend

1. GRAPHIC OR FREE HAND CURVE FITTING METHOD:

This secular trend analysis method is the simplest and the most flexible methods of time series analysis to estimating the secular trend and consists in first obtaining a histogram by plotting the time series values on a graph paper and then drawing a free-hand smooth curve through these points so that it accurately reflects the long-term tendency of the data.

Merits

–It is a very simple and time-saving method and does not require any mathematical calculations.

–It is a very flexible method in the sense that it can be used to describe all types of trend – linear as well as non-linear

Demerits

–The strongest objection to this method is that it is highly subjective in nature

–It does not help to measure trends.

–It will be dangerous to use it for forecasting or making predictions.

Example: 1

2. METHOD OF SELECTED POINTS

In this models or methods of time serries Analysis , two points considered to be the most representative or normal, are joined by straight line to get secular trend. This, again, is a subjective method since different persons may have different opinions regarding the representative points. Further, only linear trend can be determined by this method.

3. SEMI-AVERAGE METHOD

In this methods or models of time series analysis , the whole time series data is classified into two equal parts w.r.t. time.

–For example, if we are given the time series values for 10 years from 1985 to 1994 then the two equal parts will be the data corresponding to periods 1985 to 1989 and 1990 to 1994.

Having divided the given series into two equal parts, we next compute the arithmetic mean of time-series values for each half separately. These means are called semi-averages. Then these semi-averages are plotted as points against the middle point of the respective time periods covered by each part.

Merits.

–Objectivity: objectivity in the sense that it does not depend on personal judgement and everyone who uses this method gets the same trend line and hence the same trend values.

–It is easy to understand and apply as compared with the moving average or the least square methods of measuring trend

–The line can be extended both ways to obtain future or past estimates.

Limitations

–This secular trend analysis method assumes the presence of linear trend (in the time series values) which may not exist.

–The use of arithmetic mean (for obtaining semi-averages) may also be questioned because of its limitations

Example: 2

Example: 3

4. METHODS OF MOVING AVERAGES

• Method of moving averages for secular trend analysis is a very simple and flexible method of measuring trend. It consists in obtaining a series of moving averages, (arithmetic means), of successive overlapping groups or sections of the time series.
• The moving average is characterized by a constant known as the period or extent of the moving average. Thus, the moving average of period ‘m’ is a series of successive averages (A.M.’s) of m overlapping values at a time, starting with 1st, 2nd, 3rd value and so on.

Moving averages and various trends Methods or Models  of time serries Analysis

1.Moving Average and Linear Trend: If the time series data does not contain any movements except the trend which when plotted on a graph gives a straight line curve, then the moving average will reproduce the series.

2.Moving Average and Curvilinear Trend: If the data does not contain any oscillatory or irregular movements and has only general trend and the historigram (graph) of the time series gives a curve which is convex (concave) to the base, then the trend values computed by moving average method will give another curve parallel to the given curve but above (below) it.

3.Period of Moving Average: The moving average will completely eliminate the oscillatory movements if :

–The period of the moving average is equal to or a multiple of the period of oscillatory movements provided they are regular in period or amplitude, and

–The trend is linear or approximately so.

Hence, to compute correct trend values by the method of moving averages, the period or extent of the moving average should be same as the period of the cyclic movements in the series. However, if the period of moving average is less or more than the period of the cyclic movement then it (M.A.) will only reduce their effect.

4.Moving Average and Polynomial Trend:if the trend is curvilinear, the moving average values will give a distorted picture of the trend. In such a case the correct trend values are obtained by taking a weighted moving average of the given values. The weights to be used will depend on the period of the M.A. and the degree of the polynomial trend to be fitted.

5.Effect of Moving Average on Irregular Fluctuations: The optimum period of the moving average is the one that coincides with or is a multiple of the period of the cycle in the time series as it would completely eliminate cyclical variations, reduce the irregular variations and, therefore, give the best possible values of the trend.

Moving Averages Method gives a trend with a fair degree of accuracy. In this method, we take arithmetic mean of the values for a certain time span. The time span can be three-years, four -years, five- years and so on depending on the data set and our interest. We will see the working procedure of this method.

Procedure:

(i) Decide the period of moving averages (three- years, four -years).

(ii) In case of odd years, averages can be obtained by calculating,

(iii) If the moving average is an odd number, there is no problem of centering it, the average value will be centered besides the second year for every three years. 

(iv) In case of even years, averages can be obtained by calculating,

(v) If the moving average is an even number, the average of first four values will  be placed between 2 ^nd and 3^rd year, similarly the average of the second four values will be placed between 3^rd and 4^th year. These two averages will be again averaged and placed in the 3^rd year. This continues for rest of the values in the problem. This process is called as centering of the averages.

Example: 4

Example: 5

5. METHODS OF LEAST SQUARE OF TIME SERIES SECULAR TREND ANALYSIS

Models of Methods of Time Series Analysis The principle of least squares provides us an analytical or mathematical device to obtain an objective fit to the trend of the given time series. Most of the data relating to economic and business time series conform to definite laws of growth or decay and accordingly in such a situation analytical trend fitting will be more reliable for forecasting and predictions. This technique can be used to fit linear as well as non-linear trends.

The line of best fit is a line from which the sum of the deviations of various points is zero. This is the best method for obtaining the trend values. It gives a convenient basis for calculating the line of best fit for the time series. It is a mathematical method for measuring trend. Further the sum of the squares of these deviations would be least when compared with other fitting methods. So, this method is known as the Method of Least Squares and satisfies the following conditions:

(i) The sum of the deviations of the actual values of Y and Ŷ (estimated value of Y) is Zero. that is Σ(Y–Ŷ) = 0.

(ii) The sum of squares of the deviations of the actual values of Y and Ŷ (estimated value of Y) is least. that is Σ(Y–Ŷ)² is least ;

Procedure:

(i)  The straight line trend is represented by the equation Y = a + bX        …(1)

where  Y is the actual value, X is time, a, b are constants

(ii)  The constants ‘a’ and ‘b’ are estimated by solving the following two normal

Equations   ΣY = n a + b ΣX   …(2)

ΣXY = a ΣX + b ΣX²     …(3)

Where ‘n’ = number of years given in the data.

(iii) By taking the mid-point of the time as the origin, we get ΣX = 0

(iv) When ΣX = 0 , the two normal equations reduces to

The constant ‘a’ gives the mean of Y and ‘b’ gives the rate of change (slope).

(v) By substituting the values of ‘a’ and ‘b’ in the trend equation (1), we get the Line of Best Fit.

Merits

–Because of its analytical or mathematical character, this method completely eliminates the element of subjective judgement or personal bias on the part of the investigator.

–Unlike the method of moving averages , this method enables us to compute the trend values for all the given time periods in the series.

–The trend equation can be used to estimate or predict the values of the variable for any period t in future or even in the intermediate periods of the given series and the forecasted values are also quite reliable.

Demerits

–The addition of even a single new observation necessitates all the calculations to be done afresh which is not so in the case of moving average method.

–This method requires more calculations and is quite tedious and time consuming as compared with other methods. It is rather difficult for a non-mathematical person (layman) to understand and use.

–Future predictions or forecasts based on this method are based only on the long-term variations, i.e., trend and completely ignore the cyclical, seasonal and irregular fluctuations.

–It cannot be used to fit growth curves (Modified exponential curve, Gompertz curve and Logistic curve) to which most of the economic and business time series conform.

Example: 6

Therefore, the required equation of the straight line trend is given by

Y = a+bX;

Y = 45.143 + 1.036 (x-2003)

The trend values can be obtained by

When X = 2000 , Yt = 45.143 + 1.036(2000–2003) = 42.035

When X = 2001, Yt = 45.143 + 1.036(2001–2003) = 43.071,

similarly other values can be obtained.

Example: 7

Given below are the data relating to the sales of a product in a district.

Fit a straight line trend by the method of least squares and tabulate the trend values.

Solution:

Computation of trend values by the method of least squares.

In case of EVEN number of years, let us consider

METHODS FOR MEASUREMENT OF SEASONAL VARIATIONS OF TIME SERIES

i. Method of Simple Average for seasonal variation in Time Series Analysis

ii. Ratio to Trend Method seasonal variation in Time Series Analysis

iii. Ratio to Moving Average Method seasonal variation in Time Series Analysis

iv. Method of Link Relatives seasonal variation in Time Series Analysis

1. SIMPLE AVERAGE METHOD

Method of Simple Averages:

This is the simplest and easiest method for studying Seasonal Variations Analysis – Time Series Analysis. The procedure of simple average method is outlined below.

Procedure:

(i) Arrange the data by months, quarters or years according to the data given.

(ii) Find the sum of the each months, quarters or year.

(iii) Find the average of each months, quarters or year.

(iv) Find the average of averages, and it is called Grand Average (G)

(v) Compute Seasonal Index for every season (i.e) months, quarters or year is given by

(vi) If the data is given in months

Similarly we can calculate SI for all other months.

(vii) If the data is given in quarter

Example: 8

Example: 9

Example: 10

2. RATIO TO TREND METHOD:

This method of Time Series Analysis for seasonal variations analysis  is an improvement over the simple averages method and this method assumes a multiplicative model i.e

The measurement of seasonal indices by this method consists of the following steps.

• Obtain the trend values by the least square method by fitting a mathematical curve, either a straight line or second degree polynomial.
• Express the original data as the percentage of the trend values. Assuming the multiplicative model these percentages will contain the seasonal, cyclical and irregular components.
• The cyclical and irregular components are eliminated by averaging the percentages for different months (quarters) if the data are In monthly (quarterly), thus leaving us with indices of seasonal variations.
• Finally these indices obtained in step(3) are adjusted to a total of 1200 for monthly and 400 for quarterly data by multiplying them through out by a constant K which is

Advantages:

• It is easy to compute and easy to understand.
• Compared with the method of monthly averages this method is certainly a more logical procedure for measuring seasonal variations.
• It has an advantage over the ratio to moving average method that in this method we obtain ratio to trend values for each period for which data are available where as it is not possible in ratio to moving average method.

Disadvantages:

• The main defect of the ratio to trend method is that if there are cyclical swings in the series, the trend whether a straight line or a curve can never follow the actual data as closely as a 12- monthly moving average does. So a seasonal index computed by the ratio to moving average method may be less biased than the one calculated by the ratio to trend method

Example: 11

Calculate seasonal indices by Ratio to moving average method from the following data.

3. RATIO TO MOVING AVERAGE METHOD: Time Series Analysis

The ratio to moving average method – Time Series Analysis is also known as percentage of moving average method and is the most widely used method of measuring seasonal variations. The steps necessary for determining seasonal variations by this method are

• Calculate the centered 12-monthly moving average (or 4-quarterly moving average) of the given data. These moving averages values will eliminate S and I leaving us T and C components.
• Express the original data as percentages of the centered moving average values.
• The seasonal indices are now obtained by eliminating the irregular or random components by averaging these percentages using A.M or median.
• The sum of these indices will not in general be equal to 1200 (for monthly) or 400 (for quarterly). Finally the adjustment is done to make the sum of the indices to a total of 1200 for monthly and 400 for quarterly data by multiplying them through out by a constant K which is given by

Advantages:

• Of all the methods of measuring seasonal variations, the ratio to moving average method is the most satisfactory, flexible and widely used method.
• The fluctuations of indices based on ratio to moving average method is less than based on other methods.

Disadvantages:

• This seasonal variations method does not completely utilize the data. For example in case of 12-monthly moving average seasonal indices cannot be obtained for the first 6 months and last 6 months.

Example: 12

4. LINK RELATIVE METHOD:

This seasonal variations method is slightly more complicated than other methods. This method is also known as Pearson‟s method. This method consists in the following steps.

• The link relatives for each period are calculated by using the below formula

• Calculate the average of the link relatives for each period for all the years using mean or median.
• Convert the average link relatives into chain relatives on the basis of the first season. Chain relative for any period can be obtained by

the chain relative for the first period is assumed to be 100.

• Now the adjusted chain relatives are calculated by subtracting correction factor „kd‟ from (k+1)th chain relative respectively.
• Where k = 1,2,…….11 for monthly and k = 1,2,3 for quarterly data and

where N denotes the number of periods i.e. N = 12 for monthly N = 4 for quarterly

• Finally calculate the average of the corrected chain relatives and convert the corrected chain relatives as the percentages of this average. These percentages are seasonal indices calculated by the link relative method. 

Advantages:

• As compared to the method of moving average the link relative method uses data more.

Disadvantages:

• The link relative method needs extensive calculations compared to other methods and is not as simple as the method of moving average.
• The average of link relatives contains both trend and cyclical components and these components are eliminated by applying correction.

Example: 13

Business Cycle

According to Mitchell, “Business cycle are a type of fluctuation found in the aggregate economic activity of nations that organize their work mainly in business enterprises : a cycle consists of expansions occurring at about the same time in many activities, followed by general recessions, contractions and revivals which merge into the expansion phase of the next cycle; this sequence of changes is recurrent but not periodic; in duration business cycles vary from more than one year to ten or twelve years.

There are four phases of a business cycle, such as

(a) Expansion (prosperity)

(b) Recession

(c) Depression (contraction)

(d) Revival (recovery).

A cycle is measured either from trough-to-trough or from peak-to-peak. Recession and contraction are the result of cumulative downswing of a cycle whereas revival and expansion are the result of cumulative upswing of a cycle.

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