Adjustments before statistical analysis of time series data:

Before we start statistical analysis of time series data, we should see whether or not they represent a series of comparable figures over time. A series of figures may not be comparable or homogeneous for a number of reasons. For example,
Changes in population and geographical areas: The figures may relate to geographical areas which change over time. The series may relate to population which is always changing over time. The data related to geographical area or population are adjusted by expressing the data in ‘per unit’ or ‘per capita’.
Calendar variations: Industrial production data over different months are not homogeneous as the number of working days in different calendar months are not same. To make the data on production comparable, we divide the figure for each month by the number of working days in that month to express the data in ‘per working day’
Price changes: Monetary data are not comparable over time as the purchasing power of money changes. To make monetary data comparable we divide the figures of the current period by a suitable price index (say, wholesale price index number) of the current period with respect to some base period. This will necessitate dividing or deflating the current figure by the index number of prices of the current period with the chosen base period. If the index number be I(0k) (in percent form), 100 rupees in base period (0) has the same purchasing power as I(0k) rupees in the current period (k). Thus, the figure x(k) in money terms in the current period expressed in terms of base-period purchasing power would be

X(k’)=x(k)*100/I(0k)

Let us now look at the various components of a time series.
A graphical representation of a time series will reveal the changes over time. A series which exhibits no change in the period under consideration gives a horizontal line. Usually, we see continual changes over time, giving an impression of haphazard movement. A critical study of the series however may reveal that the change is not totally haphazard and at least a part of it can be accounted for. The part of the movement which can be accounted for is the systematic part and the remaining part is unsystematicor irregular. The systematic part can be attributed to several broad factors, namely
1. Secular trend
2. Seasonal variation
3. Cyclical variation
Some or all of the above components may be present in any of the given time series.

Why do we need to look at the different components of time series?

Seperation of different components of a time series is of importance as we may be interested in a particular component or we may want to study the series after eliminating the effect of a particular component. It is to be noted here that it is the systematic part of time series which can be used in forecasting.

Detailed description of different components of time series:

1. Secular Trend: on plotting the time series data we may observe that on the whole the time series tends to move either in an upward or downwarddirection over a long period of time. this type of a steady, smooth movement in either direction (upward or downward) over a long period of time is called a secular trend or simply trend. It is the general tendency of the data to increase or decrease during a long period of time. This is true for most of the series of Business and Economic Statistics. For example, an upward tendency may be seen in data pertaining to agricultural production whereas a downward tendency may be seen in no. of deaths, as a result of advancement in medical sciences, better medical facilities, literacy, etc.
It should be clearly noted that trend is the general long term average tendency. It is not necessary that the movement (increase or decrease) be in the same direction throughout the given period. The time series may show an increase, decrease or stability in different sections of time. However, the overall tendency may be upward, downward or stable in the long run. Such tendencies can be attributed to the forces which are more or less same for a long time or changes gradually and continuously for a long period of time like the changes in population, tastes, habits, customs, etc of the people in a society and so on. They do not change suddenly. They operate in an evolutionary manner. For example, the effect of population growth in the expansion of various sectors like agriculture, industry, education, etc is a continuous gradual process. the effects might be because of an interaction of forces. For example, the growth or decline in a number of economic time series may be due to interaction of forces like advances in production technology, inventions, discovery of new natural resources, large-scale production, exhaustion of existing resources, improved marketing strategies, business organization,etc-all of which are gradual processes.
It is also to be noted that it is not necessary for every time series to show an upward or downward trend. It may so happen that all the values of time series fluctuate around a certain value which does not change over time. For example, temperature of a particular place for a particular month over several years.
A time series trend may be linear or non-linear. If the values of a time series cluster more or less around a straight line then the trend is linear otherwise non-linear. in practice, linear trend is normally used although it is rarely obtained in economic and business data. In an economic or business phenomenon, the rate of growth or decline is not constant. It varies considerably with time. Usually, in the beginning, the growth is slow, then rapidwhich might be accelerated for some more time after which it becomes stationary or stable for some time and finally it decreases slowly.
(The term 'long period of time' is a relative term which is defined according to the requirements)

Detailed description of different components of time series (contd.):

Periodic changes: in the graph of a time series data, we may observe some oscillatory movements which causes the series to rise or fall at some regular interval. These oscillatory movements are superimposed on the trend curve. smaller oscillations might be superimposed on the bigger waves. the larger oscillations are due to cyclical movements whereas the smaller oscillations superimposed on the larger ones are seasonal movements.

2. Seasonal variations: Seasonal variations in a time series are due to the rythmic forces which operate in a regular periodic manner over a time span of less than 12 months and have similar pattern year after year. Seasonal variation in a time series can be observed if the data are recorded quarterly, monthly, weekly, daily, hourly, etc. the amplitude of seasonal variations may vary. Most economic time series are influenced by seasonal swings. the seasonal variations may be due to:
(i) Resulting from natural forces: Seasonal variations may be due to weather conditions and climatic changes. For example, the sale of woollens go up in winter, the demand for electric fans increase in summer, etc. This is due to natural forces namely, seasons. The production of certain commodities like sugar, etc depend on seasons. Similarly, the prices of agricultural commodities decline at the time of harvest and then pick up gradually.
(ii) Resulting from man-made conventions: These type of variations in the time series in a period of twelve months are due to habits, fashion, customs, conventions, etc of the people in a society. for example, the sale of gold and gold jewellery goes up considerably during marriage season and festivals like Diwali, Dussehra, etc. these type of variations occur in a regular spasmodic manner and keeps occuring year after year. the main objective of measurement of seasonal variation is to isolate them and study their effects. the study of seasonal patterns is extremely useful to businessmen, producers, managers, etc, in planning future operations. this is also used in formulating policies and taking decisions regarding production, purchase, inventory control, personnel requirement, etc. without knowledge of seasonal variations, a seasonal upswing may be mis-interpreted as an indicator of growing business while a seasonal slump may be taken as an indicator of deteriorating business. Hence, to understand the behavior of a time series properly, the time series data must be adjusted for seasonal variations.

3. Cyclic variation: in a time series, the oscillatory movements with a period of oscillation more than one year is termed as cyclic fluctuations. One complete period is called a 'cycle'. the cyclic movements of a time series is generally referred to as "the four phase cycle" or "Business cycle" composed of prosperity, recession, depression and recovery. A cycle usually covers a time span of more than a year. the upswings and downswings in any business depends on the cumulative nature of economic forces and the interaction between them.

Irregular component: There are some movements in a time series the reasons for which are unknown and unpredictable. Almost all time series have such factors (other than the regular variations) called random or irregular or residual fluctuations which are not accounted for by any of secular trend, seasonal variations or cyclic variations. Irregular movements are irregular, both in terms of length of time interval after they which occur and the amplitudes. These variations are completely random, erratic, unpredictable and due to numerous, non-recurring and irregular circumstances beyond the control of humans like earthquakes, floods, famines, wars, etc. These are isolated and irregular but powerful and often called episodic fluctuations. There may be cases where irregular fluctuations may not be significant but, in some cases they may be very effective giving rise to cyclic movements.
It must be noted that due to their absolute random character, it may not be possible to separate irregular variations and study them exclusively or forecast and estimate them precisely. Although, we can find rough estimates based on past behavior and experiences. Based on these estimates we may make provision for abnormalities.

Mathematical models for Time Series:

Analysis of time series essentially involves decomposition of the time series into its four components:
1. Secular trend or trend (T)
2. Seasonal variation (S)
3. Cyclic variation (C)
4. Irregular variation (I)
The objective is to estimate and separate the four types of variations, bringing out the relative impact of each on the overall behavior of the series. The following are the most common models of decomposition of a time series into its components:
(i) Additive model: According to the additive model, a time series can be expressed as y(t)=T(t)+S(t)+C(t)+I(t)

The additive model implicitly implies that the seasonal forces, cyclic forces and irregular forces operate with equal absolute effect irrespective of the trend value. Also, C(t) and S(t) will have positive or negative values according as they are above or below normal phase of the cycle and year resulting in a total of zero. I(t) too in the long run be zero. The additive model assumes that the four components are independent and have no effect on each other whatsoever. This assumption is not true in almost all cases.
(ii) Multiplicative model: If we have reasons to assume that various components of a time series operate proportionally to the general level of the series, the multiplicative model is appropriate. According to the multiplicative model, a time series can be expressed as

y(t)=T(t)*S(t)*C(t)*I(t)

The multiplicative model cannot be applied to a time series with both positive and negative values unless the time series is translated by adding suitable positive value. It can also be seen that the multiplicative decomposition of a time series is the same as the additive decomposition of logarithmic values of the original time series

log[y(t)]=log[T(t)]+log[S(t)]+log[C(t)]+log[I(t)]

In practice, most of the economic time series data conform to multiplicative model.

in addition to the additive and multiplicative models, the components of a time series can be combined in a multitude of way and termed as Mixed Models. different models designed under different assumptions yield different results. the following are a few examples of mixed models:

y(t)= T(t)*C(t)+S(t)*I(t) y(t)= T(t)+C(t)*S(t)*I(t) y(t)= T(t)+S(t)+C(