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Time Series Analysis

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A time series may be defined as a set of values of a variable collected and recorded in a chronological order of the time intervals. Time series is used by statisticians to describe the flow of economic activity. In short time series refers to the data depending on time. It refers to a set of observations concerning any activity against different periods of time. The duration of the time period may be hourly, daily, weekly, monthly or yearly. According to Morris Hamburg “A time series is a set of statistical observations arranged in chronological order”. Therefore time series is also called historical data or historical series. The study of movement of quantitative data through time is referred to as ‘time series analysis’. Time series is of great importance to the planners of economic development and economists. The success of planning depends upon making accurate forecasts of future conditions of economy. It enables the economists to foresee what is likely to come and to analyze the repercussions of past behavior. The analysis of time series enables us to understand the past behavior or performance. Time series analysis can be used to know how the data changes over time and find out probable reasons for such change.

Analysis of time series is of relevance whenever a variable is found to vary over time. Variable such as Sales, Production, Profit, Population and Employment opportunity assume different values at different points of time. The importance is given in the following points 1. Analysis of time series helps to know the past conditions: The observations made on the past few periods help to know the conditions properly 2. In assessing the present achievements 3. In predicting the future: on the basis of the past and the present conditions, the future is well predicted. It helps to forecast scientifically. 4. In comparison: various time series could be compared with regard to their movement over a long period and vital inferences could be drawn. 5. In forewarning: as it predicts the future, good or bad, eventualities could be met with the necessary preparedness. Losses if any, could be minimized, profits, if any could be increased.

A Time series may contain one or more of the following four components: 1. 2. 3. 4. Secular trend Seasonal variation Cyclical variation Irregular variation

It is ordinarily assumed that there is a multiplicative relationship amongst these four components. Mathematically it can be written as Y=TxSxCxI Where, T= trend, S= seasonal variation, C= cyclical variation, I=irregular variation. SECULAR TREND These are the changes that have occurred as a result of general tendency of the data to increase or decrease over a long period of time. This is also called long term trend or simply trend. The overall tendency may be increasing one as in population, price, number of automobiles on road and literacy. Decreasing nature may be in birth rate, infant mortality rate and poverty. Only very rarely constant nature is observed. Graphically linear trend is a straight line. Mathematically trend may be either i. ii. Linear or Non linear

SEASONAL VARIATIONS The changes that have taken place within a year as a result of change in climate, weather conditions, festivals etc. are called seasonal variations. Such change repeats itself year after year. Season is a period which is a part of one year. Certain variations are observed at some seasons and they are found to recur year after year. The factors which cause seasonal variations can be enumerated as below: i. ii. Climate and weather conditions Customs, traditions and habits

CYCLICAL VARIATIONS These are the changes that have taken place as a result of booms and depressions. Normally the period of cyclical variation is more than a year. Cyclical variation is similar to seasonal variation. If the changes take place periodically and if period is more than one year, the variations are said to be cyclical fluctuations. In business activities, there are some periods when the business activity is at its peak, while in some other periods it recedes and goes below the trend line. Cyclical variations do not follow any regular pattern. IRREGULAR VARIATIONS These are changes that have taken place because of forces that could not have been predicted like floods, earthquakes, famines etc.

One cannot say whether there would be rise or fall in a certain variable. Usually irregular variable are of smaller magnitudes. MODELS There are two types of mathematical models of time series. a) Additive model: When the changes in the data are the result of the combined impact of the four components, we can write the data as the sum of four components. i.e., Y = T + S + C + I Where, T= trend, S= seasonal variation, C= cyclical variation, I=irregular variation. b) Multiplicative model: In this mode original data, Y=TxSxCxI Where, T= trend, S= seasonal variation, C= cyclical variation, I=irregular variation. Many business and economic series agree with multiplicative model but additive model is found to suit only some series. MEASUREMENT OF TREND There are four methods to estimate the secular trend. They are i. Graphic method: - This is the simplest method for measuring trend of a time series. In this method the time series graph is drawn taking independent variable on the X axis and on the Y axis dependent variable is taken. The time values are plotted. This is called the time series graph or historiegram. Semi-averages method: - in this method the data is divided into two parts equally and the average of the values of each half together with the midpoint is plotted on the graph. The two points are so plotted and the straight line joining the two points is called trend line. Method of moving averages: - in this method the short time variations are eliminated by finding the moving averages. These moving averages indicate the trend. Method of least squares: - This is the most popular method of measuring the trend. In this method a mathematical relation is established between time and the variables which is depending on time. The relation may be linear, quadratic or exponential.


iii. iv.

MEASUREMENT OF SEASONAL VARIATIONS To obtain a statistical description of a pattern of seasonal variations it will be desirable to first free the data from the effects of trend, cycles and irregular variations. After eliminating these components the seasonal variations are measured in index form called seasonal index. Thus the measure of seasonal variations are called seasonal indices

There are various methods for measuring seasonal variations. Some of the popular methods are the following. i. ii. iii. iv. Method of Simple Averages Ratio to Moving Average Method Ratio to Trend Method Link Relative Method

MEASUREMENT OF CYCLICAL VARIATION In order to measure cyclical variations, it is necessary that the seasonal data are desesonalised and adjusted to trend. The resulting data comprises both cyclical and irregular variations. In order to isolate irregular variations from these data the irregular data must be smoothed out by using appropriate moving averages leaving only cyclic variations. MEASUREMENT OF IRREGULAR VARIATION An estimate of the irregular variations can be obtained from the following formula I= CI/C Where, I= irregular variation, CI = cyclical and irregular variation, C = cyclical variation In practice irregular variations are highly erratic and also both cyclic and irregular variations are so interwoven that it becomes extremely difficult to segregate from cyclic variation.

Time series analysis has applications in a large number of areas. Some of the applications are given below i. Understand changing world temperature patterns ii. Economic Forecasting iii. Sales Forecasting iv. Budgetary Analysis v. Stock Market Analysis vi. Yield Projections vii. Process and Quality Control viii. Inventory Studies ix. Workload Projections x. Utility Studies xi. Census Analysis

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