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Irregular variations are unprediatable and secular and seasonal variations are predretable.

In this method the actual values are plotted on a graph and smoth line or curve is drawn.

Again a line or curve is drawn a long the actual line which covers majority of the points according to personal judgment. This line is called line of best fit, is the trend line.

Correct option is  (b) Method of least squares

The statistical data collected at certain interval of time and arranged in a chronological order is called time series, e.g., population of a country, agricultural production, wholesale price index number, data from share market, the hourly temperature in a city, etc. are presented with respect to time.
Time series consists of values of a variable associated with time.

Correct option is  (c) Increase in the number of tourists during school vacation

The important reasons for the study of time series are as follows:

• The changes occuring in the values of the variable quantity of the time series indicate the situation that prevailed in the past and this knowledge becomes helpful in making decisions for the future.
• We can predict the future value or values of the variable quantity on the basis of the study of the time series.
• The policy decisions made on the basis of the study of time series, enable business firms, industrial houses and government organisations to formulate their development plans for future.
• The study of time series enable business firms, industrial houses and government organisation to compare their present performance of the economic activities with the past and make an assessment of the actual progress achieved. For example, the government can assess the changes occurred in the inflation rate on the basis of wholesale price index numbers during the five year plan period.

The effect of seasonal component indicates fluctuations repeating within a year.

Correct option is  (c) Regular

The importance of the time series are as follows:

• The direction and pattern of variation in the values of the series can be known from the past data.
• The variations in the future can be estimated.
• Important decisions can be taken for future.
• Comparative study of data can be done.
• Industrial and government policies can be easily framed.

The uses of the time series are as follows :

• Knowing the past situation to obtain the type and measure of the variation.
• To estimate the value of the variable in future.
• To take proper decision for the future.
• A comparative study can be carried out for the variations in the series.
• To find reasons for the discrepancies between the estimates and the present values.

The components of time series are :

• Long-term component or Trend,
• Seasonal component,
• Cyclical component and
• Random or Irregular component.

The seasonal components and cyclical component of time series produce short-term variations.

1. Long-Term Component or Trend: It is the measure of effect which prevails on the time series for a long period of time. Trend may be either increasing or decreasing or stable in terms of time. It represents the mathematical form and direction of variation taking place in the variable quantity y_t of a given time series. It is denoted by the symbol Tt.

2. Seasonal Component: It represents the changes taking place in the variable quantity y_t of time series at fixed period of time in the year (e.g., winter, summer, monsoon). It is short term in nature and denoted by the symbol St.

3. Cyclical Component: The variations arising due to boom and recession or business cycles in the variable quantity y_t of time series are called the cyclical component. The duration of oscillation of cyclical variation varies from 10 to 20 years. It is denoted by the symbol C_t. Eliminating trend, seasonal component and random component from the variable quantity y_t of time series, we can obtain the cyclical fluctuation.
Thus, Ct = y_t – (T_t + S_t + R_t).

4. Random or Irregular Component: The fluctuations arising in the variable quantity y_t of time series due to political, natural or random causes are called random or irregular component. It is denoted by the symbol Rr The component obtained by eliminating trend, seasonal fluctuations and cyclical fluctuations from the variable quantity y_t of time series is a random or irregular component. Thus, R_t = y_t – (T_t + S_t + C_t). This type of fluctuations are subject to uncontrollable forces and cannot be predicted in advance.

Trend is an important component of a time series. The main methods for determining trend are as follows :

1. Graphical Method: This is an easiest method for determining trend. Taking t on X-axis and time variable yt on Y-axis points are serially joined by linesegments a curve of time series is obtained. A curve passing through close to most of the points of time series curve is showing trend of the given time series. This method is not reliable for determining trend.

2. Method of Least Squares: This method for determining the trend is perfect mathematical and reliable. In this method trend is obtained for each value of t. To fit the equation of linear trend y_t = a + bt, the values of ‘a’ and ‘b’ are obtained such that the sum of the squares of errors [Σe_i² = X(y_i — ŷ_i)² = Z (y – a – bt)²] is minimum.

3. Method of Moving Average: This method determines trend values by eliminating the effect of short-term variations. The period of moving average is obtained by taking the average of oscillations of short term variations. By this method trend is not obtained for each t.

The short term variations are usually regular and have repetitions. The period of repetition of these variations are fond and their average is found for the given time series.

This average of repetitions is known as the period of moving average. We find moving total of the variables of the given time series corresponding to the period of moving average.

Keeping the average value in the centre the average obtained by dividing moving total by the period of moving average is called moving average.

Suppose, the period of moving average is 3 years, then ‘3’ yearly moving average

\(= \frac{3 ‘ yearly\, moving \,total}{ 3}\)

Since the average value lies in the centre, we get the trend values that are free from short term variation.

If the period of moving average is odd number say 3. 5, 7, …, moving average can be calculated easily. If it is even number its calculation becomes some what difficult.

The limitation of graphical method are as follows :

• In this method different people draw different curves. Hence, the uniformity is not maintained in the trend and its estimates.
• This is not a mathematical method. So it is not possible to know the reliability of the estimates.
• The exact form of the trend line of the series cannot be obtained by this method.
• When the plotted points of the time series are widely scatter from one another, then instead of drawing an unique curve of the trend of the time series, more than one curve can be drawn representing the trend.

In the variation of time series, seasonal component and cyclical component are regular component. More and above these there is effect of irregular or random component which is short-term effect.

The element that remains after eliminating the trend T_t, Seasonal component S_t, Cyclical component C_t from the term y_t of the time series is known as random or irregular component and is given by R_t = y_t – (T_t + S_t + C_t).

Random component are subject to natural forces like flood, draught, earthquake, political crisis and random causes like fire, accident, etc. Sometimes it is seen on account of technological innovations.

Random component cannot be controlled nor can be predicted in advance. Hence the variable quantity yt of time series cannot be exactly predicted by studying the time series. Sometimes, effect of random component is temporary and transient. In such situation the stipulated targets of agricultural and industrial production cannot be achieved.

Seasonal Variation.

Irregular variations.

Linear/Straight line trend/line.

Irregular/random variation.

The method of moving average is a simple arithmetic mean calculated successively by taking some specified number of values at a time (m) say 3, 4, or 5 years. The aim of averaging is to remove short term variations in the time series.

In this method a mathematical relation is developed between the time (x) and the values (y). The relation may be used to fit (a) Linear/straight line trend (6) Quadratic trend (c) Exponential trend. Such that “the sum of the squares of the deviations obtained from actual and trend vales is least”. And these relations are used for computing the trend values in the time series.

(a) Random / irregular 

(b) seasonal, 

(c) secular and 

(d) cyclical variations.

• Method of moving averages. 
• Method of least squares.