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Correct option is  (c) t

The graph of time series is called Historigram.

Correct option is  (a) Random

Correct option is  (b) y_t = T_t + S_t + C_t + R_t

The data available in the form of time series {y_t: t = 1, 2, …, n} are the bivariate data, where t is the independent variable and the variable quantity y is the dependent variable. From this data we have to obtain the linear trend that suits to the time series. According to linear regression model, we have to obtain the linear trend model y_t = α + βt + u_t, where t = 1, 2, …, n and u_t is an error variable. According to the least squares method, we determine the values of the constants α and β in such a manner that the sum of the squares of error variable, i.e., Σe² = Σ(y_t – α – βt)² is minimised. If ‘a’ and ‘b’ denote the estimated values of a and p respectively then ‘α’ and ‘β’ can be obtained by the following formulae:

\(b =\frac{ nΣty−(Σt)(Σy)}{nΣt^2−(Σt)^2}\) and a = ȳ – bt̄

where, ȳ = \(\frac{Σy}{n}\); t̄ = \(\frac{Σt}{n}\); n = No. of observations
Thus, the fitting of trend line is given by ŷ_t = a + bt. From this trend line we can forecast the variable quantity y of the time series for the values of t following the time duration after t = n.

We can obtain the estimates of trend values for all the terms of the time series by this method. If for the given time series there is no linear trend, this method is not useful.

Seasonal component is short-term in nature and its effect is seen according to the seasons. The significant variations occured in the variable quantity of time series at fixed period of time in the year say winter, summer, monsoon. For example, a spurt in demand for woolen clothes in the winter, a spurt in the sale of ice-cream and cold drinks in the summer, increase in demand for umbrellas and raincoats in the monsoon, increase in the sale of readymade garments and footwears during the religions festivals, etc. are the illustration of seasonal fluctuation.

From the study of seasonal component traders and manufacturers of seasonal goods plan for periodical stocks to take care of the demand of such goods. Generally, the duration of oscillation of seasonal fluctuations is of one year. The seasonal component is denoted by the symbol S_t.

Generally the period of oscillation of seasonal component is less than a year. To study the seasonal component it is necessary to have the value of series in short-term. If the yearly values of the variable are available then it is not possible to obtain the information of seasonal component.

The merits and limitations of the least squares method of fitting the linear trend are as follows:

Merits:

• This method of determining the trend of the time series is mathematically more compact than the graphical method.
• In this method the estimates of the trend values for all the terms of the time series can be obtained.
• From the equation of linear trend of the time series, we can forecast the variable quantity y_t for any value of t.

Limitations:

• This method of determining the trend of the time series is difficult than the graphical method from the calculation view point.
• If there is lack of linear trend in the time series, this method is not useful for fitting the linear trend.

The limitation of the method of moving average are as follows :

• If the internal for moving, average is not chosen properly, then the trend obtained by this method is not accurate.
• In this method the estimates of trend for some initial and last time periods cannot be obtained.
• In this method a specific mathematical formula is not obtained for future estimates.

The method of moving average is very useful to find trend by eliminating the effect of short-term variations.

The period of moving average : The short-term variation are usually regular and have repetition. The period of repetition of these variations can be found by observing the given time series. The average is found from the number of observations corresponding to this period which is known as the period of moving average.

Since the average value lie in the centre, the values obtained by this method show the trend.

Suppose the values of variable are y₁, y₂, …. y_n corresponding to time t = 1, 2, …, n and the period of moving average is 3 years. Then the mean of first three observations y₁, y₂, y₃ is found as \(\frac{y_1+y_2+y_3}{3}\) and it is written against the centre of these three observations which is y₂. Further, the mean of successive three observation y₂, y₃, y₄ is obtained and it written against y₃. Similarly, finding successive moving total of three observations, averages are calculated. These average are called three yearly moving averages which indicate trend.

The period of moving average not necessarily every time is year. It may be 5 days, 4 weeks, 7 months, etc.

If time period of moving average is an even number say. 4, 6, … etc., then the process of finding moving average is to be done twice.

Suppose, the period of moving average is 4 years. The four yearly successive averages

\(\frac{y_1+y_2+y_3+y_4}{4} , \frac{y_2+y_3+y_4+y_5}{4}, \frac{y_3+y_4+y_5+y_6}{4} ……,\)

are written between y₂ and y₃, y₃ and y₄, y₄ and y₅ …. respectively since, these averages are in between two years, the average of each pair of averages is found and written between two moving average. Thus, the average of the first two averages will be written against y3. The averages thus obtained are called as four yearly moving average.

When the effect of short-term variations is to be eliminated to find trend, method of moving average is more useful.

The merits of the method of moving average to measure trend are as follows :

• The effect of short-term component is eliminated to a large extent and trend values of the time series are obtained.
• In this method the calculation to find trend is easy and it is simple to understand.

The original values of time series are divided in to two equal parts (if odd number ignoring the middle value) and averages are calculated both the parts, are called semi averages the two points and extending the line which gives the trend of the time series.

Actual values are plotted on a graph which is a history gram and by plotting the semi-averages on a graph joining the two points and extending the line which gives the trend of the time series.

Suppose {y_t: t = 1, 2, …, n} is a time series and n terms of the series are y_t, y₂, …, y_n respectively. Taking the time t on X-axis and the term y_t of the time series on Y-axis, the point (1, y₁), (2, y₂) … (n, y_n) are plotted on the graph paper. Then points are joined by line segments. The graph so obtained is called the graph of the time series. A continuous curve passing from the viscinity of most of the points is drawn. The curve so obtained is called the trend line of the time series.

This is the simple and crude method of determining the trend of the time series. This method is quite easy to understand. But the mathematical form of the trend cannot be obtained by this method. When the plotted points of the time series are widely scatter from one another, it is difficult to draw an unique curve representing the trend of the time series. In such a situation, more than one curve can be drawn representing the trend of the time series. As a result it becomes difficult to determine the trend of the time series.

Correct option is  (d) Method of moving average

By substituting a = 25, 

b = 3.5 and x = 4. 

we get y = 15 + 3.5 (4) 

= 25 + 14 = 39.

Seasonal Indices are used for the measurement of seasonal variation.

ŷ = 25.1 – 1.3t
Putting t = 8, we get
ŷ = 25.1 – 1.3 (8)
= 25.1 – 10.4
= 14.7

Hence, the estimate of y for the eighth week obtain is ŷ = 14.7.

By substituting the value of x 

= 5 in the equation. 

We get y = 75.3 – 2.75 (5) 

= 75.3 – 13.75 

∴ y = 61.55.

The term secular trend refer to the tendency of a variable to increase or decrease or to remain constant over a period of time.

The second degree/parabolic trend/ equation is y = a + bx + cx^2.

Seasonal component and cyclical component differ in the following manner:
The variations occurring in the time series almost regularly over less than one year is the effect of seasonal component, while that of more than a year is the effect of cyclical component.

The period of oscillation of seasonal component is usually less than a year, while it can be 2 to 10 years and in special circumstances it can also be 10 to 15 years.

Seasonal component is the effect of natural factors and man-made factors, while cyclical component is the effect of economic situations and business cycles.

The increase in the sales of readymade garments and shoes daring festivals is an example of seasonal component while the cycles of boom and recession are the examples of cyclical component.

(1) Prosperity 

(2) Decline 

(3) Depression and 

(4) Improvement.

•  Secular trend/Trend refers to the tendency of the variable to increase, decrease, or remain steady over a given period. 
• Seasonal variations refer to the variations caused annually by the seasons of the year. Also includes the variations of any kind which are periodic in nature and whose period is shorter than one year. ( t < 1 year) 
• Cyclical variations are fluctuations spread over a period of more than one year. Most of the Time series relating to Economics and Business show some kind of cyclical variations. Cyclical variation is an oscillatory variation, which occurs in four stages, such as (1) Prosperity (2) Decline (3) Depression and (4) Improvement. 
• Irregular / Erratic variations are those changes of time series, which are irregular in nature and do not show any pattern. The causes of irregular variations are due to accidental happenings such as wars, earth quacks, floods, famine, fire, strikes, etc.,

Examples of Business and Economics such as price, sales, production etc.

The method to obtain the estimates of different components of a time series is called analysis of time series.

The methods of measuring trend are:

• Graphical method.
• Least square method and
• Moving average method.

Correct option is  (c) 36.1

Correct option is  (a) Random component

(b) Decrease in the share prices due to recession in share market

The additive model of time series is as follows :

y_t = T_t + S_t + C_t + R_t

Where, y_t = Time variable; T_t = Trend; S_t = Seasonal component; C_t = Cyclical component; R_t = Random component.

(a) Secular trend 

(b) Irregular variations

1. Fire accident in a factory 

2. Increase of woolen goods during winter 

3. Fall in the death rate due to advances in Science 

4. An era of Prosperity 

5. An increase in employment during harvest season. – (1) Random, (2) Seasonal, (3) Secular, (4) Cyclical variations (5) Seasonal variation

The components of time series are: 

1. Secular trend. 

2. Seasonal trend. 

3. Cyclical trend. 

4. Irregular / Random Variations.

Additive Model of Time Series :

= T_t + S_t + C_t + R_t
Where, y_t = Total variation of time series
T_t = Trend, S_t = Seasonal component,
C_t = Cyclical component,
R_t = Random component

• Trend ; T_t = y_t – (S_t + C_t + R_t)
• Seasonal component: S_t = y_t – (T_t + C_t + R_t)
• Cyclical component :C_t = y_t– (T_t + S_t + R_t)
• Random component: R_t = y_t – (T_t + S_t + R_t)

• ‘3’ yearly moving average = \(\frac{3′ yearly\, moving \,total }{3}\)
• ‘4’ yearly moving average =  \(\frac{Total \,of \,pairs \,of \,‘ 4 ‘\, yearly\, moving \,total }{8}\)
• Where, Total of pairs of ‘4’ yearly moving total
  = (‘4’ yearly moving total for t = 1, 2, 3, 4) + (‘4’ yearly moving total for t = 2, 3, 4, 5)

• A set of figures relating to a variable according to a time is called Time series OR chronological arrangements of the statistical data is called Time series 
• Production of a firm arranged from the year 2000 to 2008, Sales statistics of a certain article for the year 1998 to 2005.

The merit is it is simple and easy; 

Demerit is that, not all trend values can be obtained by this method.

• The equation is y = a + bx; Normal equations are

na + bΣx = Σy;

aΣx + bΣx² = Σxy

• Quadratic/second degree equation/parabolic trend equation is:

y = a + bx + cx² : Normal equations are: na + bΣx + cΣx² = Σy;

aΣx + bΣx² + cΣ = Σxy ; aΣx² + bΣx³ + cΣx⁴ = Σx²y.

Trend is the overall change taking place in the series of the data over a given period.

Moving averages are the averages obtained by finding the arithmetic means of successive values of the time series by taking say 3/4/5 years at a time by leaving the first and adding the next; this is to eliminate short-term variations/fluctuations that are present in the time series.

Trend is an important component of a time series. It is the measure of permanent 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 yt of a given time series. It is denoted by the symbol yt’. Trend is the component obtained by eliminating the effect of short-term fluctuations (namely seasonal and cyclical) from the total fluctuations in the variable quantity of time series. In the time series of human population of any country the effect of trend is likely to be more pronounced than that of other components.

From the study of trend of a time series we can predicate a value of yt of the time series for some future value of t or we can determine the missing value for some past value of t. Also two or more sets of time series data can be compared with the help of their trends. Important decisions concerning economic policy can be taken on the basis of the trend of time series of economic data.

The estimate of the trend of time series can be obtained by the following two methods :

• Graphical method and
• Least squares method of fitting linear trend.

It can be used for predicting future trend/Trend values can be obtained for all time points.

The sum of the deviations obtained from the actual and trend values is Zero ie. Σ(Y – Ȳ)= 0 and The sum of the squares of the deviations obtained from the actual and trend values is least. i.e., Σ(Y – Ȳ)² is least.

Analysis of Time series helps to understand the past behavior of the data / to predict the future trend.

1. Method of moving averages and 

2. Method of least squares.

The analysis of time series is useful in trade, science, social and political fields as follows :

• It is possible to know the past situation and use it to obtain the type and measure of variation.
• By using statistical method the estimated value of the variable in future can be obtained.
• Using the estimated values proper decisions can be taken for the future and activities can be planned accordingly.
• A comparative study can be carried out for the variations in the given variable at different times and places.
• The estimated obtained from the past data can be compared with the present values and the reasons for discrepancies between them can be investigated.

A time series is a set of observation taken at specified time periods.

Analytical study of the time series will helps to predict / plan for future.

The notation to show the cyclical component of the time series is ‘C_t’.

The variations occurring in the time series at approximately regular intervals of more than one year due to the effect of depression, recovery, boom, recession and business cycle are called cyclical variations known as cyclical component. It is denoted by ‘C_t‘.

The oscillation period for cyclical component can be 2 to 10 years. In specific circumstances it can also be 10 to 15 years.