Question 12 Marks
Describe the graphical method to measure trend.
AnswerSuppose $\left\{y_t: t=1,2,3, \ldots . n\right\}$ is a time series and $n$ terms of the series are $y_1, y_2, y_3 \ldots . . y_n$ respectively.
- Taking the time t on $X -$axis and the term $\mathrm{y}_{\mathrm{t}}$ of the time series on $Y -$axis, the point ( $\mathrm{t}, \mathrm{yt}$ ) plotted on the graph paper.
- Then points are joined by line segments.
- A continuous curve passing from the viscinity of most of the points in drawn.
- The curve so obtained is called the trend line of the time series.
View full question & answer→Question 22 Marks
State the merits of the method of moving average to measure trend.
AnswerMerits of moving average method is given below :
- We can get to the point value as it is based on principles.
- The calculation is easy to understand as it is comparatively less and simple.
- This is a moving method to find trend so, any value is added or subtracted, we can not do whole method once again.
- As average is used, uncertainty of variable removed.
- The effect of short-term component is eliminated to a large extent using the averages and the trend of the series is obtained.
View full question & answer→Question 32 Marks
AnswerA time series is a set of observations taken at specified time periods.
- The time is taken as an independent variable in the time series which will be denoted by $t$ and the dependent variable associated with it will be denoted by $y_t$.
- Thus, we shall represent the time series for different units of time as follows :
- Time $t\ 1\ 2\ 3 ....... n$
- Variable $y_t ~y_1 ~y_2 ~y_3 \ldots \ldots . . y_n$
View full question & answer→Question 42 Marks
Explain the meaning of moving average.
AnswerThe 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.
$\rightarrow $ 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.
$\rightarrow $ Keeping the average value in the centre the average obtained by dividing moving total by the period of moving average is called moving average.
$\rightarrow $ Suppose, the period of moving average is $3$ years, then $‘3’$ yearly moving average $=\frac{3 \text { yearly moving total }}{3}$
$\rightarrow $ Since the average value lies in the centre, we get the trend values that are free from short term variation.
$\rightarrow $ 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.
View full question & answer→Question 52 Marks
State the limitations of graphical method.
AnswerThe limitations of graphical method are as follows :
- It is possible that different people draw different curves.
- Hence, the uniformity is not maintained in the trend and its estimates.
- The estimates cannot be accurate as this is not a mathematical method and it is not possible to know the reliability of the estimates.
View full question & answer→Question 62 Marks
Explain the irregular component.
AnswerThe component obtained by eliminating trend, seasonal fluctuations and cyclical fluctuation from the variable quantity $y_t$ of time series is random or irregular fluctuations.
- It is denoted by the symbol $R_t$ where $R_t=y_t-\left(T_t+S_t+C_t\right)$.
- This variation occurs due to natural disasters like earthquake, floods or due to man-made predicaments like war, strike, political upheaval.
- This variation also occur due to technological invention.
- Irregular component cannot be predicted, it does repeat regularly and can not be controlled.
View full question & answer→Question 72 Marks
How does seasonal component differ from the cyclical component ?
AnswerSeasonal and Cyclical components both are short term in nature.
- Their difference is as follows:
| Point of Difference |
Seasonal Component |
Cyclical Component |
| $1.$ Meaning |
The changes taking place in the variable quantity $y_t$ of time series at fixed period of time in the year say winter, summer, monsoon is called seasonal component. |
The fluctuations arising due to boom and recession or business cycles are called cyclical component. |
| $2.$ Regularity |
Regularity is seen in seasonal component. |
Regularity is not seen in cyclical component as compared to seasonal component. |
| $3.$ Period of Oscillation |
The period of oscillation of such variations is usually less than a year. |
The period of oscillation of these variations can be $2$ to $10$ years and in specific circumstances it can also be $10-15$ years. |
| $4.$ Symbol |
We denote this component of the time series by $S_t$. |
We denote this component of the time series by $C_t$. |
| $5.$ Calculation |
It is easy to obtain seasonal component. |
It is mathematical complicated to obtain cyclical component. |
| $6.$ Example |
The demand of woollens Increases in winter, the demand of cold drinks and ice-creams increases in summer, the demand of umbrellas or raincoat increase in monsoon, the increase in sales of readymade garments or footwears in any festivals are the examples of seasonal components. |
The cycles of boom and recession, variation found in the time series of trade and financial matters. |
View full question & answer→Question 82 Marks
What is meant by Cyclical Component ?
AnswerThe fluctuations arising due to boom and recession or business cycles are called cyclical fluctuations.
- The period of oscillation of these variations can be $2$ to $10$ years and in specific circumstances it can also be $10-15$ years.
- The cyclical component of time series is denoted by ' $\mathrm{C}_{\mathrm{t}}$ '
- The component that remain after eliminating trend, $t_y$, seasonal and random component from the terms $y_t$, is called the cyclical fluctuations.
- Thus, $C_t=y_t-\left(T_t+S_t+R_t\right)$.
View full question & answer→Question 92 Marks
Describe the additive model of time series.
AnswerThe value of the variable of the time series $Y_t$ based on time $t$ is determined with the combined effect of trend $\left(T_t\right)$, seasonal component $\left(\mathrm{S}_t\right)$, cyclical component $\left(\mathrm{C}_t\right)$ and random component $\left(R_t\right)$.
- This is called additive model. This shown as follows:
$Y_t=T_t+S_t+C_t+R_t$ View full question & answer→Question 102 Marks
Using the following details fit a trend line:$n-6, \bar{y}-1020, \sum t y-21840$
View full question & answer→Question 112 Marks
The following results are obtained from the data on price indices during the time period $2012-2017$, obtain the trend line:$\sum y=648, \sum t y=2718$
Answer$\hat{y}=18.02+25.71 t$
View full question & answer→Question 122 Marks
The trend line of the time series of production is $\hat{y}=675+80.5 t$. Find the values of constants a and $b$. Obtain the trend values for $t=2$ and $t=5$,
Answer$a=675, b=80,5, \hat{y}=836 \text { for } t=2, \hat{y}=1077,5 \text { for } t=5$
View full question & answer→Question 132 Marks
The following details are obtained from the data on the units sold of $A C(y)$ during the four quarters of the year $2019 .$ Using these details fit a trend line:$\begin{aligned}&\sum y=43, \sum t y=118, \sum t_{2}=30 \\&y=5.50+2.1 t \\&\text { The following information is available q } \\&\text { months: } \\&n=9, \sum y=118, \sum t y=710.4\end{aligned}$Fit the trend line to the time series data and predict the value for $t =10$.
Answer$\hat{y}=\hat{y}=3.06+2.01 t, \hat{y}=23.16 \text { for } t=10$
View full question & answer→Question 142 Marks
For a time serles of profit, the following results are obtained from the data on profit y during the time period $2013$ to $2017 .$ Using this data fit a trend line and obtain the forecast of profit for the year $2020 .$
$n=5, \sum y=28, \sum t y=127$
Answer$\hat{y}=-7.3+4.3 t , \hat{y}=27.1 \text { for } t =8$
View full question & answer→Question 152 Marks
The following results are obtained from the data on varlable y for the first eight months of the year $2017 .$ Using this data fit a trend live and obtain the forecast for $t=$ $10, n =8, \sum y=148, \sum t y 749$.
Answer$\hat{y}=9.59+1.98 t , \hat{y}=29.39 \text { for } t =10$
View full question & answer→Question 162 Marks
The following results are obtained from the data on variable y for the first eight months of the year $2017 .$ Using this data fit a trend live and obtain the forecast for $t=$ $10 . n=8, \sum y=148, \sum t y 749$
Answer$\vec{y}=9.59+1.98 t, \hat{y}=29.39 \text { for } t=10$
View full question & answer→Question 172 Marks
Obtain the linear equation for trend for a time series with $\bar{t},=3.5 . \bar{y},=34.4$ and $n \sum t y=4461.6$
View full question & answer→Question 182 Marks
Obtain the linear equation for trend for a time series with $t=3, \sum y=214$ and $\sum t y=621$
Answer$\hat{y}=49.1-2.1 t$
View full question & answer→Question 192 Marks
Obtain the linear equation for trend for a time series with $n=6, \bar{y}=23.5$ and $b=0.09$, Find the estimated value of variable $y$ for $t=9$.
Answer$\bar{y}=23.18+0.09 t , \hat{y}=23.99 \text { for } t =9$
View full question & answer→Question 202 Marks
Obtain the linear equation for trend for a time series with $n=7, b 2.02$ and $a=12.66$. Find the estimated value of variable $y$ for $t=8$.
Answer$\hat{y}=12.66+2.02 t , \hat{y}=28.82 \text { for } t =8$
View full question & answer→Question 212 Marks
Obtain the linear equation for trend for a time series with it $=8, \bar{y}=43$ and $b =-4.9$. $\bar{y}=65.05-4.9 t$
Answer$\hat{y}=65.05-4.9 t$
View full question & answer→Question 222 Marks
Obtain the linear equation for trend for a time series with $n=7, \sum y=16.91$ and $\sum t y=62.02$
Answer$\bar{y}=3.26-0.21 t$
View full question & answer→Question 232 Marks
State the merits of the graphical method of estimating the trend of the time series.
AnswerThe merits of graphical method are as follows :
$(i)$ This is the simplest method of determining the trend of the time series.
$(ii)$ The trend can be found without any mathematical calculation.
$(iii)$ It is quite easy to understand.
$(iv)$ The mathematical form of the curve of trend of the time series can be known in general.
$(v)$ This method can be used even if the trend is not linear.
View full question & answer→Question 242 Marks
State the components of time series.
AnswerThe components are :
1. Long-term variation or Trend (T)
2. Seasonal variation (S)
3. Cyclical variation (C)
4. Random or Irregular variation (R)
View full question & answer→Question 252 Marks
State the merits of the method of moving average to measure trend.
Answer1. This method is simple to understand and easy to calculate.
2. It successfully eliminates the effect of short-term variations (seasonal and cyclical).
3. If the variation is regular, this method gives a very accurate trend.
4. It does not require any mathematical functional form for the trend.
View full question & answer→Question 262 Marks
Explain the irregular components.
AnswerIrregular components (or random variations) are short-term fluctuations in a time series that are unpredictable and do not follow a set pattern. They are caused by unforeseen events like natural disasters, strikes, wars, or sudden policy changes. They are represented by $ R_{t} $ in the additive model.
View full question & answer→Question 272 Marks
State the limitations of graphical method for determining trend.
Answer1. Different people may draw different trend lines, making it subjective.
2. It is not possible to get accurate mathematical estimates for the future.
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