## Index Numbers And Time Series Analysis – CS Foundation Statistics Notes

A time series is a collection of observations of well-defined data items obtained through repeated measurements over time. For example, measuring the value of retail sales each month of the year would comprise a time series. This is because sales revenue is well defined, and consistently measured at equally spaced intervals. Data collected irregularly or only once are not time series. Time series are best displayed in a scatter plot. The series value X is plotted on the vertical axis and time t on the horizontal axis.

1. Time series analysis
Time series analysis accounts for the fact that data points taken over time may have an internal structure The purpose of time series analysis is to

• to study the past behavior of records
• to forecast for future

2. Components of time series
Any time series can contain some or all of the following components:

• Trend (T)
• Cyclical (C)
• Seasonal (S)
• Irregular (I)

3. Trend component
The trend is the long-term pattern of a time series. A trend can be positive or negative depending on whether the time series exhibits an increasing long-term pattern or a decreasing long term pattern. If a time series does not show an increasing or decreasing pattern then the series is stationary in the mean.

4. Periodic Variations
(a) Cyclical component
Any pattern showing an up and down movement around a given trend is identified as a cyclical pattern. The duration of a cycle depends on the type of business or industry being analyzed.

(b) Seasonal component
Seasonality occurs when the time series exhibits regular fluctuations during the same month (or months) every year, or during the same quarter every year Regardless of the trend, we can observe that in each year more ice creams are sold in summer and very little in Winter season. The sales in the departmental stores are more during festive seasons than in the normal days. Retail sales peak during the month of December.

5. Irregular component
This component is unpredictable. Every time series has some unpredictable component that makes it a random variable. In prediction, the objective is to model all the components to the point that the only component that remains unexplained is the random component. Irregular fluctuations results due to the occurrence of unforeseen events like floods, earthquakes, wars, famines, etc.

6. Purpose of Time Series
The purpose of time series analysis is to decompose data of time series Methods used for decomposition are
1 Additive models For this method
Data = Seasonal effect + Trend + Cyclical + Residual For monthly data, an additive model assumes that the difference between the January and July values is approximately the same each year. In other words, the amplitude of the seasonal effect is the same each year.

2 Multiplicative models
In many time series involving quantities (e.g. money, wheat production, …), the absolute differences in the values are of less interest and importance than the percentage changes Data = Seasonal effect x Trend * Cyclical x Residual

7. Measurement of Trend
To measure the trend, the short-term variations should be removed and irregularities should be smoothed out. The following are the methods of measuring trends.

1. Graphic (or freehand curve) method
2. Semi-average method
3. Moving average method
4. Least squares method

1. Freehand method or graphic method
It is the simplest method to determine a trend. Simply, the freehand method is to create a trend line in accordance with what we see. First of all the data is plotted on a graph paper and the trend line is fitted by a line or a freehand curve by just inspecting and following the graph of the series. The curve needs to be smooth and with an almost equal number of points above and below it. By eye estimate, the sum of the vertical deviations of the given points above the trend line should approximately equal the sum of the vertical deviations of the given points below the trend line. Also, the sum of the squares of the vertical deviations of the given points from the trend line should be the minimum possible.

2. Semi average
This method is also simple and relatively objective than the freehand method. The data is divided into two equal halves and the arithmetic mean of the two sets of values of Y is plotted against the center of the relative time span. If the numbers of observations are even the division into halves will be straightforward, however, if the number of observations is odd, then the middlemost i.e.,
$$\left(\frac{n+1}{2}\right)$$th , item is dropped. The two points so obtained are joined through a straight line which shows the trend.

3 Moving Average Method: This method comprises of taking arithmetic means of the data/values for a certain span and then placing the value so calculated against the middle of the time span. The time span should be equal to the average fluctuation period. If this span is of period k, then the moving averages obtained by averaging k at a time are called Moving Averages of period or extent k. If k is even, the successive values of moving averages are placed in the center/middle of the period/span of time.

4. Least Squares Method Polynomials are one of the most commonly used types of curves in regression. The least-squares method is used to find the best linear relationship between two variables.

The least-squares line uses a straight-line method to approximate the given set of data y = a + bx The unknown coefficients aandb can therefore be obtained: Here y=production/sales etc
X=time
Please note that aandb are unknown coefficients while all xiandyi are given.
B Least square parabola
The least-squares parabola method uses a second degree curve y = a + bx + ax2 to approximate the given set of data, (x1, y1), (x2, y2), ….,(xn yn), where n ≥ 3 The least-squares parabola uses a second degree curve y- a + bx + cx2

8. Forecasting
Forecasting is a strategy used in different fields to predict the future based upon the past. This strategy is pulled from different data sources to provide a financial expert or entrepreneur with the needed information to run a business or invest more effectively and successfully. It is used by companies to determine how to allocate their budgets for an upcoming period of time.

Survey – Surveys provide a means of measuring characteristics, self-reported and observed behavior, attitudes or opinions, etc. of society. These methods are often used for forecasting sales or demand of a product Various methods of surveys are

– Complete enumeration method – in this method data is collected from all the individuals. This method is costly particularly when the survey population is large. A complete enumeration-based survey is often preferred for certain types of data, solely because it is expected that it will provide complete statistical coverage over space and time. Complete enumeration may be required as a statutory obligation, often for regulatory purposes.

– Sample survey method – this type of survey method is used when the population is large, a well-designed sample-based survey can often provide good estimates of important parameters at a fraction of the cost. Sample surveys operate on selected subsets of the target population and, using a number of assumptions regarding the distribution of the population, provide estimates of the parameters under study. Sample-based surveys involve uncertainties as to the correctness of the various assumptions used.

Some of the sample survey methods are

1. Test Marketing – it is done for the launch of a new product or if any existing product is being introduced in a new market. This technique is used during the product development or market introduction phase to determine how people respond to a product. It can be used at many different phases of development to see whether or not the public will buy the product, how the product may need to be adjusted to make it appealing to the public, and how members of the public interact with the product.

2. Expert opinion – Expert Opinion is a relatively informal technique that can be used to serve a variety of purposes, and may be used to assist in problem identification, in clarifying the issues relevant to a particular topic, and in the evaluation of products. In this, a group of experts sit together and form an opinion about the viability and success of the product

3. Delphi Technique – The Delphi technique is a group process used to survey and collect the opinions of experts on a particular subject. It is especially appropriate when it is not possible to convene experts in one meeting In this method the opinion of experts is generally taken by post. Trend projection methods – As in most other analyses, in time series analysis it is assumed that the data consist of a systematic pattern. These methods are cheaper than survey methods cause they take data from past records as a basis of analysis.

4. Time series data – we can fit mathematical trends in data for making forecasts. This analysis is more reliable for short-term forecasts.

5. Smoothing method – trend is calculated by smoothing out the fluctuation due to other components. Two main smoothing methods are a method of moving average and the method of exponential smoothing.

6. Lag technique or lead technique or Barometric technique- the forecast is done on the basis of already occurring events or currently occurring events. It is generally used for predicting business cycles situation.

9. Index numbers
In simple terms, an index (or index number) is a number showing the level of a variable relative to its level in a given base period. Edgeworth defined Index number as ‘ Index number shows by its variations the changes in a magnitude which is not susceptible either of accurate measurement in itself or of direct valuation in practice’
Further, Spiegel defined ‘index number as a statistical measure designed to show changes in variable or a group of related variables with respect of time, geographical locations or another characteristic’
Index numbers
– are time series that focuses on the relative change in a count or measurement over time.
– express the count or measurement as a percentage of the comparable count or measurement in a base period Important characteristic features of index number are

• These are expressed in percentages
• These are specialized averages
• They measure the relative change in the value of a variable or a group of related variables over a period of time

10. Uses of Index numbers

• Index number is used for measuring changes in the price level.
• Economists frequently use index numbers when making comparisons over time.
• Using an index makes quick comparisons easy.
• Index numbers are helpful in judging the changes in investment.
• They measure the level of business and economic activities and are therefore helpful in finding the economic status of the country.
• Throws Light on Economic Condition
• Importance For The Government
• Analysis of Industry
• Comparison of Developed and Under Developed Countries

11. Types of Index Numbers

1. Price Index Numbers – It indicates the relative price of a specific item. It is mostly used by statisticians, policymakers, etc. Usually, the index is assigned a value of 100 in some selected base period, and the values of the index for other periods are intended to indicate the average percentage change in prices compared with the base period. Price index numbers can be divided into Wholesale Price Index numbers and retail price index numbers. The wholesale price index number reflects the general price level in-country while the retail price index number shows the change in the retail price of various commodities.

2. Quantity index numbers – As the name suggests, these indices pertain to measuring changes in volumes of commodities like goods produced or goods consumed, etc. A quantity index is built up from information on quantities such as the number of the total weight of goods or the number of services etc.

3. Value index numbers – These pertain to compare changes in the monetary value of imports, exports, production, or consumption of commodities.

Changes in the price of an item are far easier to measure than changes in quantity (or value). The quantities (or values) of items sold in different outlets may vary enormously, depending on many factors including the size and location of the outlets. Prices, by comparison, will not vary so much from outlet to outlet. It is also much easier to find out the price of an item than the quantity that may have been sold during a month. Knowing price movements in a few outlets will be a good guide to the average movement overall. There will be a tendency for the price of similar items to change in a similar way. On the other hand, with some exceptions, quantities may vary widely and may be difficult to define.

12. Precautions in construction of Index Numbers l

• Be clear about the purpose for which the index number is used
• While selecting the base period it should be kept in mind that it is normal as well as relatively current.
• For construction arithmetic mean and geometric mean is used.
• Due importance should be given to different variables.

13. Method of construction of price index numbers

• Select one period as the base and separately calculate the movement between that period and each required period by either calculating the difference of the prices of two periods (base period and required period) or by “ calculating the ratio of the two prices of the period (base period and required period)
• Calculate the period-to-period movements and chain these

14. Calculation of the index number can be done in the following ways
Average of price relative methods- by taking suitable averages of the price of different item In this method, the price relatives for all commodities is calculated and then their average is taken to calculate the index number. Simple average of price relatives – One of the simplest types of index numbers is a price relative. It is the ratio of the price of a single commodity in a given period or point of time to its price in another period or point of time called the reference period or base period.
Price relative in percentage (of period 1 with respect to 0)
= $$\frac{p_{1}}{p_{0}}$$ × 100 ………………….. (14.1)

Price index number = Sum price relative in percentage for all item/number of items We denote price relative in percentage or without percentage Example
If the retail price of fine quality rice in the year 1980 was Rs.3.75 and that for the year 1983 was Rs.4.50, then find the price relative. Solution: Here the base period is 1980. The price of rice in the base period was Rs. 3.75. Also in the given period 1983, the price of rice was Rs.4.50. Using the formula (19.1), the required price relative is
P1980/1983 = $$\frac{\text { Rs. } 4.50}{\text { Rs. } 3.75}$$ × 100 = 120%

2 Weighted average of price relative method
An average in which each quantity to be averaged is assigned a weight. These weightings determine the relative importance of each quantity on average.
Example
the value of a commodity is as follows
Value: 10 8 5 432 1 0
weights: 2 2 1 10 8 7 68 2
Steps to calculate price index by Weighted average Multiply each value by its weight.
20,16, 5,40,24,14, 68, and 0
2. Add up the products of value times weight to get the total value.
Sum=187
3. Add the weight themselves to get the total weight.
Sum=100
4. Divide the total value by the total weight.
187/100 = 1.87

Aggregate methods – by taking ratios of averages of prices of different items. This is a simple method for constructing index numbers. In this, the total of current year prices for various commodities is divided by the corresponding base year price total and multiplying the result by 100. A simple aggregate index shows the change in the prices, quantities, or values of a group of related items. Each item in the group is treated as having equal weight for purposes of comparing group measurements over time.
Example
If the price of rice is Rs. 4 per kg in the year 1980 and Rs. 6 in the year 1983 and Rs. 8 in the year 1994 thus price relative is
P1980/1983 = $$\frac{\text { Rs. } 6}{\text { Rs. } 4}$$ × 100
P1980/1984 = $$\frac{\text { Rs. } 8}{\text { Rs. } 4}$$ × 100

In weighted agreegate method the index number is calculated by the ratio of weighted arithmetic mean of current year prices to base price

15. Nature of weights

1. Laspeyres price index – This index concentrates on measuring price changes from a base year
2. Paasche’s price index -this uses the end year quantities as weights
3. Fisher Ideal Index – It is the geometric mean of Laspeyres price index and Paasche’s price index.

16. Tests of the adequacy of Index numbers formulae

1. Unit test- It requires that the formula should be independent of the units in which or for which prices and quantities are quoted. This test is satisfied by all index number methods except the simple (unweighted) aggregative index method.

2. Time reversal test – is a test of determining whether a given method will work both ways in time forwards and backward. In the words of fisher, The test is that the formula for calculating the index number should be such that it will give the same ratio between one point of comparison and the other, no matter which of the two is taken as a base. In other words, when the data for any two years are treated by the same method, but with the bases reversed the two index numbers secured should be reciprocals of each other so that their product is unity

3. Factor Reversal Test- The factor reversal test requires that multiplying a price index and a volume index of the same type should be equal to the proportionate change in the current values. Symbolically, P01 × Q01 = = Value Index. The factor reversal test is satisfied only by the Fisher’s Ideal Index number.

4. Circular Test- It is concerned with the measurement of price changes over a period of years, when it is desirable to shift the base. If P01 represents the price change of the current year on the base year and P12, the price change of the base year on some other base and P20, the price change of the current year on this first base, then the following equation should be satisfied: P01 × P12 × P20 =1.

5. Chain base index number – this type of index number is used when the base period is too far from the current year. In such situations it may happen that the product which is in current year was of no or little importance in base year thus in such conditions a chain base index number is constructed.

Index Numbers And Time Series Analysis MCQ Questions

Question 1.
Which of the following is a measure of dispersion?
a. Mean
b. Percentiles
c. Quartiles
d. Mean absolute deviation
d. Mean absolute deviation

Question 2.
Which of the following is Not one of the uses of an index number?
a. To measure the economic well being of a country
b. Basis for comparing related series for administrative purpose
c. To measure the direction of movement of economic variables
d. To deflate series
c. To measure the direction of movement of economic variables

3. An index no. ¡s used
a. To measure changes in a variable over time
b. To measure changes in demand
c. To measure changes in price
d. To measure changes in quantity
a. To measure changes in a variable over time

Question 4.
The ratio of a new price to the base year price is called the
a. Price increase
b. Price relative
c. Price decrease
d. Price absolute
b. Price relative

Question 5.
Which of the following component is not included in a time series?
a. Regular
b. Trend
c. Seasonal
d. vinegar
a. Regular

Question 6.
A simple aggregate quantity index Is used to
a, Measure the change in the price of a product
b. Measures the overall change in the quantity of a range of products
c. Measures the overall change in the price of a range of products
d. None of the above
c. Measures the overall change in the price of a range of products

Question 7.
The price relative is a price index that is determined by
a. Price ¡n period t/base period price *00
b. Base period price/price in period *100
c Price in period t+base period price *100
d. None of the above
a. Price in period t/base period price *00

Question 8.
A simple aggregate price index
a. Compare relative quantities to relative prices
b. Compares absolute prices to absolute quantities
c. Ignores relative quantities
d. Considers reLative quantities
c. Ignores relative quantities

Question 9.
Out of the following, which is a reason for computing an index?
a. To project future sales
b. To estimates the trend in a time series
c. To check the base period
d. None of the above
d. None of the above

Question 10.
A composite price index based on the prices of a group of items is known as the
a. Laspeyres index
b. Paasche index
c. Aggregate price index
d. Consumer price index
c. Aggregate price index

Question 11.
This index measures the change from month to month in the cost of a representative basket of goods and services of the type bought by a typical household
a. Financial times index
b. Paasche price index
c. Laspeyres price index
d. Retail price index
d. Retail price index

Question 12.
Forecasts are referred to as naive if they
a. Are based only on past values of the variable
b. Are short-term forecasts
c. Are long-term forecasts
d. None of the above
a. Are based only on past values of the variable

Question 13.
A monthly price index that uses the price changes in consumer goods and services for measuring the change in consumer prices over time is known as the
a. Paasche index
b. Consumer price index
c. Producer price index
d. Laspeyres index
b. Consumer price index

Question 14.
Time series analysis is based on the assumption that
a. Random error terms are normally distributed
b. There are dependable correlations between the variable to be forecast and other independent variables
c. Past patterns in the variable to be forecast will continue unchanged into the future
d. The data do not exhibit a trend
c. Past patterns in the variable to be forecast will continue unchanged into the future

Question 15.
Though no longer the case, historically, the Dow Jones averages were aggregate price indexes showing the prices of stocks listed
a. On the American stock exchange
b. Over-the-counter
c. On the New York Stock Exchange
d. None of the above
c. On the New York Stock Exchange

Question 16.
Which of the following is Not a characteristic of a simple moving average?
a. It smoothes random variations in the data
b. It has minimal data storage requirements
c. It weights each historical value equally
d. It smoothes real variations in the data.
b. It has minimal data storage requirement

Question 17.
Which is not a characteristic of exponential smoothing?
a. Smoothes random variations in the data
b. Easily altered weighting scheme
c. Weights each historical value equally
d. None of the above
c. Weights each historical value equally

Question 18.
A composite price index where the prices of items in the composite are weighted by their relative importance is known as the
a. Price relative
b. Weighted aggregate price index
c. Consumer price index
d. None of the above
b. Weighted aggregate price index

Question 19.
A weighted aggregate price index where the weight for each item is its current period quality is called
a. Aggregate index
b. Consumer price index
c. Index of industrial production
d. Paasche index
d. Paasche index

Question 20.
A quantity index that is designed to measure changes in physical volume or production levels of industrial goods over time is known as
a. Index of industrial production and capacity utilization
b. Time index
c. Physical volume index
d. None of the above
a. Index of industrial production and capacity utilization

Question 21.
Forecasts
a. Become more accurate with longer time horizons
b. Are rarely perfect
c. Are more accurate for individual items than for groups of items
d. All of the above
b. Are rarely perfect

Question 22.
The three major types of forecasts used by business organizations are
a. Strategic, tactical and operational
b. Economic, technological, and demand
c. Exponential, tactical, seasonal
d. None of the above
b. Economic, technological, and demand

Question 23.
Which of the following is not a step in the forecasting process?
a. Determine the use of the forecast
b. Eliminate any assumptions
c. Determine the time horizons
d. All of the above
b. Eliminate any assumptions

Question 24.
The tendency of the trend to increase or decrease or stagnate over a long period of time is called
a. Periodic Variation
b. Cyclic Variation
c. Secular Trend
d. Random Variation
c. Secular Trend
Hint
The tendency of the trend to increase or decrease or stagnate over a long period of time is called a secular trend. It is a market trend with some characteristic phenomenon that is not cyclical but exists over a long period.

Question 25.
The equation Y= a+bx is used to get the value of
a. Parabolic Trend
b. Exponential Trend
c. Linear Trend
d. None of the above
c. Linear Trend
Hint
Linear trends show steady, straight-line increases or decreases where the trend-line can go up or down. Equation Y = a+bx is used to get the value of the linear trend.

Question 26.
The price index that uses current year quantities as weights is known as
a. Fisher ideal index
b. Paasche price index
c. Lasparey’s price index
d. Raman price index
b. Paasche price index
Hint
Paasche’s price index -this uses the end-year quantities as weights.

Question 27.
The test that requires that the product of Price Index & the corresponding quantity index number should be equal to the value index number is known as
a. Unit Test
b. Time Reversal Test
c. Factor Reversal Test
d. Circular Test
c. Factor Reversal Test
Hint
Factor Reversal Test- The factor reversal test requires that multiplying a price index and a volume index of the same type should be equal to the proportionate change in the current values.

Question 28.
The total sum of values of a given year divided by the sum of the values of the base year is
a. Price index
b. Quantity index
c. Value index
d. None of the above
c. Value index

Question 29.
The trend equation for the annual sale of a product is Y= 120+36x with the Year 1990 as the origin. The annual sales for the year 1992 will be
a. 156
b. 192
C. 120
d. None of the above
b. 192
Hint
Origin-1990
Annual sales of year 1992
Thus x =2
Y = 120 + 36 x
Y = 120+ 36 x 2 =192

Question 30.
The technique of estimating the probable value of phenomenon at a future date is called:
a. Interpolation
b. Extrapolation
c. Forecasting
d. Probability.
c. Forecasting
Hint
Forecasting is the process of making predictions of the future based on past and present data and analysis of trends.

Question 31.
This test of the adequacy of index number requires that the formulae for calculating an index number should give consistent results in both directions. This test is known as:
a. Time reversal test
b. Factor reversal test
c. Circular test
d. Unit test.
a. Time reversal test
Hint
Time reversal test – is a test of determining whether a given method will work both ways in time forwards and backward. In the words of fisher, The test is that the formula for calculating the index number should. be such that it will give the same ratio between one point of comparison and the other, no matter which of the two is taken as a base.In other words, when the data for any two years are treated by the same method, but with the bases reversed the two index numbers secured should be reciprocals of each other so that their product is unity

Question 32.
An index number is a
a. The measure of dispersion
b. The measure of correlation
c. The measure of regression
d. A special type of average is expressed in percentage or rate over a period of time.
d. Special type of average expressed in percentage or rate over a period of time.
Hint
Important characteristic features of index number are These are expressed in percentages

• These are specialized averages
• They measure the relative change in the value of a variable or a group of related variables over a period of time

Question 33.
Test of adequacy requires that the formulae for calculating an index number should give consistent results in both the directions. This test is satisfied by
a. Fisher Ideal Index
b. Kellys Index
c. Bowley Index
d. Walche Index.
a. Fisher Ideal Index
Hint
Time reversal test – is a test of determining whether a given method will work both ways in time forwards and backward. In the words of fisher, The test is that the formula for calculating the index number should, be such that it will give the same ratio between one point of comparison and the other, no matter which of the two is taken as a base.In other words, when the data for any two years are treated by the same method, but with the bases reversed the two index numbers secured should be reciprocals of each other so that their product is unity

Question 34.
Which of the following is/are weighted aggregative index numbers?
a. Laspeyre
b. Paasche
c. Simple Aggregative Method
d. Both (a) and (b)
d. Both (a) and (b)
Hint
Laspeyres price index – This index concentrates on measuring price changes from a base year. It is a weighted aggregative index method. In the weighted aggregate method, the index number is calculated by the ratio of weighted arithmetic mean of current year prices to base price

Question 35.
Which of the following options best suits the Laspeyres and Paasche Index?
a. Both are ideal index numbers.
b. Both are simple aggregative
c. Both are the same
d. Both are weighted aggregate indexes only
d. Both are weighted aggregate index only

Question 36.
A Limitation of Index Numbers is:
a. Tough to calculate
b. Requires software for computational purposes
c. They are Mathematical values
d. Index Number is based on a sample that may or may not be representative of the population.
d. Index Number is based on a sample that may or may not be representative of the population.
Hint
Limitations of index number – Index Number is based on a sample that may or may not be representative t of the population.

Question 37.
Which of the following is a general form of Exponential trend?
a. y = a + bt
b. Yt = a × bt
c. y = a – b
d. Yt = a + bt + ct2
b. Yt = a × bt

Question 38.
Test that requires the product of Price Index Number and Corresponding Quantity Index Number should be equal to the Value Index Number is:
a. Circular Test
b. Time reversal Test
c. Unit Test
d. Factor reversal Test
d. Factor reversal Test
Hint
Factor Reversal Test- The factor reversal test requires that multiplying a price index and a volume index of the same type should be equal to the proportionate change in the current values.

Question 39.
Which of the following is forecasting on the basis of past data?
a. Trend projection
b. Index number
c. Both trend and Index number
d. Correlation
b. Index number
Important characteristic features of index number are

• These are expressed in percentages
• These are specialized averages
• They measure the relative change in the value of a variable or a group of related variables over a period of time
Thus forecasting on the basis of past data is index number.

Question 40.
A weighted aggregate Price Index, where the weight for each item is its base period quantity, is known as:
a. Laspeyres Index
b. Producer Price Index
c. Consumer price Index
d. Paasche Index