Index Numbers: Types, Uses, and Importance in Decision-Making
Fixed Base Method
In the fixed base method, a particular year is generally chosen arbitrarily, and the prices of subsequent years are expressed as relatives of the price of the base year. Sometimes, instead of choosing a single year as the base, a period of a few years is chosen, and the average price of this period is taken as the base year’s price. The year selected as a base should be a normal year, or in other words, the price level in this year should neither be abnormally low nor abnormally high. If an abnormal year is chosen as the base, the price relatives of the current year calculated on its basis would give misleading conclusions.
For example, a year in which war was at its peak, say the year 1965, is chosen as a base year; thus, the comparison of the price level of subsequent years to the price of 1965 is bound to give misleading conclusions as the price level in 1965 was abnormally high.
In order to remove the difficulty associated with the selection of a normal year, the average price of a few years is sometimes taken as the base price. The fixed base method is used by the government in the calculation of national index numbers.
Chain Base Method
In this method, there is no fixed base period; the year immediately preceding the one for which the price index has to be calculated is assumed as the base year. Thus, for the year 1994, the base year would be 1993, for 1993, it would be 1992, for 1992, it would be 1991, and so on. In this way, there is no fixed base, and it keeps on changing.
The chief advantage of this method is that the price relatives of a year can be compared with the price levels of the immediately preceding year. Businesses mostly interested in comparing this time period rather than comparing rates related to the distant past will utilize this method.
Time Series Analysis
Time series analysis is critical for businesses to predict future outcomes, assess past performances, or identify underlying patterns and trends in various metrics. Time series analysis can offer valuable insights into stock prices, sales figures, customer behavior, and other time-dependent variables. By leveraging these techniques, businesses can make informed decisions, optimize operations, and enhance long-term strategies.
Time series analysis offers a multitude of benefits to businesses. The applications are also wide-ranging, whether it’s in forecasting sales to manage inventory better, identifying the seasonality in consumer behavior to plan marketing campaigns, or even analyzing financial markets for investment strategies. Different techniques serve distinct purposes and offer varied granularity and accuracy, making it vital for businesses to understand the methods that best suit their specific needs.
- Moving Average: Useful for smoothing out long-term trends. It is ideal for removing noise and identifying the general direction in which values are moving.
- Exponential Smoothing: Suited for univariate data with a systematic trend or seasonal component. Assigns higher weight to recent observations, allowing for more dynamic adjustments.
- Autoregression: Leverages past observations as inputs for a regression equation to predict future values. It is good for short-term forecasting when past data is a good indicator.
- Decomposition: This breaks down a time series into its core components—trend, seasonality, and residuals—to enhance the understanding and forecast accuracy.
- Time Series Clustering: Unsupervised method to categorize data points based on similarity, aiding in identifying archetypes or trends in sequential data.
- Wavelet Analysis: Effective for analyzing non-stationary time series data. It helps in identifying patterns across various scales or resolutions.
- Intervention Analysis: Assesses the impact of external events on a time series, such as the effect of a policy change or a marketing campaign.
- Box-Jenkins ARIMA models: Focuses on using past behavior and errors to model time series data. Assumes data can be characterized by a linear function of its past values.
- Box-Jenkins Multivariate models: Similar to ARIMA, but accounts for multiple variables. Useful when other variables influence one time series.
- Holt-Winters Exponential Smoothing: Best for data with a distinct trend and seasonality. Incorporates weighted averages and builds upon the equations for exponential smoothing.
State and Explain the Uses of Index Numbers
Index numbers are used for measuring changes in a specific variable or group of variables regarding location, time, or other constraints. The index number in statistics is one of the most used statistical methods for measuring changes considering specific characteristics of a variable. For example, index numbers can evaluate changes in the price of particular commodities or different geographical locations to understand inflationary and deflationary tendencies related to the product. In addition to this, index numbers reflect the barometers of the economic activities by serving as a tool for indicating agricultural production, industrial production, business activities, and other statistical information. Thus, index number can be defined as the relative measure used for comparing and describing numerical changes in the prices, quantity, and other aspects of a commodity regarding varied elements.
Key Uses of Index Numbers
Index numbers are used for the following aspects:
- Index numbers work as economic parameters that reflect changes in the economy in the form of inflationary and deflationary tendencies.
- Index numbers are used for measuring relative changes over a successive period of time for statistically measuring and depicting trends so general tendencies related to any product or service can be determined, evaluated, and forecasted.
- Index numbers are generally represented in percentages, so they are highly useful in comparing varied commodities and understanding changes with specific constraints.
- Index numbers frame business and economic policies as they depict the necessity of change in alignment with the changes noted in evaluated constraints.
- Price index numbers are used deflating as they connect the original data for comparison with notable changes and support the determination of purchasing power in the monetary unit.
Index Numbers are Economic Barometers Explain this Statement
- An index number is a number that expresses the relative change in magnitude of a variable or number of variables during a specified period. The variable may be the price of a certain commodity, the quantitative production of certain goods, or the cost of living.
- It is a statistical device to measure the level of certain phenomena in comparison with a certain period known as the base period, which may be a week, month, year, or group of years.
- The index numbers are known as economic barometers or economic indicators since they help in understanding the changes in economic conditions of the society.
Measures of Dispersion
A measure of dispersion indicates the scattering of data. It explains the disparity of data from one another, delivering a precise view of their distribution. The measure of dispersion displays and gives us an idea about the variation and the central value of an individual item.
In other words, dispersion is the extent to which values in a distribution differ from the average of the distribution. It gives us an idea about the extent to which individual items vary from one another and from the central value.
The variation can be measured in different numerical measures, namely:
(i) Range: It is the simplest method of measurement of dispersion and defines the difference between the largest and the smallest item in a given distribution. If Y max and Y min are the two ultimate items, then
Range = Y max – Y min
(ii) Quartile deviation: It is known as semi-interquartile range, i.e., half of the difference between the upper quartile and lower quartile. The first quartile is derived as Q, the middle digit Q1 connects the least number with the median of the data. The median of a data set is the (Q2)second quartile. Lastly, the number connecting the largest number and the median is the third quartile (Q3). Quartile deviation can be calculated by
Q = ½ × (Q3 – Q1)
(iii) Mean deviation: Mean deviation is the arithmetic mean (average) of deviations ⎜D⎜of observations from a central value (mean or median).
Mean deviation can be evaluated by using the formula: A = 1⁄n [∑i|xi – A|]
(iv) Standard deviation: Standard deviation is the square root of the arithmetic average of the square of the deviations measured from the mean. The standard deviation is given as,
σ = [(Σi (yi – ȳ) ⁄ n] ½ = [(Σ i yi 2 ⁄ n) – ȳ 2] ½
STATS IMPORTANT ROLE IN DECISION MAKING
Statistical analysis is collecting, organizing, and interpreting data meaningfully. Business statistics offers data to managers who help them to make successful decisions based on fundamental values rather than intuitions. Statistics is used to analyze the data and make interpretations, whether for sales estimation, introducing a new product line, making new production strategies, etc.
The primary purpose of business statistics is data collection, allowing managers to evaluate past performance, forecast future business practices, and run the organization profitably. Furthermore, it becomes the basis for risk navigation, sales prediction, market trends, changing consumer behavior, price determination, etc.
HARMONIC MEAN
The Harmonic Mean (HM) is defined as the reciprocal of the average of the reciprocals of the data values.. It is based on all the observations, and it is rigidly defined. Harmonic mean gives less weightage to the large values and large weightage to the small values to balance the values correctly. In general, the harmonic mean is used when there is a necessity to give greater weight to the smaller items. It is applied in the case of times and average rates
Since the harmonic mean is the reciprocal of the average of reciprocals, the formula to define the harmonic mean “HM” is given as follows:
If x1, x2, x3,…, xn are the individual items up to n terms, then,
Harmonic Mean, HM = n / [(1/x1)+(1/x2)+(1/x3)+…+(1/xn)]
Fisher’s method is typically applied to a collection of independent test statistics, usually from separate studies having the same null hypothesis. The meta-analysis null hypothesis is that all of the separate null hypotheses are true. The meta-analysis alternative hypothesis is that at least one of the separate alternative hypotheses is true.
The Laspeyres Price Index is a consumer price index used to measure the change in the prices of a basket of goods and services relative to a specified base period weighting. Developed by German economist Etienne Laspeyres, the Laspeyres Price Index is also called the base year quantity weighted method
Discuss the importance and use of weights in the construction of general price index numbers?
The term Weight‘ refers to the relative importance of the different items in the construction of the index. For example in daily use the importance of wheat and rice is more than jute and iron. Weights can be classified as follows –
i Explicit and Implicit weights : Explicit weights are called direct weight. They may be in the form of quantity or value. In implicit weighing a commodity or its variety is included in the index a numbers of times.
ii Fixed and Fluctuating Weight : If same weights are used from year to year they are called fixed weights. If weights are changed from time to time they are called fluctuating weights.
iii Arbitrary and Real Weights : If real quantities or units are used then those weights are called real weights.If some unrealistic weights according to the own will and assumptions of the investigator are used they are called arbitrary weights. Weights are used in construction of Index numbers because all items are not of equal importance and hence it is necessary to device some suitable method where by the varying importance f the different items is taken into account. This is done by allocating weights.
ADDITIVE AND MULTIPLICATIVE MODEL SERIES
Additive model is used when the variance of the time series doesn’t change over different values of the time series.
On the other hand, if the variance is higher when the time series is higher then it often means we should use a multiplicative models.
Additive model:
returni=pricei−pricei−1=trendi−trendi−1+seasonali−seasonali−1+errori−errori−1𝑟𝑒𝑡𝑢𝑟𝑛𝑖=𝑝𝑟𝑖𝑐𝑒𝑖−𝑝𝑟𝑖𝑐𝑒𝑖−1=𝑡𝑟𝑒𝑛𝑑𝑖−𝑡𝑟𝑒𝑛𝑑𝑖−1+𝑠𝑒𝑎𝑠𝑜𝑛𝑎𝑙𝑖−𝑠𝑒𝑎𝑠𝑜𝑛𝑎𝑙𝑖−1+𝑒𝑟𝑟𝑜𝑟𝑖−𝑒𝑟𝑟𝑜𝑟𝑖−1
If error’s increments have normal iid distributions then returni𝑟𝑒𝑡𝑢𝑟𝑛𝑖 has also a normal distribution with constant variance over time.
Multiplicative model:
If log of the time series is an additive model then the original time series is a multiplicative model, because:
log(pricei)=log(trendi⋅seasonali⋅errori)=log(trendi)+log(seasonali)+log(errori)𝑙𝑜𝑔(𝑝𝑟𝑖𝑐𝑒𝑖)=𝑙𝑜𝑔(𝑡𝑟𝑒𝑛𝑑𝑖⋅𝑠𝑒𝑎𝑠𝑜𝑛𝑎𝑙𝑖⋅𝑒𝑟𝑟𝑜𝑟𝑖)=𝑙𝑜𝑔(𝑡𝑟𝑒𝑛𝑑𝑖)+𝑙𝑜𝑔(𝑠𝑒𝑎𝑠𝑜𝑛𝑎𝑙𝑖)+𝑙𝑜𝑔(𝑒𝑟𝑟𝑜𝑟𝑖)
So the return of logarithms:
log(pricei)−log(pricei−1)=log(priceipricei−1)𝑙𝑜𝑔(𝑝𝑟𝑖𝑐𝑒𝑖)−𝑙𝑜𝑔(𝑝𝑟𝑖𝑐𝑒𝑖−1)=𝑙𝑜𝑔(𝑝𝑟𝑖𝑐𝑒𝑖𝑝𝑟𝑖𝑐𝑒𝑖−1)
must be normal with constant variance.
They do not depend on the level of the trend. With higher trends, these variations are more intensive. Though in practice the multiplicative model is the more popular, both models have their own merits.
TESTS TO BE SATISFIED BY FISCHERS IDEAL
The Fisher Price Index, also called the Fisher’s Ideal Price Index, is a consumer price index (CPI) used to measure the price level of goods and services over a given period. The Fisher Price Index is a geometric average of the Laspeyres Price Index and the Paasche Price Index. It is deemed the “ideal” price index as it corrects the positive price bias in the Laspeyres Price Index and the negative price bias in the Paasche Price Index.
The Fisher Price Index is the geometric average of the Laspeyres and Paasche Price indices, and the formula is rendered as:
Unit test requires that the formula for constructing an index number should be free from units of measurements. Practically all index numbers except simple unweighted aggregative index numbers satisfy this Fisher’s index number.
The other test suggested by Fisher is Factor Reversal Test. According to him, “Just as each formula should permit the interchange of the two times without giving inconsistent results, soit ought to permit interchanging the prices and quantities without giving inconsistent results i.e. the two results multiplied together should give the value ratio.” Fisher index number satisfies Factor Reversal Test.