IndexNumber: An R Package for Measuring the Evolution of Magnitudes

CONTRIBUTED RESEARCH ARTICLES 253 IndexNumber: An R Package for Measuring the Evolution of Magnitudes by Alejandro Saavedra-Nieves and Paula Saavedra-Nieves Abstract Index numbers are descriptive statistical measures useful in economic settings for comparing simple and complex magnitudes registered, usually in two time periods. Although this theory has a large history, it still plays an important role in modern today’s societies where big amounts of economic data are available and need to be analyzed. After a detailed revision on classical index numbers in literature, this paper is focused on the description of the R package IndexNumber with strong capabilities for calculating them. Two of the four real data sets contained in this library are used for illustrating the determination of the index numbers in this work. Graphical tools are also implemented in order to show the time evolution of considered magnitudes simplifying the interpretation of the results. Introduction The problem of reducing a large amount of available microeconomic data is common in dynamic and modern economies. Individuals of today’s societies consume services of hundred of commodities over a year, and most producers use and produce hundreds of individual products and services. This overwhelming abundance of data is usually summarized through index numbers theory. Index numbers are descriptive statistical measures useful in order to compare or measure changes in simple and complex magnitudes over time. The goal is usually to determine possible increases or decreases and, more generally, trend changes. The situations to be compared are in no way restricted; they may be two time periods (hours, days, months, or years); two places (two cities or countries); or two groups of people (economically active and inactive population). For simplicity in the exposition, we refer to temporal index numbers in this paper. The rest of the situations mentioned could be considered modifying the notation slightly. Although index numbers are considered a classical statistical tool, the problem of how to construct them is as much one of economic theory as of statistical technique, see Frisch (1936). This can be checked by analyzing their considerable history. Initial works involving index numbers date back to the early 18th century. During the creation of a college in 1440–1460, it was stipulated that any member had to leave it if his richness exceeded five pounds per year. The Anglican Bishop Fleetwood desired to know if, according to the price evolution, this promise could be kept three centuries later. Then, he studied the evolution of prices corresponding to four products (wheat, meat, drink, and clothing) from 1440 to 1700. He concluded that five pounds in 1440–1460 had the same value as 30 pounds in 1700. More details can be found in Fleetwood (1707). Dûtot (1754) studied the diminution of the money value analyzing the incomes of kings Louis XII and Louis XV. In order to know which of them had the largest disposable income, he considered the prices of several goods of different nature as such as a chicken, a rabbit, a pigeon, or the day’s work value. From the discovery of the Americas, the astronomy professor of Padoue analyzed the evolution of prices in 1764. He considered the prices of three commodities (grains, wine, and cooking oil), and he studied their variation from 1500 to 1750. Evelyn (1798) can be seen as a precursor of index numbers establishing a price index number from 1050 to 1800. In this work, notions as the year of reference and relative prices were introduced. According to Kendall (1977), the real father of index numbers is Joseph Lowe. Many problems dealing with their construction were presented in Lowe (1822). In fact, Lowe’s measurement would be known as the Laspeyres index later. In the second half of the 19th century, statisticians developed many advances in this setting. Jevons (1863) recommended considering the geometrical mean in order to construct an index number. Between 1864 and 1880, Laspeyres (1864,1871), Drobisch (1871), and Paasche (1874) worked on the evolution of prices for material goods from the approach of weighted index numbers. Palgrave (1886) proposed to weigh the relative prices by the total amount of the considered good. Fisher (1922) defined a new index number calculated as the geometric mean of the Laspeyres and the Paasche index numbers. In the same period, Marshall and Edgeworth proposed to calculate weighted means. For an in-depth review on this topic, see Kendall (1969), Allen (1982) or more recently, CPI Manual (2004), or Dodge (2008). Nowadays, economists continue to use index numbers to make comparisons over time despite their long background. In fact, the main applications of index numbers are either economic, or they occur in related fields as demography or technology. In such settings, the magnitudes to be compared through index numbers usually come in pairs, one of price and the other a matching one of quantity. This pair may be designed to account for the variation in an aggregate value, as when movements in the aggregated expenditure of consumers are analyzed into the two components of changes in The R Journal Vol. 13/1, June 2021 ISSN 2073-4859 CONTRIBUTED RESEARCH ARTICLES 254 prices and in real consumption. Some more recent contributions in index numbers theory are exposed next. Barnett (1980) focused on economic monetary aggregates from this approach. Changes in food prices were analyzed in Lamm (1980). Index numbers in chain, or more commonly chain indices, are considered in Forsyth and Fowler (1981). Boyle (1988) analyzed the volume of Irish agricultural output from 1960 to 1982. Scanner data on coffee sales are studied in de Haan and Opperdoes (1997). Hill (1999) shows how a comparison of price levels across a group of countries can be made by chaining Fisher index numbers across a spanning tree. Inequality and poverty in several regions of Thailand are studied through the construction of urban and rural cost of living and welfare indices in Kakwani and Hill (2002). Dumagan (2002) showed that the Fisher index could be numerically approximated by other superlative index numbers. In Reinsdorf et al. (2002), additive decompositions of the Fisher index are derived in order to know how much each item contributes to its overall change. From data from United States, Canada, France, Germany, Italy, Japan, and the United Kingdom, drug price and quantity index numbers are considered in Danzon and Chao (2000). Ang et al. (2004) used a generalized version of the Fisher index to analyze CO 2 emission. In Boyd and Roop (2004), the structural change in energy intensity is studied. Exploring the duality between a return to dollar definition of profit and the generalized distance function, the relationship between the Laspeyres, Paasche and Fisher productivity index numbers is established in Zofío and Prieto (2006). Hill (2006) showed an illustration on index numbers also using scanner data. An application to major crops in Manitoba is presented in Coyle (2007). According to Diewert and Nakamura (2007), Paasche, Laspeyres, or Fisher index number formula is useful in order to manage the total factor productivity growth. The importance of the hedonic imputation method in price index numbers is analyzed in Hill and Melser (2008) from a data set containing house prices for three regions in Sydney over a three years period. The impact of time aggregation on price change estimates for several supermarket item categories is considered in Ivancic et al. (2011). Białek (2012) proposed a general price index formula with the Fisher, Laspeyres, and Paasche indices as its particular cases. Białek (2014) presented an original price index, and its performance is analyzed through a simulation study where it is compared to several classical price index numbers. A generalized version of the Fisher index is considered in Su and Ang (2014) in order to analyze changes in the carbon emissions embodied in China’s exports. O’Donnell (2018) analyzed the productivity change defined as measures of output quantity change divided by measures of input quantity change. Zhen et al. (2019) constructed panel price index numbers using retail scanner data in order to compare consumption costs across space and time. This paper is focused on the description of the R package IndexNumber (Saavedra-Nieves and Saavedra-Nieves,2021), available from the Comprehensive R Archive Network at https://CRAN.R- project.org/package=IndexNumber , and its capabilities for calculating classical index numbers. It is organized as follows. Index numbers are formally defined in Section 2.2. Section 2.2.1 introduces simple index numbers. Concretely, simple index numbers in series and in chain are distinguished. In Section 2.2.2, non-weighted and weighted complex index numbers are presented. Details on the usage of IndexNumber package are considered from Section 2.3. The four real data sets contained in the library are also described briefly. Note that two of them are used in this paper in order to illustrate the calculation of index numbers. They are available on the website of the Spanish Statistical Office (INE), http://www.ine.es. Preliminaries on index numbers Index numbers are statistical measures that are useful to compare single and multiple magnitudes for the same interval of time. In both cases, this comparison is made with respect to an element of the mentioned series that is called base period or reference. Some examples of simple magnitudes are prices of a good, sold amounts of a product, or other general individual values. However, most of the time, comparing these simple quantities has not practical interest. If the goal is to analyze some real phenomena where many variables are involved, complex indices must be considered. Using these ideas, index numbers are usually classified into the following two groups: • An index number is said to be simple if it corresponds to the ratio of two values of the same variable, measured in two different instants. Therefore, a simple index number provides the variation that the single magnitude has suffered between two different time periods. For instance, a simple index number of the price will give the relative variation of the price between the current period and the period taken as reference. • Most of the time, comparing prices, amounts, or values of a single product individually is not of interest in practice. If the goal is to analyze some real situations where different variables have influence, a complex number index has to be considered. It globally summarizes the information of the different magnitudes involved in the problem. For instance, the evolution cost of life in a country is a common case where it is necessary to select a set of goods or variables that give information about it. The relative importance of each of the goods considered must be measured and taken into account. The R Journal Vol. 13/1, June 2021 ISSN 2073-4859 CONTRIBUTED RESEARCH ARTICLES 255 A wider overview of both classes of index numbers is included in the next sections. In particular, we distinguish the different subclasses belonging to each of them and their possible relationships. Simple index numbers Asimple index number is a statistical indicator of the percentage of variation of a single magnitude in two different instants. Simple index numbers are usually classified according to the element that we take as reference. In particular, we distinguish two types of simple index numbers. First, we describe simple index numbers in series, when the first value of the series is taken as the reference value, and simple index numbers in chain (or chain indices), when the reference is the immediately previous value in the serie. In what follows, we assume that X={x0 , x1 , . . . , xT} denotes the observations of the magnitude X for the T+ 1 time instants considered. Besides, x0 is usually taken as the base period. Most common simple index numbers are individually referred to variables in real-world situations as the followings: • the price of a good, denoted by p; • the amount of produced or sold product, denoted by q; or • the value of a good, denoted by v . This value is usually obtained as the product of the price and the amount variables. In this section, we illustrate the usage of simple index numbers on a real example. Table 1shows the number (thousands) of economically active women and men in Spain from the first trimester of 2002 (taken as a reference value). Remark that four trimesters of each year are denoted by T1, T2, T3 and T4, respectively. Stages 2002 (T1) 2002 (T2) 2002 (T3) 2002 (T4) 2003 (T1) 2003 (T2) ... Total of women 7442.70 7580.80 7670.20 7751.50 7868.70 7977.80 ... Total of men 11192.30 11289.40 11445.10 11472.80 11552.50 11661.40 ... Table 1: Number (thousands) of economically active women and men in Spain from first trimester of 2002. This dataset, included in IndexNumber package, can be obtained from the Economically Active Population Survey (EPA) elaborated by the Spanish Statistics National Institute (INE). Simple index numbers in series Let x0 and xt be the values of the variable X where x0 corresponds to the base period and t∈ { 0, 1, . . . , T} , respectively. Thus, the value of the simple index number in series for X in t is defined as follows: It 0(X) = xt x0 ·100. (1) For each t∈ { 0, 1, . . . , T} , this measure has a natural interpretation. Fixed a certain variable of interest X , the index number in series in t shows the percentage of variation of the magnitude in this instant of time with respect to the reference value (in this case, the initial one). Using this type of index number, some usual magnitudes can be formally defined as it is indicated next: • When prices are considered, the relative price of a product i , also called simple price index, can be determined as pt 0=pit pi0 , where pit denotes the price at instant tand pi0, the price in the base period. • The relative amount of a product ican be written as qt 0=qit qi0 , where qit denotes the produced or sold amount at instant t and qi0 , the amount for the base period. • Finally, the relative value of a product ihas the next expression: vt 0=vt v0 =pitqit pi0qi0 ·100 =pt 0·qt 0·100, where vtdenotes the value of the good at instant tand v0, the value of the base period. The R Journal Vol. 13/1, June 2021 ISSN 2073-4859 CONTRIBUTED RESEARCH ARTICLES 256 Below, we illustrate the usage of simple index numbers on the two series included in Table 1. Thus, the simple index numbers in series for the economically active women and men in Spain from the first trimester of 2002 is given in Table 2. Stages 2002 (T1) 2002 (T2) 2002 (T3) 2002 (T4) 2003 (T1) 2003 (T2) ... Index number for women 100.00 101.86 103.06 104.15 105.73 107.19 ... Index number for men 100.00 100.87 102.26 102.51 103.22 104.19 ... Table 2: Simple index numbers in series for number (thousands) of economically active women and men in Spain from the first trimester of 2002. Comparing the evolution of the index numbers in series for the population of women and men shown in Table 2has an interest, for instance, to analyze the effect of variable sex in the Spanish labor market. Note that the number of economically active women and men in the second trimester of 2003 is 7.2% and 4.2% larger than in the reference time, respectively. Therefore, women increasing is slightly larger than men. Simple index numbers in chain Below, we introduce another approach of simple index numbers. Contrary to our previous assumptions, this new setting arises when the reference value is not the initial one; rather we take the value immediately preceding. Let xt and xt−1 be two values of a variable X observed in two consecutive instants t and t− 1, being t∈ { 1, 2, . . . , T} . Thus, the value of the index number in chain or chain index (cf. Forsyth and Fowler,1981) that corresponds to an instant t , with t∈ { 1, 2, . . . , T} , is defined as follows: ICt(X) = xt xt−1 ·100. (2) Again, this index number can be naturally interpreted. It is worth mentioning that these measures the variation of the characteristic under study with respect to the previous value in a fixed instant t . For instance, these index numbers reflect the percentage variation that the variable experiments between two consecutive values in time series settings. To illustrate this definition, we take again the example considered in the previous section. For this subset of values, we obtain again the evolution of the amount of economically active people (per sex) in Spain under this new approach. Stages 2002 (T1) 2002 (T2) 2002 (T3) 2002 (T4) 2003 (T1) 2003 (T2) ... Index number for women 100.00 101.86 101.18 101.06 101.51 101.39 ... Index number for men 100.00 100.87 101.38 100.24 100.70 100.94 ... Table 3: Simple index numbers in chain for number (thousands) of economically active women and men in Spain from the first trimester of 2002. Table 3shows the evolution in time of the number (thousands) of economically active women and men in Spain from the trimester of 2002 (see Table 1). According to the obtained results, we emphasize as relevant that the number of economically active women and men in the second trimester of 2003 increases 1.4% and 0.9%, respectively, with respect to the previous trimester. Relationship between simple index numbers in series and in chain This section briefly introduces some comments on the relations between simple index numbers in series and in chain. In this way, one can be obtained from another (and vice versa) without having to use the exact values of the magnitude under study. Take xt the value of the variable X in instant t , with t∈ { 1, 2, . . . , T} . Thus, a simple index in chain can be obtained from a simple index in series due to the relation ICt(X) = xt xt−1 ·100 = xt x0 xt−1 x0 ·100 =It 0(X) It−1 0(X)·100. For example, from simple index numbers in series for women shown in Table 2, it is possible to obtain the corresponding simple index number in chain for women also for the first trimester of 2003 (contained in Table 3). That is, IC2003(T1)(X) = 7868.70 7751.50 ·100 =105.72 104.15 ·100 =101.51. Besides, simple indices in series can be equivalently obtained for each t∈ { 1, . . . , T} from indices The R Journal Vol. 13/1, June 2021 ISSN 2073-4859 CONTRIBUTED RESEARCH ARTICLES 257 in chain: It 0(X) = xt x0 ·100 =xt xt−1 xt−1 xt−2 · · · x2 x1 x1 x0 ·100 =ICt(X) 100 ·ICt−1(X) 100 · · · · · IC2(X) 100 ·IC1(X) 100 ·100. For instance, if we consider the simple index numbers in chain shown in Table 3, it is possible to obtain the simple index number in series for the first trimester of 2003 contained in Table 2. That is, we check that I2003(T1) 0(X) = 100.70 100 ·100.24 100 ·101.38 100 ·100.87 100 ·100. Variation Rate In this section, we formally introduce another measure of the evolution of a magnitude. Furthermore, we relate it to the simple index numbers previously defined. The variation rate of the observations in the instants t and t− 1, with t∈ { 1, . . . , T} , is denoted by Ratet(X)for each t∈ {1, . . . , T}. It can be calculated from the simple index in chain as follows: Ratet(X) = xt−xt−1 xt−1 ·100 =ICt(X)·100. This definition can be extended to any pair of instants in { 0, 1, 2, . . . , T} . Take t1 , t2∈ { 0, 1, 2, . . . , T} such that t1<t2 . Let xt1 be the value of a variable measure in instant t1 and let xt2 be the value of a variable measure in a later instant t2. Formally, the variation rate of Xin t2with respect to t1is Ratet2 t1(X) = xt2−xt1 xt1 ·100. Note that Ratet2 t1(X) = xt2−xt1 xt1 ·100 =xt2 xt1 −1·100 =It2 t1(X)−100, where It2 t1(X) = xt2 xt1·100, also satisfying It2 t1(X) = ICt2(X)·ICt2−1(X)·. . . ·ICt1+1(X). In particular, if the consecutive observations correspond to two different years, months, or trimesters the variation rate is called interanual variation rate, monthly variation rate, and quarterly variation rate, respectively. From the data shown in Table 2, the quarterly variation rates have been calculated in Table 4. Stages 2002 (T2) 2002 (T3) 2002 (T4) 2003 (T1) 2003 (T2) ... Rate for women 1.86 1.18 1.06 1.51 1.39 ... Rate for men 0.87 1.38 0.24 0.70 0.94 ... Table 4: Quarterly variation rate for number (thousands) of economically active women and men in Spain. Average Variation Rate Finally, we introduce a third method to measure the evolution of a given magnitude X between the instants t and t+k , with t∈ { 0, . . . , T−k} and k> 0. If xt+k denotes the observation at instant t+k and xt the corresponding to instant t , the average variation rate of the variable X between instants t and t+k is defined as the constant rate Tk that allows obtaining the observation xt+k at time t+k from observation xtat time t. Then, it is possible to write: xt+1=xt+Tk 100 ·xt=100 +Tk 100 ·xt xt+2=xt+1+Tk 100 ·xt+1=100 +Tk 100 2 ·xt xt+3=xt+2+Tk 100 ·xt+2=100 +Tk 100 2 ·xt . . . xt+k=xt+k−1+Tk 100 ·xt+k−1=100 +Tk 100 k ·xt The R Journal Vol. 13/1, June 2021 ISSN 2073-4859 CONTRIBUTED RESEARCH ARTICLES 258 Therefore, xt+k=100+Tk 100 k·xtand, as consequence, xt+k xt=100+Tk 100 k. Then, Tk=100 ·xt+k xt1/k . From data shown in Table 2, we have calculated the quarterly average variation rate in 2002 for women and men as 3 r7751.50 7442.70 −1!·100 =1.36 and 3 r11472.80 11192.30 −1!·100 =0.83, respectively. According to the results obtained, the average variation rate for women is considerably bigger than the corresponding for men in this period. Complex index numbers Most of the time, comparing variables marginally does not provide real information. There are many phenomena in the real world that involve several variables. Of course, simple index numbers described in the previous section could be naturally applied for each of these variables separately. However, comparing these magnitudes can be not realistic and, for this reason, complex index numbers have to be introduced. A complex index number summarizes the information of the different marginal index numbers related to the set of variables of interest. One of the most relevant examples that illustrates the use of complex index numbers is briefly described next. The evolution of living cost in a country is a common case where it is necessary to select a set of goods or variables that give information about it. The relative importance of each of the goods considered may be taken into account. This is the case of the Consumer Price Index (CPI) in Spain. The sets of good considered for calculating it follows the International Classification of Consumption according to Purpose (COICOP) prepared by the Statistical Division of the United Nations. This classification is also used by other countries. In this way, comparisons between different geographical areas make sense. When all variables are not of equal relevance, it is possible to add complementary information for weighting each magnitude corresponding to its degree of importance. Depending on the use of this additional information, two classes of complex index numbers are distinguished in literature: • In several practical cases, the relative weight of each involved variable has no interest. Fixed a magnitude, the class of required index numbers to be used in this setting are named non-weighted complex index numbers. • The use of weighted index numbers allow greater importance to be attached to some items. For instance, real information other than simply the change in price over time can be used. Factors as quantity sold or quantity consumed for each item can also be considered. In what follows, we assume that X= (X1 , . . . , Xn) denotes the collection of n magnitudes registered for n different products. For each j= 1, . . . , n , Xj={xj0 , xj1 , . . . , xjT} denotes the observations of the magnitude Xj for the T+ 1 instants of time considered. Analogously to the simple case, we refer xj0as the base period. In practice, the available information can be summarized in a table such as Table 5. XXXXXXXXX X Time Products 1 2 ... n 0x10 x20 ... xn0 1x11 x21 ... xn1 . . .. . .. . .. . .. . . T x1Tx2T... xnT Table 5: Evolution of a set of magnitudes Xfrom time 0 to T. The most common complex indices jointly involve (some of) the variables in real-world situations that we enumerate below: • the price of a collection of n goods, denoted by p= (p1 , . . . , pn) , where pj={pj0 , pj1 , . . . , pjT} denotes the prices for T+1 instants and for each product j=1, . . . , n; • the amount of produced or sold products, denoted by q , where qj={qj0 , qj1 , . . . , qjT} denotes the amounts of product for T+1 instants and for each product j=1, . . . , n; or The R Journal Vol. 13/1, June 2021 ISSN 2073-4859 CONTRIBUTED RESEARCH ARTICLES 259 • the value of n goods is given by v= (v1 , . . . , vn) . In this case, vj={vj0 , vj1 , . . . , vjT} denotes the values of product j, with j∈ {1, . . . , n}for T+1 instants. To illustrate the usage of complex index numbers, we take the example described in Table 6. It shows the unitary value (euros) of prices of combustibles and other energy resources for the main home in Spain from 2005 to 2015. In this case, the data source is again the Spanish Statistics National Institute (INE). Electricity Natural municipal Liquified Liquified Solid (Kwh) gas (m3) gas (kilo) combustibles (litre) combustibles (kilo) 2006 0.14 0.70 1.00 0.69 0.12 2007 0.14 0.78 0.97 0.66 0.10 2008 0.15 0.83 1.03 0.81 0.11 2009 0.16 0.87 0.93 0.68 0.10 2010 0.17 0.79 1.04 0.72 0.12 2011 0.19 0.77 1.15 0.91 0.12 2012 0.22 0.82 1.19 0.96 0.12 2013 0.23 1.00 1.34 0.99 0.13 2014 0.24 1.07 1.35 0.89 0.15 2015 0.25 1.04 1.22 0.77 0.13 Table 6: Unitary value (euros) of combustibles and other energy resources for the main home in Spain from 2005 to 2015. Table 7shows the (thousands of units) consumption of combustibles and other energy resources for the main home in Spain from 2005 to 2015. This dataset is closely related to the prices presented in Table 6. Of course, this data set can also be obtained from the Spanish Statistics National Institute (INE). Electricity Natural municipal Liquified Liquified Solid (Kwh) gas (m3) gas (kilo) combustibles (litre) combustibles (kilo) 2006 50623635 3617285 1057488 2297923 1306920 2007 51990501 3266575 1066857 2454265 1602799 2008 54990338 3473851 1210607 2274326 1556673 2009 59749470 3730349 1113642 2505711 1724222 2010 69751162 3954065 987112 2345215 1584123 2011 67574654 4466072 926824 1974662 1414234 2012 62878557 4576052 943632 2029733 1733591 2013 56017871 4116079 867695 1952593 2071152 2014 49177739 3653055 868743 2180866 2077766 2015 48541712 3795339 818183 2176533 2161208 Table 7: Consumption (thousands of units) of combustibles and other energy resources for the main home in Spain from 2005 to 2015. Next, we formally describe non-weighted and weighted index numbers. Non-weighted complex index numbers Complex index numbers analyze several magnitudes that measure the evolution of a set of goods or services. The goal here is to find a statistical measure in order to summarize the information shown, for instance, in Table 5. In particular, knowing the variation of a magnitude in time t with respect to the base period has an interest. In this sense, it is worth mentioning that non-weighted complex index numbers can be easily obtained as the arithmetic, geometric and harmonic means of the simple index numbers in series for the considered magnitudes. Anyway, their mathematical expressions are described below: • The Sauerbeck index (cf. Sauerbeck,1895) at time t for X, St(X) , is calculated as the arithmetic mean of the simple index in series for the ninvolved magnitudes at t: St(X) = 1 n n ∑ i=1 xit xi0 ·100, for each t∈ {0, · · · ,T}. (3) • The Geometric mean index at time t,Gt(X), is calculated as follows: Gt(X) = n sn ∏ i=1 xit xi0 ·100, for each t∈ {0, · · · ,T}. (4) The R Journal Vol. 13/1, June 2021 ISSN 2073-4859 CONTRIBUTED RESEARCH ARTICLES 260 Given a collection of n magnitudes, the geometric mean index at t is obtained as nth -root of the product of simple index numbers for Xat time t. See more details in Jevons (1863). • The Harmonic mean index at time t,Ht(X), is determined by Ht(X) = n ∑n i=1xi0 xit ·100, for each t∈ {0, · · · ,T}. (5) It is initially introduced in Jevons (1865) and Coggeshall (1886). • The Bradstreet-Dûtot index at time t , BDt(X) , is introduced in Walsh (1901). Its value is obtained as the ratio between the means of the magnitude in time t and the magnitude in the base period as follows: BDt(X) = ∑n i=1xit ∑n i=1xi0 ·100, for each t∈ {0, · · · ,T}. (6) If X denotes the matrix p of prices of a set of goods or services along a period of time, these index numbers are specifically considered non-weighted complex index numbers for prices. Thus, it arises the Sauerbeck index at time t for prices, St(p) ; the Geometric mean index at time t for prices, Gt(p) ; the Harmonic mean index at time t for prices, Ht(p) ; and the Bradstreet-Dûtot index at time t for prices, BDt(p) . Their usage will be illustrated on the set prices of combustibles and other energy resources for the main home in Spain from 2005 to 2015. From the information in Table 6, the four index numbers previously described are numerically shown in Table 8, and their evolution is depicted in Figure 1. t St(p)Gt(p)Ht(p)BDt(p) 2006 100.00 100.00 100.00 100.00 2007 97.48 97.06 96.64 100.00 2008 107.56 107.08 106.60 110.57 2009 102.69 101.64 100.61 103.40 2010 108.53 108.26 108.00 107.17 2011 118.52 117.76 116.99 118.49 2012 126.48 124.97 123.48 124.91 2013 138.59 137.35 136.05 139.25 2014 142.65 141.66 140.70 139.62 2015 133.81 131.37 129.14 128.68 Table 8: Non-weighted complex price indexes in seriea for the unitary value of combustibles and energy resources for the main home in Spain from 2006 to 2015. 2006 2008 2010 2012 2014 100 110 120 130 140 Sauerbeck index Geometric mean index Harmonic mean index Bradstreet−Dûtot index Figure 1: Joint evolution of the non-weighted complex price indexes in series for the unitary value of combustibles and energy resources for the main home in Spain from 2006 to 2015. However, the variation of a given magnitude in time t with respect to the previous period may also be of interest. Below, we alternatively enumerate the mathematical expressions of the previous indices based on index numbers in chain for the magnitudes. • The Carli index (cf. Carli,1804) at time t for X, Ct(X) , is calculated as the arithmetic mean of the The R Journal Vol. 13/1, June 2021 ISSN 2073-4859 CONTRIBUTED RESEARCH ARTICLES 261 simple index in chain for the ninvolved magnitudes at t: Ct(X) = 1 n n ∑ i=1 xit xi t−1 ·100, for each t∈ {1, · · · ,T}. (7) • The Jevons index at time t,Jt(X), is calculated following Jevons (1863) as follows: Jt(X) = n sn ∏ i=1 xit xi t−1 ·100, for each t∈ {1, · · · ,T}. (8) Given a collection of n magnitudes, the geometric mean index at t is obtained as nth -root of the product of simple index numbers for Xin chain at time t. • The Dûtot index at time t , Dt(X) , is introduced in Dûtot (1754). Its value is obtained as the ratio between the means of the magnitude in time tand the magnitude in time t−1 as follows: Dt(X) = ∑n i=1xit ∑n i=1xi t−1 ·100, for each t∈ {1, · · · ,T}. (9) Again, if X denotes the matrix of prices of a set of goods or services along a period of time ( p ), these index numbers are named as the Carli index at time t for prices, Ct(p) ; the Jevons index at time t for prices, Jt(p) ; and the Dûtot index at time t for prices, Dt(p) . To conclude this section, we obtain them for the set prices of combustibles and other energy resources for the main home in Spain from 2005 to 2015, detailed in Table 6. These index numbers are numerically detailed in Table 9. t Ct(p)Jt(p)Dt(p) 2006 100.00 100.00 100.00 2007 101.11 101.09 101.79 2008 113.51 113.43 115.89 2009 100.94 100.18 126.56 2010 105.75 105.27 144.02 2011 103.58 103.29 155.49 2012 110.65 110.49 168.01 2013 107.00 106.32 163.98 2014 100.72 100.39 154.09 2015 93.08 92.79 153.29 Table 9: Non-weighted complex price indexes in chain for the unitary value of combustibles and energy resources for the main home in Spain from 2006 to 2015. All of the index numbers described are easy to be computed. However, they present an important disadvantage: they do not take into account the relative importance of each product. Weighted complex index numbers For analyzing the evolution of a given magnitude X , it is very common to use an alternative magnitude Y through the value of Y in the reference or the actual period to weight complex index numbers. The information relative to this alternative variable can be summarized in a table such as Table 10. For instance, the use of the amount of production of different products or the use of prices may result of interest, depending on the setting under study. XXXXXXXXX X Time Products 1 2 ... n 0y10 y20 ... yn0 1y11 y21 ... yn1 . . .. . .. . .. . .. . . T y1Ty2T... ynT Table 10: Evolution of a set of magnitudes Yfrom time 0 to T. Next, the main weighted complex price index numbers for a given magnitude X , taking Y as weight, are formally described: • The Laspeyres index (Laspeyres,1871) analyzes the variations of X using Y as weight. In this sense, the weights considered for product i are xi0·yi0 (note that both values are referred to The R Journal Vol. 13/1, June 2021 ISSN 2073-4859 CONTRIBUTED RESEARCH ARTICLES 268 90 95 100 105 110 Stages Woman: Index Number 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Figure 4: Evolution of the simple index number in chain for the number (thousands) of economically active women in Spain. 1 0.14 0.70 1.00 0.69 0.12 2 0.14 0.78 0.97 0.66 0.10 3 0.15 0.83 1.03 0.81 0.11 4 0.16 0.87 0.93 0.68 0.10 5 0.17 0.79 1.04 0.72 0.12 6 0.19 0.77 1.15 0.91 0.12 7 0.22 0.82 1.19 0.96 0.12 8 0.23 1.00 1.34 0.99 0.13 9 0.24 1.07 1.35 0.89 0.15 10 0.25 1.04 1.22 0.77 0.13 First, we describe the R procedures that provide the variations of a magnitude in time t with respect to the base period by using non-weighted index numbers. For this purpose, we have to introduce the option base="serie" . Notice that all the values included in Table 8were obtained by using the code in R that we show in the current section. Sauerbeck index in R Next, we determine the Sauerbeck index (3) in R software. To this aim, we use the corresponding function aggregated.index.number() by adding, as option, type="arithmetic" . Recall that it corresponds to an average by stages. In this case, we also include a graphical description of the joint evolution of prices in Figure 5with opt.plot=TRUE , and a numerical summary of such magnitude (with opt.summary=TRUE). > aggregated.index.number(ECResources[,2:6],base="serie",type="arithmetic", name="Prices",opt.plot=TRUE,opt.summary=TRUE) Aggregate index number Arithmetic Summary Min.=97.4828157349897 Stage=1 Max.=142.654244306418 The R Journal Vol. 13/1, June 2021 ISSN 2073-4859 CONTRIBUTED RESEARCH ARTICLES 269 90 100 110 120 130 140 150 Stages Prices: Index Number 0123456789 Figure 5: Sauerbeck index evolution for the prices of several energy resources for the period 2005-2015. Stage=8 $Summary Min. 1st Qu. Median Mean 3rd Qu. Max. 97.48 107.55 118.52 119.59 133.81 142.65 $ ` Agg. index number ` Stages Prices 1 Prices 2 Prices 3 Prices 4 Prices 5 Agg. index number 1 0 0.14 0.70 1.00 0.69 0.12 100.00000 2 1 0.14 0.78 0.97 0.66 0.10 97.48282 3 2 0.15 0.83 1.03 0.81 0.11 107.55445 4 3 0.16 0.87 0.93 0.68 0.10 102.69110 5 4 0.17 0.79 1.04 0.72 0.12 108.52671 6 5 0.19 0.77 1.15 0.91 0.12 118.51967 7 6 0.22 0.82 1.19 0.96 0.12 126.48323 8 7 0.23 1.00 1.34 0.99 0.13 138.59089 9 8 0.24 1.07 1.35 0.89 0.15 142.65424 10 9 0.25 1.04 1.22 0.77 0.13 133.81408 Geometric mean index in R The second approach that we consider is the one given by the Geometric mean index (4). To obtain for the case of prices of the energetic resources, we slightly change the parameters of the function aggregated.index.number(). We have to introduce the parameter type="geometric" on it. aggregated.index.number(ECResources[,2:6],base="serie",type="geometric", name="Prices",opt.plot=FALSE,opt.summary=FALSE) The results have the same structure as the previous case that we have explained. For this reason, they have not already been included here. Harmonic mean index in R The third option of a non-weighted complex index is the Harmonic mean index (5). With regard to the previous cases, the main difference is the parameter type that, in this case, has to take the value "harmonic". aggregated.index.number(ECResources[,2:6],base="serie",type="harmonic", name="Prices",opt.plot=FALSE,opt.summary=FALSE) The R Journal Vol. 13/1, June 2021 ISSN 2073-4859 CONTRIBUTED RESEARCH ARTICLES 270 The scheme of showing the results maintains also in this scenario. Bradstreet-Dûtot index in R The Bradstreet-Dûtot index (6) is determined in R software by using aggregated.index.number() with the parameter type="BDutot" . We illustrate the case of obtaining the indicated index for the prices of energetic resources. Again, the output has the same structure as the previous cases. aggregated.index.number(ECResources[,2:6],base="serie",type="BDutot", name="Prices",opt.plot=FALSE,opt.summary=FALSE) Secondly, we describe examples of usage of IndexNumber package that involves non-weighted and weighted index numbers in chain. Specifically, we show those ones required for obtaining the results in Table 9. They involves the usage of base="chain". Carli index in R The Carli index (7) is obtained in R software through aggregated.index.number() with type="Carli" . The following R code determines the indicated index for the prices of energetic resources with an analogous structure for the output. aggregated.index.number(ECResources[,2:6],base="chain",type="Carli", name="Prices",opt.plot=FALSE,opt.summary=FALSE) Jevons index in R The Jevons index (8) can be determined in R software by using aggregated.index.number() with the parameter type="Jevons" . Again, we illustrate the case of obtaining the indicated index for the prices of energetic resources. The required code is the one displayed below: aggregated.index.number(ECResources[,2:6],base="chain",type="Jevons", name="Prices",opt.plot=FALSE,opt.summary=FALSE) Dûtot index in R The Dûtot index (9) is determined in R with aggregated.index.number() with additionally including type="BDutot" . Finally, we illustrate the case of obtaining this index for the prices of energetic resources. aggregated.index.number(ECResources[,2:6],base="chain",type="Dutot", name="Prices",opt.plot=FALSE,opt.summary=FALSE) Weighted complex index numbers in R Next, we enumerate the capabilities of IndexNumber package in R software to determine weighted complex index numbers. The results in Table 11 were also obtained by using the functions of R that we show below. Again, we pretend to obtain the evolution of the unitary value of prices of several energy resources for the period 2005-2015. However, in this case, we weight their values by the total amount of consumed energy resources given in Table 7. This information is also included in the data set ECResources of the package IndexNumber. > ECResources[,7:11] ElectricityConsumed NaturalGasConsumed LiquifiedGasConsumed LiquifiedCombustiblesConsumed SolidCombustiblesConsumed 1 50623635 3617285 1057488 2297923 1306920 2 51990501 3266575 1066857 2454265 1602799 3 54990338 3473851 1210607 2274326 1556673 4 59749470 3730349 1113642 2505711 1724222 5 69751162 3954065 987112 2345215 1584123 6 67574654 4466072 926824 1974662 1414234 7 62878557 4576052 943632 2029733 1733591 8 56017871 4116079 867695 1952593 2071152 9 49177739 3653055 868743 2180866 2077766 10 48541712 3795339 818183 2176533 2161208 Laspeyres index in R Using weights, the first alternative that we describe is devoted to obtain the Laspeyres index (10) in R. IndexNumber package allows this through laspeyres.index.number() by adding these as parameters: the matrix of the magnitudes to be evaluated, the matrix containing the weights, and several options of graphical and numerical representation for the results. The R Journal Vol. 13/1, June 2021 ISSN 2073-4859 CONTRIBUTED RESEARCH ARTICLES 271 > laspeyres.index.number(ECResources[,2:6],ECResources[,7:11], name="Price",opt.plot=TRUE,opt.summary=TRUE) Laspeyres index number Summary Min.=101.309108829536 Stage=1 Max.=158.535309198466 Stage=8 $Summary Min. 1st Qu. Median Mean 3rd Qu. Max. 101.3 112.1 127.8 131.2 154.0 158.5 $ ` Agg. index number ` Stages Price 1 Price 2 Price 3 Price 4 Price 5 Agg. index number 1 0 0.14 0.70 1.00 0.69 0.12 100.0000 2 1 0.14 0.78 0.97 0.66 0.10 101.3091 3 2 0.15 0.83 1.03 0.81 0.11 110.2332 4 3 0.16 0.87 0.93 0.68 0.10 112.1124 5 4 0.17 0.79 1.04 0.72 0.12 115.7457 6 5 0.19 0.77 1.15 0.91 0.12 127.7677 7 6 0.22 0.82 1.19 0.96 0.12 142.7184 8 7 0.23 1.00 1.34 0.99 0.13 153.9749 9 8 0.24 1.07 1.35 0.89 0.15 158.5353 10 9 0.25 1.04 1.22 0.77 0.13 158.2000 As before, if the option opt.plot=TRUE is considered, the output of the function also includes a graphical representation in which the joint evolution can be analyzed as Figure 6depicts. 100 120 140 160 Stages Price: Index Number 0123456789 Figure 6: Laspeyres index evolution for the prices of several energy resources for the period 2005-2015. Paasche index in R Analogously, the joint evolution of magnitudes under the Paasche index (11) can be also determined by using IndexNumber package in R software. More specifically, paasche.index.number() provides The R Journal Vol. 13/1, June 2021 ISSN 2073-4859 CONTRIBUTED RESEARCH ARTICLES 272 the mentioned weighted index number. The results of the function follows a similar structure as the function previously mentioned, and using the same graphical and numerical options. paasche.index.number(ECResources[,2:6],ECResources[,7:11], name="Price",opt.plot=TRUE,opt.summary=TRUE) Marshall-Edgeworth index in R The obtaining of the Marshall-Edgeworth index number (12) is also possible in R software through the use of IndexNumber package. In particular, we have to use the function edgeworth.index.number() with the above-mentioned options for obtaining graphics and summaries. edgeworth.index.number(ECResources[,2:6],ECResources[,7:11], name="Price",opt.plot=FALSE,opt.summary=FALSE) Fisher index in R Here, we consider the case of determining the Fisher index (13). In this case, fisher.index.number() provides a measure of the considered magnitude. Again, the options opt.plot and opt.summary allow the obtaining of additional information that may be of interest to the user. fisher.index.number(ECResources[,2:6],ECResources[,7:11], name="Price",opt.plot=FALSE,opt.summary=FALSE) A complex index for vin R The complex index for v given in (14) can be easily obtained as the Bradstreet-Dûtot index for the value in each instant of time. Recall that the value is obtained as the product of the amount of good by its price in each instant of time. Thus, the function aggregated.index.number() provides the value of this new index number as follows. aggregated.index.number(ECResources[,2:6]*ECResources[,7:11], base="serie",type="BDutot",name="Prices",opt.plot=FALSE, opt.summary=FALSE) Concluding Remarks This paper discusses the implementation in R of classical index numbers used for comparing magnitudes mainly in economic contexts. Therefore, the IndexNumber package provides R users a set of functions to calculate index numbers. Concretely, this library allows the calculation of simple index numbers in series and in chain. Furthermore, complex index numbers are also implemented. In particular, the non-weigthed index numbers included are the Sauerbeck, Geometric mean, Harmonic mean, Bradstreet-Dûtot, Carli, Jevons, and Dûtot indexes; as weighted index numbers, the Laspeyres, Paasche, Marshall-Edgeworth, Fisher, and Bradstreet-Dûtot indexes were considered. Additionally, this package contains graphical tools in order to facilitate the results visualization and four real data sets that can be used as illustrative examples. Moreover, the use of this library could be easily combined with other classical packages focused on time series analysis. Future research and development plans for forthcoming versions of the package include the addition of new index numbers already considered in the literature that can be dealt with in the framework presented above. Of course, the corresponding graphical tools should also be implemented. Acknowledgments A. Saavedra-Nieves acknowledges the financial support of FEDER/Ministerio de Ciencia, Innovación y Universidades - Agencia Estatal de Investigación under grant MTM2017-87197-C3-3-P. P. 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[p254] Alejandro Saavedra-Nieves Departamento de Estatística, Análise Matemática e Optimización Universidade de Santiago de Compostela Spain ORCID: 0000-0003-1251-6525 [email protected] Paula Saavedra-Nieves Departamento de Estatística, Análise Matemática e Optimización Universidade de Santiago de Compostela Spain ORCID: 0000-0002-3224-8858 [email protected] The R Journal Vol. 13/1, June 2021 ISSN 2073-4859