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arginal gains in accuracy of valuation from increasing the specificity of price indexes : empirical evidence… Jamal, Karim 1983

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MARGINAL GAINS IN ACCURACY OF VALUATION FROM INCREASING THE SPECIFICITY OF PRICE INDEXES: EMPIRICAL EVIDENCE FOR THE CANADIAN ECONOMY By KARIM JAMAL B.Comm.(Hon.), The University of Manitoba, 1981 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n Business A d m i n i s t r a t i o n in THE FACULTY OF GRADUATE STUDIES Faculty of Commerce and Business Administration We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 1983 (c) Karim Jamal, 1983 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date WcJlmW £ V / 8^ . DE-6 (3/81) i i ABSTRACT In this paper we present empirical estimates of marginal gains in accuracy of asset valuation from increasing the spec i f ic i ty of price indexes used to adjust Historical Cost Financial Statements. The empirical evidence strongly suggests that the accuracy function for the Canadian economy is highly convex. This implies that the marginal gain in accuracy of valuation declines sharply as the number and spec i f ic i ty of price indexes used for valuation increases. These findings are potentially valuable for Auditors, Academics and Regulatory agencies who are involved in the debate on selection of asset valuation rules. The results are of particular relevance for selection of asset valuation rules when i t is costly to use finer measurement methods. CICA handbook (Section 4510) presently allows companies to choose from several alternative methods for adjusting historical prices. In this paper we show the benefits obtained by using finer data, so that companies can made a decision as to the amount of resources they should invest to obtain finer data. i i i T A B L E O F C O N T E N T $ PAGE Abstract i i List of Tables iv List of Figures v Introduction 1 I Description of Valuation Problem 2 II Summary of Sunder, Waymire and I j i r i ' s work 4 III Purpose of Empirical Test"and Data 9 IV Empirical Evidence 12 V Discussion 15 VI Description of Tables and Figures 18 Bibliography 20 Footnotes 21 Tables 22 Figures 26 Simplified Calculation of Error for any Index System 29 E matrix 30 Computer program for Calculation MSE 31 Appendix A 36 Appendix B 43 iv LIST OF TABLES PAGE Table 1 - Search Algorithm 22 Table 2 - Components of Industry Sell ing Price Indexes 23 Table 3 - Average Error of 15 most eff i i 'c ient index combinations 24 Table 4 - Error of most ef f ic ient index combination.. 25 Table 5 - Order of combination of goods that results in the most ef f ic ient index 25 V LIST OF FIGURES PAGE Figure 1 - Average accuracy of 15 index combinations 26 Figure 2 - Most accurate index combinations 27 Figure 3 Average accuracy and most accurate index combinations 28 1 Histor ical ly two competing paradigms, the stewardship and the valuation paradigm have sought to define accounting and explain its purpose. The stewardship paradigm was original ly emphasized by Paton in the 1920's, and sought to explain the role of accounting as the provision of infor-mation to a principal so that he could assess the performance of an agent. This model requires 'hard' objective data in order to evaluate performance hence objectivity and r e l i a b i l i t y are the primary cr i ter ia for judging the value of an accounting report. The valuation paradigm was emphasized by Canning in 1930. He tr ied to relate Fishers concept of economic income to accounting. This model places a premium on relevance to decision making as the primary c r i te r -ion for judging the value of an accounting report. The two paradigms draw sharp boundaries in order to define what accounting is and how accountants should function. However, both paradigms have not been successful in explaining what accountants do and why rules requiring historical cost, conservatism, e tc . , are found in practise. At present, research is being done to develop an economic theory of information which is hol i s t ic in nature, and combines insights obtained in economics, finance and accounting. This theory is called Agency theory and the academic community has great hopes that i t wil l explain the economic forces that affect accounting. 2 In this paper we shall be dealing with the problem of valuation. A description of the valuation problem is given in section I.A Summary of Sunder, Waymire and I j i r i ' s work and a discussion of their contribution to our knowledge of the problem is given in section II.A description of the purpose of the empirical test and the data is presented in section III. The empirical evidence is presented in section IV, a discussion of the results follows in section V and a description of the tables and figures used in the study is presented in section VI. I. Various approaches to valuation have been regarded as empirical proxies for a common unobservable theoretical conception of value. The major aspect of the I j i r i , Sunder approach of research in valuation is to determine how well each method approximates the underlying value. There is no unique measure of how good a proxy i s , and several attributes such as relevance, r e l i a b i l i t y , object-i v i t y , freedom from bias and cost are mentioned in the l i terature. The major problem of valuation exists in determining the value of used assets for which an organized and ef f ic ient market does not exist. These assets can be broken down into two categories: 1) Assets whose prices are unobservable 2) Assets whose prices can be observed after undertaking a costly search for information. Continued Comparative analysis of alternate valuation rules has tradit ion-a l ly been qualitative in nature. Use of historical cost, general price level adjustments and current cost adjustments have been viewed as dist inct valuation rules rather than elements of a continuum. Historical cost represents prices that are unadjusted. General price level adjusts prices using one average economy-wide index. Current cost uses many specif ic indexes to adjust prices of assets held by a firm. Due to the qualitative nature of the analysis i t is impossible to rank these valuation rules. The current .legislation (section 4510) allows companies to choose from among many alternative methods in order to adjust historical prices. Major decisions facing managers of these companies are: a) Should indexes be used or should the company hire an appraiser. b) If indexes are used, what are the benefits gained by invest-ing resources to obtain finer data. Since the legis lat ion has been recently introduced, many companies are going through this process for the f i r s t time and are looking for guidance in order to comply with the letter and sp i r i t of the legis lat ion. In order to answer these questions we estimated the accuracy , function of the Canadian economy by assuming that i t consists of 16 assets. We empirically demonstrated Sunder's convexity result and showed that the results obtained by Sunder in the U.S. are also valid for the Canadian economy. 4 I. Continued The empirical evidence shows that gain in accuracy drops sharply as f iner information is obtained. It also shows that for a large number of industries, indexes provide a very accurate measure of the change in prices of assets owned by a firm. II. I j i r i (1967,1968) init iated an alternate approach by characteriz-ing valuation rules as aggregation functions. He defined a s ta t i s t i c "the linear aggregation coeff ic ient" which summarizes an important property of valuation rules as a single number. This was extended by Sunder (1978) who showed that most valuation rules,.for example historical cost, general price leve l , current .'. : replacement cost, exit values, e tc . , may be viewed as members of a family of asset valuation rules which he labelled "exchange valuation rules". He developed a unified scheme to algebraically represent each rule in terms of the price index configuration employed to adjust historical cost. Various results pertaining to Bias and Mean Squared Error (MSE) of valuation rules were derived. Sunder and Waymire (1983) extended these findings to examine the amount of accuracy for a given index configuration with respect to any s t r i c t l y finer index configuration. Hall (1982) provided empirical evidence in support of Sunder's results using data for e l ec t r i c a l , gas, pipeline, telephone and water u t i l i t i e s . The most important results of these papers are summarized in Appendix A. 5 II. Continued A major problem facing accountants in the past has been to c learly identify the role of accounting in providing informat-ion about the value of a firm. Is i t the accountants role to value the firm or is that task to be le f t to the stock market or appraisers? Many prominent accountants l ike I j i r i were not in favour of providing a firm's value since they did not think that accountants were appraisers. They fe l t that the valuation function should be carried out by the stock market. However, one should realize that the value reflected by the stock market is a collection of subjective values, and i t is the accountants role to provide objective information so that these subjective assessments can be made. It should also be noted that i t is more ef f ic ient to require each company to provide price level adjusted data, rather than having numerous participants in the stock market duplicating this task. Eff ic iencies of scale can be gained i f there is general acceptance about the manner in which these data should be generated. If we assume that a demand for this type of information exists we can then focus on the benefits to be gained by investing resources to obtain finer data. A well known portfolio effect in finance is that the variance of portfolio returns decline in a convex fashion as the number of securities in the portfolio increases. The accuracy of valuation (MSE) declines rapidly as more goods are added to the index since the change in prices of the goods are highly correlated. This results in a convex function, hereafter referred to as the convexity property. 6 II. Continued At an intuit ive l eve l , this property is well understood by the government for example only a few goods are used to construct the consumer price index. This property is also used extensiv-ely in the finance l i terature where a market index (TSE index), is used as a measure of the market's performance rather than using a l l the securities traded in the market. It is important to note that there are three types of assets found in the economy namely: 1) Assets that are heterogenous. These assets possess some unique properties that distinguish them from a l l other assets in the economy for example the Hotel Vancouver. At the present time Stat ist ics Canada does not publish any indexes for real estate. Since these assets are heterogen-ous, even i f an index were published, we would expect that i t would have a very low correlation with the actual change in value of an individual piece of real estate. 2) Assets that are homogenous. The government usually produces indexes for these types of assets. Since they are homogenous the convexity property holds and the index has a very high correlation with the actual change in value of an individual asset. 3) Assets that f a l l in between the two groups depicted in (1) and (2). If indexes are available for these goods, they have a medium correlation with the actual change in value of an individual asset. 7 II. Continued The major contributions of Sunder's paper are as follows: 1) The convexity property obtained from the portfolio theory, which is used extensively in finance and economics can be applied to the valuation problem. 2) If a firm has homogenous assets, the correlation between an appropriate index and the change in value of an individual asset wi l l be high. He also demonstrated the convexity property, and showed clearly that the marginal benefit from using finer indexes decreases rapidly. Thus a coarse index reflects a large portion of the change in the value of underlying assets. 3 ) He introduced the notion of stochastic dominance and f ine-ness. He showed an objective benefit that can be derived by investing resources to produce finer information. 4) He introduced the concept of a suff ic ient s tat i s t ic (MSE) which could be applied to the valuation problem. This measure reflects a l l desirable attributes such as relevance, r e l i a b i l i t y , representational faithfulIness, etc. 8 II. Continued 5) He separated the issue of quality of information from the user decision models that u t i l i ze this information. This is the concept of fineness of information. He hoped to show that we can order information systems based on f ine-ness (stochastic dominance) i .e . to show that one inform-ation system would be preferred over another for a l l purposes i f i t could be obtained at the same cost. Unfortunately we can only get a fineness relationship for a subset of a l l information systems (partial order). A major contribution of Sunder was to show that in this case the convexity property can be applied to a substant-ia l amount of goods in the economy. 6 ) Since the benefit is measured in terms of lower error (more accuracy) and cost is measured in dol lars, Sunder derived a benefit graph on which each, individual decision maker can plot his/her own cost curve. It is important to recognize that the cost curve depends on a subjective u t i l i t y for accuracy i .e . the price that each person is wil l ing to pay to obtain a lower MSE.. 9 II. Continued An important point to remember is that Sunder used American data to test his results, and that the quality and amount of inform-ation available in the U.S. is far superior than that available in Canada. Stat ist ics Canada provides a few highly aggregated price indexes but i t is wil l ing to help any group or organizat-ion that wishes to obtain finer data. See Appendix B for a chart showing the levels of aggregation of data available in the U.S. Sunder assumes that the value of the goods can be observed, thus i t is important to note that for goods that are heterogenous for example the Hotel Vancouver, i t is not appropriate to use this model. III. CICA Handbook Section 4510:Reporting the Effects of Changing  prices requires large publicly held companies to disclose the effect of changing prices on Inventory, Property Plant and Equipment (and Depreciation) as well as the gain/loss accruing to shareholders from holding net monetary assets. General Price Level data are obtained by applying a single economy-wide price index, to historical cost data. Specific price changes are obtained by applying specif ic indexes to individual goods. Considerable f l e x i b i l i t y is allowed in the choice of sources of information about current costs in order to encourage experimentation and learning. 10 Continued Section 4510A.52 says: "Detailed rules for measurement of current costs have not been provided in the recommendations. The committee recognizes that selection of appropriate techniques wil l be a matter for determination by management, taking into account the nature of the assets concerned and the circumstances of the enterprise. Practical experience with the use of various techniques for determining estimates of current cost wil l increase the present level of knowledge and may permit more detailed recommendations to be made at a later stage". Those, who choose specif ic price indexes must also decide the spec i f ic i ty of the index system by balancing the gains in accuracy of valuation against the increased costs of using a more specif ic set of indexes. Those who use other methods (appraisals) may also want to know what they can expect to gain from their efforts. 11 Continued In this paper we present empirical estimates of marginal gains in accuracy of asset valuation by increasing the speci f ic i ty of price indexes used to adjust Historical Cost. The results show that the price structure of the Canadian economy is such that marginal gains in accuracy decline sharply as the specif-i c i t y of price indexes is increased. A large proportion of total potential gain for accurate estimation of current cost is attained by a few broad indexes; additional detail adds relat ively l i t t l e to accuracy. For example increase in accuracy of valuation obtained by using 20 price indexes instead of 10 indexes is smaller than the increase in accuracy from using 10 indexes instead of 5. These results are similar to those in the finance l iterature where the marginal reduction of divers i f iable risk of a portfolio declines as the portfolio size increases. Assets of a firm as well as price indexes used to estimate their current value can be represented as portfolios of goods in the economy. The accuracy (MSE) of a given set of price indexes in approximating assets of a firm is a function of the mean vector and covariance matrix of price changes in the economy. Previous work by I j i r i (1967,1968) and Sunder (1978) provided analytical results which enable us to estimate the s tat i s t ica l accuracy of valuation rules based on various index systems direct ly from the data on price structure of the Canadian economy. Relative weights of goods' and mean and covariance of price changes from Industry Selling Price Indexes published by Stat ist ics Canada are used in this study. 12 III. Continued The Data The Manufacturing Price Index, published by Stat ist ics Canada, was used rather than the Consumer Price Index since i t provides a better valuation of industrial assets. The manu-facturing price index consists of two hierarchical levels of price indexes: 1) An overall manufacturing price index (general price level index). 2) Specific Industry price indexes. I have used fewer indexes than the total available in the data base. All the industries for whom monthly data was available for the period January 1977 - December 1982 were examined. This resulted in the selection of 16 industry specif ic price indexes out of a total of 20 industries. The weights of each index were adjusted to bring the total to 1. (see Table 2). IV. Empirical Estimates Of the Accuracy Function We estimated the accuracy function of the Canadian economy by assuming that i t consists of 16 assets, each of which is rep-resented by one of 16 industry price indexes which comprise "the manufacturing price index". The weights of these assets indicate the relative proportion of these assets in the economy. Due to the adjustment of these weights i t is assumed that these are the only goods in the economy. 13 IV. Continued Suppose a l l the firms in the economy consisted of randomly drawn bundles of these 16 assets in varying proportions which were valued using only, 10 (10=k) price indexes. Which part-icular combination of assets can be expected to yield the most-accurate valuation of individual firms on average? How accurate is this 10 price index system? Table 5 answers the f i r s t question, the curve shown in Figure 1 and Figure 2 answers the second question. I used the systematic search procedure (Table 1) to combine the 16 assets into 10 price indexes in f ifteen different ways. Point X (Figure 1) shows the average accuracy for these f ifteen 10-index values. Point Y (Figure 2) depicts the values of the most accurate of these f ifteen 10-index values. Note that in this 16-asset economy, 16 price indexes (k=16), one for each good, automatically yields the most accurate current cost of the hypothetical bundle of 16 assets and therefore the mean squared error for a 16 index system is zero in Figure 1 and Figure 2. At the other end of the scale, valuation based on a single price index (k=l), "the manufacturing price index", is least accurate. Note that the curve in Figure 1 and Figure 2 have the same end points (at k=l and k=16). There is only one way of combining 16 assets into a single index or 16 individual indexes. Figure 1 and Figure 2 show values of X and Y for a l l possible values of k (k=l-16). 14 Continued The X-curve shows the accuracy of the average k-index systems that can be created from these 16 assets. There are two points of interest: 1) The X-curve accuracy function is convex. The marginal gain in accuracy by increasing k (the number of price indexes in this economy) keeps declining. More than half the gain in accuracy can be realized from using only four price indexes and there is hardly any gain beyond nine indexes. 2) X is an upward biased estimate of the accuracy function because further experimentation with this algorithm (and more computer resources), or with more ef f ic ient algorithms we can expect to obtain even better k-index system for different values of k. The Y-curve shows the values of the most accurate k-index system created from the 16 •.available.-assets. It is interesting to note that the accuracy function Y, is even more convex than the accuracy function X. Since the estimated accuracy function shows further improvement for a given k as n increases, the accuracy function for , the actual C a n a d i a n economy is l i ke ly to be highly convex. 15 IV. Continued The analysis shows that i f this 16 goods index (manufacturing) price index) is standardized for current valuation, we can construct finer indexes than those produced by Statist ics Canada. This concept is intu i t ive ly apparent since Stat ist ics Canada produces data for a variety of users. Therefore modification of this data for valuation of assets only, should yield a more accurate valuation. V. Discussion: The empirical evidence strongly suggests that the accuracy function of the Canadian economy is highly convex. This implies that the marginal gain in accuracy of valuation declines sharply as the number and spec i f ic i ty of price indexes used for valuat-ion increases. These findings are valuable for Auditors, Managers, Academics and regulatory agencies who are involved in the debate on selection of asset valuation rules. Especially in a world where i t is costly to employ more finer measurement methods. Prior to practical implementation of the analytical model used in this paper, several research issues need to be resolved. F i r s t , i t remains to be shown that in general, the accuracy function is convex. If not, under what conditions is i t not convex? 16 V. Continued This is important because prior to any policy decision on asset valuation rules, i t would be useful to know whether the economy under consideration is characterized by a convex accuracy function. If a given economy does not possess this function, then accuracy of valuation wil l not be improved substantially by using only a few price indexes. Second, to what extent is the convexity of the accuracy function related to parameters u and 2 (the mean and covariance) which describe the underlying process generating relative price changes. By identifying a relationship between the parameters u and^and the convexity curve of the accuracy function, i t would be possible to determine the partitions along the accuracy function without employing costly search procedures for the set of valuation rules. Any policy process attempting to exploit this framework would • have to implement i t on an ex-ante basis. Thus, additional information is needed to determine whether u a n d £ are stable over time and can be rel iably predicted. If i t is not possible to derive the relationship between u and^F and the convexity of the accuracy function, can we identify superior search algorithms for the accuracy function? Note that this is a one period model and we have also assumed that there are no changes in the quantity of goods during the year. The impact of changes in quantities as well as over time would have to be assessed prior to practical implementation of this model. 17 V. Continued The government can take a more active role by col lecting data and providing finer information. The government, however, has to provide information for a variety of users and the CICA should conduct more research to determine: 1) For which specif ic industries does the convexity property apply? 2) What specif ic adjustments need to be made to data provided by the government to improve accuracy of asset valuation? 3) Identify a l l the alternatives available for valuation of heterogenous assets. After spec i f ica l ly defining the information required the CICA and Stat ist ics Canada should work together to provide the required information and the CICA can then provide more guid-ance to companies on how to adjust historical prices. 18 VI Description of Tables and Figures Table 1 - describes the search Algorithm used in this study to find the MSE of Index Systems and identify finer partitions of the Index Combinations. Any individual who wishes to reproduce the results presented in this paper, or who wishes to do a similar analysis using different data, should follow the steps set out in Table 1. Table 2 - describes the Components of Industry Selling Price Indexes published by Stat ist ics Canada that were used in this paper. Al l Components of the Index for whom monthly data was available for the period January 1977 - December 1982 were included in the study. The original weights of these goods and their adjusted weights are also shown in th.e table. The weights are determined by Stat ist ics Canada based on surveys, questionnaires and collection of data by other means at their disposal. Their weights are revised on a periodic basis by Stat ist ics Canada. Table 3 - shows the Average MSE obtained of the 15 most ef f ic ient Combinations we could find for k, where k is the number of Indexes (k=l-16). For example, for k=10, there are a very large number of ways of combining 16 goods into 10 Indexes. Using the Systematic Search Procedure in Table 1, the 15 most ef f ic ient Combinations were identif ied and we ca lcul -ated an Average for these Combinations hence Average MSE for k=10 is 55. This table shows the decrease in MSE on average as k increases. 19 Description of Tables and Figures (Continued) Table 4 - shows the MSE obtained from the most-efficient Index Combination we could find for k (k=l-16) using the Systemic Search procedure described in Table 1. The third column shows the marginal gain obtained from increasing the Speci f ic i ty of Indexes Used. Table 5 - shows the order in which goods should be combined in order to generate the most ef f ic ient Index Systems. For example, we found that using our search procedure, i f you wish to combine three goods into an Index than combining goods 6,8, and 12 results in the lowest MSE. Figure 1 - is a graphical representation of the data presented in Table 3. The point X on the graph shows the average MSE of the 15 most ef f ic ient Index Configurations for k=10. This graph shows the average gain in accuracy as k increases. Figure 2 - is a graphical representation of the data presented in Table 4. The point Y on the graph shows the most ef f ic ient Index Combination we could identify using the Systematic Search procedure for k=10. This graph shows the gain in accuracy for the most ef f ic ient combination as k increases. Figure 3 - is a graphical representation of the data in Table 3 and Table 4. This graph enables us to see the marginal gains for the most ef f ic ient Index Configuration and on Average as k increases. The graph shows that both curves are convex. 20 Bibliography 1) Canadian Institute of Chartered Accountants CICA Handbook Section 4510 - "Reporting the Effects of Changing  Prices". 1982. 2) Financial Accounting Standards Board Statement of Financial Accounting Standards No. 33: Financial  Reporting and Changing Prices. 1979 3) Hal l , T.W. "An Empirical Test of The Effect of Asset Aggregation on Valuation Accuracy". Journal of Accounting Research (Spring 1982) pages 139-151. 4) I j i r i , Y The Foundations of Accounting Measurement : A Mathematical, Economic and Behavioural Inquiry. Englewood c l i f f s , New Jersey: Prentice-Hall, 1976. 5) Sunder, S. "Accuracy of Exchange Valuation Rules". Journal of Accounting Research (Autumn 1978) pages 341-367. 6) Sunder, S and G. Waymire. "Accuracy of Exchange Valuation Rules :Additivity of Accuracy and Estimation Problems: Journal of Accounting Research (Forthcoming). 7) Sunder, S. and G. Waymire. "Marginal Gains in Accuracy of Valuation From Increasingly Specific Price Indexes: Empirical evidence for the U.S. Economy. (Unpublished working paper-1983). 21 Footnotes k-1 1. Sunder (1978, p. 347). 2. Note that mean square error is an inverse measure of accuracy. The valuation system becomes increasingly accurate as the mean squared error decreases. 22 Table 1  Search Algorithm Step : 1 ) Collect Data 2) Compute Relative Price Change (P 1 -P°) /P° 3) Find Mean (u) and variance-covariance matrix (^) Of relative price changes 4) w is given 5) S=2>UU" 6) E^w. (S . i + S . . -2S . J ) 7) Combine assets using E matrix as a guide to identify goods that have a low mean squared error when combined and compute error for the index. 23 TABLE 2 Components of Industry Sei1ing Price Indexes: Monthly Data 1877-1982 (72 observations) Code Description Original Weights % Revised Weights % 1 D500001 Food & Beverage 19.016 23.6 2 D511200 Tobacco Products 1 .084 1 .4 3 D511500 Rubber & Plastics 2.417 3.0 4 D513400 Leather Industries .840 1 .0 5 D514500 Textiles 3.369 4.2 6 D516600 Knitting Mil ls .846 1 .0 7 D519100 Wood 4.515; 5.6 8 D523200 Furniture 1 .539 1 .9 9 D524200 Paper 7.809 9.7 10 D527100 Primary Metals 7.970 9.9 11 D529400 Metal Fabrication 7.169 8.9 12 D532900 Machinery Industries 4.162 5.2 13 D537300 Electr ical Products 6.470 8.0 14 D541400 Non-Metallic Industries 3.043 3.8 15 D544000 Petroleum & Coal 4.044 5.0 16 D5452Q0 Chemical 6.270 7.8 80.563 100.0 24 TABLE 3 Average Error of Marginal Gain from Number of 15 Most Eff ic ient Increase in Specif ic i ty Indexes (K) Combinations of Indexes Used K = 1 1 ,161 . -K = 2 1 ,076. 85. K = 3 799. 277. K = 4 583. 216. K = 5 391 . 192. K = 6 250. 141 . K = 7 163. 87. K = 8 114. 49. K = 9 83. 31 . K = 10 55. 28. K = 11 41 . 14. K = 12 27. 14. K = 13 17. 10. K = 14 10. 7. K = 15 5. 5. K = 16 0 5. 25 TABLE 4 Error of Marginal Gain from Number of Most Eff ic ient Increase in Specif ic i ty Indexes (K) Combination of Indexes Used K = 1 1161 .0 -K = 2 894.5 266.5 K = 3 647.6 247.0 K = 4 410.0 237.6 K = 5 255 .7 154.3 K = 6 167.0 88.7 K - 7 113.0 54.0 K = 8 87.0 26.0 K = 9 62.6 24.0 K = 10 46.0 17.0 K = 11 32.0 14.0 K = 12 18.6 13.0 K 13 10.5 8.0 K = 14 5.5 5.0 K = 15 2.0 3.5 K = 16 0 2.0 TABLE 5 Most e f f ic ient Index Systems are generated by combining goods in the following order: 6, 8, 12, 3, 5, 13, 4, 11, 2, 16, 14, 9, 10, 15, 7. Best Index Containing 3 Goods i s : (6, 8, 12) Best Index Containing 5 Goods i s : (6, 8, 12, 3, 5) Best Index Containing 10 Goods i s : (6, 8, 12, 3, 5, 13, 4, 11, 2, 16) 26 o RL F I G U R E 1 NUMBER OF GOODS IN INDEX 27 " F I G U R E 2 2 8 F I G U R E 3 A V E R A G E ACCURACY OF EACH I N D E X RND MOST A C C U R A T E I N D E X C O M B I N A T I O N F I G U R E F I G U R E 2 NUMBER'OF GOODS IN INDEX 29 Simplified Calculation of Error for any Index System Suppose an index system contains 4 goods 1,2,3,4. Contribution of this index to the mean squared error is : E 1 2 + E 1 3 + E 1 4 + E 2 3 + E 2 4 + E 3 4 where E.. = w. . = w.w. Tvarlr. - r. N + (u, - u.) 2 I i j i j i j v—i i J) + 1 J J E = w.w. \ cf., + d.. - 2 <f. . + u. 2 + u. 2 - 2 u.u. E = w . w . S . . + S . . - 2 s . . T J L . n JJ 1 J J where S.. is the i j element of S = ^ + uu'. Add up the contribution of each index to obtain the total mean squared error of the index system. For a detailed Mathematical Analysis, interested readers should consult Sunder (1978) and Sunder and Waymire (1983). 30 E MATRIX 0.0 L 8.845 7.727 4.091 , 11.719 1 2.320 1 67.265 i 4.890 . 43.296 . 73.212 . 19.193 . 12.232 . 17.183 . 22.725 . 71.485 , . 28.687 8.845 0.0 0.680 0.400 1.411 0.293 5.214 0.517 4.101 5.818 2.471 1.327 2.326 1.882 4.905 2.361 7.727 0.680 0.0 0.556 0.514 0.116 9.222 0.200 3.336 9.223 1.182 0.408 0.967 2.073 7.866 1.025 4.091 0.400 0.556 0.0 0.702 0.143 2.746 0.268 2.373 3.572 1.386 0.781 1.238 0.965 3.571 1.498 11.719 1.411 0.514 0.702 0.0 0.113 12.310 0.380 3.950 12.186 1.512 0.674 1.421 3.035 10.717 1.405 2.320 0.293 0.116 0.143 0.113 0.0 2.621 0.057 1.310 3.091 0.319 0.122 0.241 0.584 2.661 0.364 67.265 5.214 9.222 2.746 12.310 2.621 0.0 5.535 38.044 39.670 26.953 14.613 21.787 13.669 30.417 26.248 4.890 0.517 0.200 0.268 0.380 0.057 5.535 0.0 2.604 5.994 0.734 0.263 0.773 1.077 5.181 0.811 43.296 4.101 3.336 2.373 3.950 1.310 38.044 2.604 0.0 32.451 10.282 5.983 11.008 11.482 29.761 9.383 73.212 5.818 9.223 3.572 12.186 3.091 39.670 5.994 32.451 0.0 27.884 16.646 23.005 15.886 40.431 24.118 19.193 2.471 1.182 1.386 1.512 0.319 26.953 0.734 10.282 27.884 0.0 1.530 2.968 6.072 23.495 3.884 12.232 1.327 0.408 0.781 0.674 0.122 14.613 0.263 5.983 16.646 1.530 0.0 1.241 3.350 12.661 1.976 17.183 2.325 0.367 1.238 1.421 0.241 21.787 0.773 11.008 23.005 2.968 1.241 0.0 6.305 22.026 4.431 22.725 1.882 2.073 0.965 3.035 0.584 13.669 1.077 11.482 15.886 6.072 3.350 6.305 0.0 11.424 3.840 71.485 4.905 7.866 3.571 10.717 2.661 30.417 5.181 29.761 40.431 23.495 12.661 22.026 11.424 0.0 19.204 28.687 2.361 1.025 1.498 1.405 0.364 26.248 0.811 9.383 24.118 3.884 1.976 4^431 3.840 19.204 0.0 The E Matrix shows a l l possible combinations of 2 Goods i n an Index. The numbers divided by the weights of the respective Goods re s u l t i n the MSE of combining two goods i n an Index. Table E helps us f i n d f i n e r p a r t i t i o n s . The reasoning i s that a low E value result's i n a low MSE 31 COMPUTER PROGRAM FOR CALCULATING ERROR 2 020 7 030 50 100 110 120 130 140 150 180 185 190 200 210 300 Dimension W(l6) ,E(16,16) ,Index(16,l6) ,Kn(16) ,Error(l6) ,WK(,16) Dimension Dummy(16) Read (2,020) (W(I),1=1,16) Format (F 11.5) Read (7,030) (E (I,J) J=l,16) (Dummy(J) J=l,16)) 1=16) Write (6,30) (E (I,J) J - l ,16)1=1,16) Format (16F 7.3) Write (6,100) Format ('How Many Indexes?') Read (5,110) K If (K.EQ.O) Go to 300 Format (12) Write (6,120) Format ('Please enter Number of Variables and their Identification in Each Index:') ERR=0 DO 200 1=1,K Write (6,130) I Format ('Index No. 1, 15. '?') Read (5,140) KN (I), (Index (1,11) 11=1,16) Format (1712) WK (I)=0 KNK=KN(I) DO 150 11=1, KNK 12=Index (1,11) WK(I) = WK(I)+W(I2)/100 Error (I)=0 If (KN(I).EQ.l) Go to 185 KKK = KN(I)-1 Do 180 11=1,KKK III = 11+1 Do 180 12=111,KNK 13 = Index (1,11) 14 = Index (1,12) Error (I) = ERR0R(I)+E(I3,I4)/WK(I) Write (6,190)1,WK(I),ERROR (I) Format ('For Index', 15,Weight ' .F 11.5' ERROR ' .F 11.5) ERR = ERR = ERROR (I) Continue Write (6,210) ERR Format ('Total Error For This Index System = 'F 11.5') Go To 50 Stop END This Computer Program enables the User to log-on to the Comput-er and Use the Compu-ters Computational Speed and memory cap-acity to combine Various goods in an Index and Compute the MSE of the Index. 32 COMPUTER OUTPUT  INDEX CONTAINING 2 GOODS This Printout is an Example of a run on the Computer. Where the reasoning Implicit in the E matrix was tested. 1 Represents the number of goods Combined in the Index. 2 The two specif ic goods in the Index ( i.e.) asset 6 and as set8 . 3 The Weights of goods 6 and 8, Used to divide their E matrix Value of .057. 4 The MSE Obtained by Combining these two goods. How Many Indexes? Please Enter Number of Variables and Their Identification in Each Index. Index No. 2 68 For Index Index No. 2 63 For Index Index No. 2 65 For Index Index No. 2 6 For Index Index No. 2 6 For Index Index No. 2 6 For Index 1? 1 2? 2 3? 4? 12 4 5? 13 5 6? 11 6 WEIGHT WEIGHT WEIGHT WEIGHT WEIGHT 0.2900 0.4000 0.5200 0.6200 0.0900 ERROR ERROR ERROR ERROR ERROR 1.96552 2.90000 2.17338 1.96774 2.67776 WEIGHT 0.09900 ERROR 3.22227 Index No. 7? 2 616 For Index 7 WEIGHT Index No. 8? 2 64 For Index 8 WEIGHT Index No. 9? 2 62 For Index 9 WEIGHT Index No. 10? 2 61 For Index 10 WEIGHT Index No. 11? 2 83 For Index 11 WEIGHT Index No. 12? 2 85 For Index 12 WEIGHT Index No. 13? 2 812 For Index 12 WEIGHT Index No. 14? 2 813 For Index 14 WEIGHT Index No. 15? 2 8 11 For Index 15 WEIGHT Index No. 16? 2 816 For Index 16 WEIGHT 33 COMPUTER OUTPUT INDEX CONTAINING 2 GOODS (Cont.) 0.08800 ERROR 4.13637 0.0200 ERROR 7.1500 0.02400 ERROR 12.20834 0.24600 ERROR 9.43089 0.04900 ERROR 4.08163 0.06100 ERROR 6.22951 0.07100 ERROR 3.70423 0.9900 ERROR 7.80808 0.10800 ERROR 6.79630 0.09700 ERROR 8.36083 34 COMPUTER OUTPUT  INDEX CONTAINING 2 GOODS (Cont.) TOTAL ERROR FOR THIS INDEX SYSTEM = 84.81244 How Many Indexes? 16 Please Enter Number of Variables and Their Identification in Each Index: Index No. 1? 2 84 For Index 1 WEIGHT 0.02900 ERROR 9.24138 Index No. 2? 2 8 2 For Index 2 WEIGHT Index No. 3? 2 8 11 For Index 3 WEIGHT Index No. 4? 2 3 r 5 For Index 4 WEIGHT Index No. :"5? 2 312 For Index 5 WEIGHT Index No. 6? 2 313 For Index 6 WEIGHT Index No. 7? 2 311 For Index 7 WEIGHT 0.03300 ERROR 15.66667 0.25500 ERROR 19.17645 0.07200 ERROR 7.13889 0.08200 ERROR 4.97561 0.1100 ERROR 8.79091 0.11900 ERROR 9.93278 35 Index No. 8? 2 316 For Index 8 WEIGHT Index No. 9? 2 34 For Index 9 WEIGHT Index No. 10? 2 32 For Index 10 WEIGHT Index No. 11? 2 31 For Index 11 WEIGHT Index No. 12? 2 512 For Index 12 WEIGHT Index No. 13? 2 513 For Index 13 WEIGHT Index No. 14? 2 511 For Index 14 WEIGHT Index No. 15? 2 516 For Index 15 WEIGHT Index No. 16? 2 54 For Index 16 WEIGHT COMPUTER OUTPUT  INDEX CONTAINING 2 GOODS (Cont.) 0.10800 ERROR 9.49074 0.0400 ERROR 13.9000 0.04400 ERROR 15.45455 0.26600 ERROR 29.04886 0.09400 ERROR 7.17021 0.12200 ERROR 11.64754 0.13100 ERROR 11.54193 0.12000 ERROR 11.70834 0.05200 ERROR '13.500 TOTAL ERROR FOR THIS INDEX SYSTEM = 198.38486 36 Appendix A In this Appendix we summarize the important analytical results derived by Sunder (1978), and by Sunder and Waymire (1983). Interested readers should consult the original art ic les for a more thorough analysis. Consider an economy with n assets. Let q be the vector of quantities of the n assets contained in a given bundle. Suppose that under a given rule, valuation of the bundle is p^  at time 0 and p^  at time 1; the relative price change is R = (P^-P^/P^. Let r be the n-vector of relative price changes from time 0 to time 1 for the n assets. If valuation of each of the n assets in the bundle is arrived at by applying a specif ic price index to each asset, the resulting value of R is the relative price change in current value of the bundle (Rev) and is defined to be the principal aggregation: R c v = w ' r (1) where 0 * PU q, W. = i ;i for i = l,2...,n . n O * V P. q. J = 1 and P? = Unit Price of asset i at time 0, W represents the vector of relative weights characterizing a given bundle of assets, i . e . the firm. 37 Appendix A (Continued) R is used as a generic symbol for valuation rules and two modifiers are added to identify a specif ic rule. Rki represents a valuation rule which uses k (k<*n) different price indexes to adjust the beginning of the period valuation of a l l n assets. Since, in most cases, there is more than one way of forming k price indexes from the n assets, Rki represents the valuation obtained by using the ith of Lk possible configurations of k indexes^. Since there is only one way of forming n indexes from n assets Rcv=Rnl. Sunder (1978) defined the accuracy of a valuation rule to be the economy-wide average of the mean squared error of valuation for individual firms, Rki, with respect to the principal aggregation Rnl (=Rcv) . Each individual bundle of assets (representing one firm) is characterized by its vectors of relative weights, W, which is assumed to be generated randomly from the economy-wide bundle of relative proportions w, using a constant number (p) of multinomial t r i a l s . This accuracy measure (denoted Aki) is given by: A(Rki) = E E (Rki-Rnl) 2 =l/p (w'(€+u)-Jw7w'e (TLuu + uu')w) (2) where e = vector of unit elements of appropriate length w=E(w);..n-vector of relative weights of n assets in the economy ,w*'e-l u=E(r); n-vector of expected relative price changes for n assets, u = n-vector of squared elements of u, E = E(r-u) (r-u) ',n x n covariance matrix of relative price changes for n assets, 38 Appendix A (Continued) <f = n-vector of diagonal elements of "5L k = number of price indexes used in the valuation rule. The set Of n assets is partitioned into k nonempty subsets and a price index is constructed for each subset.w,u and ^ u u are the sub-vectors and submatrix respectively corresponding to tne uth of the k subsets. p = number of multinomial t r i a l s by which the bundle of assets for Individual firms is randomly drawn from the economy wide bundle defined by.w. Let Tki denote the ith of Lk dist inct partitions that can be used to form k price indexes for n assets. Similarly, Tk+m j is the jth of Lk+m, partitions that can be used to form (k+m) indexes from the n assets, m= 1,2 ,n-k. Sunder (1978) proved that i f Tk+m j is a s t r i c t l y finer partition of the n assets than Tk i , then A(Rk+m j) must be less than A(Rki): Tk+m j c Tki ^ A(Rk+m j) A(Rki). In other words, economy-wide average of mean squared error of valuation is a monotonically decreasing function of the fineness of the partitions used to form price indexes for n goods. Aki, however, is not monotonic in k, the number of price indexes employed. * For each value of k, let Rk denote the most accurate of the Lk valuation rules ( i.e.) Rk has the smallest economy-wide average of mean squared error. A(R*k) A(Rki) i = l ,2,... ,Lk. 39 Appendix A (Continued) Let the corresponding partition of the n assets be denoted by Tk. Thus, for every value of k there exists a partition Tk* which yields the best accuracy ( i.e.) smallest mean squared error, through the valuation rule R*k. We define H(k) to be the accuracy  function which gives the accuracy for the most accurate k-index valuation rule: H(k) = A (R*k), k=l,2,...n. (3) The remainder of this paper is devoted s t r i c t l y to examining the properties of H(k) for the Canadian economy. If one wishes to use k specif ic price indexes for valuation of n(>k) assets, what is the most accurate partition of n assets for constructing k price indexes from n assets and how accurate is this method. Speci f ica l ly , we are concerned with whether H(k) is convex in k. We know already from the results in Sunder (1978) that H(k) is s t r i c t l y decreasing in k. For any partition Tk* (for any k 4. n) along the accuracy function, we can generate a s t r i c t l y finer partition of the n assets into k+1 indices (denoted Tk+1 j ) . Since T*k and Tk+1 j are comparable with respect to fineness, A(Rk+l j) must be less than A(Rk*). By apply-ing this argument for every value of k s t r i c t l y less than n, i t is concluded that H(k) is s t r i c t l y decreasing in k. 40 Appendix A (Continued) Convexity of the accuracy function, H(k) implies that the marginal gain in accuracy ( i.e.) reduction in mean squared error, declines as k. increases. The convexity properties of the accuracy function are important from a practical viewpoint. A highly convex accuracy function implies that use of only a few broad indexes achieves a large proportion of total potential gain towards accurate estimates of current cost and further gains in accuracy may not be worth the additional costs associated with using a more detailed set of price indexes or direct measurement of current cost of individual assets. II Estimation of Accuracy Function H(k) Two problems must be solved to estimate the accuracy function, H(k), from data. F i r s t , accuracy measure A(Rki) for index config-uration Tk i , is a function of u and £ the mean vector and covari-ance matrix, respectively, of relative price changes for n assets in the economy. Since these parameters are unknown, they must be estimated from data. Sampling errors bias estimates of A(Rki) upward i f (2) is applied. We present an estimator which corrects for the bias. Second, even for moderate values of n, the set of a l l possible valuation rules is very large making exhaustive search over the set infeasible. Estimated accuracy function depends on search procedures used to identify the most accurate price index systems for each value of k. We present an algorithm which system-at ica l l y searches the set to identify relat ively accurate index systems. 41 II Estimation of Accuracy Function H(k) - Continued Consider f i r s t the issue of sampling errors in u and . If A ^ unbiased parameter estimates (denoted u and"2.) are employed in (2), the estimated accuracy for valuation Rki based on partition Tki i s : y\ /\ A ^ ( £ -~ A ) A(R k i ) = w'(<r+ u) - X w^ u ( > + u u' ) w (4) u=l w'e (^"uu ) Sunder and Waymire (1983) show that the presence of sampling errors in u biases (4) upward, while sampling errors in estimating the covariance matrix, "3> have no effect. The estimator to correct for this bias in (4) is henceforth referred to as the unbiased estimator where T represents the number of observations of relative price changes used to estimate u and From inspection of Equation (5), i t is evident that the unbiased est-imator differs from equation (4) only by the term (T-l)/T which is multiplied by the diagonal elements in the covariance matrix and the covariance submatrix for each of k indexes. Estimator (4) converges to the unbiased estimator (5) as T«^ tP and is therefore asymptotic-a l ly unbiased. The values of A(Rki) reported in this paper have been calculated using the unbiased estimator (5). 42 II Estimation of Accuracy Function H(k) - Continued The second estimation problem concerns selection of an appropriate procedure for searching over the set of valuation rules. Identif ic-ation of the accuracy function H requires that for each value of k=2,...,n-l, the index configuration with best accuracy ( i.e.) minimun mean squared error, be identif ied and an unbiased estimate of i ts accuracy be obtained by using (5). This requires optimizat-ion of (5) with respect to alternative index systems defined by k partitions of the set of n assets, a problem that has not been solved analytical\y. The total number Lk of alternative k-partit-ions of the set of n elements is extremely large, even for moderate values of n. Exhaustive search over such a large set is infeasible and ef f ic ient search algorithms must be devised to obtain an approx-imation of the accuracy function H(k) under the constraint of limited computer resources. In order to estimate H(k) we have used the procedure described in Table 1. This systematic search procedure exploits previous analy-t ica l results to identify those index partitions which are l ike ly candidates for inclusion in the accuracy function. Likely candid-ates for the most accurate k-partition are those which are s t r i c t l y coarser than the most accurate (k+1) partition and s t r i c t l y finer than the most accurate (k-1) part it ion. The search procedure for-cuses on the neighbourhood around the finest part it ion. P r o d u c e r P r i c e Index Rubber P r o d u c t s (07U Rubber and P l a s t i c Produc t s (07). Crude Rubber (0711) T i r e s and Tubes (0712) M i s c e l l a n e o u s (0713) P l a s t i c C o n s t r u c t i o n (0721) P r o d u c t s PI a s t i c P r o d u c t s (072T U n s u p p o r t e d P l a s t i c Film'(0722) and S h e e t i n g S i n g l e Index Two D i g i t Index Three Dip.it Indexes \ Laminated P l a s t i c (0723) Sheets Four D i g i t I ndexes F i g u r e 1 Pr o d u c e r P r i c e Index (PPT) C l a s s i f i c a t i o n Example 

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