Superlative Index Numbers Research Articles

Index Numbers and Revealed Preference Rankings

For previously identified weakly separable blockings of goods and assets, we construct aggregates using four superlative index numbers, the Fisher, Sato-Vartia, Tornqvist and Walsh, two non-superlative indexes, the Laspeyres and Paasche and the atheoretical simple summation. We conduct several tests to examine how well each of these aggregates “fit” the data. These tests are how close the aggregates come to solving the revealed preference conditions for weak separability, how often each aggregate gets the direction of change correct and how well the aggregates mimic the preference ranking from revealed preference tests. We find that, as the number of goods and assets being aggregated increases, the problems with simple summation manifest.

INDEX NUMBERS AND REVEALED PREFERENCE RANKINGS

For previously identified weakly separable blockings of goods and assets, we construct aggregates using four superlative index numbers, the Fisher, Sato-Vartia, Törnqvist, and Walsh, two non-superlative indexes, the Laspeyres and Paasche, and the atheoretical simple summation. We conduct several tests to examine how well each of these aggregates “fit” the data. These tests are how close the aggregates come to solving the revealed preference conditions for weak separability, how often each aggregate gets the direction of change correct, and how well the aggregates mimic the preference ranking from revealed preference tests. We find that, as the number of goods and assets being aggregated increases, the problems with simple summation manifest.

TFP estimation at firm level: The fiscal aspect of productivity convergence in the UK

This paper reviews the state of the art in firm-level Total Factor Productivity (TFP) estimation employing an unbalanced panel of 7400 manufacturing firms in the UK during 2004–2011. The five methodologies considered are Superlative Index Numbers, System-GMM, Olley and Pakes (1996), Levisohn and Petrin (2003) and Ackerberg et al. (2015). Each of these estimators assumes different underlying properties for some inputs, which potentially affect the TFP measurements. We analyze the role of corporate taxation within a TFP catch-up model, providing new insights on the unexplored fiscal aspect of the UK productivity puzzle. Our findings are summarized as: (i) the Ackerberg et al. (2015) algorithm is the cutting-edge estimation technique with most plausible results, while GMM system (GMM-SYS) is the second best estimator; (ii) the global financial crisis in 2009 impacted negatively on TFP; (iii) corporate tax adversely affects TFP growth as it induces distortive effects on productivity enhancing investment; (iv) the adverse effect is found to be severe in the groups of R&D and exporting firms, suggesting that the distortive nature of corporate tax affects disproportionately the firms that are more financially constrained and more exposed to risk.

Consumer Substitution, Cost of Living Indices and Exchange Rate in Argentina - First Part: 1960-1995

This paper reviews the theory of the true cost of living index and its approximation through superlative index numbers, as well as the Vartia quantum theory of bias in index numbers as applied to substitution bias in consumer price indices. Estimated adjustments are proposed for the Argentine Consumer Price Index on account of consumer substitution, for the period 1960-1995. Annual substitution is shown to be a relatively minor proportion of annual inflation, but to accumulate a sizable effect over the years. The absolute effect varies with the absolute size of annual inflation. Substitution bias in the CPI is shown to have a strong effect on estimates of the historical inflation record and the PPP exchange rates in the 1990s.

An Index Number Formula Problem: The Aggregation of Broadly Comparable items

Index number theory informs us that if data on matched prices and quantities are available, a superlative index number formula is best to aggregate heterogeneous items, and a unit value index to aggregate homogeneous ones. The formulas can give very different results. Neglected is the practical case of broadly comparable items. This paper provides a formal analysis as to why such formulas differ and proposes a solution to this index number problem.

An Index Number Formula Problem: The Aggregation of Broadly Comparable Items

Index number theory informs us that if data on matched prices and quantities are available, a superlative index number formula is best to aggregate heterogeneous items, and a unit value index to aggregate homogeneous ones. The formulas can give very different results. Neglected is the practical case of broadly comparable items, for which price dispersion can be decomposed into a quality component, say due to product differentiation, and a component that is stochastic or due to price discrimination. This paper analyses why such formulas differ and proposes a solution to this index number problem. JEL Classification Numbers: C43, C81, E31, L11, L15.

Operational identification of the complete class of superlative index numbers: An application of Galois theory

We provide an operational identification of the complete class of superlative index numbers to track the exact aggregator functions of economic aggregation theory. If an index number is linearly homogeneous and a second order approximation in a formal manner that we define, we prove the index to be in the superlative index number class of nonparametric functions. Our definition is mathematically equivalent to Diewert's most general definition. But when operationalized in practice, our definition permits use of the full class, while Diewert's definition, in practice, spans only a strict subset of the general class. The relationship between the general class and that strict subset is a consequence of Galois theory. Only a very small number of elements of the general class have been found by Diewert's method, despite the fact that the general class contains an infinite number of functions. We illustrate our operational, general approach by proving for the first time that a particular family of nonparametric functions, including the Sato-Vartia index, is within the superlative index number class.

The Theory of Exact and Superlative Index Numbers Revisited

This paper proposes to clarify some important questions that are still open in thefield of index number theory. The main results are the following: (i) the so-called Quadratic Identity on which the superlative index numbers are based can be applied in more general cases than those traditionally considered; (ii) it is not only the Tornqvist index number that does not rely on separability restrictions, but also some other indicators of absolute or relative changes are not based on such restrictions; (iii) in practice, however, all the index numbers or indicators that are considered to be superlative in Diewert’s (1976) sense generally fail by construction to be really superlative; (iv )t hese hybrid index numbers may be far from providing the expected secondorder approximation to the true index and may be found beyond the Laspeyres-Paasche interval even in the homothetic case. In conclusion, it would be more appropriate to construct a range of alternative index numbers (including even those that are not superlative) rather than follow the common practice of searching for only one optimal formula.

Superlative index numbers: not all of them are super

This paper shows that just because superlative index numbers approximate each other to the second order does not necessarily imply that they are numerically similar. In fact, the spread between the largest and smallest superlative indexes sometimes even exceeds that between Paasche and Laspeyres indexes. This result has significant implications for the index number literature. It shows that the economic approach does not by itself solve the index number problem, since it does not tell us which superlative index should be used. It may be necessary to combine the economic and axiomatic approaches to arrive at an answer.

Luenberger Productivity Indicators: Aggregation Across Firms

Two methods of aggregating Luenberger productivity indicators across firms are considered. One method makes use of some rather implausible allocative efficiency assumptions. The second method, a superlative index number approach, relies on more palatable assumptions and is judged to be the more promising of the two methods.

The quadratic approximation lemma and decompositions of superlative indexes

It was shown in 1976 that a difference in a quadratic function of N variables evaluated at two points is exactly equal to the sum of the arithmetic average of the first order partial derivatives of the function evaluated at the two points times the differences in the independent variables. In the present paper, this result is generalized and the resulting generalized quadratic approximation lemma is used to establish all of the superlative index number formulae that were derived in Diewert [4]. In addition, some new exact decompositions of the percentage change in the Fisher and Walsh superlative indexes into N components are derived. Each component in this decomposition represents the contribution of a change in a single independent variable to the overall percentage change in the index. Finally, these components are given economic interpretations.

The measurement of inflation: Aggregation at the basic level

The advent of bar-code, retail scanner data provides an alternative data source for the compilation of consumer price indexes. This paper outlines the nature and merits of such data for this purpose. The data allows aggregation via superlative index number formulae at a very elementary level and provides superior coverage to conventional sources. However, it has been argued that aggregation bias may still arise if the unit values which feed into the superlative indexes are defined for product items, as opposed to product items for given outlets. We explore for six products the nature and extent of unit value bias by outlet type. For three products we found the level of aggregation did not matter. However, we also found it can matter and established the extent of its effects which were always the aggregation by model only, falling less than by model and outlet. This should help provide a basis for an understanding of the issues at stake in the use of scanner data for the compilation of consumer price indexes.

Measuring Effective Protection as a Superlative Index Number: An Application to European Agriculture

AbstractIn this study duality and the theory of superlative index numbers are used to obtain a theoretically consistent Effective Protection Rate (EPR) index that imposes no a priori restrictions on input‐output substitution possibilities. Within this framework, EPRs are estimated for European agriculture over the period 1973 to 1989. The empirical results demonstrate that maintaining zero input/output substitutability—a restriction embedded in all common protection measures—can lead to false results about the level of protection both across countries and time.

Constant-market-shares analysis and index number theory

This paper examines the constant market-shares (CMS) analysis of a country's export growth within the context of index number theory and is aimed at finding a satisfactory solution to the problems encountered by the traditional CMS decomposition procedures. The last ones perform the task of disentangling the variations of each accounting factor only partially, because they are based on linear approximations to non-linear functions. The residual “interaction” term left out in the current CMS analyses does not appear if a more flexible CMS decomposition based on the so-called superlative index numbers is used. Moreover, the discussion of the basic identities has clarified the difference in meaning and levels of the accounting components between the alternative versions of CMS analysis.

Flexible Consumer Demand Functions and Linear Estimation: Reply

We are grateful to Wohlgenant for pointing out that the approximation of utility by real income requires that the underlying (flexible) utility function be homothetic. Indeed Diewert's proof of correspondence between utility changes and real income changes (as measured by superlative index numbers) relies on homotheticity. We thus concur with Wohlgenant that the three functional forms proposed by us are best regarded as statistical demand functions integrable into an arbitrary neoclassical utility function only at some base point. However, it is not clear that this is the reason for implausible expenditure elasticities obtained in the TL model as we move from sample means. This functional form (and the other two used in the study) provide a flexible approximation to some true consumption-expenditure relationship. It is unlikely to give reasonable estimates of elasticities at expenditure levels which are less than half the mean or nearly twice as high. What is interesting is that elasticities for all the food grains decline as expenditure levels rise.

A general approach to intertemporal and interspatial productivity comparisons

The purpose of this paper is to introduce a general methodology for analysing the sources of intertemporal or interspatial differences in outputs and costs, general in the sense that our methodology allows the productivity analyst to ‘break out’ of the quadratic ‘straightjacket’ imposed by the class of superlative index number comparisons. Starting fromTaylor's series expansions about the two points to be compared, we develop a general growth accounting equation which can be approximated to any desired degree of accuracy, depending on the information available. The theoretical framework is applied to two recent examples of interspatial comparisons which use the Tornqvist superlative index. In the first example, we show that the biases in regional Canadian total manufacturing cost-efficiency comparisons which result from the use of this index are negligible. However, in the second example, it is shown that the Tornqvist index imparts a substantial bias in United States-Japan total domestic economy productivity comparisons. The index consistently overestimates the relative productivitylevel of the U.S. economy and misses the turning point, when the Japanese economy becomes more efficient, by two years.

Multilateral Comparisons of Output, Input, and Productivity Using Superlative Index Numbers

Early in this century economists began to give serious attention to making comparisons using index number techniques. There was extensive debate as to which index number formulas were the most appropriate for carrying out comparisons.1 The debate was extensive in no small part due to the lack of agreement as to criteria for preferring one formula over another. In recent decades there has been a resurgence of interest in index numbers, resulting from discoveries that the properties of index numbers can be directly related to the properties of the underlying aggregator functions that they represent. The underlying functions - production functions, utility functions, etc. - are the building blocks of economic theory, and the study of relationships between these functions and index number formulas has been referred to by Samuelson and Swamy (I974) as the economic theory of index numbers.2 A key development in the economic theory of index numbers has been the demonstration that numerous index number formulas can be explicitly derived from particular aggregator functions. This development provides a powerful new basis for selecting an index number procedure. Rather than starting the selection process with a number of plausible index number formulas, one can specify an aggregator function with desirable properties and derive the corresponding index number procedure. The resulting index is termed exact for that particular aggregator function. Diewert (I976) makes a strong case for limiting the consideration of aggregator functions to those which are flexible, i.e. those which can provide a second order approximation to an arbitrary aggregator function. He has termed index numbers that are exact for flexible aggregator functions 'superlative '. There are two superlative index numbers that are of particular interest - the Fisher Ideal index and the Tornqvist-Theil-translog index. Fisher (I 922) dubbed the following index Ideal since it best satisfied his several criteria for choosing among index numbers:

Post-War Productivity in Canadian Manufacturing

Post-War Productivity in Canadian Manufacturing

Superlative Index Numbers and Consistency in Aggregation

Superlative Index Numbers and Consistency in Aggregation