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Exact and Superlative Index Number
Abstract
The paper rationalizes certain functional forms for index numbers with functional forms for the underlying aggregator function. An aggregator functional form is said to be ‘flexible’ if it can provide a second order approximation to an arbitrary twice diffentiable linearly homogeneous function. An index number functional form is said to be ‘superlative’ if it is exact (i.e., consistent with) for a ‘flexible’ aggregator functional form. The paper shows that a certain family of index number formulae is exact for the ‘flexible’ quadratic mean of order r aggregator function, defined by Den and others. For r equals 2, the resulting quantity index is Irving Fisher's ideal index. The paper also utilizes the Malmquist quantity index in order to rationalize the Törnqvist-Theil quantity indexin the nonhomothetic case. Finally, the paper attempts to justify the Jorgenson-Griliches productivity measurement technique for the case of discrete (as opposed to continuous) data.
... As mentioned above, scanner data allow the calculation of any price index formula, including superlative indices (Diewert 1976), which are considered the best approximation of the COLI. Superlative indices (e.g., the Fisher index (Fisher 1922)) are unattainable in traditional data collection because there is no knowledge of current period consumption. ...
... where the parameters a ik satisfy a ik = a ki for any i and k, and r ≠ 0 . Diewert (1976) demonstrate that the utility function f r is a flexible functional form, i.e., it can approximate an arbitrary twice-differentiable, linearly homogeneous functional form to the second order. Let us define the quadratic mean of order r quantity index as follows: ...
... Since the quantity index Q r is exact for the functional form f r , following to Diewerts's terminology (see (Diewert 1976) and also earlier work (Fisher 1922)), Q r is a superlative price index for each r ≠ 0 . For each quantity index (5), we can define the corresponding implicit quadratic mean of order r price index as follows: ...
Scanner data are electronic transaction data that specify turnover and the number of items sold by barcodes, e.g., the Global Trade Article Number. These data are of particular value and interest to theorists and practitioners who wish to measure the Cost of Living Index or the Consumer Price Index, since their complete content makes it possible to compute any price index formula, including superlative indices or CES (Constant Elasticity Substitution) indices. Since the CES index requires the estimation of the elasticity of substitution, this paper focuses on verifying various methods of estimating this parameter based on scanner data. The paper considers both algebraic methods and methods based on the panel regression approach. The main achievement of the paper is the separation of the main factors that affect the estimated value of the elasticity of substitution, i.e., the type of data filter used and the level of data aggregation. The paper also verifies how the elasticity of substitution estimates affect the differences between the values of the CES indices based on these estimates.
... Official monetary aggregates reported by most central banks utilize simple sum aggregate, i.e., the simple-unweighted-sum of monetary assets. These aggregates have long been criticized as inaccurate measures of both the monetary serves flow (Barnett, 2000a(Barnett, , 2000bDiewert, 1976;Binner et al., 2018) and the money stock (Kelly, 2009). However, the official simple sum aggregates do express the market value of the current portfolio of monetary aggregates. ...
... Divisia M3 measures the flow of monetary services from money holdings by weighting each monetary asset based on expenditure shares, (Barnett, 2000a(Barnett, , 2000bDiewert, 1976). Divisia aggregates, rooted in microeconomic aggregation theory, offer an advanced approach that considers the level of monetary services provided by each asset. ...
This study examines whether or not Swiss monetary aggregates enhance inflation forecasting in Switzerland during the out-of-sample period, December 2008 to November 2019. We use a state-of-the-art multi-recurrent neural network endowed with a sluggish state-based memory to approximate a non-linear auto-regressive moving average model. Conventional monetary aggregates have been shown to lose dynamic information, potentially explaining why many deem traditional measures of the money supply to have minimal economic relevance. Our findings suggest that when conventional monetary aggregates, Divisia money measures, and a short-term interest rate are combined, forecasts of Swiss inflation over the 12, 24 and 36-month forecasting horizons are significantly improved compared to a model that excludes a measure of the money supply.
... A solu9ao para a questao da substitutibilidade dos ativos surgiu no final da decada de 70 e se difundiu na teoria monetaria com os trabalhos de Diewert (1976), Bamett (1980Bamett ( , 1982Bamett ( e 1984 e Bamett, Offenbacher e Spindt (1984). A relevancia da agrega9ao ponderada pode ser buscada na questao levantada por Bamett (1982) Esse fato ilustra bem a deficiencia da agregado monetaria feita pela simples soma de ativos financeiros heterogeneos. ...
This paper investigates causality among money and the variables inflation rate and level of income in the period that spans from 1980 until the Real Plan edition, when Brazilian economy faced a great economic instability. It was used, as money concept, monetary aggregates weighted by the Divisia index as well as obtained by simple sum of assets. It was accomplished the causality test for cointegrated variables proposed by Engle and Granger (1987). The results indicate that the causality runs simultaneously from money, in its several definitions, to inflation and to level of income. This can be explained by the passivity that characterized the monetary policy during the period. In the relationship between money and price level, the statistically more significant result was observed in the causality prices-money and it was evidenced when money was widely defined.
... The productivity index in (6) is very similar to the one in (4), with the difference that the weights are output distance elasticities and negative input distance elasticities, respectively. Using Diewert (1976) 's Quadratic Identity Lemma, 7 it is easy to show that ...
We examine an extension of the latent class stochastic frontier model (LCSFM) to productivity estimation and the decomposition of productivity change into technical change, output-oriented technical efficiency change, and scale change. We base our productivity estimation on a Multi-class Grifell-Tatjé, Lovell & Orea Malmquist (GLOM) index. An advantage of this new productivity index is to account for classes' posterior probabilities to derive individual farm parameters. In addition, we extend our analysis to estimate a metafrontier GLOM productivity index to explore potentialities when all firms use the best available technologies. An empirical application to a sample of French sheep and goat farms observed between 2002 and 2021 confirms the necessity to account for technological heterogeneity when measuring productivity change. Among the two classes of farms identified by the LCSFM, the intensive class experiences TFP gains, while the extensive class sees its TFP worsening. However, the gap between intensive and extensive technologies seems to reduce over time. Finally, the multi-class GLOM reveals technical change as the primary driver of productivity for French goat and sheep farms.
... It constitutes a temporal modified trans-logarithmic production function. TFP indices have gained significant recognition through Divisia index after approximating Tornqvist-Theil discretion, involving consistent aggregation and linear homogenous trans-logarithmic production function (Diewert 1976;1978). ...
This paper proposes an index number method called Asanya-Jibasen-Mbaga index, the method satisfied Laspeyre’s and Paasche’s bounding test which is unbiased compared to existing methods. The proposed index methods used expenditure/quantity and rank as weights and it satisfied all the tests of consistency of elementary index formulae. The proposed method was found to approximate the Fisher’s Ideal Price index (PF). The proposed method is an improvement on Jibasen –Gazali-Asanya rank price index. Numerical analyses conducted at low and high levels of aggregation revealed that the Jibasen-Gazali-Asanya rank price Index is biased and also failed Laspeyre’s and Paasche's bounding test. Thus, based on the results of the findings, Asanya-Jibasen-Mbaga index method is recommended as the appropriate functional form of the elementary index method among other existing elementary index methods
This paper studies monetary policy and welfare in a sticky wage New Keynesian model with heterogeneous firms and endogenously variable markups. We show that stabilizing nominal wages is optimal only when product creation is based on an instantaneous zero‐profit condition and when the aggregate markup is constant. A constant markup requires strong selection effects generated by a Pareto firm‐level productivity distribution. When product creation is based on a dynamic zero‐profit condition optimal monetary policy accounts for the distribution of firms and the welfare loss from stabilizing nominal wages is between and 0.2% of steady‐state consumption.