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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Törnqvist index ： ウィキペディア英語版 Törnqvist index In economics, the Törnqvist index is a price or quantity index. In practice ,Törnqvist index values are calculated for consecutive periods, then these are strung together, or "''chained''". Thus, the core calculation does not refer to a single base year. == Computation == The price index for some period is usually normalized to be 1 or 100, and that period is called "base period." A Törnqvist or Törnqvist-Theil price index is the weighted geometric average of the price relatives using arithmetic averages of the value shares in the two periods as weights.〔(Glossary ), OECD〕 The data used are prices and quantities in two time-periods, (t-1) and (t), for each of ''n'' goods which are indexed by ''i''. If we denote the price of item ''i'' at time t-1 by $p\_$, and, analogously, we define $q\_$ to be the quantity purchased of item ''i'' at time t, then, the Törnqvist price index $P\_t$ at time t can be calculated as follows:〔(“Törnqvist Index and other Log-change Index Numbers” ), Statistics New Zealand: Glossary of Common Terms.〕 :$\backslash frac^\backslash left(\backslash frac\}\backslash right)^\; \backslash left(\backslash frac\}\backslash left(p\_q\_\backslash right)\}\backslash right)\}$ The denominators in the exponent are the sums of total expenditure in each of the two periods. This can be expressed more compactly in vector notation. Let $p\_$ denote the vector of all prices at time t-1 and analogously define vectors $q\_$, $p\_t$, and $q\_t$. Then the above expression can be rewritten: : :$ln\; \backslash frac\; \backslash sum\_^\; \backslash left\; (\backslash frac\; \}\}\; +\; \backslash frac\; \}\; \backslash right)\; ln\backslash left\; (\backslash frac\}\; \backslash right)$ A Törnqvist quantity index can be calculated analogously using prices for weights. Quantity indexes are used in computing aggregate indexes for physical "capital" summarizing equipment and structures of different types into one time series. Swapping p's for q's and q's for p's gives an equation for a quantity index: :$\backslash frac^\backslash left(\backslash frac\}\backslash right)^\; \backslash left(\backslash frac\}\backslash left(p\_q\_\backslash right)\}\backslash right)\}$ If one needs matched quantity and price indexes they can be calculated directly from these equations, but it is more common to compute a price index by dividing total expenditure each period by the quantity index so the resulting indexes multiply out to total expenditure. This approach is called the ''indirect'' way of calculating a Törnqvist index,〔 and it generates numbers that are not exactly the same as a direct calculation. There is research on which method to use based partly on whether price changes or quantity changes are more volatile.〔Allen, Robert C.; W. Erwin Diewert. 1981. Direct versus Implicit Superlative Index Number Formulae. ''The Review of Economics and Statistics'', 63:3 (Aug., 1981), 430-435. ((on jstor ))〕 For multifactor productivity calculations, the indirect method is used. Törnqvist indexes are close to the figures given by the Fisher index.〔〔Diewert, W. E. 1978. "Superlative Index Numbers and Consistency in Aggregation", ''Econometrica'', 46(4): 883-900.〕〔Dumagan, Jesus Castanos. 2002. ("Comparing the superlative Törnqvist and Fisher ideal indexes" ), ''Economics Letters'' 76:2, 251-258.〕 The Fisher index is sometimes preferred in practice because it handles zero-quantities without special exceptions, whereas in the equations above a quantity of zero would make the Törnqvist index calculation break down. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Törnqvist index」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.