

Unfortunately the inequality which Levell quotes in proposition 1 is not quite correct as simple examples can show. The correct upper and lower bounds are given in two papers which I have just posted in the library. However, as I note there, the difference between upper and lower bounds can be quite large, so they do not convey much information. Moreover, while it may often be true that broader product categories produce a greater formula effect, it cannot be assumed always to be the case. It depends on the numerical details in each instance. The subsequent comments in that paragraph, relating to the clothing sector, seem to involve still further assumptions which I have not seen demonstrated anywhere. GJ
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 Original Message: Sent: 28062013 15:23 From: Andrew Baldwin Subject: Re: Peter Levell's paper
Thank you to Gareth for posting Peter Levell's paper in the library. The issue of lower level aggregation formula is only very briefly touched on in Jill's paper on uprating indexes, and quite rightly so, since there is little reason to believe that an upratings index need differ from a macroeconomic index in its lower level aggregation formulas.
Peter Levell's paper contains much of value. I liked the inequality showing that the difference between the Jevons and the Carli index is always greater than or equal to the variance in the price relatives. Also, he lays out very well one of the critical weaknesses of the Dutot formula: "If we were to double the base and current period price of one particular item (by for instance, measuring the price of a pair of gloves rather than a single glove), then the Dutot index would change, while the Jevons and Carli would be unaffected. This comes about because the level of the Dutot index depends on the value of base period prices relative to their mean." It was concern about this very property that led Statistics Canada to replace the Dutot formula with the Jevons formula with the updating to a 1992 basket for the January 1995 update. Table 3.2 shows both the Dutot and the Jevons formula satisfying all 14 axioms, although the passage just quoted shows very clearly that the Dutot formula fails the invariance to changes in units test which both the Carli and Jevons formula pass.
Mr. Levell seems somewhat too blasé for my taste about the Carli formula failing both the time reversal and the circularity test. He writes that "it is important to remember that it only really makes sense to talk about bias with respect to some target index. The expression in equation (5) only gives the bias of the chained Carli if we think that the unchained Carli is the target." But is this really so? The significance of the Dutot and Jevons formulas both passing the circularity tests is that if prices are the same in period t as in period 0, no matter how many links lie between them, the chain Dutot index or the chain Jevons index will always show no price change between periods 0 and t, as will their direct indexes. The chain Carli index by contrast, will not generally show no price change between periods 0 and t, and in all too many cases the chain Carli index will show a spectacular price increase, even as its direct working shows no price change. Under these conditions it seems like verbal quibbling to say that the chain Carli index is not upward biased. Wherever prices circle back to their original level, a wide variety of elementary aggregate formulas will show no price change besides the direct Carli formula. They include, but are not limited to, the direct harmonic mean, the direct Dutot, the direct CSWD, the direct Dutot, the direct Jevons, the chain Dutot and the chain Jevons formulas. When all these formulas will show no price change while the chain Carli may show a substantial price increase, is it really inappropriate to speak of an upward bias?
Mr. Levell also writes: "While the Carli's failure to satisfy time reversibility is indeed a problem, it is important to realise that the index numbers into which these elementary aggregates eventually feed (the RPI and CPI) are themselves not timereversible, and nor would they be even if the elementary aggregates were timereversible." This is true but surely doesn't mean we should not try to apply SATIRE formulas (formulas which satisfy the time reversal property) at the elementary aggregate level, where we can do so without violating the convention against revision of official consumer price indexes. Most countries use a chain Lowe formula to calculate official consumer price series at the basic aggregation level and up, because it is compatible with a nonrevision policy, even though the chain Lowe formula does not satisfy the time reversal policy. Nevertheless, the practice is not universal. Statistics Sweden calculates its official CPI as a chain Walsh index since 2005. The US has been publishing a chain Törnqvist index, the Chained Consumer Price Index for All Urban Consumers (CCPIU) since August 2002. Both formulas are SATIRE formulas, and both indexes have a 24month revision period.
Of course, it is an open question whether the benefits from eliminating the upper level substitution bias imposed by using a chain Lowe formula outweigh the costs of giving up the longstanding norevision policy. But surely there is no similar argument against eliminating the lower level bias imposed by using a nonSATIRE formula like the Carli formula by replacing it with something more suitable, like the Jevons formula. If the convention of a nonrevision policy prevents a national statistical institute from removing one source of bias, why should that prevent it from removing other sources of bias?
 Original Message: Sent: 26062013 08:00 From: Gareth Jones Subject: Re: Peter Levell's paper
I have put a copy of this paper in the library for ease of reference as it is referred to in Jill Leyland's paper on an uprating index.
I have some comments on that part of the paper which deals with maximisation of entropy and which equates to use of the Jevons formula (geometric mean).
In particular I refer to the last paragraph of section 6 of the paper.
This shows that the same considerations imply a CobbDouglas utility function to represent consumer preferences. In turn this is equivalent to assuming elasticities of substitution of exactly unity for all products in all time periods.
There is therefore quite a lot of information available to test these hypotheses which are all effectively equivalent.
In particular I refer to ONS' paper CPAC(12)15 + Annex A which was lodged in the library of this website on 7th June 2012. This is an empirical study of data on the alcohol sector.
It shows that elasticities of substitution vary considerably. Some are approximately unity while others are significantly higher or lower. Moreover there is no guarantee that those near unity would remain so over time. This effectively shows that the CobbDouglas utility function and assumptions equivalent to it are not valid.
ONS' paper also compares the performance of different elementary formulae relative to a target index using several different loss functions.
Under this analysis, both the Carli and Dutot formulae outperform the Jevons formula under all loss functions. Given that consumer substitution clearly takes place in the alcohol sector and that the Carli and Dutot formulae make no allowance for it, this is a remarkable finding. It implies that the form of consumer substitution implied by the Jevons formula is so far from reality that formulae representing no substitution do better in representing reality.
This point is confirmed by a paper of my own entitled "Implausibility of the Geometric Mean ......." lodged in the library on 9th March 2012. This shows that the Jevons formula does not represent substitution towards cheaper products but towards products whose rate of price rise is lower (even if the products are more expensive).
I think these papers effectively refute the Jevons formula, the CobbDouglas utility function and the maximisation of entropy.
GJ


