Inflation measurement in the presence of stockpiling and consumption smoothing

von Auer, Ludwig Article — Published Version Inflation measurement in the presence of stockpiling and consumption smoothing Review of Income and Wealth Provided in Cooperation with: John Wiley & Sons Suggested Citation: von Auer, Ludwig (2024) : Inflation measurement in the presence of stockpiling and consumption smoothing, Review of Income and Wealth, ISSN 1475-4991, Wiley Periodicals, Inc., Hoboken, NJ, Vol. 71, Iss. 1, https://doi.org/10.1111/roiw.12682 This Version is available at: https://hdl.handle.net/10419/313732 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc-nd/4.0/ bs_bs_banner Review of Income and Wealth Series 71, Number 1, February 2025; e12682 DOI: 10.1111/roiw.12682 INFLATION MEASUREMENT IN THE PRESENCE OF STOCKPILING AND CONSUMPTION SMOOTHING BY LUDWIG VON AUER* Universität Trier A chained price index is said to suffer from chain drift bias if it indicates an overall price change, even though the prices and quantities in the current period have reverted back to their levels of the base period. The empirical relevance of this bias is well documented in studies that apply sub-annual chaining to scanner data. There it is shown that stockpiling can lead to downward chain drift bias. The present paper draws attention to the fact that smoothing consumption causes substantial upward chain drift. In addition, this study introduces a stochastic simulation approach that is consistent with both, stockpiling and consumption smoothing. A “stress test” is conducted that examines whether rolling window variants of multilateral indices (GEKS, TPD, and GK) effectively curtail the chain drift problem. JEL Codes: C43, E31, E52 Keywords: bias, chain drift, multilateral, price index, simulation 1. INTRODUCTION The conventional approach to compute the price change between a base period 0 and a comparison period Tuses only the prices and quantities of these two periods. Such indexformulasareknown asbilateral price indices (or directpriceindices). In the following, they are denoted by PT∕0. As an alternative to the bilateral index, PT∕0, one may compute the price change between periods 0 and Tfrom a sequence of overlapping adjacent bilateral price indices. The elements of this sequence can be linked by multiplication: P1∕0P2∕1…PT∕T−1. Such products are known as chain indices and its factors as chain links. When the item universe remains constant over time and all prices and quantities in the current period Trevert back to their levels in the base period 0, a price index comparing the current period with the base period should indicate that no price change occurred. All reasonable bilateral price indices, PT∕0,satisfy this requirement. However, chained bilateral price indices usually violate it. In the present paper, this violation is denoted as chain drift bias. It is usually attributed Note: A former version of this study was presented at the “Ottawa Group Meeting, 2019” in Rio de Janeiro and at the conference “Messung der Preise, 2019” in Erfurt. Helpful comments from participants are gratefully acknowledged. The paper also benefitted from valuable suggestions by Sebastian Weinand as well as by three anonymous referees and the editor. *Correspondence to: Ludwig von Auer, Universität Trier, Fachbereich IV–VWL, Universitätsring 15, 54296 Trier, Germany ([email protected]). © 2024 The Authors. Review of Income and Wealth published by John Wiley & Sons Ltd on behalf of International Association for Research in Income and Wealth. This is an open access article under the terms of the Creative Commons Attribution-NonCommercialNoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. 1of20 Review of Income and Wealth, Series 71, Number 1, February 2025 to sales triggering nonstandard substitution behavior of consumers. Feenstra & Shapiro (2003, p. 135) examine scanner data on canned tuna and compile a weekly chained Törnqvist index that exhibits upward chain drift caused by sales, while de Haan (2008, p. 19), studying scanner data on detergents, finds that sales lead to downward chain drift, even though both studies apply the same index formula and frequency of chaining. These contradictory results suggest analyzing the causes of chain drift in a more systematic way. The present paper puts its focus on chained bilateral indices that include the quantities of both, the base and the current period (e.g. the indices of Törnqvist, Fisher, Marshall–Edgeworth, and Walsh). It is shown that the chain drift of such chained bilateral price indices is caused by quantity and price changes that are not perfectly synchronized in time.1Such asynchronous price and quantity changes can arise from different forms of intertemporal optimization behavior of consumers. A well-known example of such consumer behavior is stockpiling. During a sale, the consumers increase both their consumption and their stocks. Therefore, the purchases “overshoot” consumption. As soon as the price returns to normal, the purchases overshoot consumption in the downward direction because the consumers first use up their extra stocks. When in the subsequent periods the price remains on its normal level, purchased quantities gradually align with consumed quantities. For simplicity, these consequences of stockpiling are denoted here as overshooting quantity reactions to price changes.2Also temporary price spikes trigger overshooting quantity reactions. There are other forms of intertemporal optimization behavior leading to asynchronous price and quantity changes. Probably the most important ones are delayed quantity adjustments to price changes. Often, this smoothing is caused by search and adjustment costs. Even so the price of a consumer’s favorite product may have permanently increased or the price of a competing product may have permanently fallen, the consumer may show no or only a moderate immediate change in her usual purchasing behavior. Only at some later point of time, after acquiring information about the qualitative features or the handling of alternative products, she may partly or completely switch over to such alternatives. Other reasons for delayed quantity reactions are consumption habits caused by harmful addictions (e.g. nicotine) or by past investments in increased enjoyment from consumed goods (e.g. ability to cook tasty and healthy food).3For simplicity, all delayed quantity adjustments are denoted here as sticky quantity reactions to price changes. Several studies of scanner data (e.g. de Haan, 2008, p. 18; de Haan & van der Grient, 2011, p. 43) show that overshooting quantities cause downward chain drift. 1Hill (2006, pp. 314-315) reaches a similar conclusion for the chained Laspeyres index and the chained Paasche index. 2Hayashi (1985) identifies a similar effect for durables: “A higher level of expenditure means a larger stock of consumption, which will depress expenditure in the next period if households behave in a way to smooth out consumption (rather than expenditure) over time (p. 1092).” 3Muellbauer (1988) emphasizes the relevance of habit formation for macroeconomic models. His empirical evidence appears to favor myopic habits, that is, consumers who are not aware of the impact of current consumption decisions on future consumption decisions. © 2024 The Authors. Review of Income and Wealth published by John Wiley & Sons Ltd on behalf of International Association for Research in Income and Wealth. 2of20 Review of Income and Wealth, Series 71, Number 1, February 2025 The chain drift arising from sticky quantities, however, went largely unnoticed.4 Therefore, the present paper’s first contribution is to show that sticky quantities lead to upward chain drift. As a solution to the chain drift problem, Ivancic et al. (2011) advocate a rolling window variant of the Gini–Éltet˝ o–Köves–Szulc (GEKS) approach. The GEKS index is free of chain drift. However, its rolling window variant (R-GEKS) involves a mechanism that links the current window to past windows. This special form of chaining is usually denoted as splicing. It cannot be ruled out that splicing generates chain drift. By now, several variants of such R-GEKS indices have been developed.5 Besides R-GEKS indices, many other multilateral approaches have been proposed to overcome the chain drift problem. These include rolling window variants of the time-product dummy method (R-TPD) and of the Geary–Khamis approach (R-GK).6Again, these methods require some form of splicing. Consequently, they might also suffer from chain drift bias. How can one investigate this suspicion? A meaningful examination requires an unassailable benchmark such that the deviation of an index number from that benchmark represents chain drift bias. This benchmark directly follows from the definition of chain drift: when all prices and quantities revert back to their former levels, a price index should indicate that no price change occurred. Unfortunately, the prices and quantities of real world data (e.g. scanner data) never return to their original levels. Therefore, an analysis of chain drift that uses real world data must do without this natural benchmark. By contrast, in a simulation approach the researcher can generate price– quantity scenarios where all prices and quantities return to their original levels. However, the simulation must ensure that the price–quantity scenarios reflect features of real world purchasing behavior. This requires price scenarios with sales and ordinary price changes as well as quantity scenarios that capture overshooting and sticky quantities arising from consumer behavior such as stockpiling and smoothing of consumption. Using such a simulation approach, this study’s second contribution is a quantitative analysis of the chain drift bias of the R-GEKS, R-TPD, and R-GK approaches. It is shown that these index methods reduce chain drift bias, but they cannot eliminate it. Furthermore, some variants are more effective than others. As a first step, Section 2introduces the notion of chain drift and explains why overshooting quantities and sticky quantities cause different directions of chain drift. Section 3briefly discusses the implications of chain drift for price measurement relying on scanner data and explains why a simulation approach is appropriate for a systematic examination of the various index methods’ resilience to chain drift bias. Section 4presents a simulation-based “stress test” that examines whether the 4A notable exception is Triplett (2003, p. 152) who points out that storage, search, and information cost may generate measurement problems for scanner data price indices. 5For compact surveys, see, for example, Diewert & Fox (2022, pp. 357-359), Fox et al. (2022, pp. 9-12), Van Loon & Roels (2018, pp. 7-8), or Online Appendix Bof the present paper. 6For recent surveys of the various methods see, for example, Chessa et al. (2017), Chessa (2019), de Haan & Krsinich (2014), or Diewert & Fox (2022). © 2024 The Authors. Review of Income and Wealth published by John Wiley & Sons Ltd on behalf of International Association for Research in Income and Wealth. 3of20 Review of Income and Wealth, Series 71, Number 1, February 2025 R-GEKS, R-TPD, and R-GK methods curtail chain drift and whether some variants are better suited than others. Section 5concludes. 2. CHAIN DRIFT AND ITS SOURCES Let the integers i=1,…,Nrepresent the Nitems of an economy. All items are available during the base period (t=0) and the comparison period (t=T)and also during all intermediate periods (0 <t<T). The period tvector of prices is (pt 1,…,pt N)and the corresponding vector of quantities is (xt 1,…,xt N). It is customary to interpret a bilateral price index, P, as a mapping of the N-dimensional vectors (p0 1,…,p0 N),(x0 1,…,x0 N),(pT 1,…,pT N),and(xT 1,…,xT N)into a single positive number, PT∕0, that measures the “overall price change” between periods 0 and T. All bilateral price indices considered in this study are listed in the Online Appendix A. Consider some sequence of time periods, t=0,…,T. Definition 1. A chain index that compares period Tto the base period 0 is defined by (1)  PT∕0=P1∕0P2∕1…PT−1∕T−2PT∕T−1. When the prices and quantities in period Treturn to their levels of period 0, the chain index,  PT∕0, should give the same index number as the bilateral index, PT∕0, namely 1 (e.g. Ivancic et al., 2011, p. 26). Accordingly, in this specific price–quantity scenario, the deviation from unity is an appropriate measure of the extent of chain drift bias (e.g. Ribe, 2012, p. 3; Diewert, 2022; Diewert & Fox, 2022, p. 557). This interpretation of chain drift bias can be formalized in the following way: Definition 2. The chain drift test of the bilateral price index, Pt∕t−1, postulates that (2) P1∕0P2∕1…PT−1∕T−2P0∕T−1=1. Condition (2) can be found in Walsh (1901, p. 401).7Note that for T=1 the chain drift test simplifies to the time reversal test: P1∕0=1∕P0∕1.8 7Walsh later calls condition (2) the circularity test (see Diewert, 1993, p. 40). In the current price index literature, this label is reserved for the condition P2∕0=P2∕1P1∕0. This condition is stricter than the alternative formalization of chain drift because it does not prescribe that the prices and quantities of periods 0 and 2 coincide. In the context of spatial price comparisons, circularity is usually denoted as transitivity. Diewert (1993, p. 40) coins the chain drift test as the “multiperiod identity test”. Note, however, that the chain drift (or multiperiod identity) test considers the case where the quantities reverse to their base period values, while in the identity test the evolution of the quantities is completely irrelevant. In fact, a chain drift test without the quantity reversal postulate is considered in de Haan (2008, p. 10). He calls it the “invariance to price bouncing test”. 8An anonymous referee pointed out that for bilateral price indices that satisfy the time reversal test, one gets PT∕T−1=P0∕T−1=1∕PT−1∕0and condition (2) becomes (P1∕0P2∕1…PT−1∕T−2)∕(PT−1∕0)= 1. According to this alternative formalization, a bilateral price index suffers from chain drift when the ratio of the chained index and the corresponding bilateral index,  PT−1∕0∕PT−1∕0, deviates from unity. Several writers use this alternative formalization of chain drift (e.g. Forsyth & Fowler, 1981, p. 234; Frisch, 1936, p. 8; Hill, 2006, pp. 14-15; Lent, 2000, p. 314; Persons, 1921, p. 109; Persons, 1928, p. 101). © 2024 The Authors. Review of Income and Wealth published by John Wiley & Sons Ltd on behalf of International Association for Research in Income and Wealth. 4of20 Review of Income and Wealth, Series 71, Number 1, February 2025 Figure 1. Synchronous Quantity Reactions to Price Changes (Scenario 1) In standard microeconomic consumer theory, consumers instantaneously adjust their current purchases to the current prices. The usual assumption is that consumers substitute away from products that have become relatively more expensive. The larger the price elasticity of demand, the more pronounced are these quantity reactions. Figure 1translates this standard theory into a highly stylized example. The figure depicts the prices and quantities of some item during nine consecutive periods, t=0,…,8. In the depicted Scenario 1, the prices of the item can take only the values “low”, “normal”, and “high”, and the quantities sold can be “small”, “normal”, or “large”. Price changes are marked by the weight icons. The price starts at a normal level, drops in period 1 to the lower level, stays there for another period, returns to normal in period 3, and stays there also during period 4. In period 5, the price increases to high, stays there for another period, before it drops back to normal during period 7, and remains there during period 8. The quantities purchased move exactly inversely to the prices; that is, the consumers’ quantity reactions are perfectly synchronous to the price changes. As soon as the price returns to normal, the quantity also returns to normal. In times of constant prices also the quantities remain constant. When the price elasticity of demand is less than unity, expenditure shares and prices are positively correlated. Many price indices are expenditure-weighted averages of intertemporal price ratios (pT i∕p0 i) where the impact of the two periods 0 and Ton the expenditure weights is symmetric (e.g. Törnqvist, Walsh, Marshall–Edgeworth, Theil, Sato–Vartia).9Due to this symmetric impact, Scenario 1 generates weights such that the price decline between periods 0 and 1 (its weight is related to the expenditures during periods 0 and 1) and the price increase between periods 2 and 3 (its 9The Fisher index and the generalized unit value (GUV) indices of Banerjee, Davies, and Lehr (von Auer, 2014, pp. 848–852) exhibit a similar type of “intertemporal symmetry”. A more complete classification of “symmetric bilateral price indices” is provided by von Auer & Shumskikh (2024, sect. 2). © 2024 The Authors. Review of Income and Wealth published by John Wiley & Sons Ltd on behalf of International Association for Research in Income and Wealth. 5of20 Review of Income and Wealth, Series 71, Number 1, February 2025 Figure 2. Downward Chain Drift Bias from Overshooting Quantity Reactions Caused by Sales or Price Spikes (Scenario 2) weight is related to the expenditures during periods 2 and 3) exactly offset each other. Graphically, this balanced weighting is indicated by the equal-sized weight icons located at the price decline between periods 0 and 1 and at the price increase between periods 2 and 3. The same is true for the price increase between periods 4 and 5 and the price decline between periods 6 and 7. As a consequence of this balanced weighting of price increases and price declines, symmetrically weighted indices are immune to chain drift from scenarios like Scenario 1 and one gets P1∕0⋅P2∕1⋅…⋅P8∕7=1.10 Scenario 1 corresponds to standard consumer theory as presented in introductory microeconomics textbooks. Realworld consumer behavior, however, is more complex. One important aspect ignored by standard consumer theory is stockpiling. For example, sales usually lead to increased purchases, part of which are stored. If in the next period the price returns to its normal level, the purchased quantity falls below its normal level, because consumers first use up their extra stock. Only after the extra stock is depleted, the purchased quantity returns to its normal level. In the following, such a scenario is denoted as an overshooting quantity response to sales.11 During the sales period, acquisitions exceed consumption (i.e. stocksincrease),while right after the sales period, consumption exceeds acquisitions (stocks decrease). Periods 0–3 of Scenario 2 (see Figure 2) depict this case in a highly stylized form. In such a scenario, symmetrically weighted indices exhibit chain drift. As a result of the overshooting quantity response, the weight attached to the price reduction between periods 0 and 1 is larger than the weight attached to the price increase 10Chain drift arises, however, for the Laspeyres and Paasche index. This issue is addressed in various studies including Forsyth & Fowler (1981, pp. 234-235), Szulc (1983, pp. 540-541), and Hill (2006, pp. 314-315). 11This type of scenario is described, for example, in Ivancic et al. (2009, p. 4), de Haan & van der Grient (2011, p. 39), and Ribe (2012,p.3). © 2024 The Authors. Review of Income and Wealth published by John Wiley & Sons Ltd on behalf of International Association for Research in Income and Wealth. 6of20 Review of Income and Wealth, Series 71, Number 1, February 2025 between periods 1 and 2. Therefore, the price index level of period 2 is below that of period 0. At the end of period 2, the customers’ inventory is back to its normal level. Therefore, the purchases in period 3 increase to their normal level, even though the price is constant. In the literature, there is a rather broad consensus that every valid bilateral index formula must satisfy the identity test. This test says that, in the absence of any price changes, the price index is unity, regardless of any quantity changes. All price indices listed in the Online Appendix Asatisfy the identity test. Therefore, the price index level of period 3 remains on the level of period 2 and, thus, below that of period 0. In other words, downward chain drift arises. Periods 0–3 of Scenario 2 describe a storable item that consumers keep in stock. In times of unusually low prices (sales) the consumers add to their ordinary stock some extra stock and they deplete that extra stock when the price reverts to its normal level. This is the standard narrative of stockpiling. However, stockpiling is relevant not only in times of sales but also in times of price spikes. During such spikes, consumers can plunder their ordinary stock and restock it as soon as the price returns to normal. Again, overshooting quantity reactions arise. This is depicted by periods 4–8 of Scenario 2. The price increases in period 5. Many consumers switch to using up their stocks. This leads to a negative quantity reaction that is larger than for items that cannot be stored. In period 6, the price returns to its normal level. This gives the consumers the opportunity to refresh their inventories. At the same time, they return to their normal consumption. Therefore, the total quantity purchased exceeds the normal quantity. As a result of these overshooting quantity responses, the weight attached to the price increase between periods 4 and 5 is smaller than that attached to the price reduction between periods 5 and 6. This leads again to downward chain drift. At the end of period 6, the stock is back to its standard level, such that the purchases in period 7 return to their normal level even though no price change occurs between periods 6 and 7. In sum, overshooting quantities triggered by sales or by price spikes work in the same direction. Both generate downward chain drift. Overshooting quantities, however, are only one driver of chain drift. Another driver operates in the opposite direction; that is, it causes upward chain drift. In the field of industrial organization there is extensive literature on search and adjustment costs and their implications for markets.12 Also in the field of price measurement it is well known that search and adjustment costs are relevant in realworld consumption decisions and that they create problems for price measurement purposes (e.g. Reinsdorf, 1994, p. 137; Triplett, 2003, p. 152). Such costs can delay the consumers’ substitution behavior, such that part of the quantity response or the complete quantity response happens in a later period than the underlying price change. This is particularly true when the length of a period is relatively short (e.g. 1 week or 1 month). Another cause of delayed quantity responses are harmful or beneficial consumption habits that the consumers have developed over time. Delayed quantity responses are denoted here as sticky quantities. 12A survey of this literature is Fisher Ellison (2016). © 2024 The Authors. Review of Income and Wealth published by John Wiley & Sons Ltd on behalf of International Association for Research in Income and Wealth. 7of20 Review of Income and Wealth, Series 71, Number 1, February 2025 Figure 3. Upward Chain Drift Bias Caused by Sticky Quantity Reactions to Price Changes (Scenario 3) Figure 3illustrates the consequences of sticky quantities. In the depicted Scenario 3, demand is completely price inelastic in the short run but price elastic in the long run. More specifically, the complete quantity reaction to each price change is delayed by one period. In period 1 the price drops, while the quantity remains unchanged. Therefore, the observed expenditure during period 1 is smaller than it would be with the usual quantity reaction of an elastic demand. As a consequence, also the weight attached to the price decline is smaller than it would be with an elastic demand. Graphically, this diminished weight is indicated by the small weight icon at the price decline between periods 0 and 1. The complete quantity reaction to the reduced price in period 1 occurs one period delayed, that is, in period 2. The price in period 2 remains on the level of period 1. All popular price indices satisfy the identity test. Therefore, their price index level does not change between periods 1 and 2. Because demand is inelastic in the short run, the price increase between periods 2 and 3 occurs without a quantity reduction. Therefore, the weight attached to this price increase is larger than it would be with the usual quantity reduction. The large weight icon at the transition from periods 2 to 3 highlights this inflated weight. The price increase between periods 4 and 5 is analogous to that between periods 2 and 3. Again, the price increase receives an inflated weight. The price decline between periods 6 and 7 shows the same pattern as that between periods 0 and 1. This price decline receives a diminished weight. Overall, this unbalanced weighting of price declines and price increases leads to upward chain drift. Theoretically, one can think of quantity reactions that are antedated by one period. This could be regarded as a “negative delay”. The weighting effects are exactly opposite to those of Scenario 3. Price increases receive a diminished weight, whereas price declines receive an inflated weight. As a result, downward chain drift would arise. In reality, such anticipating consumption behavior is unlikely unless stockpiling is involved. Then stocks are depleted in anticipation of the sale. This tends to aggravate the overshooting quantities effect and the resulting downward chain drift. Anticipated price spikes have the same effect. © 2024 The Authors. Review of Income and Wealth published by John Wiley & Sons Ltd on behalf of International Association for Research in Income and Wealth. 8of20 Review of Income and Wealth, Series 71, Number 1, February 2025 TABLE 1 CHAIN DRIFT OF SEVERAL BILATERAL PRICE INDICES WITH SYMMETRIC EXPENDITURE WEIGHTS (IN %) Overshooting Sticky Hybrid Törnqvist −8.51 9.63 4.90 Marshall–Edgeworth −7.96 9.13 4.69 Walsh −8.68 9.46 4.86 Theil −8.60 9.48 4.86 Sato–Vartia −8.60 9.48 4.86 TABLE 2 CHAIN DRIFT OF SOME ADDITIONAL BILATERAL PRICE INDICES (IN %) Overshooting Sticky Hybrid Fisher −7.95 9.12 4.68 Banerjee −8.60 9.48 4.86 Davies −7.96 9.13 4.69 Lehr −9.19 9.60 4.96 large downward chain drift that varies between the indices. Also the upward chain drift values arising from sticky quantities are similar to those listed in Table 1. The tables show that chained bilateral indices generate the expected results. Overshooting quantities lead to considerable downward chain drift, while sticky quantities generate considerable upward chain drift. Overall, the bilateral price indices of Marshall–Edgeworth, Fisher, and Davies slightly outperform the other ones. This is true for overshooting quantities as well as for sticky quantities. GEKS indices are transitive and, thus, immune to chain drift bias arising from overshooting quantities. Therefore, R-GEKS indices have been proposed as a remedy for chain drift bias. However, these indices involve some form of splicing which, in turn, can cause chain drift bias. The performance of the various R-GEKS indices may depend on the choice of the splicing method and the window length (e.g. Melser, 2018, p. 517). Therefore, Table 3presents the results for various splicing methods and for window lengths of 4, 8, 12, and 24 periods. These results are derived from R-GEKS indices that use the Törnqvist index as their bilateral base index. The index numbers in the first two rows of Table 3show that the direct mean splice and the mean splice produce virtually the same R-GEKS index numbers. Overall, the numbers reinforce the arguments of Diewert & Fox (2022, pp. 560-561) and de Haan (2015, pp. 25-26) in favor of a more “balanced” splicing approach than the movement splice or window splice. With a sufficiently large window, the R-GEKS approach in conjunction with the direct mean splice, mean splice, or half splice effectively curtails chain drift bias arising from overshooting quantities (upper left part of Table 3). The half splice with a window length of 12 months performs best. The upper middle part of Table 3reveals that the previous findings carry over to the scenario of sticky quantities. In the hybrid scenario, three quarters of the households are smoothing households. The resulting R-GEKS index numbers are listed in the upper right part of © 2024 The Authors. Review of Income and Wealth published by John Wiley & Sons Ltd on behalf of International Association for Research in Income and Wealth. 15 of 20 Review of Income and Wealth, Series 71, Number 1, February 2025 TABLE 3 CHAIN DRIFT BIAS OF R-GEKS, R-TPD, AND R-GK INDICES (IN %) FOR DIFFERENT SCENARIOS,SPLICING METHODS,AND WINDOW LENGTHS (4,8,12,AND 24 MONTHS) Overshooting Sticky Hybrid Window length 4 8 12 24 4 8 12 24 4 8 12 24 R-GEKS Direct mean −2.41 −0.60 −0.35 −0.14 3.49 0.99 0.61 0.25 2.03 0.60 0.38 0.16 Mean −2.41 −0.60 −0.34 −0.13 3.49 0.99 0.59 0.24 2.03 0.60 0.37 0.15 Movement −2.56 −1.28 −0.80 −0.33 3.38 1.47 0.97 0.41 1.90 0.79 0.54 0.23 Half −2.11 −0.23 −0.01 −0.15 3.72 0.59 0.20 0.28 2.27 0.39 0.15 0.18 Window −2.56 −1.28 −0.78 −0.32 3.38 1.47 0.96 0.39 1.90 0.79 0.53 0.22 R-TPD Direct mean −1.65 −0.21 −0.01 0.22 3.14 0.65 0.33 −0.03 2.00 0.44 0.25 0.03 Mean −2.44 −0.60 −0.33 −0.12 3.33 0.98 0.58 0.23 1.96 0.60 0.36 0.14 Movement −0.16 0.09 0.46 0.68 2.70 0.60 0.17 −0.29 1.97 0.47 0.24 −0.06 Half −2.21 −0.23 −0.01 −0.14 3.45 0.61 0.22 0.27 2.16 0.40 0.16 0.17 Window −4.91 −2.62 −2.01 −1.29 3.86 2.30 1.71 1.04 1.76 1.10 0.81 0.47 R-GK Direct mean −4.03 −1.34 −1.24 −1.22 1.23 −0.45 −0.91 −1.46 −0.04 −0.67 −0.99 −1.41 Mean −2.67 −0.64 −0.35 −0.13 3.63 1.03 0.60 0.23 2.13 0.62 0.37 0.14 Movement −6.88 −4.18 −4.00 −3.35 −3.48 −3.58 −4.17 −4.23 −4.34 −3.74 −4.13 −4.02 Half −2.30 −0.22 0.03 −0.16 3.79 0.60 0.19 0.27 2.38 0.40 0.14 0.16 Window 1.36 1.46 2.35 2.75 11.12 6.96 6.46 5.26 8.77 5.62 5.46 4.65 © 2024 The Authors. Review of Income and Wealth published by John Wiley & Sons Ltd on behalf of International Association for Research in Income and Wealth. 16 of 20 Review of Income and Wealth, Series 71, Number 1, February 2025 Table 3. The chain drift bias caused by the smoothing households seems to dominate the chain drift bias caused by the stockpiling households. However, this effect is driven by the parameters that determine the households’ desire for stockpiling and smoothing as well as by the share of stockpiling households. When that share or the desire for stockpiling is sufficiently increased or the desire for smoothing is sufficiently reduced, the dominance would be reversed (not shown in the table). The other qualitative results are not affected by such changes. The direct mean splice, mean splice, and half splice remain the least biased options when applied with a sufficiently large window length. R-GEKS indices are not the only approach to curb the chain drift problem. Among the alternatives are the R-TPD and the R-GK approaches. Therefore, the same stress test with the same splicing options and window lengths has been conducted for the R-TPD and the R-GK approaches. Table 3also presents these results. For the R-GK approach, the choice of window length and splicing method causes more variation than for the R-GEKS and the R-TPD approaches.27 When the window splice is avoided, the R-TPD approach slightly outperforms the R-GEKS approach which, in turn, outperforms the R-GK approach. When the mean splice or the half splice is applied, the choice between R-GEKS, R-TPD, and R-GK is of minor relevance.28 It should be kept in mind that in the applied simulation no item attrition occurs. With item attrition, large windows may generate assignment or assortment bias (e.g. von Auer, 2017, pp. 83–84). In a context of item attrition, Melser & Webster (2021, pp. 777-783) identify in their own simulations life-cycle pricing and, in particular, run-out sales as an important driver of chain drift. The simulation framework developed in the present study could be modified and elaborated to gain further insights into the issue of item attrition and its effect on price measurement. To this end, it would be useful to identify the basic structure of product turnover in realworld scanner data and to transfer this structure to the simulations. Such a simulation may also address another important issue. To mitigate chain drift bias, one may try to aggregate the weekly prices into monthly or even quarterly unit values (say, e.g. Diewert, 2007, p. 3). In the simulation one could systematically study the effects of such a strategy. 5. CONCLUDING REMARKS Sales and the associated stockpiling give rise to “overshooting quantity” movements. It is well known that overshooting quantities create problems for sub-annual chaining of bilateral price indices because such quantities generate downward chain drift bias. The present study argues that overshooting quantities 27This is in line with findings in Fox et al. (2022, pp. 17-22), Lamboray (2021, pp. 13-17) and Van Loon & Roels (2018, pp. 9-13). Note that the former study uses household-level scanner data, while the lattertwostudies use point-of-salesscannerdata.None ofthese studies differentiates betweenstockpiling and consumption smoothing. 28The comparable performance of the R-GEKS and R-GK approaches is also reported in Fox et al. (2022). © 2024 The Authors. Review of Income and Wealth published by John Wiley & Sons Ltd on behalf of International Association for Research in Income and Wealth. 17 of 20 Review of Income and Wealth, Series 71, Number 1, February 2025 are only part of the chain drift problem. Other important causes of chain drift bias are search and adjustment costs as well as habits. They imply that price changes lead to delayed quantity changes. The resulting “sticky quantities” generate upward chain drift. In the literature, R-GEKS, R-TPD, and R-GK approaches have been proposed as a remedy for chain drift bias. However, it is unclear whether these approaches merely reduce chain drift bias or even eliminate it. Furthermore, some approaches may be more effective than others. The present paper answers all these questions. To this end, it develops a novel simulation framework that is consistent with households that have a desire for stockpiling (overshooting quantities) and/or for smoothing quantity adjustments over time (sticky quantities). Building on this framework, a stress test that examines the resilience of different price indices against chain drift is developed. 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