Price indices theory and practice

Transcription

1 Price indices theory and practice 1 st EMOS Spring School Dr Jens Mehrhoff Deutsche Bundesbank, Statistics Department Trier, 23 March 2015

2 Contents 1. Introduction 2. Basket approach for target indices Laspeyres and Paasche price indices Deflation and quantity/volume indices 3. Sub-indices Aggregation Implicit weights 4. Harmonised Index of Consumer Prices Definitions terminology Recommendations and examples Dr Jens Mehrhoff Price indices 23 March

3 1. Introduction Objective Key macroeconomic indicators, such as consumer prices and gross domestic product (at previous-year prices), are widely used in monitoring the state and development of the economy. Movements of such price and volume measures are used in short-term business cycle analysis not least at central banks. The most recent financial crisis highlights again the need for high-quality data to inform monetary policy-makers and macroprudential authorities. The objective of this course is to acquaint participants with price and volume measurements in official statistics. Both index theory and its application in practice are covered. Dr Jens Mehrhoff Price indices 23 March

4 1. Introduction CPI methodology A price index is a measure of the proportionate, or percentage, change in a set of prices over time. A consumer price index measures changes in the prices of goods and services that households consume. Such changes affect the real purchasing power of consumers incomes and their welfare. As the prices of different goods and services do not all change at the same rate, a price index can only reflect their average movement. Dr Jens Mehrhoff Price indices 23 March

5 1. Introduction Scope and uses of CPIs While there may be interest in a CPI which is as broadly defined as possible, covering all the goods and services consumed by all households, there are many other options for defining CPIs covering particular sets of goods and services, which may be more useful for particular analytical or policy purposes. There is no necessity to have only a single CPI. When only a single CPI is compiled and published, there is a risk that it may be used for purposes for which it is not appropriate. More than one CPI could be published in order to meet different analytic or policy needs. It is important to recognise, however, that the publication of more than one CPI can be confusing to users who view consumer inflation as a pervasive phenomenon affecting all households equally. The coexistence of alternative measures could undermine their credibility for many users. Dr Jens Mehrhoff Price indices 23 March

6 1. Introduction Purposes of CPIs There are three main purposes for which it is desirable to measure the average rate of change in consumer prices going from a base period 0 to a comparison period 1 for a well-defined household or group of households that pertains to a well-specified value aggregate: 1. As a summary measure of the overall rate of price change that the specified group faces over the two periods being compared for the value aggregate under consideration. 2. As a deflator for the value aggregate under consideration. Deflating the value ratio by the price index gives us the rate of growth of the corresponding quantity aggregate and this rate of growth can often be given a welfare change interpretation. 3. As a compensation or indexation measure. Governments or private employers may want to index benefit or salary levels to ensure that the indexed entitlement or salary level in the current period is equivalent in some sense to the base level of entitlement or salary. Dr Jens Mehrhoff Price indices 23 March

7 1. Introduction Practical CPI construction It can be seen that the first two uses are really complimentary aspects of the same problem, which is to decompose a ratio of value aggregates for two periods into a price change component times a quantity change component. At the outset, it should be recognised that no consumer price index will be conceptually perfect: data limitations and cost considerations will prevent the perfect index from being produced. However, it will be useful to introduce the various idealised types of index that have been suggested in the literature on index number theory over the past 200 years. These idealised indices are called target indices. Of the main approaches to the determination of the functional form for a target price index that compares the prices (and associated quantities) between two periods, we will cover the basket approach. Dr Jens Mehrhoff Price indices 23 March

8 1. Introduction Percentages Change from previous period in per cent: Change during the current month or quarter. Examples: 2014 July compared to 2014 June or 2014 Q3 compared to 2014 Q2. Used with seasonally adjusted time series in order to help revealing the news. Annual percentage change: Change in current month or quarter over the year. Examples: 2014 July compared to 2013 July or 2014 Q3 compared to 2013 Q3. Proxy for comparing like with like but not able to eliminate working-day variations. Exponential growth (e.g. compound interest): X T X 0 p1 ` Rq T, where XT and X 0 are the values at times T and 0, respectively, and R is the (annual) growth rate; p1 ` Rq is the (annual) growth factor. X 0 X T p1 ` Rq T. R Ta X T {X 0 1. Dr Jens Mehrhoff Price indices 23 March

9 1. Introduction Means (Microsoft Excel) Arithmetic mean: (=AVERAGE(X 1 :X n )) sx a 1 nÿ X i. n Weighted mean: (=SUMPRODUCT(X 1 :X n,ω 1 :ω n )/SUM(ω 1 :ω n )) i 1 i 1 sx ω X i ω i i 1 ω. i Geometric mean: (=GEOMEAN(X 1 :X n )) g ffe nź 1 nÿ sx g n X i exp ln X i. n i 1 i 1 Harmonic mean: (=HARMEAN(X 1 :X n )) n sx h i 1 X 1 ď X s g ď X s a. i Dr Jens Mehrhoff Price indices 23 March

10 1. Introduction Setting the stage When the comparison of price changes is restricted to two periods, the price index that constructs an average measure of price change is called a bilateral index number formula. But of course, if one period is held fixed, then a bilateral price index formula can be used to make a sequence of price comparisons over a number of subsequent periods. We specify two accounting periods, t P t0, 1u, for which we have micro price and quantity data for n commodities pertaining to transactions by a consumer (or a well-defined group of consumers). Denote the price and quantity of commodity i P t1,..., nu in period t by p t i and q t i, respectively. For t P t0, 1u, define the value of transactions in period t as: V t nÿ pi t qi t p t q t, (15.1) i 1 where p t pp t 1,..., pt nq is the period t price vector, q t pq t 1,..., qt nq is the period t quantity vector and p t q t denotes the inner product of these two vectors. Dr Jens Mehrhoff Price indices 23 March

11 1. Introduction Index number problem We attempt to decompose a value ratio for the two periods under consideration into a price change component P times a quantity change component Q. Thus, we now look for two functions of 4 n variables, Ppp 0, p 1, q 0, q 1 q and Qpp 0, p 1, q 0, q 1 q, such that: V 1 V 0 p1 q 1 p 0 q 0 Ppp0, p 1, q 0, q 1 q Qpp 0, p 1, q 0, q 1 q. (15.3) We think of Ppp 0, p 1, q 0, q 1 q as being a weighted average of the price ratios p 1 1 {p0 1,..., p1 n{p 0 n. Dr Jens Mehrhoff Price indices 23 March

12 2. Basket approach for target indices Lowe price index A very simple approach to the determination of a price index over a group of commodities is the (fixed) basket approach. In this approach, we are given a basket of commodities that is represented by the positive quantity vector q. Given the price vectors for periods 0 and 1, p 0 and p 1, respectively, we can calculate the cost of purchasing this same basket in the two periods, p 0 q and p 1 q. Then, the ratio of these costs is a very reasonable indicator of pure price change over the two periods under consideration, provided that the basket vector q is representative. Thus, define the Lowe (1823) price index, P Lo, as follows: P Lo pp 0, p 1, qq i 1 p1 i q i p1 q i 1 p0 i q i p 0 q. (15.15) Dr Jens Mehrhoff Price indices 23 March

13 2. Basket approach for target indices Laspeyres and Paasche price indices As time passed, economists and price statisticians demanded a bit more precision with respect to the specification of the basket vector q. There are two natural choices for the reference basket: the period 0 commodity vector q 0 or the period 1 commodity vector q 1. These two choices lead to the Laspeyres (1871) price index P L defined by Equation (15.5), if we choose q q 0, and the Paasche (1874) price index PP defined by Equation (15.6), if we choose q q 1 : P L pp 0, p 1, q 0, q 1 q i 1 p1 i q 0 i i 1 p0 i q 0 i P P pp 0, p 1, q 0, q 1 q i 1 p1 i q 1 i i 1 p0 i q 1 i p1 q 0 p 0 q 0, (15.5) p1 q 1 p 0 q 1. (15.6) Dr Jens Mehrhoff Price indices 23 March

14 2. Basket approach for target indices Fisher price index The Paasche and Laspeyres price indices are equally plausible but often give different estimates of the amount of aggregate price change between periods 0 and 1. Taking an evenly weighted average of these basket price indices as a single estimator of price change between the two periods leads to symmetric averages. The geometric mean, which leads to the Fisher (1922) price index, P F, is defined as: P F pp 0, p 1, q 0, q 1 q b P L pp 0, p 1, q 0, q 1 q P P pp 0, p 1, q 0, q 1 q. (15.12) Dr Jens Mehrhoff Price indices 23 March

15 2. Basket approach for target indices Expenditure share weights The Laspeyres price index (15.5) can be rewritten in an alternative manner that is more useful for statistical agencies as follows: P L pp 0, p 1, q 0, q 1 q i 1 p1 i q 0 i i 1 p0 i q 0 i nÿ p 1 i p 0 i 1 i nÿ p 1 i p 0 i 1 i p 0 i q 0 i j 1 p0 j q 0 j s 0 i, (15.8) where the period 0 expenditure share on commodity i, si 0, is defined as pi 0 qi 0 { j 1 p0 j qj 0 for i P t1,..., nu. The Laspeyres price index P L can thus be written as an arithmetic average of the n price ratios, pi 1{p0 i, weighted by base period expenditure shares. Dr Jens Mehrhoff Price indices 23 March

16 2. Basket approach for target indices Deflation approach The first approach to index number theory can be regarded as the problem of decomposing the change in a value aggregate, V 1 {V 0, into the product of a part that is attributable to price change, Ppp 0, p 1, q 0, q 1 q, and a part that is attributable to quantity change, Qpp 0, p 1, q 0, q 1 q. This approach to the determination of the price index is the approach that is taken in the National Accounts, where a price index is used to deflate a value ratio in order to obtain an estimate of quantity change. Thus, in this approach to index number theory, the primary use for the price index is as a deflator. Note that once the functional form for the price index Ppp 0, p 1, q 0, q 1 q is known, then the corresponding quantity or volume index Qpp 0, p 1, q 0, q 1 q, is completely determined by P, i.e. rearranging Equation (15.3): Qpp 0, p 1, q 0, q 1 q V 1 {V 0 Ppp 0, p 1, q 0, q 1 q. (15.4) Conversely, if the functional form for the quantity index Qpp 0, p 1, q 0, q 1 q is known, then the corresponding price index Ppp 0, p 1, q 0, q 1 q is completely determined by Q. Dr Jens Mehrhoff Price indices 23 March

17 2. Basket approach for target indices Quantity indices The quantity index that corresponds to the Laspeyres price index using Equation (15.3) is the Paasche quantity index, i.e. if P in Equation (15.4) is replaced by P L defined by Equation (15.5), then the following quantity index is obtained: Q P pp 0, p 1, q 0, q 1 q i 1 p1 i q 1 i i 1 p1 i q 0 i p1 q 1 p 1 q 0. (15.10) The quantity index that corresponds to the Paasche price index using Equation (15.3) is the Laspeyres quantity index, i.e. if P in Equation (15.4) is replaced by P P defined by Equation (15.6), then the following quantity index is obtained: Q L pp 0, p 1, q 0, q 1 i 1 q p0 i qi 1 i 1 p0 i qi 0 p0 q 1 p 0 q 0. (15.11) Note that Q L is the (hypothetical) value of the period 1 quantity vector valued at the period 0 prices, i 1 p0 i qi 1 p 0 q 1, divided by the value of the period 0 quantity vector valued at period 0 prices, i 1 p0 i qi 0 p 0 q 0. Thus, the period 0 and 1 quantity vectors, q 0 and q 1, are valued at the same set of prices, the base period prices, p 0. Dr Jens Mehrhoff Price indices 23 March

18 2. Basket approach for target indices Deflation It is good practice that deflators be applied at a detailed product group level: v 1 i P i p1 i q 1 i p 1 i {p0 i p 0 i q 1 i, where pi 0 qi 1 is the volume in the ith product group for i P t1,..., nu. Consistent aggregation of volumes yields the Laspeyres volume index: i 1 p0 i qi 1 i 1 V 0 p0 i qi 1 i 1 v i 0 i 1 p0 i qi 1 i 1 p0 i qi 0 The implicit Paasche deflator is thus defined: Q L. V 1 {V 0 Q L i 1 v 1 i i 1 p0 i q 1 i i 1 p1 i qi 1 i 1 p0 i qi 1 P P. Dr Jens Mehrhoff Price indices 23 March

19 3. Sub-indices Aggregation A higher-level index is an index in which the sub-indices are averaged using a set of pre-determined weights. The formula can be written as follows: nÿ P P i s i, i 1 where P denotes the overall CPI, or any higher-level index, from period 0 to 1; s i is the weight attached to each of the sub-indices; and P i is the corresponding sub-index. Example: Laspeyres price index. P L nÿ P L,i si 0, where s 0 i is the period 0 expenditure share on the lower-level sub-aggregate i for i P t1,..., nu. i 1 Dr Jens Mehrhoff Price indices 23 March

20 3. Sub-indices Core inflation In the case of two sub-indices only, we have: Example: core inflation. P L P L,1 s 0 ` P L,2 p1 s 0 q, PL : Total consumer price index. PL,1 : Core consumer price index less food and energy. PL,2 : Sub-index for food and energy. s 0 : Expenditure share on total consumer price index excluding food and energy. Note that rounding affects these calculation results. Dr Jens Mehrhoff Price indices 23 March

21 3. Sub-indices Example German national CPI, 2010 = 100, final July 2014 data: Calculate the total consumer price index (PL ) from the core consumer price index less food and energy (P L,1 ), the sub-index for food and energy (P L,2 ) and the expenditure share (s): P L P L,1 s 0 ` P L,2 p1 s 0 q Calculate the core consumer price index less food and energy (P L,1 ) from the total consumer price index (P L ), the sub-index for food and energy (P L,2 ) and the expenditure share (s): P L,1 pp L P L,2 p1 s 0 qq{s Calculate the the sub-index for food and energy (P L,2 ) from the total consumer price index (P L ), the core consumer price index less food and energy (P L,1 ) and the expenditure share (s): P L,2 pp L P L,1 s 0 q{p1 s 0 q Calculate the expenditure share (s) from the total consumer price index (P L ), the core consumer price index less food and energy (P L,1 ), the sub-index for food and energy (P L,2 ): s 0 pp L P L,2 q{pp L,1 P L,2 q Dr Jens Mehrhoff Price indices 23 March

22 3. Sub-indices Implicit weights Implicit weighting of rates of change between periods τ and t based on the Laspeyres price index as opposed to weighting of sub-indices: PL 0:t PL 0:τ 1 looomooon rate of change P0:t L,1 PL,1 0:τ 1 loooooomoooooon rate of change 1 PL,1 0:τ PL 0:τ s 0 looomooon implicit weight 1 looooooooooooooomooooooooooooooon contribution to growth 1 P0:t L,2 ` PL,2 0:τ 1 loooooomoooooon rate of change 2 P0:τ L,2 P 0:τ p1 s 0 q. loooooooomoooooooon L implicit weight 2 loooooooooooooooooomoooooooooooooooooon contribution to growth 2 Dr Jens Mehrhoff Price indices 23 March

23 3. Sub-indices Example Calculate the contribution of the core consumer price index less food and energy and the contribution of the sub-index for food and energy to the rate of change of the total consumer price index. Total CPI Core CPI Sub-index Expenditure share 1, Index July Index July Rate of change (%) `0.8 ` Implicit weight Contribution to growth (percentage points) ` Note that rounding affects these calculation results. Dr Jens Mehrhoff Price indices 23 March

24 4. Harmonised Index of Consumer Prices Problem statement When a time series of Laspeyres(-type) price indices is calculated using a fixed set of weights (quantities), these become progressively out of date and increasingly irrelevant to the later periods for which prices are being compared. This phenomenon reflects the fact that consumers typically react to price changes by substituting goods or services that have become relatively cheaper to those that have become relatively dearer (the relative price effect). The base period, in which weights are set, has to be updated sooner or later and the new index series chain-linked to the old. Hence, chaining is inevitable in the long run. Dr Jens Mehrhoff Price indices 23 March

25 4. Harmonised Index of Consumer Prices Problem statement Also associated with the regular process of chain-linking is sample rotation (or re-initiation), i.e. reviewing and updating the Harmonised Index of Consumer Prices (HICP) basket. In COICOP classes where new products are continuously appearing and old ones disappearing, the sample of products can quickly become outdated and be unrepresentative of what consumers are purchasing. In such a case, the sample of varieties within each elementary aggregate can be totally resampled to reflect current spending patterns. The same holds true for the sample of outlets. Last but not least, major methodological changes can be implemented as part of this process. Dr Jens Mehrhoff Price indices 23 March

26 4. Harmonised Index of Consumer Prices Definitions terminology The concepts of weighting, price, and index reference period should be clearly distinguished. In particular, the term base period is sometimes used for different concepts. Having regard to Commission Regulation (EC) No 1708/2005, the terminology used is as follows: Weighting reference period for the 12-month period of consumption or expenditure from which the weights are estimated for the compilation of the latest HICP index figures: average year y 1 (price-updated to December). Price reference period for the period from which the current price change is measured and for which prices are used as denominators in the index calculations: December of the previous year. Index reference period for the period for which the index is set to 100 index points: currently Dr Jens Mehrhoff Price indices 23 March

27 4. Harmonised Index of Consumer Prices Definitions terminology Let p m,y i and q m,y i denote the price and the quantity, respectively, of item i in month m of year y. Then, the short-term monthly Laspeyres price index is defined as: P 12,y 1:m,y L i 1 pm,y i sq y 1 i p12,y 1 i 1 i sq y 1 i i 1 ř 12 qm,y 1 12 m 1 i nÿ i 1 p m,y i p 12,y 1 i p 12,y 1 i sq y 1 i p12,y 1 i 1 i sq y 1, i where sq y 1 is the simple arithmetic average of the quantities of item i in the months of y 1. is a Laspeyres price index for month m of year y with average year y 1 as weighting reference period and December of the previous year as price reference period. P 12,y 1:m,y L Dr Jens Mehrhoff Price indices 23 March

28 4. Harmonised Index of Consumer Prices Definitions terminology The long-term annually chain-linked monthly Laspeyres price index is thus defined: rp 0:m,Y L r P 0:12,Y 1 L 12,Y 1:m,Y PL, where P r0:m,y L measures the overall change from the average of year 0 (the index reference year) to month m of year Y. Dr Jens Mehrhoff Price indices 23 March

29 4. Harmonised Index of Consumer Prices Recommendations Chain-linking means constructing long-run price measures by cumulating movements in short-term indices with different price reference periods. The technique of one-month overlaps as defined above requires compiling estimates for the overlap month (December of year Y ) at the average quantities of the current year, i 1 p12,y i sq i Y, in addition to estimates at the average quantities of the previous year, i 1 p12,y i sq Y 1 i. The ratio between the estimates for the linking month at the average quantities of the current year and at the average quantities of the previous year then provides the linking factor to scale the monthly data up or down. Dr Jens Mehrhoff Price indices 23 March

30 4. Harmonised Index of Consumer Prices Recommendations Infra-annual chain linking using the one-month overlap technique Short-term monthly Laspeyres price indices 100 Long-term annually chain-linked monthly Laspeyres price index 100 J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M A M J Y 2 Y 1 Y Y+1 Deutsche Bundesbank S3IN0374.Chart Dr Jens Mehrhoff Price indices 23 March

31 4. Harmonised Index of Consumer Prices Recommendations Hence, the set of weights is kept fixed for 13 consecutive months, from December of the previous year to December of the current year. As outlined above, this results in two estimates being produced for the overlap month. Likewise, this is the very moment when a new HICP basket (and sample of outlets) can be introduced as well as major methodological changes be incorporated. In these circumstances, both the old basket and the previous methodology have to be applied alongside the new ones in December over and above of the erstwhile and updated weights. Frequently, this entails double price collection for the overlap month. The former estimate and the new one, hence, have different measurement targets (e.g. another sample of outlets), although they refer to the same reporting month. Dr Jens Mehrhoff Price indices 23 March

32 4. Harmonised Index of Consumer Prices Examples Using data on the HICP for the changing composition of the euro area from the ECB s Statistical Data Warehouse, aggregation and disaggregation of chain-linked price indices are demonstrated. The time series consist of monthly chain-linked price indices and annual weights for the all-items HICP, sub-indices on energy ( NRGY 00 ) and unprocessed food ( FOODUN ) as well as the total excluding unprocessed food and energy ( XEFUN0 ). Dr Jens Mehrhoff Price indices 23 March

33 4. Harmonised Index of Consumer Prices Examples Harmonised index of consumer prices, changing composition of the euro area 2005 = 100, log scale % Total Year-on-year change lin scale Energy Unprocessed food Total excluding unprocessed food and energy Year-on-year change lin scale % Source: Eurostat. Deutsche Bundesbank S3IN0375.Chart Dr Jens Mehrhoff Price indices 23 March

34 4. Harmonised Index of Consumer Prices Examples The approach to (dis-)aggregation of chained sub-indices is: 1. Unchaining, 2. (Dis-)Aggregation, and 3. Chaining. Dr Jens Mehrhoff Price indices 23 March

35 4. Harmonised Index of Consumer Prices Examples The first thing to do is to unchain all chained sub-indices. When using one-month overlaps, this means the chained series for the current month have to be divided by the chained series for December of the previous year, thus, reverting the chain-linking as outlined above: rp 0:m,y L rp 0:12,y 1 L P 12,y 1:m,y L. The short-term link in the all-items HICP, based on the three aggregates considered here, can be written as: P 12,y 1:m,y L P 12,y 1:m,y L,NRGY 00 sy 1 NRGY 00 ` P 12,y 1:m,y L,FOOODUN sy 1 FOODUN ` P12,y 1:m,y L,XEFUN0 sy 1 XEFUN0, where s y 1 denotes the (expenditure share) weights used for the compilation of the latest HICP index figures. Dr Jens Mehrhoff Price indices 23 March

36 4. Harmonised Index of Consumer Prices Examples Rather than weighting together the all-items HICP, the total excluding unprocessed food and energy is derived by means of being weighted apart: P 12,y 1:m,y L,XEFUN0 P 12,y 1:m,y L P 12,y 1:m,y L,NRGY 00 sy 1 NRGY 00 P12,y 1:m,y L,FOOODUN sy 1 s y 1 XEFUN0 FOODUN After the desired aggregates are calculated, they need to be chained again and the index reference period let equal Dr Jens Mehrhoff Price indices 23 March