An econometric analysis of the Italian cultural supply
AAn econometric analysis of the Italian cultural supply
Consuelo R. Nava , Maria Grazia Zoia Abstract
Price indexes in time and space is a most relevant topic in statistical analysisfrom both the methodological and the application side. In this paper a priceindex providing a novel and effective solution to price indexes over several periodsand among several countries, that is in both a multi-period and a multilateralframework, is devised. The reference basket of the devised index is the unionof the intersections of the baskets of all periods/countries in pairs. As such, itprovides a broader coverage than usual indexes. Index closed-form expressions andupdating formulas are provided and properties investigated. Last, applicationswith real and simulated data provide evidence of the performance of the index atstake.
Keywords: multi-period price index, multilateral price index, stochastic approach,updating process, OLS.
JEL code: C43; E31; C01. Department of Economics and Political Sciences, University of Aosta Valley, Italy and Department ofEconomic Policy, Catholic University of the Sacred Heart, Milan, Italy, [email protected] Department of Economic Policy, Catholic University of the Sacred Heart, Milan, Italy,[email protected] a r X i v : . [ ec on . E M ] M a y Introduction
Multi-period and multilateral price indexes, used to compare sets of commodities over timeand across countries, respectively, are of prominent interest for statisticians (see, e.g., Biggeriand Ferrari, 2010). Several approaches to the problem have been carried out in the literature.One of them is the axiomatic approach (see, e.g., Balk, 1995, and the references quotedtherein), which rests on the availability of both quantities and prices, dealt with as inde-pendent random variables, and aims at obtaining price indexes, to enjoy suitable properties(Fisher, 1921, 1922).A second approach hinges on the economic theory (see, among others, Diewert, 1979;Caves et al., 1982, for a review) and rests on the idea that consumption choices come fromthe optimization of a utility function under budget constraints. Here, prices play the roleof independent variables, while quantities arise as solutions to an optimization problem inaccordance with the decision maker’s preference scheme.A third approach is the stochastic one (see Clements et al., 2006; Diewert, 2010, for areview), which can be traced back to the works of Jevons (1863, 1869) and Edgeworth (1887,1925). Thanks to Balk (1980); Clements and Izan (1987), this approach has recently beenreappraised, and its role in inflation measurements duly acknowledged (see, e.g., Asghar andTahira, 2010, and references quoted therein). In this framework, prices are assumed to beaffected by measurement errors whose bias effect must be duly minimized. The stochasticapproach (hereafter, SA) turns out to be somewhat different from other approaches, insofaras it is closely related to regression theory (Theil, 1960; Clements and Izan, 1987). In fact,the SA enables the construction of tests and confidence intervals for price indexes, whichprovide useful pieces of information (Clements et al., 2006). Furthermore, the SA has lesslimits than other approaches and clears the way to further extensions, as shown in Diewert(2004, 2005); Silver (2009); Rao and Hajargasht (2016).In this paper, we devise a multi-period/multilateral price index, MPL index henceforth,within the stochastic framework. The derivation of the MPL index, which is the solution toan optimization problem, calls for quantities and values of the commodities (not prices), likeWalsh (1901). The reference basket, namely the set of commodities for all periods/countries, This approach is also known as preference field approach or functional approach (Divisia, 1926). The SA, differently from the index number theory originating from Theil (1967) does not need to accountfor the economic importance of single prices.
2s made up of the union of the intersections of all the couples of year/country baskets inpairs. Namely, the price index of a commodity can be always computed once the latteris present in at least two periods/countries. Thus, the reference basket turns out to bemore representative than the ones commonly used by the majority of statistical agencies,which either align the reference basket to that of the first period, or make it tally with theintersection of the commodity sets of all periods/countries. Eventually, such a reference basketis likely to be scarcely representative of the commodities present in each period/country. Inthis sense, just like hedonic (Pakes, 2003), GESKS (Balk, 2012) and country/time-product-dummy (CPD/TPD) approaches with incomplete price tableaus (Rao and Hajargasht, 2016),the MPL index does not drop any observation on the account of having no counterpart inthe reference basket. The lack of a commodity in a period/country t requires putting bothits quantity and value equal to zero in that period/country. However, unlike the aforesaidapproaches, the MPL index is built on quantities and values, not on prices. Neither anypreliminary computation of binary price indexes, as in the GESKS approach (Ivancic et al.,2011), nor the use of any type of weighting matrix for dealing with missing values or quantities,as in the case of CPD/TPD indexes, are needed.The updating of the MPL index is easy to accomplish and suitable formulas, tailored tothe multi-period or multilateral nature of the data, are provided. In fact, while the inclusionof fresh values and quantities, of a set of commodities corresponding to an extra period, doesnot affect the previous values of the MPL index, the inclusion of a new country affects allformer MPL indexes. Hence, two updating formulas have been proposed for the MPL index:one for the multi-period case and another for the multilateral case. A comparison of the saidindex to CPD/TPD index – a multilateral/multi-period index that, like the MPL one, can beread as a solution to an optimization problem – provides evidence of an easier implementationand greater accuracy of the former index.To sum up, a threefold novelty characterizes the paper. First, a price index, which proveseffective either for the multi-period or the multilateral case, is devised. Second, updatingformulas tailored to the multilateral and the multi-period version of the index are provided.Third, the grater simplicity of use and accuracy of the said index is highlighted in comparisonwith well-known standard multilateral/multi-period indexes.The paper is organized as follows. In Section 2, we briefly go over the SA to price indexes3nd point out how several indexes are solutions to optimization problems. In Section 3,within the SA, we devise the MPL index according to a minimum-norm criterion as well asits updating formulas for the multi-period and the multilateral cases, respectively. Section 4is devoted to the properties of the MPL index. Section 5 provides an application of theMPL index to the Italian cultural supply data to shed light on its potential as both a multi-period and a multilateral index. To gain a better insight into the performance of the MPLindex, a comparison is made of the CPD/TPD indexes by using both real and simulateddata. Section 6 completes the paper with some concluding remarks and hints. For the sakeof easier readability, an Appendix has been added with proofs and technicalities. In this section, we review the main features of the SA and show how several well-known priceindexes can be obtained within this framework.The SA works out price indexes as solutions to an optimization problem consisting infinding a line (more generally a plane or a hyperplane) which lies as closely as possible to thepoints whose coordinates are the N commodity prices in the T periods under examination.Following Theil (1967) and Selvanathan and Rao (1994, Ch. 3) and by assuming for expositionpurposes T = 2 , the idea underlying the SA is that, in both the periods taken into account,all prices move almost proportionally. Namely, p i ≈ λ p i ∀ i = 1 , , . . . , N (1)where p it is the price of commodity i in period t ( t = { , } ) and λ is a scalar factor actingas price index. Eq. (1) can be conveniently rewritten as follows p i p i = λ + η i ∀ i = 1 , , . . . , N (2)where η i are error terms which, as a rule, are assumed to be non-systematic and uncor-related between commodities with variances, that can be either constant (with respect tocommodities), that is var ( η i ) = σ , ∀ i = 1 , , . . . , N, (3)4r not. In the latter case, variances are frequently specified as inversely proportional to thecommodity budget share, namely var ( η i ) = σ w i , ∀ i = 1 , , . . . , N. (4)Here w i = p i q i p (cid:48) q and p (cid:48) q is the total expenditure in the base period ( t = 1 ), being p t and q t the vectors of prices and quantities at time t . Eq. (2) can be written in compact notationas follows p ∗ p − H = λ u + η (5)where u is an N -dimensional unit vector, p − H is defined as the vector of the reciprocals ofthe non-null entries of p and zeros otherwise, that is p − H N, = [ p i ] − H = p − i if p i (cid:54) = 00 otherwise (6)and η is a vector of N non-systematic and spherical random variables, unless otherwisespecified. Taking the Hadamard product of both sides of Eq. (5) by p , yields the following p ∗ p − H ∗ p = λ u ∗ p + η ∗ p = λ p + ς . (7)The appeal of the SA lays in the possibility of evaluating, besides point estimates, alsoprice index standard errors which increase as the relative price variability increases. Thecomputation of price index standard errors allow to verify the intuitive notion that the lessprices move proportionally, the less precise are price index estimators. Further, standarderrors prove useful to build confidence intervals for price indexes.The following theorem shows that several well-known price indexes can be seen as offspringof Eq. (7). The error terms ς of Eq. (7) are no longer homoschedastic. In particular, if the assumption in Eq. (3)holds, then the dispersion matrix of ς takes the form E ( ςς (cid:48) ) = σ D where D = [( p ∗ p ) u (cid:48) ] ∗ I N where u is the unit vector and I N denotes the N -dimensional identity matrix. If Eq. (4) holds, then E ( ςς (cid:48) ) = σ p (cid:48) q ˜ D where ˜ D = [( p ∗ q ) − H u (cid:48) ] ∗ I N . heorem 1. Let p t be a vector of the prices of N commodities at time t and q t be the vectorof the corresponding quantities. The Laspeyeres, Paasche, Marschall-Edgeworth and Walshindexes are solutions to an optimization problem of the form min λ || p − λ p || N = min λ || e || N (8) where || · || N = ( e (cid:48) Ae ) stands for a (semi)norm of the reference price index and A is aproperly chosen non-negative definite matrix . Proof.
See Appendix A.1.
In this section, a multi-period/multilateral price index is derived in the wake of the SAintroduced in the previous section. The construction of this kind of index, MPL indexhereafter, poses several issues, like the choice of the reference basket and its updating. Whenthe prices of commodity sets in two periods/countries are compared, the reference basket, K τ , is generally set to be a subset of the commodities of the first period/country ( K τ = K ),which is also assumed to be representative for the other period/country.The simplest solution is to take the “intersection” of the two baskets as reference basket,( K τ = K ∩ K ). When the comparison is among more than two periods/countries, statisticalagencies generally align the reference basket to that of the first period/country ( K τ = K ),which turns out to play the role of base period/country. Of course, this choice is somewhatarbitrary as it leaves open the basket updating problem and it does not take into account linksamong baskets corresponding to couples, triples, . . . , of periods. An alternative approach setsthe reference basket as the intersection of all the commodities considered in each single period( K τ = (cid:84) t K t ). As a result, the reference basket is likely to be partially representative of thosecommodities which are peculiar to each period.Taking the SA as the reference frame, we derive a multi-period/multilateral price index,satisfying a minimum-norm criterion, whose reference basket – over a set of periods or acrossa set of countries – is the union of the intersections of the commodity baskets of variousperiods/countries, taken in pairs. Such a reference basket proves to be an effective solution Some similarities arise with the chaining rule (Forsyth and Fowler, 1981; von der Lippe, 2001). This is t simply entails settingboth its quantity and value equal to zero in that period/country. Figure 1 shows the refer-ence basket corresponding to the aforementioned approaches for the case of two and threeperiods/countries. The MPL index hinges, first of all, on the idea that in each time/countryFigure 1: The left-hand side panel shows the reference basket corresponding to the “tradi-tional” approach. The right-hand side panel shows the reference basket corresponding to theMPL index. t , the N commodity prices move proportionally to a set of N reference prices, that is p t ( N, ≈ λ t ˜ p ( N, ∀ t = 1 , , . . . , T (9)or p t ( N, = λ t ˜ p ( N, + ς t ( N, ∀ t = 1 , , . . . , T. (10)Here p t is the actual price vector of the N commodities at period/country t , ˜ p is the vectorof (unknown (time invariant) reference prices, λ t is a scalar factor acting as price index at a specific type of temporal aggregation method based on the use of the complete time series when computinga price index from to t . The resulting price index is a measure of the cumulated effect of adjacent periodsfrom 0 to 1, 1 to 2, . . . , t − to t . However, chain indexes leave unresolved the reference basket updating andare not applicable in a multilateral perspective, differently from the MPL index. Chain indexes compare thecurrent and the previous periods in order to evaluate the evolution over many periods. For a comparison ofthis approach with the fixed base one see Diewert (2001) t and ς t is a vector of error terms. Eq. (10) can be viewed as a generalizationof Eq. (7) for two periods/countries. As per Eq. (10), in each period/country t , the N prices( p t ) can be represented by a point in a N -dimensional space. Accordingly, N prices in T periods/countries, P = [ p , ..., p t , .., p T ] , can be represented by T points in a N -dimensionalspace. If all prices move proportionally, these points would lie on a hyperplane, ˜ p , and, inparticular, on a straight line crossing the origin for T = 2 .In general, this is only approximately true and the price “line” crossing the origin enjoysthe property of fitting the observed price points, according to a criterion which minimizes thedeviations of the data from the “line”. In compact notation, Eq. (9) can be more convenientlyreformulated as follows P ( N,T ) ≈ Π ( N,T ) = ˜ p ( N, λ (cid:48) (1 ,T ) (11)where λ is the vector of the T price indexes and ˜ p is the vector of the N (unknown) referenceprices. According to Eq. (11), the problem of determining a set of T price indexes can be readas the problem of approximating the price matrix, P , with a matrix of unit rank, Π , definedas the product of a vector of price indexes by a (unknown) reference price vector. Movingfrom prices to values, in light of Eq. (11), the matrix V of the values of N commodities in T periods/countries has the representation V ( N,T ) = P ∗ Q ≈ Π ∗ Q ≈ ( ˜ p ( N, λ (cid:48) (1 ,T ) ) ∗ Q ( N,T ) (12)where Q is the matrix of the quantities of N commodities in T periods/countries. Eq. (12)can be rewritten as [ v , ..., v t , ..., v T ] ≈ [ λ ˜ p ∗ q , .., λ t ˜ p t ∗ q t , ..., λ T ˜ p T ∗ q T ] (13)where v t and q t are the t th columns of V and Q respectively, and λ t is the t th element of λ t .Eq. (13) entails the following v t = λ t ˜ p ∗ q t + ε t ∀ t = 1 , , . . . , T (14)where ε t is a non-systematic stochastic term. The above formula, taking into account the8dentity ˜ p ( N, ∗ q t ( N, = D ˜ p ( N,N ) q t ( N, , can be re-written as δ t v t ( N, = D ˜ p ( N,N ) q t ( N, + ε t ( N, ∀ t = 1 , , . . . , T (15)where δ t = ( λ t ) − takes the role of the deflator, and D ˜ p is a diagonal matrix with diagonalentries equal to the elements of ˜ p defined as in Eq. (10). Eq. (15) expresses the value, v it ,of each commodity i at time t (discounted by a factor δ t ) as the product between the (timeinvariant) reference price, ˜ p i , and the corresponding quantity, q it , plus an error term, ε it .Note that λ t = δ − t plays the role of the multi-period/multilateral price index in Eq. (15).Over T periods/countries, the model can be written as V ( N,T ) D δ ( T,T ) = D ˜ p ( N,N ) Q ( N,T ) + E ( N,T ) (16)where D δ is a T × T diagonal matrix with diagonal entries equal to the elements of δ (seeFootnote 7). Without lack of generality, we assume that the first period is the base period(that is δ = λ = 1 ), and write the first equation separately from the others T − . Accordingly,the system in Eq. (16) can be written as (cid:34) v , ( N, V N,T − (cid:35) (cid:48) (1 ,T − ( T − , ˜ D δ ( T − ,T − = D ˜ p (cid:34) q , ( N, Q N,T − (cid:35) + (cid:34) ε , ( N, E N,T − (cid:35) (17)or as v = D ˜ p q + ε V ˜ D δ = D ˜ p Q + E (18) The matrix D ˜ p is defined as follows: D ˜ p = ˜ p . . .
00 ˜ p . . . ... ... . . . ... . . . ˜ p N . , can be also expressed as v = ( q (cid:48) ⊗ I N ) R (cid:48) N ˜ p + ε ( N ( T − , = ( I T − ⊗ ( − V )) R (cid:48) T − δ + ( Q (cid:48) ⊗ I N ) R (cid:48) N ˜ p + η . (19)Here, η = vec ( E ) , δ is a vector whose elements are the diagonal entries of ˜ D δ as specified inEq. (17) and R j denotes the transition matrix from the Kronecker to the Hadamard product. The vector δ , whose (non-null) entries are the reciprocals of the elements of the price index λ , can be obtained by estimating Eq. (19) with the ordinary least squares (OLS), underthe assumption that E ( µµ (cid:48) ) = σ I N ⊗ I T = σ I NT where µ = (cid:20) ε η (cid:21) (cid:48) . (20)In this connection, we state the following result. Theorem 2.
The MPL index can be obtained as reciprocal of the OLS estimate of the deflatorvector, δ , in the system in Eq. (19) . This OLS estimate is given by ˆ δ ( T − , = (cid:110) ( I T − ∗ V (cid:48) V ) − ( Q (cid:48) ∗ V (cid:48) ) (cid:2) ( q q (cid:48) + Q Q (cid:48) ) ∗ I N (cid:3) − ( Q ∗ V ) (cid:111) − ( Q (cid:48) ∗ V (cid:48) ) (cid:2) ( q q (cid:48) + Q Q (cid:48) ) ∗ I N (cid:3) − ( q ∗ v ) . (21) Proof.
See Appendix A.2.The case where T = 2 is worth considering because it sheds light on the index structure. Inthis case, the price index turns out to be a ratio of weighted price averages, with weights de-pending on the harmonic means of the squared quantities as stated in the following corollary. Use has been made of the relationships vec ( ABC ) = ( C (cid:48) ⊗ A ) vec ( A ) and vec ( D a ) ( N , = R (cid:48) N a ( N, where D a is a diagonal matrix whose diagonal entries are the elements of the vector a and R N is the transitionmatrix from the Kronecker to the Hadamard product (Faliva, 1996). The matrix R (cid:48) j is defined as follows R (cid:48) j ( j × j ) = (cid:20) e ,j ) ⊗ e ,j ) e j, ⊗ e j, . . . e jj, ⊗ e j (1 ,j ) (cid:21) , where e i represents the N dimensional i -th elementary vector. The reciprocal of the deflator vector, δ , is defined in Eq. (6). orollary 1. When T = 2 , the MPL index, (cid:98) λ , becomes ˆ λ = (cid:80) Ni =1 p i π i (cid:80) Ni =1 p i π i (22) where π i = 2 p i q i q i q i + q i and p it = v it q it . The price index in Eq. (22) can be also expressed as aconvex linear combination of prices, that is ˆ λ = N (cid:88) i =1 p i p i ˜ π i (23) with weights ˜ π i given by ˜ π i = v i v i q i q i q i + q i (cid:80) Ni =1 v i v i q i q i q i + q i . (24) Furthermore, by setting ˜ q t = D − / q t with D specified as follows D = q + q . . . q + q . . . ... ... . . . ... . . . q N + q N , (25) the M P L index can be also written in compact form as follows ˆ λ = ( ˜ q ∗ v ) (cid:48) ( ˜ q ∗ v )( ˜ q ∗ v ) (cid:48) ( ˜ q ∗ v ) . (26) Proof.
See Appendix A.3.As a by-product of Theorem 2, we state the following result.
Corollary 2.
The variance-covariance matrix of the deflator vector, (cid:98) δ , given in Theorem 2is V ar ( ˆ δ ) = σ [ I T − ∗ V (cid:48) V ] − . (27) The t th diagonal entry of the above matrix provides the variance of the deflator in the t th period/country, given by var (ˆ δ t ) = σ [ v (cid:48) t v t ] − (28) where v t denotes the t th column of the matrix V . roof. See Appendix A.4.As for the deflator vector ˆ δ , its moments and confidence intervals can be easily obtainedwithin the theory of linear regression models, from the result given in Corollary 2. As theprice index vector ˆ λ turns out to be the reciprocal of the said deflator (see Eq. 6), its statisticalbehavior can be derived from the former, following the arguments put forward, for example,in Geary (1930); Curtiss (1941) and Marsaglia (1965), merely to quote a few, on ratios (inparticular reciprocals) of random variables.The following corollary provides an approximation of the variance of ˆ λ t , obtained by usingthe first Taylor expansion of the variance of a ratio of two random variables. Corollary 3.
The variance of the MPL index ˆ λ t is var (ˆ λ t ) ≈ var (ˆ δ t ) E (ˆ δ t ) = (cid:98) σ [ v (cid:48) t v t ] E (ˆ δ t ) ∀ t = 1 , . . . , T (29) In the above equation (cid:98) σ = (cid:98) µ (cid:48) (cid:98) µ NT − ( N + T − where (cid:98) µ are the OLS residuals of the equation system (19) .Proof. See Stuard and Ord (1994, p. 351) and Elandt-Johnson and Johnson (1980, p. 69).The following two theorems provide updating formulas for the price index ˆ λ . The formerproves suitable when the index is used as a multilateral price index, while the latter isappropriate when it is employed as a multi-period index. In the former case, values andquantities of the commodities included in the reference basket are assumed available for anadditional T + 1 country. In the latter case, it is supposed that values and quantities of thecommodities included in the reference basket become available at time T + 1 . Theorem 3.
Should the values and quantities of N commodities of a reference basket becomeavailable for a new additional country, say the T + 1 -th, then, the updated multilateral versionof the MPL index, ˆ λ , turns out to be the vector of the reciprocals, as defined in Eq. (6) , ofthe following deflator vector ˆ δ ( T, == I T − ∗ V (cid:48) V v (cid:48) T +1 v T +1 − Q (cid:48) ∗ V (cid:48) v (cid:48) T +1 ∗ q (cid:48) T +1 (cid:104) ( q q (cid:48) + Q Q (cid:48) + q T +1 q (cid:48) T +1 ) ∗ I N (cid:105) − (cid:104) Q ∗ V v T +1 ∗ q T +1 (cid:105)(cid:111) − Q (cid:48) ∗ V (cid:48) v (cid:48) T +1 ∗ q (cid:48) T +1 (cid:104) ( q q (cid:48) + Q Q (cid:48) + q T +1 q (cid:48) T +1 ) ∗ I N (cid:105) − ( q ∗ v ) . (30) ere the symbols are defined as in Theorem 2 and v T +1 , q T +1 denote the vector of valuesand quantities of N commodities of the new T + 1 -th country, respectively.Proof. See Appendix A.5.
Theorem 4.
Should the values and quantities of N commodities of a reference basket becomeavailable for time T + 1 , then, the updated value ˆ λ T +1 of the multi-period version of the MPLindex at time T + 1 turns out to be the reciprocal of the deflator value at time T + 1ˆ δ T +1(1 , = (cid:110) v (cid:48) T +1 v T +1 − (cid:0) q (cid:48) T +1 ∗ v (cid:48) T +1 (cid:1) (cid:2) ( QQ (cid:48) + q T +1 q (cid:48) T +1 ) ∗ I N (cid:3) − ( q T +1 ∗ v T +1 ) (cid:111) − (cid:0) q (cid:48) T +1 ∗ v (cid:48) T +1 (cid:1) (cid:2) ( QQ (cid:48) + q T +1 q (cid:48) T +1 ) ∗ I N (cid:3) − ( Q ∗ V ) ˜ δ (31) where ˜ δ (cid:48) = [1 , ˆ δ ] (cid:48) and ˆ δ is defined as in Eq. (21) .Proof. See Appendix A.6.Figure 2 highlights the difference between the updating process of the deflator, and thus ofthe price index, depending on whether it is used in the multilateral or in the multi-periodcase.
Let us assume for simplicity T = 2 and denote with ˆ λ ( p , p , q , q ) a generic index numberwhere p t and q t are prices and quantities at time t . Without lack of generality, t = 1 isassumed as the base period.Following Predetti (2006) and Martini (1992), the main propertiesof an index number can be summarized as follows:1. Strong identity : ˆ λ ( p , p , q , q ) = 1 .2. Commensurability : ˆ λ ( γ ∗ p , γ ∗ p , γ − ∗ q , γ − ∗ q ) = ˆ λ ( p , p , q , q ) with γ ( N, > where γ − ∗ q t = [ q t /γ . . . q Nt /γ N ] (cid:48) .3. Proportionality : ˆ λ ( p , α p , q , q ) = α ˆ λ ( p , p , q , q ) with α > .4. Dimensionality : ˆ λ ( α p , α p , q , q ) = ˆ λ ( p , p , q , q ) with α > .5. Monotonicity : ˆ λ ( p , k ∗ p , q , q ) > ˆ λ ( p , p , q , q ) and ˆ λ ( k ∗ p , p , q , q ) < ˆ λ ( p , p , q , q ) with k ( N, > u where u is the unit vector.13 he price index can also be written in a compact form as follows ˆ = ( ˜ q ⇤ f ) ( ˜ q ⇤ f )( ˜ q ⇤ f ) ( ˜ q ⇤ f ) . (28) Proof.
See Appendix A.3.As a by-product of Theorem 2, we state the following result.
Corollary 2.
The variance-covariance matrix of the multi-period/multi-lateral price index givenin Theorem 2 and under assumption in eq. (3) is V ar ( ˆ ) = [ I T ⇤ F F ] . (29) The t -th diagonal entry of the above matrix provides the variance of the price index in the t -thperiod, given by var (ˆ t,t ) = [ f t f t ] (30) where f t denotes the t -th row of the matrix F .Proof. See Appendix A.4.According to the aim of this contribution, Theorems 3 and 4 provide two updating formulasfor the price index given in Theorem 2. The former proves suitable when the index is used asa multi-lateral price index, while the latter when it is employed as a multi-period one. In theformer case, values and quantities of commodities in the reference basket are assumed availablefor an additional T + 1 country. In the latter case, it is supposed that values and quantities ofthe commodities included in the reference basket become available at time T + 1 . Theorem 3.
Should the values and quantities of N commodities of a reference basket becomeavailable for a new additional country, say the T + 1 one, then, the updated multi-lateral priceindex can be obtained as a vector of reciprocals, defined as in eq. (6) , of the following vector ˆ ( T, == (" I T ⇤ F F f T +1 f T +1 " Q ⇤ F f T +1 ⇤ q T +1 ( q q + Q Q + q T +1 q T +1 ) ⇤ I N i h Q ⇤ F f T +1 ⇤ q T +1 i) " Q ⇤ F f T +1 ⇤ q T +1 ( q q + Q Q + q T +1 q T +1 ) ⇤ I N i ( q ⇤ f ) . (31) Here the symbols are defined as in Theorem 2 and f T +1 , q T +1 denote, respectively, the vectorof values and quantities of N commodities of the reference basket for the new T + 1 country.Proof. See Appendix A.5. 12 and X u y u ( N + T, = ( T, q ⇤ f N, . (66)Upon nothing that ˆ ( T , ˆ T +1(1 , = I T ( T,N ) ˆ ( N + T, = ⇤ ( q ⇤ f ) (67)where ⇤ is the upper diagonal block of the inverse matrix ( X u X u ) , partitioned inversionformulas lead to ⇤ T,N ) = (" I T ⇤ F F f T +1 f T +1 " Q ⇤ F f T +1 ⇤ q T +1 ( q q + Q Q + q T +1 q T +1 ) ⇤ I N i h Q ⇤ F f T +1 ⇤ q T +1 i) " Q ⇤ F f T +1 ⇤ q T +1 ( q q + Q Q + q T +1 q T +1 ) ⇤ I N i (68) which, post-multiplied by ( q ⇤ f ) , yields the estimator ˆ ( T , ˆ T +1(1 , . The reciprocal of thisestimator provides the required updated formula.24 = NT + 1 V D = D ˜ p Q + Ev T +1 T +1 = D ˜ p q T +1 + " T +1 ! v = D ˜ p q + " F ˜ D = D ˜ p Q + E v T +1 T +1 = D ˜ p q T +1 + " T +1 v T +1 q T +1 NT + 1 f = ( q ⌦ I N ) R N ˜ p + " ( N ( T , = ( I T ⌦ ( V )) R T + ( Q ⌦ I N ) R N ˜ p + ⌘ = v T +1 T +1 + ( q T +1 ⌦ I N ) R N ˜ p + " T +1 . y u ( NT + N, = X u ( NT + N,N + T ) u ( N + T, + µ u ( NT + N, y u ( NT + N, = v N, ( N ( T , ( N, , X u ( NT + N,N + T ) = ( N,T ( N, ( q ⌦ I N ) R N ( N,N ) ( I T ⌦ V ) R T N ( T ,T ( N ( T , ( Q ⌦ I N ) R N ( N ( T ,N ) ( N,T v T +1( N, ( q T +1 ⌦ I N ) R N ( N,N ) u ( N + T, = ( T , T +1(1 , ˜ p ( N, , µ u ( NT + N, = " N, ⌘ ( N ( T , " T +1( N, . X u X u ( N + T,N + T ) = I T ⇤ V V Q ⇤ V v T +1 v T +1 v T +1 ⇤ q T +1 Q ⇤ V v T +1 ⇤ q T +1 (˜ q ˜ q + ˜ Q ˜ Q + q T +1 q T +1 ) ⇤ I N A.5 Proof of Theorem 3
Proof.
When the values and the quantities of N commodities in a reference basket becomeavailable for a new additional country, say the T + 1 one, the model we refer to for updating themulti-lateral base index becomes V D = D ˜ p Q + Ev T +1 T +1 = D ˜ p q T +1 + " T +1 ! v = D ˜ p q + " V ˜ D = D ˜ p Q + E v T +1 T +1 = D ˜ p q T +1 + " T +1 (60)with v T +1 and q T +1 denoting the vector of values and quantities of the N commodities for the T + 1 country. After some computations , the model can be also expressed as v = ( q ⌦ I N ) R N ˜ p + " ( N ( T , = ( I T ⌦ ( V )) R T + ( Q ⌦ I N ) R N ˜ p + ⌘ = v T +1 T +1 + ( q T +1 ⌦ I N ) R N ˜ p + " T +1 . (61)The above system in compact form ca be written as y u ( NT + N, = X u ( NT + N,N + T ) u ( N + T, + µ u ( NT + N, (62)where y u ( NT + N, = v N, ( N ( T , ( N, , X u ( NT + N,N + T ) = ( N,T ( N, ( q ⌦ I N ) R N ( N,N ) ( I T ⌦ V ) R T N ( T ,T ( N ( T , ( Q ⌦ I N ) R N ( N ( T ,N ) ( N,T v T +1( N, ( q T +1 ⌦ I N ) R N ( N,N ) (63)and u ( N + T, = ( T , T +1(1 , ˜ p ( N, , µ u ( NT + N, = " N, ⌘ ( N ( T , " T +1( N, . Then, by following the same argument of Theorem 2, we obtain X u X u ( N + T,N + T ) = I T ⇤ V V Q ⇤ V v T +1 v T +1 v T +1 ⇤ q T +1 Q ⇤ V v T +1 ⇤ q T +1 (˜ q ˜ q + ˜ Q ˜ Q + q T +1 q T +1 ) ⇤ I N (64)31 Theorem 4.
Should the values and quantities of N commodities of a reference basket becomeavailable for the time T + 1 , then, the updated multi-period price index can be obtained as thereciprocal, defined as in eq. (6) , of the following ˆ T +1(1 , = n f T +1 f T +1 q T +1 ⇤ f T +1 ⇥ ( QQ + q T +1 q T +1 ) ⇤ I N ⇤ ( q T +1 ⇤ f T +1 ) o q T +1 ⇤ f T +1 ⇥ ( QQ + q T +1 q T +1 ) ⇤ I N ⇤ ( Q ⇤ F ) ˆ (32)where ˆ is defined as in eq. (23). Proof.
See Appendix A.6.The comparison among the multi-lateral and multi-period updating process can be seen also infigure 2. Main properties of the constructed multi-period/multi-lateral price index are derivedin the following section. M ULTI -P ERIOD /M ULTI -L ATERAL P RICE I NDEX M ULTI -P ERIOD /M ULTI -L ATERAL P RICE I NDEX
Figure 2: Graphical representation, respectively, of the multi-lateral (Theorem 3) and multi-period (Theorem 4) updating approaches des.
For exposition purposes, let denote with ˆ ( p , p , q , q ) a generic index number according toprices and quantities of the two observed periods: the base year ( p and q ) and a second year( p and q ). Following Martini (1992), main properties that an index number could satisfy canbe summarized as follow:1. Strong identity : ˆ ( p , p , q , q ) = 1 ;2. Commensurability : ˆ ( ⇤ p , ⇤ p , ⇤ q , ⇤ q ) = ˆ ( p , p , q , q ) with ( N, > ;3. Homogeneity : ˆ ( p , ↵ p , q , q ) = ↵ ˆ ( p , p , q , q ) with ↵ > ;13 NT + 1 " V ( N,T ) v T +1( N, D ⇤ ( T,T ) = D ˜ p ( N,N ) " Q ( N,T ) q T +1( N, + " E ( N,T ) " T +1( N, v T +1 q T +1 NT + 1 D ⇤ D ⇤ ( T +1 ,T +1) = ˆ D ( T,T ) ( T, (1 ,T ) T +1(1 , . ˆ D D ˆ D ( T,T ) = (1 , (1 ,T ( T , ˆ · · ·
00 ˆ · · ·
00 0 · · ·
00 0 · · · ˆ T = " D ˆ ˆ , . . . , ˆ T ˆ v = D ˜ p q + " V ˆ˜ D = D ˜ p Q + E v T +1 T +1 = D ˜ p q T +1 + " T +1 ! V ˆ D = D ˜ p Q + Ev T +1 T +1 = D ˜ p q T +1 + " T +1 .vec ˜ v = ( Q ⌦ I N ) R N ˜ p + = v T +1 T +1 + q T +1 ⌦ I N R N ˜ p + " T +1 ˜ v = vec ( V ˆ D ) = ( I T ⌦ V ) R T ˜ ˜ = [1 , ˆ ] vec ( E ) NT + 1 V D = D ˜ p Q + Ev T +1 T +1 = D ˜ p q T +1 + " T +1 ! v = D ˜ p q + " F ˜ D = D ˜ p Q + E v T +1 T +1 = D ˜ p q T +1 + " T +1 v T +1 q T +1 NT + 1 f = ( q ⌦ I N ) R N ˜ p + " ( N ( T , = ( I T ⌦ ( V )) R T + ( Q ⌦ I N ) R N ˜ p + ⌘ = v T +1 T +1 + ( q T +1 ⌦ I N ) R N ˜ p + " T +1 . y u ( NT + N, = X u ( NT + N,N + T ) u ( N + T, + µ u ( NT + N, y u ( NT + N, = v N, ( N ( T , ( N, , X u ( NT + N,N + T ) = ( N,T ( N, ( q ⌦ I N ) R N ( N,N ) ( I T ⌦ V ) R T N ( T ,T ( N ( T , ( Q ⌦ I N ) R N ( N ( T ,N ) ( N,T v T +1( N, ( q T +1 ⌦ I N ) R N ( N,N ) u ( N + T, = ( T , T +1(1 , ˜ p ( N, , µ u ( NT + N, = " N, ⌘ ( N ( T , " T +1( N, . X u X u ( N + T,N + T ) = I T ⇤ V V Q ⇤ V v T +1 v T +1 v T +1 ⇤ q T +1 Q ⇤ V v T +1 ⇤ q T +1 (˜ q ˜ q + ˜ Q ˜ Q + q T +1 q T +1 ) ⇤ I N Figure 2: The top panel shows the ratio of the updating formula for the multilateral versionof the MPL index (see Theorem 3); the bottom panel shows the ratio of the updating formulaof the the multi-period version of the MPL index (see Theorem 4).We now prove the above properties for the MPL index in Eq. (23), which will be denotedwith the extended notation ˆ λ ( p , p , q , q ) .1. Strong identity : taking the price vectors as equal in the two periods, it can be easilyproved that ˆ λ ( p , p , q , q ) = (cid:80) Ni =1 p i π i (cid:80) Ni =1 p i π i = 1 where π i = 2 p i q i q i q i + q i . (32)2. Commensurability : this property is trivially verified upon noting that, when each el-ement p it of the price vectors are multiplied by a positive constant γ i and the corre-sponding quantity q it is divided by γ i , then the weights ˜ π i become ˜ π i = π i γ i . Accordingly, ˆ λ ( γ ∗ p , γ ∗ p , γ − ∗ q , γ − ∗ q ) = (cid:80) Ni =1 γ i p i ( γ − i π i ) (cid:80) Ni =1 γ i p i ( γ − i π i ) = (cid:80) Ni =1 p i π i (cid:80) Ni =1 p i π i . (33)14. Proportionality : this property is satisfied upon noting that ˘ π i = απ i . Accordingly, ˆ λ ( p , α p , q , q ) = (cid:80) Ni =1 αp i ˘ π i (cid:80) Ni =1 p i ˘ π i = α (cid:80) Ni =1 p i π i (cid:80) Ni =1 p i π i . (34)4. Dimensionality : the proof of this property follows the same argument used to provehomogeneity. Upon noting that ˘ π i = απ i , it follows that ˆ λ ( α p , α p , q , q ) = (cid:80) Ni =1 αp i ˘ π i (cid:80) Ni =1 αp i ˘ π i = (cid:80) Ni =1 p i π i (cid:80) Ni =1 p i π i . (35)5. Monotonicity : if we consider k > u , then the associated price index is ˆ λ ( p , k ∗ p , q , q ) = (cid:80) Ni =1 k i p i π i (cid:80) Ni =1 p i π i > (cid:80) Ni =1 p i π i (cid:80) Ni =1 p i π i = ˆ λ ( p , p , q , q ) if k i > k j and k j > k i for every i (cid:54) = j given i, j = 1 , . . . , N . While if we consider k ∗ p ,taking into account that the weight vector π is independent from k , the following holds ˆ λ ( k ∗ p , p , q , q ) = (cid:80) Ni =1 p i π i (cid:80) Ni =1 k i p i π i < (cid:80) Ni =1 p i π i (cid:80) Ni =1 p i π i = ˆ λ ( p , p , q , q ) . It is worth noticing that the MPL index enjoys also the following properties: • Positivity : this property follows straightforward given that the index is the sum ofratios of non-negative quantities ( π i , p i , p i ∀ i = 1 , , . . . , N ). • Inverse proportionality in the base period : let consider a vector of prices in the baseperiod α p with α > . Under this case, π i turns out to be independent from α andthe associated index price proves to be proportional to ˆ λ ( p , p , q , q )ˆ λ ( α p , p , q , q ) = (cid:80) Ni =1 p i π i (cid:80) Ni =1 αp i π i = 1 α (cid:80) Ni =1 p i π i (cid:80) Ni =1 p i π i = 1 α ˆ λ ( p , p , q , q ) . • Commodity reversal property : it follows straightforward that the index price is invariantwith respect to any permutation ( i ) : ˆ λ ( p , p , q , q ) = (cid:80) ( N )( i )=1 p ( i )2 π ( i ) (cid:80) ( N )( i )=1 p ( i )1 π ( i ) = (cid:80) Ni =1 p i π i (cid:80) Ni =1 p i π i . Quantity reversal test : a change in the quantity order only affects π i that remainsinvariant ∀ i = 1 , . . . , N . Therefore the index price ˆ λ does not change. • Base reversibility (symmetric treatment of time) : for one commodity, that is N =1 , it is easy to prove that ˆ λ ( p , p , q , q ) = p p and, thus, ˆ λ ( p , p , q , q ) = p p =ˆ λ ( p , p , q , q ) − . • Transitivity : for N = 1 , ˆ λ ( p , p , q , q ) = p p , ˆ λ ( p , p , q , q ) = p p , ˆ λ ( p , p , q , q ) = p p therefore, ˆ λ ( p , p , q , q ) = ˆ λ ( p , p , q , q ) · ˆ λ ( p , p , q , q ) . • Monotonicity : if p = β p then ˜ π i = β (cid:18) p i q i q i q i + q i (cid:19) = βγ i , (36)and the associated index price turns out to be equal to β ˆ λ ( p , β p , q , q ) = (cid:80) Ni =1 β p i γ i (cid:80) Ni =1 βp i γ i = β (cid:80) Ni =1 p i γ i (cid:80) Ni =1 p i γ i = β. In this section, we provide an application of the MPL index to Italian cultural supply data,such as revenues and the number of visitors to museums (i.e. monuments, archeologicalsites, museum circuits, . . . ). The availability of temporal and geographical data on Italianculture provides a stimulating basis for ascertaining the potential of the MPL price-indexmethodology set forth in this paper. The flexibility of the MPL index paves the way tomoving beyond ISTAT (and similar) analyses, which are confined to price indexes on thesupply of data on Italian culture like access to museums and entertainment sectors, aggregatedat the national level (ISTAT, 2018). In addition, to evaluate the performance of the MPLindex, we have made a comparison with the CPD/TPD price indexes (Diewert, 2005; Raoand Hajargasht, 2016), using both real and simulated data. Reference has been made tothis approach because, under the log-normality assumption of the error term, the maximumlikelihood estimator of the said price index tallies with the least square one, likewise with theMPL index. 16s for the nature of the data, note that Italian cultural heritage is at the top of variousworld-class lists and plays a key role in the Italian economy (see, e.g., Alderighi and Gaggero,2018). Lately, local cultural supply has evolved significantly. Indeed, most of the Italian mu-seum circuits were founded relatively recently. Figure 3 shows the MPL price index togetherwith its annual percentage variations for the period 2004–2017: 2004 being the base year and2017 the year used for updating the index. We can note that in the early years of our newMillennium, when important investments started being made in the Italian cultural sector,the prices of museums (and the like) tickets grew (Figure 3). Thereafter, the price dynamicbecame more moderate and then tapered in 2009 and 2014 when, the so called “W” recession,namely the international financial and debt crisis in European peripheral countries, hit Italy.For the sake of further evidence from an empirical standpoint, a comparison of the MPL andthe TPD index has been conducted. First TPD price indexes have been computed only forthose museums whose prices are available at all times. This has led to a drop in the numberof museums/monuments/archeological sites from 36 to 17. Figure 4 (first panel) shows boththe MPL and TPD price indexes together with their 3 σ confidence bounds. The result is that The Italian heritage supply chain accounts for 4,976 museums and the like; it generated almost 200million euro of revenues in 2017 and employs more than 45,000 people (ISTAT, 2016). Approximately 2,300 sites (45.5%) of the Italian cultural supply chain were opened between 1960 and1999, while 2,200 sites (38.6%) were opened in 2000, taking advantage of the investments for economic recoveryand infrastructure enhancement made for Italian cultural heritage sites (ISTAT, 2016). All analyses in this investigation have been made with our own codes, written in R. l l l l l l l l l l l l l Year M P L i nde x l l l l l l l l l l l l l Year % v a r i a t i on M P L i nde x Figure 3: MPL index and percentage annual change.MPL indexes always fall within the confidence bounds of TPD indexes. This result highlightstheir alignment with the latter, and provides evidence of their greater accuracy, due to theirlower standard errors. In the case when not all items (museums) are priced in all periods,TPD estimates have been obtained by using the time version of the weighted CPD (Rao andHajargasht, 2016, pp. 420-421). Figure 4 (second panel) shows both MPL and TPD indexestogether with their σ confidence bounds. The same comments of the complete price tableaucase apply here. l l l l l l l l l l l l l l Year P r i c e i nde x type l MPLTPD l l l l l l l l l l l l l l
Year P r i c e i nde x type l MPLTPD
Figure 4: In the first panel the MPL index is compared to the TPD index with data from17 museums always ranked in the top 30; in the second panel the MPL index is compared tothe weighted TPD index with data from 36 museums ranked in the top 30 at least twice.The availability of data on visitors and revenues in 2017 for museums, monuments, archaeo-logical sites, and museum circuits in the North-West, North-East, Centre and South (whichincludes the two islands Sicily and Sardinia) has allowed the computation of the multilat-eral version of the MPL index. Looking at the data, we see that almost half (46.3%) arelocated in the North, while 28.5% in the Centre, and 25.2% in the South and Islands. TheRegions with the highest number of cultural institutions are Tuscany (11%), followed byEmilia-Romagna (9.6%), Piedmont (8.6%) and Lombardy (8.2%) (ISTAT, 2016). However,alongside the more famous attractions, Italy is home to a wide and rich array of notablelocations of cultural interest. A considerable percentage of these places (17.5%) are found inmunicipalities with less than 2,000 inhabitants, but which can have up to four or five cultural18ites in their small area. Almost a third (30.7%) are distributed in 1,027 municipalities witha population varying from 2,000 to 10,000, and a bit more than half (51.8%) are situated in712 municipalities with a population of 10,000 to 50,000. Italy is, therefore, characterizedby a strongly polycentric cultural supply distributed throughout its territory, even in areasconsidered as marginal from a geographic stance. Table 1 reports, in the first row, the MPLindex computed applying Eq. (22) to the first three areas (North-West, North-East and Cen-tre) considering the Centre as base area. The second row shows the updated values of theMPL index when the South-Islands are added to the data-set. As for the multi-period case,a comparison of the MPL estimates with the CPD ones is provided. The third row of Table 1shows CPD estimates in the case of full price tableau, as all commodities are priced in thefour geographic areas. Figure 5 shows both the MPL and CPD indexes together with their3 σ confidence bounds. Once again, the estimates of the MPL index turn out to be moreaccurate than those provided by the CPD approach, as the former have standard errors lowerthan the latter. As in the comparison with the TPD index, the confidence bounds of CPDindexes always include MPL estimates, thus suggesting the compatibility of both indexes. Itis worth noting that in 2017, access to cultural sites in Southern Italy cost the most: almosttwice as much as in the North-Eastern area. While the disparity could be ascribed to severalfactors, such as different costs of managing museums and similar institutions, tourism flows,etc: that type of analysis goes beyond the scope of the current investigation.Table 1: Updated MPL index compared to the CPD index (standard error in parentheses).North West North East Centre SouthMPL 1.072 (0.185) 0.621 (0.190) 1.000Updated MPL 1.070 (0.164) 0.622 (0.170) 1.000 1.142 (0.086)CPD 1.524 (0.337) 1.283 (0.284) 1.000 1.021 (0.226) Area P r i c e i nde x type CPDMPL
Figure 5: MPL and CPD indexes with their 3 σ confidence bounds.Finally, in order to investigate more thoroughly the performance of the MPL index as com-19ared to the TPD one, a simulation analysis has been performed. One thousand simulationswere carried out by using perturbed values (and prices as a by-product) and assuming fixedquantities (i.e. equal to the original ones). Next, the simulated values (and prices) were usedto compute MPL and TPD indexes in different settings: with and without missing valuesand/or quantities (and accordingly prices). The final MPL and TPD indexes were obtainedas averages of all indexes computed on simulated values and prices. Two types of simulationswere carried out. First simulated values from the nd to the T th period (base period values, v , being kept fixed) were obtained from the original ones ( V ) by adding random termsdrawn from Normal laws with different means and variances. Plots in Figure 6 show boththe MPL and the TPD indexes, for complete and incomplete price tableau cases, togetherwith the associated σ confidence bounds. l l l l l l l l l l l l l l Year P r i c e i nde x type l MPLTPD l l l l l l l l l l l l l l
Year P r i c e i nde x type l MPLTPD
Figure 6: MPL and TPD indexes obtained from simulated data generated by adding to V random terms drawn from a Normal law with a mean equal to 20000 and a standard errorvarying randomly from 0 to 1000. The first and second panel respectively refer to a completeand an incomplete price tableau scenario.Then, simulated values, from the nd to the T th period (base period values, v , beingkept fixed), were obtained from simulated values of the previous period with the addition oferror terms drawn from Normal laws with given means and variances. Plots in Figure 7 showboth the MPL and the TPD indexes, for the case of complete and incomplete price tableaus,together with the associated σ confidence bounds. In both cases, the MPL estimates arein line with the TPD ones, but are more accurate than the latter as their tighter confidencebounds show. The paper works out a novel price index that can be used either as a multi-period or as amultilateral index. This index, called MPL index, is obtained as a solution to an “ad hoc”20 l l l l l l l l l l l l l
Year P r i c e i nde x type l MPLTPD l l l l l l l l l l l l l l
Year P r i c e i nde x type l MPLTPD
Figure 7: MPL and TPD indexes obtained from simulated values υ t obtained by addingstochastic terms drawn from a Normal law with a mean equal to -5000 and a standard errorvarying randomly from 0 to 800 to υ t − , for t = 2 , ...T . The first and second panel respectivelyrefer to a complete and an incomplete price tableau scenario.minimum-norm criterion, within the framework of the stochastic approach. The computa-tion of the MPL index does not require the knowledge of commodity prices, but only theirquantities and values. The reference basket of the MPL index, over periods or across coun-tries, is more informative and complete than the ones commonly used by statistical agencies,and easy to update. The updating process is twofold depending on the multi-period or themultilateral use of the index. An application of the MPL index to the Italian cultural supplydata provides proof of its positive performance. A comparison between the MPL and theCPD/TPD index on both real and simulated data provides evidence of the greater accuracyof the MPL estimates. 21 eferences Alderighi, M. and Gaggero, A. A. (2018). Flight availability and international tourism flows.
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The measurement of general exchange-value , volume 25. Macmillan.25 ppendixA Proofs of Theorems and Corollaries
A.1 Proof of Theorem 1
Proof.
The optimization problem is min λ e (cid:48) Ae . (A.1)The first order condition for a minimum are ∂ e (cid:48) Ae ∂λ = 0 which lead to the solution λ = p (cid:48) Ap p (cid:48) Ap . (A.2)Setting A = ( q q (cid:48) ) in Eq. (A.2) yields the Laspeyeres index λ L = (cid:80) Ni =1 p i q i (cid:80) Ni =1 p i q i = p (cid:48) q p (cid:48) q . (A.3)Setting A = ( q q (cid:48) ) in Eq. (A.2) yields the Paasche index λ P = (cid:80) Ni =1 p i q i (cid:80) Ni =1 p i q i = p (cid:48) q p (cid:48) q . (A.4)Setting A = ( q + q ) ( q + q ) (cid:48) in Eq. (A.2) yields the Marshall-Edgeworth index λ M − E = (cid:80) Ni =1 p i ( q i + q i ) (cid:80) Ni =1 p i ( q i + q i ) = p (cid:48) ( q + q ) p (cid:48) ( q + q ) . (A.5)Finally, setting A = ( ˜ q ˜ q (cid:48) ) in Eq. (A.2), where ˜ q is a vector whose elements are the squareroots of the entries of the vector ( q ∗ q ) , leads to the Walsh index λ W = (cid:80) Ni =1 p i ( q i q i ) (cid:80) Ni =1 p i ( q i + q i ) = p (cid:48) ˜ qp (cid:48) ˜ q . (A.6)26 .2 Proof of Theorem 2 Proof.
In compact form, the model in Eq. (19) can be written as y ( NT, = X ( NT,N + T − β ( N + T − , + µ ( NT, (A.7)where y ( NT, = v N, ( N ( T − , , X ( NT,N + T − = ( N,T − ( q (cid:48) ⊗ I N ) R (cid:48) N ( N,N ) ( I T − ⊗ ( − V )) R (cid:48) T − N ( T − ,T − ( Q (cid:48) ⊗ I N ) R (cid:48) N ( N ( T − ,N ) (A.8)and β ( N + T − , = δ ( T − , ˜ p ( N, , µ ( NT, = ε N, η ( N ( T − , . The ordinary least square estimator of the vector β is given by ˆ β = ( X (cid:48) X ) − X (cid:48) y (A.9)where ( X (cid:48) X ) ( N + T − ,N + T − = R T − ( I T − ⊗ V (cid:48) V ) R (cid:48) T − R T − ( Q (cid:48) ⊗ ( − V (cid:48) )) R (cid:48) N R N ( Q ⊗ ( − V )) R (cid:48) T − R N (( q q (cid:48) + Q Q (cid:48) ) ⊗ I N ) R (cid:48) N = I T − ∗ V (cid:48) V − Q (cid:48) ∗ V (cid:48) − Q ∗ V ( q q (cid:48) + Q Q (cid:48) ) ∗ I N (A.10)and X (cid:48) y ( N + T − , = ( T − , R N ( q ⊗ I N ) v N, = ( T − , q ∗ v N, . (A.11) Note that R N ( q ⊗ I N ) v = R N ( q ⊗ I N )( I ⊗ v ) R (cid:48) = R N ( q ⊗ v ) R (cid:48) = q ∗ v where R = e (cid:48) , ⊗ e (cid:48) , = 1 . Use has been made of the following relationship between the Kronecker and the Hadamard product(Faliva, 1996) A ( N,M ) ∗ B ( N,M ) = R N ( A ⊗ B ) R (cid:48) M ˆ δ = (cid:20) I T − ( T − ,N ) (cid:21) ˆ β ( N + T − , = Λ ( q ∗ v ) (A.12)where Λ is the upper off diagonal block of the inverse matrix ( X (cid:48) X ) − N + T − ,N + T − = Λ = Λ T − ,T − Λ T − ,N ) Λ N,T − Λ N,N ) . Partitioned inversion (see Faliva and Zoia, 2008) leads to Λ = (cid:110) ( I T − ∗ V (cid:48) V ) − ( Q (cid:48) ∗ V (cid:48) ) (cid:2) ( q q (cid:48) + Q Q (cid:48) ) ∗ I N (cid:3) − ( Q ∗ V ) (cid:111) − ( Q (cid:48) ∗ V (cid:48) ) (cid:2) ( q q (cid:48) + Q Q (cid:48) ) ∗ I N (cid:3) − and this yields Eq. (21). A.3 Proof of Corollary 1
Proof.
When T = 2 , Q = q N, and V = v N, . Accordingly, the following holds ( I T − ∗ V (cid:48) V ) = v (cid:48) v . (A.13)Then, upon noting that (cid:2) ( q q (cid:48) + q q (cid:48) ) ∗ I N (cid:3) − = q + q . . . q + q . . . ... ... . . . ... . . . q N + q N to obtain the right-hand sides of Eq. (A.10) and (A.11). q it denotes the quantity of the i th good at time t , some computations prove that ˆ δ = (cid:110) ( v (cid:48) v ) − ( q (cid:48) ∗ v (cid:48) ) (cid:2) ( q q (cid:48) + q q (cid:48) ) ∗ I N (cid:3) − ( q ∗ v ) (cid:111) − ( q (cid:48) ∗ v (cid:48) ) (cid:2) ( q q (cid:48) + q q (cid:48) ) ∗ I N (cid:3) − ( q ∗ v )= v (cid:48) v − [ q v , · · · , q N v N ] q + q . . . q + q . . . ... ... . . . ... . . . q N + q N q v ... q N v N − [ q v , · · · , q N f N ] q + q . . . q + q . . . ... ... . . . ... . . . q N + q N q v ... q N v N == (cid:40) N (cid:88) i =1 v i − N (cid:88) i =1 q i v i q i + q i (cid:41) − (cid:40) N (cid:88) i =1 q i v i q i v i q i + q i (cid:41) = (cid:40) N (cid:88) i =1 q i v i q i + q i (cid:41) − (cid:40) N (cid:88) i =1 q i v i q i v i q i + q i (cid:41) . The reciprocal of the deflator ˆ δ yields the intended price index ˆ λ = (cid:40) N (cid:88) i =1 q i v i q i + q i (cid:41) (cid:40) N (cid:88) i =1 q i v i q i v i q i + q i (cid:41) − = (cid:80) Ni =1 p i π i (cid:80) Ni =1 p i π i where π i = p i q i q i q i + q i . (A.14)Multiplying and dividing the numerator of Eq. (A.14) by p i , the index can be also writtenas N (cid:88) i =1 p i p i ˜ π i (A.15)that is as a convex linear combination of prices with weights given by ˜ π i = v i v i q i q i q i + q i (cid:80) Ni =1 v i v i q i q i q i + q i . (A.16)Further, the index ˆ λ can be also rewritten in a compact form as follows ˆ λ = ( q ∗ v ) (cid:48) D − ( q ∗ v )( q ∗ v ) (cid:48) D − ( q ∗ v ) (A.17)29here D = q + q . . . q + q . . . ... ... . . . ... . . . q N + q N . (A.18)By setting D − / q t = ˜ q t and by making use of the properties of the Hadamard product ,the intended price index may be eventually written as follows ˆ λ = ( ˜ q ∗ v ) (cid:48) ( ˜ q ∗ v )( ˜ q ∗ v ) (cid:48) ( ˜ q ∗ v ) . (A.19) A.4 Proof of Corollary 2
Proof.
The variance-covariance matrix of the estimator (cid:98) β , given in Eq. (A.9), is V ar ( (cid:98) β ) = σ [ X (cid:48) X ] − (A.20)where ( σ I NT ) is the variance-covariance matrix of the stochastic vector µ given in Eq. (A.37).According to Eq. (A.20), the variance-covariance matrix of the vector ˆ δ , as in Eq. (A.12),turns out to be V ( ˆ δ ) = σ [ I T − , T − ,N ][ X (cid:48) X ] − I T − N,T − (A.21)which, with some computations and taking into account Eq. (A.10), can be worked out asfollows V ( ˆ δ ) = σ [ I T − ∗ V (cid:48) V ] − . (A.22)The t th diagonal entries of the above matrix is var (ˆ δ t,t ) = σ (cid:2) v (cid:48) t v t (cid:3) − , (A.23)where v t denotes the t th column of the matrix V . Let D be a diagonal matrix. Then, simple computations show that D ( a ∗ b ) = ( Da ∗ b ) . .5 Proof of Theorem 3 Proof.
When the values, v T +1 , and the quantities, q T +1 , of N commodities in a referencebasket become available for the ( T + 1) th additional country, the reference equation systemfor updating the MPL index becomes V D δ = D ˜ p Q + Ev T +1 δ T +1 = D ˜ p q T +1 + ε T +1 → v = D ˜ p q + ε V ˜ D δ = D ˜ p Q + E v T +1 δ T +1 = D ˜ p q T +1 + ε T +1 . (A.24)After some computations, the above system can be also written as v = ( q (cid:48) ⊗ I N ) R (cid:48) N ˜ p + ε ( N ( T − , = ( I T − ⊗ ( − V )) R (cid:48) T − δ + ( Q (cid:48) ⊗ I N ) R (cid:48) N ˜ p + η = − v T +1 δ T +1 + ( q (cid:48) T +1 ⊗ I N ) R (cid:48) N ˜ p + ε T +1 (A.25)or, in compact form, as y u ( NT + N, = X u ( NT + N,N + T ) β u ( N + T, + µ u ( NT + N, (A.26)where y u ( NT + N, = v N, ( N ( T − , ( N, , X u ( NT + N,N + T ) = ( N,T − ( N, ( q (cid:48) ⊗ I N ) R (cid:48) N ( N,N ) ( I T ⊗ − V ) R (cid:48) T − N ( T − ,T − ( N ( T − , ( Q (cid:48) ⊗ I N ) R (cid:48) N ( N ( T − ,N ) ( N,T − − v T +1( N, ( q (cid:48) T +1 ⊗ I N ) R (cid:48) N ( N,N ) (A.27)and β u ( N + T, = δ ( T − , δ T +1(1 , ˜ p ( N, , µ u ( NT + N, = ε N, η ( N ( T − , ε T +1( N, . Then, following the same argument of Theorem 2, we obtain that31 (cid:48) u X u ( N + T,N + T ) = I T − ∗ V V (cid:48) − Q (cid:48) ∗ V (cid:48) v (cid:48) T +1 v T +1 − v (cid:48) T +1 ∗ q (cid:48) T +1 − Q ∗ V − v T +1 ∗ q T +1 ( q q (cid:48) + Q Q (cid:48) + q T +1 q (cid:48) T +1 ) ∗ I N (A.28)and X (cid:48) u y u ( N + T, = ( T, q ∗ v N, . (A.29)Upon nothing that ˆ δ ( T − , ˆ δ T +1(1 , = (cid:20) I T ( T,N ) (cid:21) ˆ β u ( N + T, = Λ ( q ∗ v ) (A.30)where Λ is the upper off diagonal block of the inverse matrix ( X (cid:48) u X u ) − ( X (cid:48) u X u ) − N + T,N + T ) = Λ = Λ T,T ) Λ T,N ) Λ N,T ) Λ N,N ) , partitioned inversion leads to Λ T,N ) == I T − ∗ V (cid:48) V v (cid:48) T +1 v T +1 − Q (cid:48) ∗ V (cid:48) v (cid:48) T +1 ∗ q (cid:48) T +1 (cid:104) ( q q (cid:48) + Q Q (cid:48) + q T +1 q (cid:48) T +1 ) ∗ I N (cid:105) − (cid:104) Q ∗ V v T +1 ∗ q T +1 (cid:105)(cid:111) − Q (cid:48) ∗ V (cid:48) v (cid:48) T +1 ∗ q (cid:48) T +1 (cid:104) ( q q (cid:48) + Q Q (cid:48) + q T +1 q (cid:48) T +1 ) ∗ I N (cid:105) − . (A.31) Then, pre-multiplying ( q ∗ v ) by Λ yields the estimator ˆ δ ( T − , ˆ δ T +1(1 , . The reciprocal of the(non-null) elements of this estimator provides the values of the updated multilateral versionof the MPL index. 32 .6 Proof of Theorem 4 Proof.
When the values, v T +1 , and the quantities, q T +1 , of N commodities of a referencebasket become available at time T + 1 , the updating of the multi-period version of the MPLindex must not change its past values with meaningful computational advantages. In orderto get the required updating formula, let us rewrite Eq. (16) as follows (cid:34) V ( N,T ) v T +1( N, (cid:35) D ∗ δ ( T,T ) = D ˜ p ( N,N ) (cid:34) Q ( N,T ) q T +1( N, (cid:35) + (cid:34) E ( N,T ) ε T +1( N, (cid:35) (A.32)where D ∗ δ is specified as follows D ∗ δ ( T +1 ,T +1) = ˆ D δ ( T,T ) ( T, (cid:48) (1 ,T ) δ T +1(1 , . (A.33)Here ˆ D δ denotes the estimate of D δ , defined as in Eq. (17), that is ˆ D δ ( T,T ) = (1 , (cid:48) (1 ,T − ( T − , ˆ δ · · ·
00 ˆ δ · · ·
00 0 · · ·
00 0 · · · ˆ δ T = (cid:48) D δ (A.34)where the entries ˆ δ , ˆ δ , . . . , ˆ δ T are the elements of the vector ˆ δ given in Eq. (21). System inEq. (A.32) can be also written as v = D ˜ p q + ε V ˆ˜ D δ = D ˜ p Q + E v T +1 δ T +1 = D ˜ p q T +1 + ε T +1 → V ˆ D δ = D ˜ p Q + Ev T +1 δ T +1 = D ˜ p q T +1 + ε T +1 . (A.35)The application of the vec operator to the first block of equations in Eq. (A.35) yields ˜ v = ( Q (cid:48) ⊗ I N ) R (cid:48) N ˜ p + η = − v T +1 δ T +1 + (cid:0) q (cid:48) T +1 ⊗ I N (cid:1) R (cid:48) N ˜ p + ε T +1 (A.36)33here ˜ v = vec ( V ˆ D δ ) = ( I T ⊗ V ) R (cid:48) T ˜ δ with ˜ δ (cid:48) = [1 , ˆ δ ] (cid:48) and η is equal to vec ( E ) .Eq. (A.36) can be written in vector form as y u ( N ( T +1) , = X u ( N ( T +1) ,N +1) β u ( N +1 , + µ u ( N ( T +1) , (A.37)where y u ( N ( T +1) , = ˜ v , X u ( N ( T +1) ,N +1) = ( Q (cid:48) ⊗ I N ) R (cid:48) N − v T +1 ( q (cid:48) T +1 ⊗ I N ) R (cid:48) N (A.38)and β u ( N +1 , = δ T +1 ˜ p , µ u ( N ( T +1) , = ηε T +1 . The OLS estimator of the vector β u is given by ˆ β u = ( X (cid:48) u X u ) − X (cid:48) u y u (A.39)where X (cid:48) u X u ( N +1 ,N +1) = v (cid:48) T +1 v T +1 − v (cid:48) T +1 (cid:0) q (cid:48) T +1 ⊗ I N (cid:1) R (cid:48) N − R N ( q T +1 ⊗ ( I N )) v T +1 R N (( QQ (cid:48) + q T +1 q (cid:48) T +1 ) ⊗ I N ) R (cid:48) N = v (cid:48) T +1 v T +1 − v (cid:48) T +1 ∗ q (cid:48) T +1 − q T +1 ∗ v T +1 (( QQ (cid:48) + q T +1 q (cid:48) T +1 ) ∗ I N ) (A.40)and X (cid:48) u y u ( N +1 , = (1 , R N ( Q ⊗ I N )˜ v N, = (1 , ( Q ∗ V ) ˜ δ ( N, . (A.41)Now, upon nothing that ˆ δ T +1(1 , = (cid:20) (cid:48) (1 ,N ) (cid:21) ˆ β u ( N +1 , = Λ ( Q ∗ V ) ˜ δ , (A.42) Note that, differently from the proof of Theorem 3, the vector δ does not enter in the updating estimationprocess as it is considered given. Λ is the upper off diagonal block of the inverse matrix ( X (cid:48) u X u ) − ( X (cid:48) u X u ) − N +1 ,N +1) = Λ = Λ , Λ ,N ) Λ N, Λ N,N ) , partitioned inversion leads to Λ = (cid:110) v (cid:48) T +1 v T +1 − (cid:0) q (cid:48) T +1 ∗ v (cid:48) T +1 (cid:1) (cid:2) ( QQ (cid:48) + q T +1 q (cid:48) T +1 ) ∗ I N (cid:3) − ( q T +1 ∗ v T +1 ) (cid:111) − (cid:0) q (cid:48) T +1 ∗ v (cid:48) T +1 (cid:1) (cid:2) ( QQ (cid:48) + q T +1 q (cid:48) T +1 ) ∗ I N (cid:3) − . (A.43)Then, pre-multiplying ( Q ∗ V ) ˜ δ by Λ yields the estimator of ˆ δ T +1+1