A Novel Multi-Period and Multilateral Price Index
AA Novel Multi-Period and Multilateral PriceIndex
Consuelo R. Nava , Maria Grazia Zoia ∗ Department of Economics and Statistics “Cognetti de Martiis”, Univeristà degli Studi diTorino, Lungo Dora Siena 100A, Torino, Italy. Email: [email protected] Department of Economic Policy, Università Cattolica del Sacro Cuore, Largo Gemelli 1,20123, Milano, Italy. Corresponding Author. Tel:+390272342948; fax: +390272342324. Email:[email protected] ∗ Corresponding author
Abstract
A novel approach to price indices, leading to an innovative solution in botha multi-period or a multilateral framework, is presented. The index turnsout to be the generalized least squares solution of a regression model linkingvalues and quantities of the commodities. The index reference basket, whichis the union of the intersections of the baskets of all country/period taken inpair, has a coverage broader than extant indices. The properties of the indexare investigated and updating formulas established. Applications to bothreal and simulated data provide evidence of the better index performance incomparison with extant alternatives.
JEL code: C43; E31; C01.Keywords: multi-period index, multilateral index, GLS solution, updatingformulas, country-product dummy index.1 a r X i v : . [ ec on . E M ] F e b Introduction
Multi-period and multilateral price indices, used to compare sets of commodities overtime and across countries respectively, are of prominent interest for statisticians (see,e.g., Biggeri and Ferrari, 2010). Several approaches to the problem have been carried outin the literature.One of these is the axiomatic approach (see, e.g., Balk, 1995, and the referencesquoted therein), which rests on the availability of both quantities and prices and aims atobtaining price indices enjoying 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 comefrom the optimization of a utility function under budget constraints. Here, prices playthe role of independent variables, while quantities arise as solutions to an optimizationproblem in accordance with the preference scheme OF decision makers.A third approach is the stochastic one (see Clements et al., 2006; Diewert, 2010, fora review), which can be traced back to the works of Jevons (1863, 1869) and Edgeworth(1887, 1925). Thanks to Balk (1980) and Clements and Izan (1987), this approach hasbeen recently reappraised, and its role in inflation measurements duly acknowledged(see, e.g., Asghar and Tahira, 2010, and references quoted therein). In this framework,prices are assumed to be affected by measurement errors whose bias effect must be dulyminimized.The stochastic approach (hereafter, SA) turns out to be somewhat different from otherapproaches, insofar as it is closely related to regression theory (Theil, 1960; Clements andIzan, 1987). In fact, the SA enables the construction of tests and confidence intervalsfor price indices, which provide useful pieces of information (Clements et al., 2006).Furthermore, the SA has less limits than other approaches and clears the way to further This approach is also known as preference field approach or functional approach (Divisia, 1926). The SA, differently from the index number theory does not need to account for the economicimportance of single prices.
In this section, taking the SA as the reference frame, we derive a multi-period/multilateralprice index whose reference basket – over a set of periods or across a set of countries – is theunion of the intersections of the commodity baskets of various periods/countries, takenin pairs. Such a reference basket proves to be an effective solution for several reasons.First, it is broader and more representative than the ones built on the intersection ofcommodities which are present in all periods/countries. Second, the price index is welldefined, provided each commodity is present in at least two baskets. The computation Some similarities arise with the chaining rule (Forsyth and Fowler, 1981; von der Lippe, 2001) wherethe price index is a measure of the cumulative effect of adjacent periods from 0 to 1, 1 to 2, . . . , t − to t . Thus, chain indices compare the current and the previous periods in order to evaluate the evolutionover many periods (for a comparison of this approach with the fixed base one see Diewert (2001)).However, chain indices, unlike the MPL index, leave unresolved the reference basket updating and are
5f the MPL price index hinges on quantities and values of the commodities, not prices.The lack of a commodity in a given period/country t entails that both its quantity andvalue vanish in that period/country.Figure 1 shows the reference basket corresponding to the usual approach as comparedwith that devised in the paper for the case of two and three periods/countries.[Figure 1 about here.]According to the SA, the MPL index is worked out as solution of an optimizationproblem consisting in finding an hyperplane lying as close as possible to the points whosecoordinates are the N commodities prices in T periods. In fact, it hinges on the ideathat in each time/country t , the N commodity prices move proportionally to a set of N reference prices to within “small” discrepancies, that is p t ( N, ≈ λ t ˜ p ( N, ∀ t = 1 , , . . . , T (1)or p t ( N, = λ t ˜ p ( N, + ς t ( N, ∀ t = 1 , , . . . , T. (2)Here p t is the actual price vector of the N commodities at period/country t , ˜ p is thevector of the unknown (time invariant) reference prices, λ t is a scalar factor acting asprice index at period/country t and ς t is the discrepancy vector, that is a vector of errorterms. As per Eq. (2), in each period/country t , the N prices p t can be representedby a point in a N -dimensional space. Accordingly, the N prices in T periods/countries,namely 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,in particular, on a straight line crossing the origin for T = 2 . In general, this is onlyapproximately true and a price “line” crossing the origin is chosen with the property offitting the observed price points, by minimizing the deviations of the data from the “line”. not applicable in a multilateral perspective.
6n compact notation, Eq. (1) can be more conveniently reformulated as follows P ( N,T ) ≈ Π ( N,T ) = ˜ p ( N, λ (cid:48) (1 ,T ) (3)where λ is the vector of the T price indices and ˜ p is the vector of the N (unknown)reference prices. According to Eq. (3), the problem of determining a set of T priceindices can be read as the problem of approximating the price matrix, P , with a matrixof unit rank, Π , defined as the outer product of a vector of price indices by a virtual pricevector. Moving from prices to values, the matrix V of the values of N commodities in T periods/countries has the following representation V ( N,T ) = P ∗ Q ≈ Π ∗ Q ≈ ( ˜ p ( N, λ (cid:48) (1 ,T ) ) ∗ Q ( N,T ) (4)where Q is the matrix of the quantities of N commodities in T periods/countries and ∗ stands for the Hadamard (element-wise) product. According to Eq. (4), the values of N commodities at time t or for the t -th country can be represented as v t = λ t ˜ p ∗ q t + ε t ∀ t = 1 , , . . . , T (5)where v t and q t are the t -th columns of V and Q respectively, and ε t is added to embodythe error term inherent in the model specification of Eq. (4). The above formula, takinginto account the identity ˜ 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 (6)where δ t = ( λ t ) − takes the role of the deflator, and D ˜ p denotes a diagonal matrix with7iagonal entries equal to the elements of ˜ p . Eq. (6) expresses the value, v it , of eachcommodity 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 .By assuming δ t (cid:54) = 0 , its inverse λ t = δ − t tallies with the price index in Eq. (5). 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 ) (7)where D δ is a T × T diagonal matrix with diagonal entries equal to the elements of δ .With no 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 − . The systemtakes the form (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) (8)or equivalently, the form v = D ˜ p q + ε V ˜ D δ = D ˜ p Q + E . (9) The matrix D ˜ p is defined as follows: D ˜ p = ˜ p . . .
00 ˜ p . . . ... ... . . . ... . . . ˜ p N . , the system can be more conveniently rewritten in the form 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 . (10)Here, η = vec ( E ) , δ is a vector whose elements are the diagonal entries of ˜ D δ as specifiedin Eq. (8), and R j denotes the transition matrix from the Kronecker to the Hadamardproduct. An estimate of the vector δ can be obtained by applying generalized leastsquares (GLS), by taking E ( µµ (cid:48) ) = Ω ( NT,NT ) = diag [ ϑ ] = Ω N,N ) ( N,N ( T − ( N,N ) Ω ∗ ( N ( T − ,N ( T − (11)where µ (cid:48) = [ ε , η (cid:48) ] and Ω ∗ is a block diagonal matrix Ω ∗ = Ω N,N ) ( N,N ) ... ( N,N ) ( N,N ) Ω N,N ) ... ( N,N ) ... ( N,N ) ( N,N ) ... Ω T T ( N,N ) . (12)In this connection, we have the following result. 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 thetransition matrix 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. heorem 1. The GLS estimate of the deflator vector δ is given by (cid:98) δ GLS = I T − ∗ (cid:101) V (cid:48) V − ( Q (cid:48) ∗ (cid:101) V (cid:48) ) (cid:34) T (cid:88) j =1 q j q (cid:48) j ∗ Ω − j,j (cid:35) − ( Q ∗ (cid:101) V ) − ( Q (cid:48) ∗ (cid:101) V (cid:48) ) (cid:34) T (cid:88) j =1 q j q (cid:48) j ∗ Ω − j,j (cid:35) − ( q ∗ (cid:101) v ) (13) where (cid:101) V is the matrix whose j -th columns is Ω − j,j v j and (cid:101) v = Ω − , v . The vector (cid:98) λ ,whose entries are the reciprocals of non-null elements of the GLS estimate of δ and zerootherwise, is the MPL index of N commodities over T periods or between T countries . Proof.
See Appendix 1.1.The following corollaries provide estimates of the deflator vector for special cases ofinterest.
Corollary 1.
Let us assume that the error terms of the system in Eq. (10) are stationary.Accordingly, the diagonal blocks, Ω j,j of the matrix Ω are the same, i.e. Ω j,j = (cid:101) Ω for all j . Then, the GLS estimate of the deflator vector, δ , is (cid:98) δ GLS = (cid:26) ( I T − ∗ (cid:98) V (cid:48) V ) − ( Q (cid:48) ∗ (cid:98) V (cid:48) ) (cid:104) ( q q (cid:48) + Q Q (cid:48) ) ∗ (cid:101) Ω − (cid:105) − ( Q ∗ (cid:98) V ) (cid:27) − · ( Q (cid:48) ∗ (cid:98) V (cid:48) ) (cid:104) ( q q (cid:48) + Q Q (cid:48) ) ∗ (cid:101) Ω − (cid:105) − ( q ∗ (cid:98) v ) (15) where (cid:98) V is the matrix whose j − th column is (cid:101) Ω − v j and (cid:98) v = (cid:101) Ω − v . Proof.
The proof is given in Appendix 1.2.The case T = 2 is worth considering because it sheds light on the index structure. Whenassuming stationary and uncorrelated error terms, the matrix (cid:101) Ω reduces to a diagonal The vector λ is defined as follows λ = [ λ t ] with λ t = (cid:40) δ − t if δ t (cid:54) = 00 otherwise . (14) (cid:101) Ω = diag ( θ ) = [ θ i ] , and the price index turns out to be simply the ratio ofweighted price averages, as stated in the following corollary. Corollary 2.
Let the matrix (cid:101) Ω be diagonal and T = 2 . Then the MPL index (cid:98) λ GLS for aset of N commodities is (cid:98) λ GLS = (cid:80) Ni =1 p i π i (cid:80) Ni =1 p i π i = p (cid:48) πp (cid:48) π . (16) Here p t is the price vector of the N commodities at time t and π is the weight vectorwhose i -th entry is π i = 1 ϑ i p i q i q i q i + q i . (17) Proof.
See Appendix 1.3.In case the diagonal entries ϑ i of (cid:101) Ω are all equal, say equal to 1, then Eq. (16) wouldtally with the OLS version of the MPL index. In this case the weights of the index wouldbe function of the harmonic mean of the squared quantities π i = p i q i q i q i + q i . (18)Two considerations are worth making about Eq. (16). The first is that most well-knownprice indices can be viewed as particular cases of the MPL index. In fact, they can beobtained from Eq. (16) for particular choices of the scalars ϑ i , as proved in the followingcorollary. Corollary 3.
By taking ϑ i = p i q i q i q i + q i , (19) then π i = q i and the MPL index tallies with the Laspeyeres index, (cid:98) λ L , i.e. (cid:98) λ GLS = (cid:98) 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 . (20)11 y taking ϑ i = p i q i q i q i + q i , (21) then π i = q i and the MPL index turns out to tally with the Paasche index, (cid:98) λ P , i.e. (cid:98) λ GLS = (cid:98) 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 . (22) By taking ϑ i = p i q i q i ( q i + q i )( q i + q i ) , (23) then π i = ( q i + q i ) and the MPL index turns out to tally with the Marshal-Edgeworthindex, (cid:98) λ ME i.e. (cid:98) λ GLS = (cid:98) λ ME = (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 ) . (24) By taking ϑ i = p i ( q i q i ) / q i + q i , (25) then π i = ( q i q i ) / and the MPL index turns out to tally with the Walsh index, (cid:98) λ W , i.e. (cid:98) λ GLS = (cid:98) 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) (cid:101) qp (cid:48) (cid:101) q , (26) where (cid:101) q is a vector whose entries are the square roots of those of the vector ( q ∗ q ) .By taking ϑ i = 1 p i (27) and considering the square roots of the quantities in the index computation, then π i = q i q i q i + q i and the MPL index turns out to tally with the Geary-Khamis index (Drechsler,1973), (cid:98) λ GK , i.e. (cid:98) λ GLS = (cid:98) λ GK = (cid:80) Ni =1 p i q i q i q i + q i (cid:80) Ni =1 p i q i q i q i + q i = p (cid:48) (cid:98) qp (cid:48) (cid:98) q , (28)12 here (cid:98) q is a vector whose i -th entry is q i q i q i + q i .Proof. The proof is simple and follows straight forward. Therefore, it is omitted.The other consideration is that the index (cid:98) λ GLS can be obtained as solution of an opti-mization problem specified as in the following corollary.
Corollary 4.
With reference to the following model p = λ p + η (29) where η is a vector of random terms, the index (cid:98) λ GLS is solution of the optimizationproblem min λ (cid:107) p − λ p (cid:107) = min λ | e (cid:107) (30) where (cid:107) e (cid:107) = ( e (cid:48) Ae ) / is a (semi)norm of e and A is given by A = ππ (cid:48) (31) with π is a vector whose the i -th entry, π i , is specified as in Eq. (17) .Proof. See Appendix 1.4.Eq. (29), together with Figure 1, is useful to show that the index basket is the unionof the intersection of the commodities that are present in at least two periods/countriesand, accordingly, it does not include commodities which are not present in at least twoperiods/countries. In fact, simple computations prove that weights pertaining commodi-ties not fulfilling this condition turn out to be null. Consequently, the value of the indexdoes not change if computed leaving out them, as shown by the following example.
Example 1.
Let us consider the case of three commodities in two periods and assumethat the first commodity is missing in both periods (i.e. q = q = 0 ). Then, let us13ssume that ϑ i = 1 for each i for simplicity. Simple computations prove that the weight,as defined in Eq. (18), associated to the first coefficient turns out to be null π = p q q q + q = p i (cid:32) q i + 1 q i (cid:33) = p + = 0; The same happens if the first commodity is missing in just one period, for instance thesecond one ( q = 0 ) As in the previous case, the coefficient π vanishes π = p q q q + q = 0; Accordingly, in both cases the first commodity can be left out from the computation ofthe index. In fact, the value of the latter does not change if computed by consideringonly the second and the third commodities.As a by-product of Theorem 1, we state the following result.
Corollary 5.
The variance-covariance matrix of the deflator vector, (cid:98) δ , given in Theo-rem 1 is V ar ( (cid:98) δ ) = σ κ − (32) where κ = I T − ∗ (cid:101) V (cid:48) V − ( Q (cid:48) ∗ (cid:101) V (cid:48) ) (cid:34) T (cid:88) j =1 q j q (cid:48) j ∗ Ω − j,j (cid:35) − ( Q ∗ (cid:101) V ) . The t -th diagonal entry of the above matrix provides the variance of the deflator in the t -th period/country, given by var ( (cid:98) δ t ) = σ (cid:101) v (cid:48) t v t − ( q (cid:48) t ∗ (cid:101) v (cid:48) t ) (cid:34) T (cid:88) j =1 q j q (cid:48) j ∗ Ω − j,j (cid:35) − ( q t ∗ (cid:101) v t ) − = σ κ − t (33) where (cid:101) v t , v t and q t denote the t -th column of the matrix (cid:101) V , V and Q , respectively.Proof. See Appendix 1.5. 14s for the deflator vector (cid:98) δ , its moments and confidence intervals can be easily obtainedwithin the theory of linear regression models, from the result given in Corollary 5. As theprice index vector (cid:98) λ turns out to be the reciprocal of the said deflator (see Eq. 13), itsstatistical behavior 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 (in particular reciprocals) of random variables. The following corollary providesan approximation of the variance of (cid:98) λ t , obtained by using the first Taylor expansion ofthe variance of a ratio of two random variables. Corollary 6.
The variance of the MPL index (cid:98) λ t is var ( (cid:98) λ t ) ≈ var ( (cid:98) δ t ) E ( (cid:98) δ t ) = (cid:98) σ κ t E ( (cid:98) δ t ) ∀ t = 1 , . . . , T. (34) In the above equation (cid:98) σ = (cid:98) µ (cid:48) (cid:98) Ω − (cid:98) µ NT − ( N + T − where (cid:98) µ are the GLS residuals of Eq. (10) and (cid:98) Ω is an estimate of Ω .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 (cid:98) 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 foran additional T + 1 country. In the latter case, it is supposed that values and quantitiesof the commodities included in the reference basket become available at time T + 1 . It isworth noting that the approach used to update the index guarantees the temporal fixityissue requiring that its historical values must not be affected by the inclusion of valuesand quantities pertaining a new period. Spatial fixity, demanding that results for a core15et of countries must be unaffected by the inclusion of new countries, is not preserved bythe updating method here proposed. This property can be easily fulfilled by updatingthe index with the same approach used for the multi-period index. Theorem 2.
Should the values and quantities of N commodities of a reference basketbecome available for a new additional country, say the T + 1 -th, then, the updated mul-tilateral version of the MPL index, (cid:98) λ , turns out to be the vector of the reciprocals, asdefined in Eq. (13) , of the following deflator vector (cid:98) δ ( T, = I T − ∗ (cid:101) V (cid:48) V (cid:101) v (cid:48) T +1 v T +1 − Q (cid:48) ∗ (cid:101) V (cid:48) (cid:101) v (cid:48) T +1 ∗ q (cid:48) T +1 (cid:34) T +1 (cid:88) j =1 q j q (cid:48) j ∗ Ω − jj (cid:35) − (cid:20) Q ∗ (cid:101) V (cid:101) v T +1 ∗ q T +1 (cid:21)(cid:27) − Q (cid:48) ∗ (cid:101) V (cid:48) (cid:101) v (cid:48) T +1 ∗ q (cid:48) T +1 (cid:34) T +1 (cid:88) j =1 q j q (cid:48) j ∗ Ω − jj (cid:35) − ( q ∗ (cid:101) v ) . (35) Here the symbols are defined as in Theorem 1. The terms v T +1 , q T +1 denote the vectorof values and quantities of N commodities of the new T + 1 -th country, respectively and (cid:101) v T +1 = Ω − T +1 ,T +1 v T +1 with Ω T +1 ,T +1 variance-covariance matrix of the disturbances attime T + 1 .Proof. See Appendix 1.6.
Theorem 3.
Should the values and quantities of N commodities of a reference basketbecome available for time T + 1 , then, the updated value (cid:98) λ T +1 of the multi-period versionof the MPL index at time T + 1 turns out to be the reciprocal of the deflator value at time T + 1 (cid:98) δ T +1(1 , = (cid:101) v (cid:48) T +1 v T +1 − (cid:0) q (cid:48) T +1 ∗ (cid:101) v (cid:48) T +1 (cid:1) (cid:34) T +1 (cid:88) j =1 q j q (cid:48) j ∗ Ω j,j (cid:35) − ( q T +1 ∗ (cid:101) v T +1 ) − (cid:0) q (cid:48) T +1 ∗ (cid:101) v (cid:48) T +1 (cid:1) (cid:34) T +1 (cid:88) j =1 q j q (cid:48) j ∗ Ω j,j (cid:35) − (cid:16) Q ∗ (cid:101) V (cid:17) ˜ δ (36) where ˜ δ (cid:48) = [1 , (cid:98) δ ] (cid:48) and (cid:98) δ is defined as in Eq. (13) . roof. See Appendix 1.7.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. [Figure 2 about here.]
Let us assume for simplicity T = 2 and denote with (cid:98) λ ( p , p , q , q ) a generic indexnumber where p t and q t are prices and quantities at time t . Without lack of generality, t = 1 is assumed to be the base period. Following Predetti (2006), Martini (1992) andFattore (2010), the main properties of an index number can be summarized as follows:P.1 Strong identity : (cid:98) λ ( p , p , q , q ) = 1 .P.2 Commensurability : (cid:98) λ ( γ ∗ p , γ ∗ p , γ − ∗ q , γ − ∗ q ) = (cid:98) λ ( p , p , q , q ) where γ ( N, is a vector with non-null entries and γ − is the vector of reciprocals of the entriesof γ .P.3 Proportionality : (cid:98) λ ( p , α p , q , q ) = α (cid:98) λ ( p , p , q , q ) with α > .P.4 Dimensionality : (cid:98) λ ( α p , α p , q , q ) = (cid:98) λ ( p , p , q , q ) with α > .P.5 Monotonicity : (cid:98) λ ( p , k ∗ p , q , q ) > (cid:98) λ ( p , p , q , q ) and (cid:98) λ ( k ∗ p , p , q , q ) < (cid:98) λ ( p , p , q , q ) with k ( N, > u where u is the unit vector.Moreover, also the following properties are worth mentioning:P.6 Positivity : (cid:98) λ ( α p , p , q , q ) ≥ , with α positive constant;P.7 Inverse proportionality in the base period : (cid:98) λ ( α p , p , q , q ) = α (cid:98) λ ( p , p , q , q ) ;17.8 Commodity reversal property : invariance of the index with respect to any commod-ity permutation;P.9
Quantity reversal test : a change in the quantity order does not affects π i thatremains invariant ∀ i = 1 , . . . , N . Therefore the index price (cid:98) λ does not change.P.10 Base reversibility (symmetric treatment of time) (cid:98) λ ( p , p , q , q ) = (cid:98) λ ( p , p , q , q ) − ;P.11 Transitivity (cid:98) λ ( p , p , q , q ) (cid:98) λ ( p , p , q , q ) = (cid:98) λ ( p , p , q , q ) ;P.12 Monotonicity : If p = β p , then (cid:98) λ ( p , β p , q , q ) = β . Proposition 1.
The MPL index satisfies all properties.Proof.
See Appendix 1.8.
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 wayto moving beyond ISTAT (and similar) analyses, which are confined to price indices onthe supply of data on Italian culture like access to museums and entertainment sectors,aggregated at the national level (ISTAT, 2020). In addition, to evaluate the performanceof the MPL index, we have made a comparison with the CPD/TPD price indices (Diewert,2005; Rao and Hajargasht, 2016), using both real and simulated data. Reference has beenmade to this approach because, under the log-normality assumption of the error term, themaximum likelihood estimator of the said price index tallies with the least square one,18ikewise the MPL index. As for the nature of the data, note that Italian cultural heritageis at the top of various world-class lists and plays a key role in the Italian economy (see,e.g., Symbola, 2019). Lately, local cultural supply has evolved significantly. Indeed, mostof the Italian museum circuits were founded relatively recently. Figure 3 shows the MPL price index together with its annual percentage variations forthe period 2004–2017: 2004 being the base year and 2017 the year used for updating theindex. Here the computation of the index has been done by assuming non spherical, and inparticular heteroschedastic and uncorrelated (GLS-d) error terms. Looking at the graph,we can note that in the early years of the new Millennium, when important investmentsstarted being made in the Italian cultural sector, the prices of museums (and the like) The Italian heritage supply chain accounts for 4,889 museums and the like; it generated almost 200million euro of revenues in 2017 and employs 38.300 people (ISTAT, 2019). 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 economicrecovery and 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. σ confidencebounds. In the left panel the price indices have been computed only for those museumswhose prices are available at all times. This has led to a drop in the number of museum-s/monuments/archaeological sites from 36 to 17 (note that this case corresponds to the“standard” reference basket). The right panel shows these indices computed with datafrom 36 museums ranked in the top 30 at least twice together with their σ confidencebounds. In this case, when not all items (museums) are priced in all periods, TPD es-timates have been obtained by using the time version of the weighted CPD (Rao andHajargasht, 2016, pp. 420-421). Looking at both panels we see that the two indicesare aligned, but that the MPL one always fall within the confidence bounds of TPDindices. This result provides evidence of the MPL greater efficiency, due to its lowerstandard errors. For completeness, in Appendix 2.1 the estimated reference prices, underthe different error specifications, have been displayed.20Figure 4 about here.]The availability of data on visitors and revenues in 2017 for museums, monuments,archaeological sites, and museum circuits in the North-West, North-East, Centre andSouth (which includes the two islands Sicily and Sardinia) has allowed the computationof the multilateral version of the MPL index. Looking at the data, we see that almosthalf (46.3%) are located in the North, while 28.5% in the Centre, and 25.2% in theSouth and Islands. The Regions with the highest number of cultural institutions areTuscany (11%), followed by Emilia-Romagna (9.6%), Piedmont (8.6%) and Lombardy(8.2%) (ISTAT, 2016). However, alongside the more famous attractions, Italy is home toa wide and rich array of notable locations of cultural interest. A considerable percentageof these places (17.5%) are found in municipalities with less than 2,000 inhabitants, butwhich can have up to four or five cultural sites in their small area. Almost a third (30.7%)are distributed in 1,027 municipalities with a population varying from 2,000 to 10,000,and a bit more than half (51.8%) are situated in 712 municipalities with a populationof 10,000 to 50,000. Italy is, therefore, characterized by a strongly polycentric culturalsupply distributed throughout its territory, even in areas considered as marginal from ageographic stance.Table 1 reports, in the first three rows, the MPL index, computed under the assump-tion of stationary (GLS-s), non-spherical (GLS-d) and spherical (OLS) error terms, forthree areas (North-West, North-East and Centre) considering the Centre as base area.The following three rows show updated values of the MPL index, computed for the GLS-s, GLS-d and OLS specifications of the error terms, when the South-Islands are added tothe data-set. As for the multi-period case, a comparison of the MPL estimates with theCPD ones is provided. The last row of Table 1 shows CPD estimates in the case of fullprice tableau, as all commodities are priced in the four geographic areas. Figure 5 showsboth the MPL and CPD indices together with their σ confidence bounds. The compari-son of the CPD and MPL indices computed under other specifications for the error terms21GLS-f and OLS) is provided in Appendix Appendix 2.2. Once again, the estimates ofthe MPL index turn out to be more accurate than those provided by the CPD approach,as the former have standard errors lower than the latter. As in the comparison withthe TPD index, the confidence bounds of CPD indices always include MPL estimates,thus suggesting the compatibility of the MPL index with the estimates provided with theCPD one. It is worth noting that in 2017, access to cultural sites in Southern Italy costthe most: almost twice as much as in the North-Eastern area. While the disparity couldbe ascribed to several factors, such as different costs of managing museums and similarinstitutions, tourism flows, etc: that type of analysis goes beyond the scope of the currentinvestigation. [Table 1 about here.][Figure 5 about here.]Finally, in order to investigate more thoroughly the performance of the MPL indexas compared to the TPD one, a simulation analysis has been performed based on theperturbation of the original value matrix V . In particular, one thousand simulations havebeen carried out by using perturbed values (and prices as a by-product) and assumingfixed quantities (i.e. equal to the original ones). Next, the simulated values (and prices)have been used to compute MPL and TPD indices in different settings: with and withoutmissing quantities (and accordingly prices in the TPD model). The final MPL and TPDindices have been obtained as averages of all indices computed on simulated values (andprices). Two types of simulations have been carried out. First, simulated values from the -nd to the T -th period (base period values, v , being kept fixed) have been obtainedby adding random perturbations, drawn from Normal laws with different means andvariances, to the original values of V . The plots in Figure 6 show both the MPL andthe TPD indices obtained by using these simulated data together with the associated σ confidence bounds. As before the MPL index has been computed under the assumptionof heteroschedastic and uncorrelated error terms (GLS-d).22Figure 6 about here.]Then, following another approach, simulated values, from the -nd to the T -th period(base period values, v , being kept fixed), have been obtained from simulated values of theprevious period with the addition of perturbation terms, drawn from Normal laws withgiven means and variances. Plots in Figure 7 show both the MPL, and the TPD indices,for the case of complete and incomplete price tableau, together with the associated σ confidence bounds. As before the MPL index has been computed under the assumptionof heteroschedastic and uncorrelated error terms (GLS-d). Looking at these figures, wesee that in both cases, the MPL estimates are in line with the TPD ones, but are moreaccurate than the latter as their tighter confidence bounds show.[Figure 7 about here.]In Appendix 2.3, a comparison of the TPD and the MPL indices, always computed ondata simulated and by assuming other specifications for the error terms, (GLS-f, GLS-sand OLS), is provided. To further illustrate the potentialities of the MPL index, the latter has been computedby using a simulated data set built as follows. For given q t , ˜ p t and λ t , the values v t have been computed according to Eq. 5. The random terms (cid:15) t of this equation have beengenerated from a standard Normal distribution. These elements represent the ingredientsof the system in Eq. 10 used to work out the MPL index. The aim of this simulation isto see the capability of the MPL and TPD indices to reproduce the values of the “true”index λ . Let’s assume that the quantities, Q , the reference prices, ˜ p and the values of23he price index, λ , of four commodities from 2015 to 2020 are specified as follows Q = , λ = . . . . . . , ˜ p = . . . . . Then, with these data at hand, the values computed as in Eq. 5 result to be V = .
29 19 .
10 24 .
14 23 .
36 38 .
17 53 . .
78 30 .
51 34 .
40 17 .
31 28 .
40 19 . .
08 26 .
43 31 .
29 37 .
17 34 .
83 22 . .
15 10 .
01 10 .
18 21 .
73 33 .
28 47 . while the price matrix, needed to compute the TPD index, λ T P D hereafter, is P = .
86 2 .
39 2 .
41 2 .
34 2 .
54 2 . .
52 1 .
69 1 .
72 1 .
73 1 .
89 1 . .
94 0 .
98 1 .
04 1 .
06 1 .
16 1 . .
43 2 .
00 2 .
04 2 .
17 2 .
22 2 . . The matrices Q and V have been used to compute both the MPL and the TPD in-dixes, in a multi-period perspective. In particular, the MPL index has been computedunder several specification of the error terms and, more precisely, stationary (GLS-s),heteroschedastic and uncorrelated (GLS-d), spherical (OLS) error terms.We propose three different examples in which the MPL is compared with the TPD inthe following cases1. complete price tableau, implying a reference basket including the complete set of24he four commodities;2. incomplete price tableau, assuming missing the second and forth commodity, (thatis q = v = 0 and q = v = 0 ), with a “standard” reference basket, (see theleft-hand side of Figure 1) that, accordingly includes only the first and the thirdcommodities;3. incomplete price tableau assuming missing the second and forth commodity, (thatis q = v = 0 and q = v = 0 ), with the “MPL” reference basket, (see the right-hand side of Figure 1), that includes commodities present in at least two periods,namely all the four commodities.The rows of Table 2 provides the sum of the squares of the differences between theestimated indices (GLS-s, GLS-d, OLS, and TPD) ˆ λ and the index λ for the three cases .Looking at this table, we see that the MPL index, whatever is the specification assumedfor the error terms, provides always the best fit to the index λ . Thus, for any specificationof the error terms, the MPL index exhibits an higher performance than the TPD, exceptfor the GLS-d in the third case. In Appendix 3, the graphs of both the TPD and MPLindixes are provided under different specifications for the error terms (GLS-s, GLS-d;OLS). In all cases the MPL estimates turn out to be more accurate, as they have lowervariances and, consequently, they are always included in a σ confidence band of the TPD(see Figure 16).These examples are also particularly interesting to highlight the role played by thereference prices ˜ p , which are the prices that consumers are expected to pay for the com-modities in the base period/country. Reference prices prove useful to obtain estimatesof the prices of those commodities which, being missing in the basket, can not be deter-mined. This case occurs in the third example with incomplete price tableau and referencebasket including the complete set of commodities. In this case, if a commodity, say j , ismissing in a period, say i , then its price, even if different from zero, turns out to be un-detectable. However, in the MPL approach, it can be determined, through the estimates25f both its associated reference price ˆ˜ p j and price index ˆ λ i , as follows ˆ˜ p j ˆ λ i . This strategyhas been used to estimate the prices of the second and fourth commodity in the thirdcase.To assess the goodness of the estimates ˆ˜ p j ˆ λ i , j = 2 , , i = 1 , ..., in reproducing thereal prices ˜ p j λ i , j = 2 , , i = 1 , ..., , the sum of the squares between observed and es-timated prices have been computed for all commodities, either included in the basket ormissing. The results, reported in Table 3 in Appendix 3, are very satisfactory. Lookingat Figure 17 in Appendix 3, which compares the estimates of the reference prices withthe “real” ones, it is clear that the MPL proves able to suitably estimate the referenceprices for all commodities, also for the missing ones. Figure 18 compares the values ofthe MPL and TPD indexes ˆ λ to the real price index λ of this experiment. Interestingly,differently from the TPD, the MPL better captures the trend of the “real” price indexover time, avoiding a TPD overestimation issue (see Table 2) in all cases (except for theGLS-d incomplete price tableau with the novel basket).Finally, in Table 4 in Appendix 3 “real” prices P have been compared with theirestimates obtained by using the MPL and the TPD index, given by ˆ P MP L = ˜ p ˆ λ (cid:48) and (cid:98) P T P D = ˜ p ˆ λ (cid:48) T P D for the former and the latter, respectively. The better performance ofthe MPL compared to the TPD one in reproducing the sequences of prices emerges fromTable 4, providing the sum of the squares of the differences between the real prices andthe ones estimated by the two indexes.[Table 2 about here.][Figure 8 about here.]
The paper works out a novel price index that can be used either as a multi-period oras a multilateral index. This index, called MPL index, is obtained as a solution to an26ad hoc” minimum-norm criterion, within the framework of the stochastic approach. Thecomputation of the MPL index does not require the knowledge of commodity prices, butonly their quantities and values. The reference basket of the MPL index, over periodsor across countries, is more informative and complete than the ones commonly used bystatistical agencies, and easy to update. The updating process is twofold depending onthe multi-period or the multilateral use of the index. An application of the MPL index tothe Italian cultural supply data provides proof of its positive performance. A comparisonbetween the MPL and the CPD/TPD index on both real and simulated data providesevidence of the greater efficiency of the MPL estimates.Thus, the MPL index is very promising both in the multi-period and the multilateralperspective, also considering the simple and efficient way of updating the index series(Diewert and Fox, 2020).The approach here proposed can be extended along several paths. For instance,the application of the MPL index for a multilateral comparison across countries usingdifferent currencies would require a suitable adjustment to ensure comparability amongcountries. This could be done by using a set of purchasing power parities to convert thedifferent currencies into a common one. In this case, the MPL index could be built byemploying “international” quantities and “country volumes” as suggested by (Balk, 1996).Furthermore, the MPL approach could be employed to construct price indexes acrossboth space and time as in (Hill, 2004). Both these research lines are being investigated.
Acknowledgements
We sincerely thank Prof. E. Diewert for his valuable and constructive suggestions, aswell as precious comments during the development of this article.All remaining errors, typos or inconsistencies of this work are of our own.27 eferences
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Table 1: Updated MPL index assuming spherical (OLS) and non-spherical (GLS-d andGLS-s) error terms compared to the CPD index (standard error in parentheses).North West North East Centre SouthMPL-GLS-s 1.070 (0.021) 0.634 (0.018) 1.000MPL-GLS-d 1.162 (0.165) 0.632 (0.011) 1.000MPL-OLS 1.072 (0.200) 0.621 (0.194) 1.000Updated MPL-GLS-f 1.048 (0.078) 0.665 (0.087) 1.000 1.107 (0.036)Updated MPL-GLS-d 1.148 (0.174) 0.631 (0.008) 1.000 1.143 (0.001)Updated MPL-OLS 1.070 (0.176) 0.622 (0.174) 1.000 1.142 (0.153)CPD 1.524 (0.337) 1.283 (0.284) 1.000 1.021 (0.226)Table 2: Sum of square of the differences between the estimated indices and the real one.Data GLS-s GLS-d OLS TPD1. Complete pricetableau 0.00308 0.00311 0.00333 0.043272. Incomplete pricetableau (“classical basket”) 0.00053 0.00327 0.00126 0.005203. Incomplete pricetableau (novel basket) 0.00219 0.00352 0.00212 0.0030132 igures
Figure 1: The left-hand side panel shows the reference basket corresponding to the “tra-ditional” approach. The right-hand side panel shows the reference basket correspondingto the MPL index. the 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 multilateralversion of the MPL index (see Theorem 2); the bottom panel shows the ratio of theupdating formula of the the multi-period version of the MPL index (see Theorem 3).33 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 −0.050.000.050.100.15 2005.0 2007.5 2010.0 2012.5 2015.0 2017.5 Year % v a r i a t i on M P L i nde x Figure 3: MPL index and percentage annual change assuming heteroschedastic and un-correlated error terms (GLS-d). 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 GLS−dOLSTPD 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 GLS−dOLSTPD
Figure 4: The left panel compares the MPL index, computed for the GLS-d and OLSspecification for the error terms, to the TPD index. Both the indices have been builtby using data from 17 museums always ranked in the top 30; The right panel comparesthe MPL index, computed for the GLS-d and OLS specification for the error terms, toweighted TPD index. Here data from 36 museums ranked in the top 30 at least twicehave been used to compute both indices.
Area P r i c e i nde x type CPDGLS−d
Figure 5: GLS-d and CPD indices with their σ confidence bounds.34 l l l l l l l l l l l l l Year P r i c e i nde x type l MPL (GLS−d)TPD 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 MPL (GLS−d)TPD
Figure 6: MPL and TPD indices obtained by using simulated data obtained by addingrandom variables, generated from a Normal law (with a mean equal to 20000 and astandard error varying randomly from 0 to 1000), to the values of V . The left and rightpanels refer to the complete and incomplete price tableau scenario, respectively. 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 MPL (GLS−d)TPD 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 MPL (GLS−d)TPD
Figure 7: MPL and TPD indices obtained, at each time t , for t = 2 , . . . , T , by addingperturbation terms (drawn from a Normal law with a mean equal to -5000 and a standarderror varying randomly from 0 to 800) to the simulated values υ t − . The left and rightpanels refer to the complete and incomplete price tableau scenario respectively. l l l l l l Year P r i c e i nde x type l GLS−dGLS−sOLSRealTPD l l l l l l
Year P r i c e i nde x type l GLS−dGLS−sOLSRealTPD l l l l l l
Year P r i c e i nde x type l GLS−dGLS−sOLSRealTPD
Figure 8: Comparison across different version of the MPL and TPD without confidencebands are proposed for the three examples: complete price tableau, incomplete pricetableau “standard” reference basket, incomplete price tableau novel reference basket, re-spectively. 35 ppendix1 Proofs of Theorems and Corollaries
Proof.
In compact form, the model in Eq. (10) can be written as y ( NT, = X ( NT,N + T − β ( N + T − , + µ ( NT, (A.1)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 ) and β ( N + T − , = δ ( T − , ˜ p ( N, , µ ( NT, = ε N, η ( N ( T − , . The generalized least square estimator of the vector β is given by (cid:98) β GLS = ( X (cid:48) Ω − X ) − X (cid:48) Ω − y (A.2)where, Ω is as defined in Eq. (11). ome computation prove that, X (cid:48) Ω − X ( N + T − ,N + T − = R T − (cid:2) ( I T − ⊗ V (cid:48) ) Ω ∗− ( I T − ⊗ V ) (cid:3) R (cid:48) T − − R T − (cid:2) ( I T − ⊗ V (cid:48) ) Ω ∗− ( Q (cid:48) ⊗ I N ) (cid:3) R (cid:48) N − R N (cid:2) ( Q ⊗ I N ) Ω ∗− ( I T − ⊗ V ) (cid:3) R (cid:48) T − R N ( q q (cid:48) ⊗ Ω − ) R (cid:48) N + R N ( Q ⊗ I N ) Ω ∗− ( Q (cid:48) ⊗ I N ) R (cid:48) N = R T − (cid:104)(cid:80) T − j =1 ( e j e (cid:48) j ⊗ V (cid:48) Ω − j +1 ,j +1 V ) (cid:105) R (cid:48) T − − R T − (cid:104)(cid:80) T − j =1 ( e j ⊗ V (cid:48) Ω − j +1 ,j +1 )( q (cid:48) j ⊗ I N ) (cid:105) R (cid:48) N − R N (cid:104)(cid:80) T − j =1 ( e (cid:48) j ⊗ Ω − j +1 ,j +1 V )( q j ⊗ I N ) (cid:105) R (cid:48) T − R N (cid:104)(cid:80) Tj =1 q j q (cid:48) j ⊗ Ω − j,j (cid:105) R (cid:48) N = (cid:104)(cid:80) T − j =1 ( e j e (cid:48) j ∗ V (cid:48) Ω − j +1 ,j +1 V ) (cid:105) − (cid:104)(cid:80) T − j =1 ( e j q (cid:48) j +1 ∗ V (cid:48) Ω − j +1 ,j +1 ) (cid:105) − (cid:104)(cid:80) T − j =1 ( e (cid:48) j q j +1 ∗ Ω − j +1 ,j +1 V ) (cid:105) (cid:104)(cid:80) Tj =1 q j q (cid:48) j ∗ Ω − j,j (cid:105) = I T − ∗ (cid:101) V (cid:48) V − Q (cid:48) ∗ (cid:101) V (cid:48) − Q ∗ (cid:101) V (cid:80) Tj =1 q j q (cid:48) j ∗ Ω − j,j = A BC D where e j is the T − dimensional j -th elementary vector and (cid:101) V is a N × ( T − matrix whose j -th column is Ω − j,j v j .Furthermore X (cid:48) Ω − y ( N + T − , = ( T − , R N ( q ⊗ I N )( Ω − v ) ( N, = ( T − , q ∗ Ω − v N, . (A.3) Note that R N ( q ⊗ Ω − ) v = R N ( q ⊗ Ω − )( I ⊗ v ) R (cid:48) = R N ( q ⊗ Ω − v ) R (cid:48) = q ∗ Ω − 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 to obtain the right-hand sides of Eq. (A.3). ccordingly, (cid:98) δ = (cid:20) I T − ( T − ,N ) (cid:21) (cid:98) β ( N + T − , = Λ ( q ∗ (cid:101) v ) (A.4)where (cid:101) v = Ω − v and Λ is the upper off diagonal block of the inverse matrix and ( X (cid:48) Ω − X ) − N + T − ,N + T − = Λ = Λ T − ,T − Λ T − ,N ) Λ N,T − Λ N,N ) . (A.5)Partitioned inversion (see Faliva and Zoia, 2008) leads to Λ = − (cid:0) A − BD − C (cid:1) − BD − = I T − ∗ (cid:101) V (cid:48) V − ( Q (cid:48) ∗ (cid:101) V (cid:48) ) T (cid:88) j =1 q j q (cid:48) j ∗ Ω − j,j − ( Q ∗ (cid:101) V ) − · ( Q (cid:48) ∗ (cid:101) V (cid:48) ) T (cid:88) j =1 q j q (cid:48) j ∗ Ω − j,j − . This, together with Eq. (A.4), yields Eq. (13).
Proof.
The proof follows the same lines of Theorem 1. Let us assume Ω ( NT,NT ) = E ( µµ (cid:48) ) = (cid:101) Ω ( N,N ) ( N,N ( T − ( N ( T − ,N ) I T − ⊗ (cid:101) Ω ( N ( T − ,N ( T − . Then, some computations prove that X (cid:48) Ω − X ( N + T − ,N + T − = R T − ( I T − ⊗ V (cid:48) (cid:101) Ω − V ) R (cid:48) T − − R T − ( Q (cid:48) ⊗ ( V (cid:48) (cid:101) Ω − )) R (cid:48) N − R N ( Q ⊗ ( (cid:101) Ω − V )) R (cid:48) T − R N (( q q (cid:48) + Q Q (cid:48) ) ⊗ (cid:101) Ω − ) R (cid:48) N = I T − ∗ V (cid:48) (cid:101) Ω − V − Q (cid:48) ∗ V (cid:48) (cid:101) Ω − − Q ∗ (cid:101) Ω − V ( q q (cid:48) + Q Q (cid:48) ) ∗ (cid:101) Ω − nd X (cid:48) Ω − y ( N + T − , = ( T − , R N ( q ⊗ (cid:101) Ω − ) R v N, = ( T − , q ∗ (cid:101) Ω − v N, . Accordingly, (cid:98) δ GLS = (cid:20) I T − ( T − ,N ) (cid:21) (cid:98) β GLS ( N + T − , = Λ ( q ∗ (cid:101) Ω − v ) (A.6)where Λ is Λ = (cid:26) ( I T − ∗ V (cid:48) (cid:101) Ω − V ) − ( Q (cid:48) ∗ V (cid:48) (cid:101) Ω − ) (cid:104) ( q q (cid:48) + Q Q (cid:48) ) ∗ (cid:101) Ω − (cid:105) − ( Q ∗ (cid:101) Ω − V ) (cid:27) − · ( Q (cid:48) ∗ V (cid:48) (cid:101) Ω − ) (cid:104) ( q q (cid:48) + Q Q (cid:48) ) ∗ (cid:101) Ω − (cid:105) − and this yields Eq. (15). Proof.
When T = 2 , Q = q N, and V = v N, . Then, by assuming stationary disturbances, i.e Ω j,j = (cid:101) Ω for j = 1 , , the following holds X (cid:48) Ω − X ( N +1 ,N +1) = v (cid:48) (cid:101) Ω − v − q (cid:48) ∗ v (cid:48) (cid:101) Ω − − q ∗ (cid:101) Ω − v ( q q (cid:48) + q q (cid:48) ) ∗ (cid:101) Ω − (A.7)and X (cid:48) Ω − y ( N +1 , = (1 , q ∗ (cid:101) Ω − v N, . Accordingly, the GLS estimate of the deflator δ GLS turns out to be (cid:98) δ GLS − S = (cid:26) ( v (cid:48) (cid:101) Ω − v ) − ( q (cid:48) ∗ v (cid:48) (cid:101) Ω − ) (cid:104) ( q q (cid:48) + q q (cid:48) ) ∗ (cid:101) Ω − (cid:105) − ( q ∗ (cid:101) Ω − v ) (cid:27) − ·· ( q (cid:48) ∗ v (cid:48) (cid:101) Ω − ) (cid:104) ( q q (cid:48) + q q (cid:48) ) ∗ (cid:101) Ω − (cid:105) − ( q ∗ (cid:101) Ω − v ) . hen, by denoting with q it and v it the quantity and the value of the i -th good at time t and byassuming diagonal the matrix (cid:101) Ω , with diagonal entries ϑ i , some computations yield (cid:98) δ GLS = (cid:40) N (cid:88) i =1 q i v i ϑ i q i + q i (cid:41) − (cid:40) N (cid:88) i =1 q i v i q i v i ϑ i q i + q i (cid:41) . (A.8)The reciprocal of Eq. (A.8) yields the GLS estimate of the index. Proof.
The optimization problem is min λ e (cid:48) Ae . Thus, the first order conditions 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.9)Setting A = ( π π (cid:48) ) , with π as defined in Corollary 2, in Eq. (A.9) yields the MPL index. Proof.
The variance-covariance matrix of the vector ˆ δ GLS , as defined in Eq. (A.6), turns out tobe V ( (cid:98) δ GLS ) = σ [ I T − , T − ,N ][ X (cid:48) Ω − X ] − I T − N,T − = Λ where Λ is specified as in Eq. (A.5). Thus, partitioned inversion rules lead to Λ = − (cid:0) A − BD − C (cid:1) − = I T − ∗ (cid:101) V (cid:48) V − ( Q (cid:48) ∗ (cid:101) V (cid:48) ) T (cid:88) j =1 q j q (cid:48) j ∗ Ω − j,j − ( Q ∗ (cid:101) V ) − . .6 Proof of Theorem 2 Proof.
When the values, v T +1 , and the quantities, q T +1 , of N commodities in a reference basketbecome available for the ( T +1) -th additional country, the reference equation system for updatingthe 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 . 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 or, in compact form, as y u ( NT + N, = X u ( NT + N,N + T ) β u ( N + T, + µ u ( NT + N, where y u ( NT + N, = v N, ( N ( T − , ( N, , 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, 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 ) ith E ( µ u µ (cid:48) u ) = Ω u ( N ( T +1) ,N ( T +1)) = Ω , ( N,N ) ( N,N ( T − ( N,N ) ( N ( T − ,N ( T − Ω ∗ ( N ( T − ,N ( T − ( N ( T − ,N ) ( N,N ) ( N,N ( T − Ω T +1 ,T +1 ( N,N ) . Here Ω , and Ω ∗ are as defined in Eq. (11) and Ω T +1 ,T +1 = E ( (cid:15) T +1 (cid:15) (cid:48) T +1 ) .Following the same argument of Theorem 1, we obtain that X (cid:48) u Ω − u X u ( N + T,N + T ) = I T − ∗ (cid:101) V V (cid:48) − Q (cid:48) ∗ (cid:101) V (cid:48) (cid:101) v (cid:48) T +1 v T +1 − (cid:101) v (cid:48) T +1 ∗ q (cid:48) T +1 − Q ∗ (cid:101) V − (cid:101) v T +1 ∗ q T +1 ( (cid:80) T +1 j =1 q j q (cid:48) j ∗ Ω − j,j ) and X (cid:48) u Ω − u y u ( N + T, = ( T, q ∗ (cid:101) v N, where (cid:101) V and (cid:101) v are defined as in Theorem 1 and (cid:101) v T +1 = Ω − T +1 ,T +1 v T +1 .Then, upon nothing that (cid:98) δ ( T − , (cid:98) δ T +1(1 , = (cid:20) I T ( T,N ) (cid:21) (cid:98) β u ( N + T, = Λ ( q ∗ (cid:101) v ) where Λ is the T × N upper off diagonal block of the inverse matrix ( X (cid:48) u Ω − u X u ) − , partitionedinversion leads to Λ T,N ) == I T − ∗ (cid:101) V (cid:48) V (cid:101) v (cid:48) T +1 v T +1 − Q (cid:48) ∗ (cid:101) V (cid:48) (cid:101) v (cid:48) T +1 ∗ q (cid:48) T +1 (cid:20)(cid:16)(cid:80) T +1 j =1 q j q (cid:48) j ∗ Ω − j,j (cid:17)(cid:21) − (cid:20) Q ∗ (cid:101) V (cid:101) v T +1 ∗ q T +1 (cid:21)(cid:27) − Q (cid:48) ∗ (cid:101) V (cid:48) (cid:101) v (cid:48) T +1 ∗ q (cid:48) T +1 (cid:20)(cid:16)(cid:80) T +1 j =1 q j q (cid:48) j ∗ Ω − j,j (cid:17)(cid:21) − . hen, pre-multiplying ( q ∗ (cid:101) v ) by Λ yields the estimator (cid:98) δ ( T − , (cid:98) δ T +1(1 , . The reciprocal of the(non-null) elements of this estimator provides the values of the updated multilateral version ofthe MPL index. Proof.
When the values, v T +1 , and the quantities, q T +1 , of N commodities of a reference basketbecome available at time T + 1 , the updating of the multi-period version of the MPL indexmust not change its past values with meaningful computational advantages. In order to get therequired updating formula, let us rewrite Eq. (7) 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.10)where D ∗ δ is specified as follows D ∗ δ ( T +1 ,T +1) = (cid:98) D δ ( T,T ) ( T, (cid:48) (1 ,T ) δ T +1(1 , . Here (cid:98) D δ denotes the estimate of D δ , defined as in Eq. (8), that is (cid:98) D δ ( T,T ) = (1 , (cid:48) (1 ,T − ( T − , (cid:98) δ · · · (cid:98) δ · · ·
00 0 · · ·
00 0 · · · (cid:98) δ T = (cid:48) (cid:99) ˜ D δ here the entries (cid:98) δ , (cid:98) δ , . . . , (cid:98) δ T are the elements of the vector (cid:98) δ given in Eq. (13). The systemin Eq. (A.10) can be also written as v = D ˜ p q + ε V (cid:98) ˜ D δ = D ˜ p Q + E v T +1 δ T +1 = D ˜ p q T +1 + ε T +1 → V (cid:98) D δ = D ˜ p Q + Ev T +1 δ T +1 = D ˜ p q T +1 + ε T +1 . (A.11)The application of the vec operator to the left hand-side block of equations in Eq. (A.11) 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.12)where v = vec ( V (cid:98) D δ ) = ( I T ⊗ V ) R (cid:48) T ˜ δ with ˜ δ (cid:48) = [1 , (cid:98) δ ] (cid:48) and η is equal to vec ( E ) .It is worth noting that Eq. (A.12) can also 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) , 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 and β u ( N +1 , = δ T +1 ˜ p , µ u ( N ( T +1) , = ηε T +1 with E ( µ u µ (cid:48) u ) = Ω u = Ω , , Ω T +1 ,T +1 where Ω is as defined in Eq. (11) and Ω T +1 ,T +1 = E ( (cid:15) T +1 (cid:15) (cid:48) T +1 ) .The GLS estimator of the vector β u is given by (cid:99) β u = ( X (cid:48) u Ω − u X u ) − X (cid:48) u Ω − u y u Note that, differently from the proof of Theorem 2, the vector δ does not enter in the updatingestimation process as it is considered given. here X (cid:48) u Ω − u X u ( N +1 ,N +1) = (cid:101) v (cid:48) T +1 v T +1 − (cid:101) 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 )) (cid:101) v T +1 R N (( Q ⊗ I N ) Ω ( Q (cid:48) ⊗ I N )) R (cid:48) N + R N ( q T +1 q (cid:48) T +1 ⊗ I N ) R (cid:48) N = (cid:101) v (cid:48) T +1 v T +1 − (cid:101) v (cid:48) T +1 ∗ q (cid:48) T +1 − q T +1 ∗ (cid:101) v T +1 (cid:80) T +1 j =1 q j q (cid:48) j ∗ Ω j,j and X (cid:48) u Ω − u y u ( N +1 , = (1 , R N ( Q ⊗ I N ) v N, = (1 , (cid:16) Q ∗ (cid:101) V (cid:17) ˜ δ ( N, where (cid:101) V is a N × T matrix whose j -th column is Ω j,j v j .Now, upon nothing that (cid:98) δ T +1(1 , = (cid:20) (cid:48) (1 ,N ) (cid:21) (cid:98) β u ( N +1 , = Λ ( Q ∗ V ) ˜ δ , where Λ is the × N upper off diagonal block of the inverse matrix ( X (cid:48) u Ω − u X u ) − , partitionedinversion leads to Λ = (cid:101) v (cid:48) T +1 v T +1 − (cid:0) q (cid:48) T +1 ∗ (cid:101) v (cid:48) T +1 (cid:1) T +1 (cid:88) j =1 q j q (cid:48) j ∗ Ω j,j − ( q T +1 ∗ (cid:101) v T +1 ) − · (cid:0) q (cid:48) T +1 ∗ (cid:101) v (cid:48) T +1 (cid:1) T +1 (cid:88) j =1 q j q (cid:48) j ∗ Ω j,j − . Then, pre-multiplying (cid:16) Q ∗ (cid:101) V (cid:17) ˜ δ by Λ yields the estimator of (cid:98) δ T +1 given in Theorem 3. Thereciprocal of this estimator provides the updated value of the multi-period version of the MPLindex. Proof.
In this Appendix the main properties enjoyed by the MPL index, (cid:98) λ GLS , as defined inEq. (16), are proved. To this end, let (cid:98) λ ( p , p , q , q ) denote the MPL price index with p t and q t vectors of N prices and quantities at time t . .1 Strong identity : Simple computation prove that (cid:98) λ ( p , p , q , q ) = p (cid:48) πp (cid:48) π = 1 . where the weights π are as defined in (16)P.2 Commensurability : Let p ∗ j = p j ∗ γ and q ∗ j = q j ∗ γ − and denote with p ∗ ij , q ∗ ij and γ i the i -th entries of p ∗ j and q ∗ j and γ respectively.Then, simple computations prove that the following holds for the i − th weight, π ∗ i , of theMPL index π ∗ i = 1 ϑ i p ∗ i q ∗ i q ∗ i q ∗ i + q ∗ i = 1 γ i π i . where π i is as defined in (17). This means that π ∗ = γ − ∗ π and, accordingly (cid:98) λ ( γ ∗ p , γ ∗ p , γ − ∗ q , γ − q ) = p ∗ π ∗ p ∗ π ∗ = ( p ∗ γ ) (cid:48) ( γ − ∗ π )( p ∗ γ ) (cid:48) ( γ − ∗ π ) = p (cid:48) πp (cid:48) π . P.3
Proportionality : Simple computations prove that when p ∗ = α p , the weights of the MPLindex become π ∗ = α π . (A.13)Thus (cid:98) λ ( p , α p , q , q ) = p ∗ (cid:48) π ∗ p (cid:48) π ∗ = α p (cid:48) π α p (cid:48) π = α p (cid:48) πp (cid:48) π . P.4
Dimensionality : When p ∗ = α p and p ∗ = α p the weights of the MPL index are as inEq. (A.13). Accordingly (cid:98) λ ( α p , α p , q , q ) = p ∗ (cid:48) π ∗ p ∗ (cid:48) π ∗ = α p (cid:48) π α p (cid:48) π = p (cid:48) πp (cid:48) π . P.5
Monotonicity : Let k be a vector with entries greater than 1, that is k > u where u is theunit vector. Let p ∗ = p ∗ k and note that the entries of p ∗ are greater than those of p ,namely p ∗ > p . Furthermore, when p ∗ are the prices of the second period, the weightsof the MPL index become π ∗ i = π i ∗ k i where k i is the i -th element of k . Accordingly, ∗ = π ∗ k and the MPL index becomes (cid:98) λ ( p , k ∗ p , q , q ) = p ∗ (cid:48) π ∗ p (cid:48) π ∗ > p (cid:48) πp (cid:48) π . Similarly setting p ∗ = p ∗ k , with k specified as before, the weights of the MPL index donot change and the index becomes (cid:98) λ ( k ∗ p , p , q , q ) = p (cid:48) πp ∗ (cid:48) π < p (cid:48) πp (cid:48) π as the entries of p ∗ are greater than those of p , namely p ∗ > p .It is worth noticing that the MPL index enjoys also the following properties:P.6 Positivity : When p ∗ = α p , the MPL index becomes (cid:98) λ ( α p , p , q , q ) = 1 α p (cid:48) πp (cid:48) π ≥ . (A.14)P.7 Inverse proportionality in the base period : For the proof see Eq. (A.14).P.8
Commodity reversal property : It follows straightforward that the index price is invariantwith respect to any permutation ( i ) : (cid:98) 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 . P.9
Quantity reversal test : Simple computations prove that the index price (cid:98) λ does not changeas a consequence of a change in the quantity which affects only the weights π i .P.10 Base reversibility (symmetric treatment of time) : This property is satisfied by the MPLindex under a suitable choice of the weights. Setting ϑ i = p i , or ϑ i = z i p i where z i is avariable or more simply a constant term z i = z , leads to weights, π ∗ i , of the MPL indexthat does not depend on prices π ∗ i = 1 z i q i q i q i + q i , i = 1 , , . . . , n. ence, by denoting with π ∗ the vector whose i -th entry is π ∗ i , simple computations provethat (cid:98) λ ( p , p , q , q ) = p (cid:48) π ∗ p (cid:48) π ∗ = (cid:18) p (cid:48) π ∗ p (cid:48) π ∗ (cid:19) − . P.11
Transitivity : For a particular choice of θ i and by using the square roots of the quantities,we have proved that the MPL index tallies with the GK index which satisfies all tests formultilateral comparison proposed by Balk (1996) except for the proportionality one.P.12 Monotonicity : If p = β p then the following holds for the weights of the MPL index π ∗ i = β (cid:18) p i ϑ i q i q i q i + q i (cid:19) = βp i γ i . Hence (cid:98) λ ( p , β p , q , q ) = β p (cid:48) π ∗ p (cid:48) π ∗ = β. Empirical application of MPL: graphics [Figure 9 about here.][Figure 10 about here.][Figure 11 about here.] .2 MPL index and the cultural supply: a multilateral perspec-tive [Figure 12 about here.] [Figure 13 about here.][Figure 14 about here.] Simulations [Figure 15 about here.][Figure 16 about here.][Table 3 about here.][Figure 17 about here.][Figure 18 about here.][Table 4 about here.] ables Table 3: Sum of squares of the differences between the estimated reference prices and thereal ones. Data GLS-s GLS-d OLS1. Complete price tableau(4 commodities) 0.0152 0.0134 0.01652. Incomplete price tableau“classical” basket (2 commodities) 0.0016 0.0005 0.00083. Incomplete price tableaunovel basket (4 commodities) 0.0083 0.0038 0.0055Table 4: Sum of squares of the differences between the real prices and the estimated oneswith the TPD and MPL index. For the latter different specifications of the error termshave been consideredData GLS-s GLS-d OLS TPD1. Complete pricetableau 0.1337 0.0940 0.1324 0.76022. Incomplete pricetableau (“classical basket”) 0.0243 0.0291 0.0227 0.06613. Incomplete pricetableau (novel basket) 0.0772 0.0660 0.0724 0.147152 igures l 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 l Year M P L i nde x l 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 l Year M P L i nde x l 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 l Year M P L i nde x Figure 9: GLS-f, GLS-s, and OLS versions of the MPL index are shown in the top, middleand bottom panel, respectively for the case of incomplete (left) and complete (right) pricetableau. 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 GLS−fGLS−sOLSTPD 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 GLS−fGLS−sOLSTPD
Figure 10: Comparison of the GLS-f, GLS-s and OLS versions of the MPL index withthe TPD for the case of incomplete (left) and complete (right) tableau.53 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
246 0 10 20 30
Museums R e f e r en c e p r i c e s type l GLS−dGLS−fGLS−sOLS
Reference prices with missing l l l l l l l l l l l l l l l l l l l l
246 5 10 15 20
Museums R e f e r en c e p r i c e s type l GLS−dGLS−fGLS−sOLS
Reference prices without missing
Figure 11: Comparison of the GLS-f, GLS-s, GLS-d and OLS versions of the MPL refer-ence prices for the case of incomplete (left) and complete (right) tableau.
Area P r i c e i nde x type CPDGLS−s 0.00.51.01.52.02.5 North.East North.West South
Area P r i c e i nde x type CPDOLS
Figure 12: MPL and CPD indices with their σ confidence bounds. Left panel shows theGLS-s and the right one the OLS version of the MPL index. 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 GLS−fGLS−sOLSTPD 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 GLS−fGLS−sOLSTPD
Figure 13: Comparison of the GLS-f, GLS-s and OLS versions of the MPL index withthe TPD index for the case of complete (left panels) and incomplete (right panels) pricetableau. The MPL indices have been obtained on simulated data by adding to V randomterms drawn from a Normal law with a mean equal to 20000 and a standard error varyingrandomly from 0 to 1000. Confidence bands are also plotted.54 l l l l l l l l l l l l l Year P r i c e i nde x type l GLS−fGLS−sOLSTPD 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 GLS−fGLS−sOLSTPD
Figure 14: Comparison of the GLS-f, GLS-s and OLS versions of the MPL index withthe TPD index for the case of complete (left panels) and incomplete (right panels) pricetableau. The MPL indices have been obtained on simulated data by adding to υ t , for t = 2 , . . . , T , stochastic terms, drawn from a Normal law with a mean equal to -5000 anda standard error varying randomly from 0 to 800, to υ t − . Confidence bands are alsoplotted. l l l l l l Year P r i c e i nde x type l GLS−sSIM l l l l l l
Year P r i c e i nde x type l GLS−sSIM l l l l l l
Year P r i c e i nde x type l GLS−sSIM l l l l l l
Year P r i c e i nde x type l GLS−dSIM l l l l l l
Year P r i c e i nde x type l GLS−dSIM l l l l l l
Year P r i c e i nde x type l GLS−dSIM l l l l l l
Year P r i c e i nde x type l OLSSIM l l l l l l
Year P r i c e i nde x type l OLSSIM l l l l l l
Year P r i c e i nde x type l OLSSIM
Figure 15: Comparison of the GLS-s (first row panels), GLS-d (second row panels) andOLS (third row panels) versions of the MPL index with the "real" index λ assigned inthe simulation (SIM), for the case of complete (left panels) and incomplete price tableauwith “standard” (central panels) and MPL (right panels) reference basket. Confidencebands are also plotted. 55 l l l l l Year P r i c e i nde x type l GLS−dGLS−sOLSTPD l l l l l l
Year P r i c e i nde x type l GLS−dGLS−sOLSTPD l l l l l l
Year P r i c e i nde x type l GLS−dGLS−sOLSTPD
Figure 16: Comparison of different versions of the MPL index with the TPD one for eachof the three cases considered in the simulation. Confidence bands are also plotted. l l l l
Commodities R e f e r en c e p r i c e s type l GLS−dGLS−sOLSReal
Reference prices without missing l ll l
Commodities R e f e r en c e p r i c e s type l GLS−dGLS−sOLSReal
Reference prices with missing l l l l
Commodities R e f e r en c e p r i c e s type l GLS−dGLS−sOLSReal
Reference prices with missing