Elementary Aggregation Using the Country Product Dummy (CPD) Method

Transcription

1 Chapter 23 Elementary Aggregation Using the Country Product Dummy (CPD) Method he country product dummy (CPD) regression has been used by the International Comparison Program (ICP) since the program was in its infancy. he need for an index that could deal with multiple gaps in price data was obvious because of the many missing prices. he CPD index was first introduced by Summers (1973) for use in elementary aggregation (i.e., at the level of basic heading) and for filling gaps in prices. It can be presented in two equivalent forms, with the intercept and without. 1 he regression equation for the CPD can be written as ln p cp = y cp = x cp β + ε cp, (23.1) where P cp is the price of product p in country c; Dc j and Dp i are country and product dummies, respectively; and Nc are number of products and countries, respectively; and and x = [ Dc... Dc DpDp... Dp ] cp 2 Nc Nc 1 2 β = [ α... α γ γ... γ ]. (23.2) In matrix notation, by stacking individual observations, equation (23.1) can be rewritten as y = X β+ε. (23.3) he first country dummy is dropped from the system because matrix X is of rank ( + Nc 1). (In fact, one could drop any variable from the system; dropping the first country's dummy simply makes it the base.) he solution is given (under the conditions of independently and identically distributed random disturbances) by ˆ 1 β=( X X) X y. (23.4) In addition, one could drop one product variable (e.g., the first product dummy) and introduce the intercept. his is the second form of the CPD. In this case, 2 x cp = Dc2... DcNc 1 Dp2... Dp β = α2... α Nc cinterceptγ 2... γ and (23.5) y = X β +ε. (23.6) he solutions of systems (23.4) and (23.6) the country and product price relatives are identical up to a scalar. In the case with the intercept, αj = α ; j g 1 = c intercept ; and γ i = γ j + cintercept, for i = 2..., j = 2... Nc, where γ i and α i are the product coefficients for product i and country j, respectively. Practical Considerations in Computing the CPD System (23.6) is quite computer-intensive in general. his chapter presents a simpler, mathematically equivalent version of system (23.6). 437

2 In the case of the full price matrix, matrix (X X) becomes X X = Nc Nc Nc. (23.7) However, computation of matrix (X X) in the general case with a sparse matrix can still be simplified. Moreover, in solving for ˆ X 1 β = ( X) X y, there is no need to invert matrixes; 3 it is only important to simplify (X X) in order to significantly speed up the computations. In general, the matrix in expression (23.7) corresponds to the case in which the price matrix is full, and thus the solution of the CPD system becomes a simple geometric mean. his means there is no need to solve system (23.4), and thus this case is not interesting. In the case with missing price observations, expression (23.7) becomes i ( c2) δ ( c j) 0 j i δ δ j δ j 1 i 0 0 ( c ) Nc δnc δnc δ Nc X X = 1 δ 1 j δ 1 Nc Nc( p1) 0 0 i i i δ j δ Nc 0 Nc( pi) 0 δ j δ Nc 0 0 Nc( p) (23.8) where (c j ) is the number of products that country j priced; Nc(p i ) is the number of countries that priced product i; and δ i j is equal to 1 if the price of product i is observed in country j; otherwise, it is 0. Country Product Representative Dummy (CPRD) he CPRD was first introduced by Cuthbert and Cuthbert (1988) to adjust for biases arising from the varying representativity of the products being compared in different countries. Matrix X becomes in this case and xcp = Dc2 DcNcDp1 DpRcp β = α2 αnc γ1 γ ρ (23.9) y = X β + ε, (23.10) where R cp is the representativity dummy, and r is its respective regression coefficient. Expression (23.8) can be augmented with one extra row and one extra column to yield i ( c2) δ 2 Nrep( c ) ( cj) 0 j i δ δ j δ j Nrep( c ) j 1 i 0 0 ( cnc ) δnc δnc δ Nc Nrep( cnc ) X X = δ δ δ Nc( p ) 0 0 Nrep( p ) j 1 Nc 1 1 i i δ j i δ Nc 0 Nc( pi) 0 Nrep( pi) δ j δ 0 0 Nc( p ) Nrep( p ) Nc Nrep( c2) Nrep( cj) Nrep( cnc) Nrep( p1) Nrep( pi) Nrep( p) Nrep (23.11) where Nrep(c j ) is the number of representative products priced by country j; Nrep(p i ) is the number of countries in which product i was found to be representative; and Nrep is the total number of representative observations. It is important to note that the CPRD can be easily generalized to incorporate generic weights and not just the representativity dummies. Weighted Country Product Dummy (CPD-W) he Diewert formulation of the weighted CPD (CPD-W) can be written in this notation as X X = δ s δ s δ s S i i j j i j i j j j 1 1 i i Nc Nc Nc Nc Nc Nc δ s δ s δ s 11 δ s δ s δ s δ s δ s δ s j 1 j 1 Nc 1 Nc s s S( p1) i i i i i i 2 2 j j Nc Nc s s S( pi ) 0 0 δ s δ s δ s 0 S 0 s2 δ j sj δnc s Nc 0 0 Ss s S( p ) (23.12) where S(p i ) is a set of observations for product i, and s i j is weight of product i in country j (see Diewert 2004). Sample code for the CPRD and CPD-W systems is presented in annexes A, B, and C. 438 Operational Guidelines and Procedures for Measuring the Real Size of the World Economy

3 Numerical Example of CPD Computation able 23.1 contains a set of prices in which eight prices are missing. he log price vector and matrix X are written as in table 23.2 (the country and product dummies are designated by c1 to c4 and p1 to p5). he first column is the logarithm of price. he first entry in matrix X corresponds to the price in country 1 of item 1. hus the dummy variables are 1 for c1 and p1, and other lines are computed in the same way. he result (vector β) in accordance with equation (23.4) can be computed, for example, in Excel using Worksheet Function LinEst, 4 or with a corresponding function from any other statistical package (see table 23.3 for the result). able 23.1 Original Price Data, Five Items and Four Countries Country 1 Country 2 Country 3 Country 4 Item Item Item Item Item Source: ICP, Note: = not available. able 23.2 Matrix X ln(p) c1 c2 c3 c4 p1 p2 p3 p4 p Source: ICP, able 23.3 Result, Vector β Country 1 Country 2 Country 3 Country 4 PPP Source: ICP, Elementary Aggregation Using the Country Product Dummy (CPD) Method 439

4 Annex A Pseudo-Code (Algorithm) of Simple Country Product Representative Dummy (CPRD) Calculation, According to Equation (23.4) 'function body, Prices and Weights are inputs, CPD coefficients are outputs Function CPD(Prices, Weights) 'SEP 1: finding dimensions of regression by counting the number of non-zero prices 'Nc is number of countries, is number of products For j = 1 o Nc For i = 1 o If (Prices(i, j) > 0) hen n = n + 1 'SEP 2: building matrix X [dummies] and vector Y [log prices] for regression k = 1 For j = 1 o Nc For i = 1 o If Prices(i, j) > 0 hen Y(k) = Log(Prices(i, j)) If j > 1 hen X(k, j - 1) = 1 If i > 1 hen X(k, Nc i - 1) = 1 X(k, Nc ) = Weights(i, j) k = k + 1 'SEP 3: calling Regressor function with intercept [specific to environment: WorksheetFunction.LinEst in Excel, [X\Y] in MALAB, regress in SAA, etc.] Regression = Regressor(Y(), X(), intercept) 'SEP 4: preparing output vector of regression coefficients, inserting zeroes for one item and one product dummy coefficient Output() <- Regression() + zeros for two coefficients 'function returns results CPD = Output End Function Annex B Actual VBA/Excel Code of Simple Country Product Representative Dummy (CPRD) Calculation, According to Equation (23.4) 'general settings Option Base 1 Option Explicit 'function body Function CPD(Prices, Weights) 'local definitions of variables Dim P#(), W#(), Output#(), Regression, X#(), Y#(), Nc&, &, i&, j&, k&, n& 'finding the boundaries of arrays = UBound(Prices, 1): Nc = UBound(Prices, 2) 'dimensioning arrays of prices and weights ReDim P(, Nc), W(, Nc) 'SEP 1: assigning data to working arrays and finding dimensions of regression For j = 1 o Nc For i = 1 o P(i, j) = Prices(i, j) W(i, j) = Weights(i, j) If (P(i, j) > 0) hen n = n + 1 'dimensioning arrays for regression ReDim X(n, Nc ), Y(n, 1), Output(1, Nc + + 2) 'SEP 2: building matrix X and vector Y for regression k = 1 For j = 1 o Nc For i = 1 o If P(i, j) > 0 hen Y(k, 1) = Log(P(i, j)) If j > 1 hen X(k, j - 1) = 1 If i > 1 hen X(k, Nc i - 1) = 1 X(k, Nc ) = W(i, j) k = k + 1 'SEP 3: calling built-in Excel regressor Regression = WorksheetFunction.LinEst(Y(), X(), 1, 0) 440 Operational Guidelines and Procedures for Measuring the Real Size of the World Economy

5 'SEP 4: preparing output vector of regression coefficients For j = 1 o Nc - 1: Output(1, j + 1) = Regression(Nc + - j): For i = 1 o - 1: Output(1, Nc + i + 1) = Regression( - i + 1): Output(1, Nc + + 2) = Regression(Nc + ) Output(1, Nc + + 1) = Regression(1) 'function returns results CPD = Output End Function Annex C Pseudo-Code (Algorithm) of Simple Weighted Country Product Dummy (CPD-W) Calculation, According to Equation (23.12) 'function body, Prices and Weights are inputs, CPD coefficients are outputs Function CPDW(Prices, Weights) 'SEP 1: finding dimensions of regression by counting the number of non-zero prices 'Nc is number of countries, is number of products For j = 1 o Nc For i = 1 o If (Prices(i, j) > 0) hen n = n + 1 'SEP 2: building matrix Xty and vector XtX according to ˆ 1 β = ( X X) X y k = 1 For j = 1 o Nc For i = 1 o yy = P(i, j): xx = W(i, j) If (yy > 0) hen yy = Math.Log(yy) Xty(i + Nc - 1) = Xty(i + Nc - 1) + yy * xx Xt X(Nc i, Nc i) = XtX(Nc i, Nc i) + xx If j > 1 hen Xty(j - 1) = Xty(j - 1) + yy * xx XtX(j - 1, j - 1) = XtX(j - 1, j - 1) + xx XtX(j - 1, Nc i) = xx 'SEP 3: calling Regressor function with intercept [specific to environment: WorksheetFunction.LinEst in Excel, [X\Y] in MALAB, regress in SAA, etc.] Regression = Regressor(Xty(), XtX(), intercept) 'SEP 4: preparing output vector of regression coefficients, inserting zeroes for one item and one product dummy coefficient Output() <- Regression() + zeros for two coefficients 'function returns results CPDW = Output End Function Notes 1. he variant with intercept appears in chapter 11 of the ICP Methodological Handbook (World Bank 2008). Rao (2004) and Diewert (2004) use one without the intercept. 2. Note that the sign (') does not mean transpose. he sign ( ) is used for that purpose. 3. Modern methods of computational matrix algebra find solving a linear system via matrix inversion very inefficient. Other methods such as Cholesky decomposition and LU decomposition are judged to be superior. 4. If the log price vector is located in range (O3:O14) and matrix X is in range (Q3:X14), then the formula will be invoked as = EXP(LINES(O3:O14,Q3:X14,0,0)). Note that the result is in logs and needs to be exponentiated. References Cuthbert, J., and M. Cuthbert "On Aggregation Methods of Purchasing Power Parities." Working paper no. 56, Organisation for Economic Co-operation and Development, Paris, November. Elementary Aggregation Using the Country Product Dummy (CPD) Method 441

6 Diewert, W. E "On the Stochastic Approach to Linking the Regions in the ICP." Paper presented at Workshop on Estimating Production and Income across Nations, Institute of Governmental Affairs, University of California at Davis, April. Rao, P "he Country-Product-Dummy Method: A Stochastic Approach to the Computation of Purchasing Power Parities in the ICP." Paper presented at SSHRC Conference on Index Numbers and Productivity Measurement, Vancouver, July. Summers, R "International Price Comparisons Based upon Incomplete Data." Review of Income and Wealth 19: World Bank "Estimation of PPPs for Basic Headings within Regions." ICP Methodological Handbook Operational Guidelines and Procedures for Measuring the Real Size of the World Economy