Australian Bureau of Statistics
6427.0.55.005 - Implementation of the Review of the Producer and International Trade Price Indexes , 2012
Latest ISSUE Released at 11:30 AM (CANBERRA TIME) 28/09/2012 First Issue
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APPENDIX 2 METHODOLOGY USED FOR DERIVING STAGE OF PRODUCTION WEIGHTS
10 The weights for the intermediate and preliminary components of the SOP indexes are derived from the following relationships:
where (refer to Diagram 1 for locations of these variables in the I-O table):
y = the vector of final demand quantities (i.e. list of Final demand output quantities by industry)
= the matrix of technology coefficients produced from the Input-Output tables
= 1,…,1284 subscript referring to input products (rows in the I-O table)
= 1,…,1284 subscript referring to output products (columns in the I-O table)
= final demand for product
= total production of product
= the flow of product into product as part of intermediate input into product
the amount of product directly required to produce one unit of output of product . Known as the industry technology coefficient or the direct requirements coefficient.
= total use of product
= "adjusted" final demand for product , that is final demand less change in inventories
intermediate consumption of product across all products
11 To derive the SOP weights, three tables are required from the Supply-Use Section: IOPC level Use table (at basic prices -BP) with indirect allocation of imports, IOPC level Supply table (at BP), and IOPC Import table (at BP). The respective tables provide details in a 1284 products by 112 industry array.
12 The weighting process requires a number of phases and steps which are described below.
Phase 1 - Creation of weights for total output including imports
Step 1. Using the Use table with indirect allocation of imports ('U matrix'), create the technology coefficients matrix ('B matrix'). This is achieved by dividing the product elements in each industry in U by the total industry outputs for each industry (detailed as vector g).
Step 2. Using the Supply table ('S matrix'), create the technology coefficients matrix ('D matrix'). This requires a sub-step that transposes the S matrix. Create the industry coefficients for the Supply table ('D matrix') by dividing the product elements in each industry in S by the total product outputs (vector q). Note, vector q is obtained by transposing vector q.
Step 3. Using matrix multiplication, create a technology coefficient 'A matrix' by multiplying 'B matrix' by the 'D matrix' (A=BD). While not required for weighting purposes, the 'A matrix' can be used to create a PxP I-O table.
Step 4. Calculate an adjusted final demand 'y vector'. This requires adjusting final demand for inventories, and in the case of weights for domestic demand, removing exports.
Step 5. Using matrix multiplication, multiply the 'A matrix' by 'y vector' to generate Intermediate demand (i.e. Intermediate demand = Ay).
Step 6. Using matrix multiplication, multiply A by Ay (i.e. multiply the 'A matrix' by the Intermediate demand vector derived in step 5) to get Preliminary demand (Preliminary demand = A2y).
Step 7. Using the final demand classifications from the Use table and the results of steps 5 and 6, calculate the required SOP weights.
Matrix calculations summary:
= Use matrix
= Supply matrix
Final demand = y
Intermediate demand = Ay
Preliminary demand = A2y
Phase 2 - Creation of weights for domestic demand excluding imports
Step 1. Create the Use table for domestic demand by subtracting the Import table from the Use table based on the indirect allocation of imports. The resulting table is a Domestic Use table with direct allocation of imports.
Step 2. Follow the steps in Phase 1 using the newly created Domestic Use table instead of the original total Use table used in Phase 1. However, instead of using the total industry outputs from the Direct Use table as the denominator for the coefficients, the total industry outputs from the original total Use table in Phase 1 are to be used as the denominator in the creation of the technology coefficients for Phase 2. By doing this, the production of Domestic output as a proportion of total output including imports is obtained. Ensure that the newly created Domestic A matrix is multiplied by the Final demand y vector from Phase 1 to obtain the Intermediate demand and Preliminary demand for Domestic Use.
Phase 3 - Creation of weights for imports
Step 1 Subtract the Domestic A matrix from the A matrix from Phase 1 to create the Imports A matrix.
Step 2 Use the Imports A matrix and vector y to create the vector Ay and A2y for imports, and the subsequent weights as per steps 5 to 7 in Phase 1. Ensure that the Imports A matrix is multiplied by the Final demand y vector from Phase 1 to obtain the Intermediate demand and Preliminary demand for Imports.
Step 1. Using the Use table with direct allocation of imports from Phase 2, create a new technology B matrix. However, instead of using the total industry outputs from the Direct Use table as the denominator for the coefficients, the total industry outputs from the Indirect Use table are to be used as the denominator in the creation of the technology coefficients for Phase 3. By doing this, the production of Domestic output as a proportion of total output including imports is obtained. This will allow the coefficients for the proportion of domestic production excluding imports to be subtracted from the coefficients of domestic output including imports to provide the coefficients of technology for the import component of outputs.
Step 2. Subtract the resulting modified Direct A matrix from the Indirect A matrix from Phase 1 to create the Imports A matrix.
Step 3. Use the Imports A matrix and Indirect vector y to create the vector Ay and A2y for imports, and the subsequent weights as per steps 5 to 7 in Phase 1.
13 An example of how to derive SOP weights is available on request.
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This page last updated 27 September 2012