APPENDIX 2 METHODOLOGY USED FOR DERIVING STAGE OF PRODUCTION WEIGHTS


INTRODUCTION

1 This appendix sets out the current Product by Product (PxP) approach used for deriving the weights for the Stage of Production (SOP) indexes.


BACKGROUND

2 The last re-weighting of the SOP indexes was undertaken in 2002. At the time, the collection of price indexes was concentrated in the agricultural and manufacturing industries, with only a small representation of services industries. Since 2002, there has been a greater emphasis placed on the need to capture services industries. The 2012 Producer Price Indexes (PPIs) and International Trade Price Indexes (ITPIs) acknowledged this need, and recommended an economy wide construction of the SOP indexes to reflect all industries of the economy. To achieve this, an efficient means of producing weights for all Input Output Product Codes (IOPC) is required.

3 In 2002, the weights were based on an Industry-by-Industry (IxI) approach. The IxI approach bases the initial calculation of weights on an industry by industry formatted Input-Output (I-O) table. In 2012, this would use a 112 x 112 industry Use matrix from the IxI I-O table (in 2002 there was a 107 x 107 industry matrix). The industry weights, once obtained, are then converted to product weights. In contrast, the PxP approach directly calculates all product weights from a product by product I-O table, and uses a 1284 x 1284 product matrix from the PxP I-O table.

4 The PxP is seen as the most efficient method for producing SOP weights.


ASSESSMENT OF INDUSTRY-BY-INDUSTRY VERSUS PRODUCT-BY-PRODUCT APPROACH

5 The 2002 weighting methodology employed an IxI approach to formulating weights at the IOPC level. This method used the four digit industry level industry I-O tables. The weights were then disaggregated using the eight digit IOPC relative weights within each four digit level industry weight, to generate individual weights for each eight digit IOPC.

6 The current PxP approach directly creates weights for each IOPC without the need for the product disaggregation stage used in the IxI approach. This allows a faster weights compilation method.

7 Mathematically, the IxI and PxP approaches for creating I-O tables are nearly identical. The only difference is the order of matrix multiplication of the Use and Supply matrices to create the technology coefficients. For the ABS, the difference results in a 112 x 112 IxI matrix within the I-O table versus a 1284 x 1284 PxP matrix within the I-O table. Apart from this one calculation, all steps in computing the IxI I-O table and the PxP I-O table are the same.


THE PRODUCT-BY-PRODUCT METHODOLOGY

8 The principles underlying the compilation of the weights for the SOP indexes are provided in the Appendix of Chapter 13: Producer and International Trade Price Indexes: Concepts, Sources and Methods, 2006 (cat. no. 6429.0).

9 The SOP weighting methodology is based on the construction of PxP I-O tables. The procedure for creating the PxP symmetric I-O table is detailed in the United Nations Input Output Handbook (see pp 86-90).

10 The weights for the intermediate and preliminary components of the SOP indexes are derived from the following relationships:

Stage	Description	Value used for weighting
Final demand	Final demand, products with no further processing	y
Intermediate demand	Products consumed as inputs into the production of Final demand	Ay
Preliminary demand	Products consumed as inputs into the production of intermediate demand	A2y

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

and

= 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.