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Reconciling China’sRegional Input-Output Tables
David Roland-Holst and Muzhe YangUC Berkeley
Lecture I Presented to theDevelopment Research CentreState Council of the PRCBeijing, 6 June 2005
6 June 2005 Roland-Holst and Yang Slide 2
Lectures on Data and Model Development
1. Regional Data Reconciliation2. Multi-regional Trade Flow Estimation3. Integrated Micro-Macro Modeling
6 June 2005 Roland-Holst and Yang Slide 3
Objectives
• Implement an efficient econometric methods for reconciling provincial Input-output tables with national accounts.
• Establish coherent national standards for data harmonization
6 June 2005 Roland-Holst and Yang Slide 4
Motivation
• Provincial Input-output are available for China, but they exhibit a variety of consistency problems– Among the more serious of these is
inconsistency with national-level tables, individually and collectively
• Consistent individual and aggregate tables are essential to implement detailed economic analysis within and across provinces and regions
6 June 2005 Roland-Holst and Yang Slide 5
Foundation – PRC Provincial IO Tables
• Already available• Nationally comprehensive and
consistent in terms of account definitions
• This work supports efforts already under way at the provincial and national (NBS) level, and also builds on existing DRC capacity for SAM and CGE research
6 June 2005 Roland-Holst and Yang Slide 6
Proposed Approach
• Using Bayesian econometric techniques to incorporate prior information when updating and reconciling economic accounts
• We show how to estimate a consistent provincial table with additional prior information at the national level.
• The estimation begins with a consistent national table that is assumed (for convenience only) to be known with certainty.
6 June 2005 Roland-Holst and Yang Slide 7
Overview of the Estimation Problem
The set-up of this matrix balancing problem follows Golan, Judge and Miller (1996). We focus on balancing schemes for provincial table.
The proposed approach is an extension of that usually applied to a national table.
6 June 2005 Roland-Holst and Yang Slide 8
Estimation Strategy
Consider one province, { }1, 2, ,g G∈ L , a K -sector
economy, represented by an input-output table, IO(g) , where each entry indicates a payment by a column account to a row account:
( )( ) ( )
( )( ) ( )1 10
g gg
gK K
TIO
′+ × +
⎡ ⎤= ⎢ ⎥⎢ ⎥⎣ ⎦
f
v
where ( )gT is a K K× matrix of intermediate sales, ( )gf is a K -vector of final demands, and ( )gv is a K -vector of sectoral value added. The table IO ( )g is therefore a ( ) ( )1 1K K+ × + matrix,
where corresponding column and row sums are equal.
6 June 2005 Roland-Holst and Yang Slide 9
Estimation 2
Assume: (1) Intermediate demands are determined by a K K× fixed coefficient matrix ( )gA ;
(2) A K -vector, ( )gx , represents sectoral sales to both intermediate and final demanders. Then, we have the following standard Leontief input-output model:
( ) ( ) ( ) ( )g g g gA + =x f x Define ( ) ( ) ( )g g g≡ −y x f , as the sectoral sales to intermediate demanders. This
transaction has double meanings: the column vector of ( )gy represents sectoral
intermediate expenditures, while the row vector of ( )gy represents sectoral intermediate receipts.
6 June 2005 Roland-Holst and Yang Slide 10
Estimation 3
Now we transform the matrix balancing problem into the econometric problem of identifying the ( )g
ija elements of the ( )gA
matrix, based on the available economic information contained in the row and column sums IO table. This strategy takes the form
( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( ) ( )
( ) ( )
( ) ( )
( )11 1 1 1
1
( )
( ) ( )1 1
1, ,
, 1, ,
, 1, ,
g g g
K g ggj jjK K
Kg g gi ij jj
g ggij ij j
K Kgg gij i jij j
A
A x j K
y a x i j K
T a x
T y T i j K
=× × ×
=
= =
=
= =
⇒ = =
=
⇒ = = =
∑
∑
∑ ∑
y x
y L
L
Q
L
6 June 2005 Roland-Holst and Yang Slide 11
Identification Strategy
To proceed, we transform the national table in precisely the same way [omit the (g) superscript in the last three slides].
Now we use the entropy principle to recover A and A(g) from the top down, under the row-column linear restrictions and the micro-macro consistency requirement.
6 June 2005 Roland-Holst and Yang Slide 12
Balancing Scheme for the National Table
Consider the standard formulation y = Ax, where y and x are K
-dimentional vectors of known data and A is an unknown K K× matrix that must satisfy the following three conditions: (1) Consistency:
( )11 1, ,K
ijia j K
== =∑ L
(2) Adding up:
( )1 1, ,K
ij j ija x y i K
== =∑ L
(3) Non-negativity:
( )0 , 1, ,ija i j K≥ = L
6 June 2005 Roland-Holst and Yang Slide 13
Maximum Entropy Principle
Given the three conditions, the problem of identifying the aij
elements of the A matrix is formulated as:
1 10max ln
ij
K Kij iji ja
a a= =>
−∑ ∑
subject to:
( )( )
1
1
1 1, ,
1, ,
Kiji
Kij j ij
a j K
a x y i K=
=
= =
= =
∑∑
L
L
The solution to this problem is denoted as
MEija)
.
6 June 2005 Roland-Holst and Yang Slide 14
Balancing Scheme for Provincial Tables
Consider the previous formulation for province { }1, 2, ,g G∈ L , i.e. ( ) ( ) ( )g g gA=y x
where ( )gy and ( )gx are K -dimensional vectors of known data and ( )gA is an unknown K K× matrix that must satisfy:
(1) Consistency:
( )( )1
1 1, ,K giji
a j K=
= =∑ L
(2) Adding up: ( )( ) ( ) ( )
1 1, ,K g g g
ij j ija x y i K
== =∑ L
(3) Non-negativity: ( )( ) 0 , 1, ,g
ija i j K≥ = L
6 June 2005 Roland-Holst and Yang Slide 15
Specification of Prior Information
The national level estimates provide information that may be used in recovering estimates of provincial SAMs. This information can be stated as a series of prior restrictions on estimating the new provincial IO. We give six examples: (1) Links between national and provincial accounts:
( ) ( )( ) ( )
1
1
1, ,
1, ,
G gj jg
G gi ig
x x j K
y y i K
=
=
= =
= =
∑∑
L
L
(2) Properties of national level estimates
MEija)
:
( ) ( )
ME
1
(1 above) ME
1 1 1
Kij j ij
G K Gg gij j ig j g
a x y
a x y
=
= = =
=
⇔ =
∑
∑ ∑ ∑
)
)
6 June 2005 Roland-Holst and Yang Slide 16
Priors 3-5
(3) Maximum entropy estimates for each provincial IO: ME( )gija)
(4) Provincial adding-up restrictions:
( )
ME( ) ( ) ( )1
ME( ) ( ) ( )1 1 1
(2 above) ME( ) ME( )1 1 1 1
gK g gij j ij
gG K Gg gij j ig j g
gG K G K ggij ijj jg j g j
a x y
a x y
a x a x
=
= = =
= = = =
=
=
⇔ =
∑∑ ∑ ∑
∑ ∑ ∑ ∑
)
)
) )
(5) If the set of provincial IO tables is not complete, we can assume the following:
( )ME( ) ME( )1 1
gK K ggij ijj jj j
a x a x= =
=∑ ∑) )
6 June 2005 Roland-Holst and Yang Slide 17
Prior 6
(6) Assume that the activity accounts represent homogeneous technologies in each province, with the technological
coefficients denoted as aij . Therefore, we have:
( )
( )
( )
ME ME( ) ME( )( ) ( ) ( )1 1 1 1
ME ME( )( ) ( )1 1 1
ME ME( ) ( )1 1
( )1
A
A
A
A
A
A
g gK K K Kg g g gij ij ij ijj j j jj j j j K
gK K Kgg gij ij ijj j jj j j K
gg gK Kij ijj j Kj j
i K gj j
a x a x a x a x
a x a x a x
a x a xa
x
= = = = +
= = = +
= = +⋅
=
= = +
⇒ = −
−∑ ∑⇒ =
∑
∑ ∑ ∑ ∑∑ ∑ ∑
) ) )
) )
) )
6 June 2005 Roland-Holst and Yang Slide 18
Other Prior Information
In addition to the examples given here, any specific prior information about the accounts or underlying technical relationships. These include:
1. Cell inequality or boundary constraints (><0, etc.)
2. Institutional budget constraints.3. Fixed values or variance constraints.
6 June 2005 Roland-Holst and Yang Slide 19
Summation
With these conditions in mind, we can express prior information for estimating each provincial table as follows:
( )( )( )0( )
ME( )
1, ,1, ,
, ,i Ag
ij gAij
j Kaa i K
j K Ka
⋅⎧ =⎪= =⎨ =⎪⎩
LL) L
where
ME( )gija)
comes from the first-round maximum entropy estimation for each provincial IO table.
6 June 2005 Roland-Holst and Yang Slide 20
Cross Entropy Principle
Finally, the problem of identifying the ( )gija elements of provincial
the ( )gA matrix is formulated as:
( )
( ) ( ) ( ) 0( )1 1 1 10
min ln lng
ij
K K K Kg g g gij ij ij iji j i ja
a a a a= = = =>
−∑ ∑ ∑ ∑
subject to:
( )( )
( )1
( ) ( ) ( )1
1 1, ,
1, ,
K giji
K g g gij j ij
a j K
a x y i K=
=
= =
= =
∑∑
L
L
the solution to which is denoted by
CE( )gija)
.
6 June 2005 Roland-Holst and Yang Slide 21
How to Proceed
I recommend four steps:1. Encode the estimation methods
(STATA, MATLAB)2. Select a sample province to establish
estimation standards3. Apply to all provincial tables4. Iterate with NBS, provincial sources
6 June 2005 Roland-Holst and Yang Slide 22
References
1. Golan, A. (2003), “Image Reconstruction of Incomplete and Ill-Posed Data: An Information -Theoretic Approach,” Working Paper, Department of Economics, American University, January.
2. Golan, A., G. Judge, and D. Miller (1996), Maximum Entropy Econometrics: Robust Estimation with Limited Data (John Wiley & Sons, New York).