Reconciling China’s Regional Input- Output Tablesare.berkeley.edu/~dwrh/Slides/DRC-DRH-I.pdf ·...

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

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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

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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

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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.

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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.

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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.

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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.

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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

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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)

.

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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

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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

=

= = =

=

⇔ =

∑

∑ ∑ ∑

)

)

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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= =

=∑ ∑) )

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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

= = = = +

= = = +

= = +⋅

=

= = +

⇒ = −

−∑ ∑⇒ =

∑

∑ ∑ ∑ ∑∑ ∑ ∑

) ) )

) )

) )

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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.

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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.

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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)

.

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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

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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).