Provincial Trade Flow Estimation for China
David Roland-Holst and Muzhe YangUC Berkeley
Lecture II Presented to theDevelopment Research CentreState Council of the PRCBeijing, 6 June 2005
6 June 2005 Roland-Holst and Yang Slide 2
Objectives
• Implement an efficient procedure for estimating a multi-regional trade flows across China
• Integrate this with a complete set of consistent provincial SAMs to form a national Multi-regional Social Accounting Matrix (MrSAM)
6 June 2005 Roland-Holst and Yang Slide 3
Motivation
• Single-region IO tables are already accessible, but neither mutually consistent not integrable
• MRSAM is of interest for its own sake, but can also support more integrated policy analysis– CGE– Economic integration studies
• Generate a prototype data set as a template for more standardized regional data reporting and management
6 June 2005 Roland-Holst and Yang Slide 4
Foundation – PRC Provincial IO Tables
• Already available• Nationally comprehensive and
consistent in terms of account definitions
• Builds on DRC capacity for SAM and CGE research at the national level
6 June 2005 Roland-Holst and Yang Slide 5
Consistency Issues
• Provincial trade statistics are maintained independently
• Domestic imports and exports are not consistently distributed across other sub-national regions
• There is very little accounting of margins arising from distribution costs and administrative measures
6 June 2005 Roland-Holst and Yang Slide 6
Proposed Approach
• Uses a new gravity specification to estimate bilateral trade econometrically
• Integrates the steps necessary to – Generate the interregional trade flow
portions of the China MrSAM, while– insuring the consistency of the province
accounts, regional aggregations, and the national system as a whole
6 June 2005 Roland-Holst and Yang Slide 7
Procedure
• Definitional Framework– Define the provinces– Define sectoral classifications and
detail • Generate single-region and
national tables• Estimate interregional trade
distributions by commodity
6 June 2005 Roland-Holst and Yang Slide 8
Overview of the Estimation Problem
• Extending prior DRC work (He and Li: 2004) we propose a new gravity model specification of bilateral trade.
• We then propose three alternative estimators.• Each of these can be implemented with
standard statistical software, and the most attractive estimates used for multi-regional analysis
6 June 2005 Roland-Holst and Yang Slide 9
Schematic Trade Matrix
Industry Commod Factor Institution Industry Commod Factor Institution Industry Commod Factor InstitutionDomestic
TradeForeign Trade
Industry
Commodity
Factor
Institution
Industry
Commodity
Factor
Institution
Industry
Commodity
Factor
Institution
Domestic Trade
Foreign Trade
Region 2
Region 3
Region 1
Region 1 Region 2 Region 3
6 June 2005 Roland-Holst and Yang Slide 10
Estimation Technique
• The gravity type model has been commonly used in estimating trade flows in international economics.
• We apply this approach to modeling and predicting regional trade flows with a variation of an international strategy proposed by Mátyás (1997).
6 June 2005 Roland-Holst and Yang Slide 11
Generic Gravity Model
• Consider the following specification ( ) ( ) ( )
1 2 3ln ln lni i imnt m n t mt nt mn mnty Y Y dα γ λ β β β ε= + + + + + +
where: ( )i
mnty is the volume of commodity i 's trade (exports) from region m to
region n at time t ; ( )i
mtY is the GDP for commodity i in region m at time t , and the same for ( )i
ntY for region n ;
dmn is the distance between the regions m and n ; mα is the home regional effect, nγ is the foreign regional effect, and tλ is the time effect; 1, ,m N= L , 1, , 1, 1, , 1n i i N= − + +L L , where the 1N + -th element is the rest of the world, 1, ,t T= L ; 1, ,i I= L , the number of tradable goods; mntε is a white noise disturbance term.
6 June 2005 Roland-Holst and Yang Slide 12
Comments
From an econometric point of view, the α , γ and λ specific effects can be treated as either random effects or fixed effects. In this analysis, we assume those specific effects associated with regions are time-invariant, and adopt the fixed effects approach.
Also note that our main goal is prediction, so the parameter estimates for α , γ , λ , 1β , 2β , 3β only bear the meaning of best linear predictor, not estimates for latent structural parameters.
In addition, we could also add other terms to the right hand side, such as ln POP mt , and ln POP mt , the population for
region m and region n at time t respectively.
6 June 2005 Roland-Holst and Yang Slide 13
Estimating Bilateral Trade Flows
Consider commodity 1, 2, ,i I∈ L , the explained variable, ( )iy , in the model (1-1) is an N N T× × -vector of
observations, arranged in the form:
( ) ( )( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )121 12 131 13 11 1 1 1 1, , , , , , , , , , , , ,i i i i i i i i i
T T N N T N N N N Ty y y y y y y y′
+ +=y L L L L L L
The explanatory variables are arranged accordingly:
( ) ( ) ( ), , , , ,i i imt nt mnX D D D Y Y dα γ λ⎡ ⎤= ⎣ ⎦
where Dα , Dγ and Dλ are dummy variable matrices for
α , γ and λ .
6 June 2005 Roland-Holst and Yang Slide 14
Trade Flow Estimation 2
Then we stack these I ( 1, ,i I= L ) vectors to construct an
I -good trade-flow (demand) system:
( )( )
( ) ( )
(1) (2) ( )
1
(1)
(2)
( )6
, ,
0 0 00 0 0
0 0
I
N N T I
IN N T I I
Y
XX
X
X
′′ ′ ′
× × × ×
× × × × ×
=
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
y y yL
M M O M
L
6 June 2005 Roland-Holst and Yang Slide 15
Three Alternative Estimators
Modern econometrics has developed a large set of alternative estimation strategies, each performing differently under different data conditions.
For the present application, we recommend three alternative estimators, to be evaluated ex post in terms of statistical performance:
1. Ordinary Least Squares (OLS)– the most traditional approach
2. Seemingly Unrelated Regressors (SUR) – an generalization of OLS that imposes less prior assumptions on the data structure
3. Generalized method of moments (GMM)
6 June 2005 Roland-Holst and Yang Slide 16
Ordinary Least Squares 1
OLS estimates for Y and X above can be obtained as follows:
Regressand: vector ( )iy consists of bilateral trade flows of
commodity 1, 2,i I∈ L between region m ( 1, ,m N= L )
and region ( 1, , 1, 1, , 1n i i N= − + +L L ) sorted by t ( 1, ,t T= L ).
( ) ( )( )( )( )
( ) ( ) ( ) ( ) ( ) ( ) ( )121 12 11 1 1 1 1
(1) (2) ( )
1
, , , , , , , , , ,
, , (where 1, , )
i i i i i i iT N N T N N N N T
I
N N T I
y y y y y y
Y i I
′
+ +
′′ ′ ′
× × × ×
=
= =
y
y y y
L L L L L
L L
6 June 2005 Roland-Holst and Yang Slide 17
OLS 2
Regressors: matrix ( )iX consists of dummy variables for home region m ,
foreign region n and time t :
1 for region 0 else
1 for region 0 else
1 for time 0 else
mD
nD
tD
α
γ
λ
⎧= ⎨⎩⎧
= ⎨⎩⎧
= ⎨⎩
( )
( ) ( )
( ) ( ) ( )
6
(1)
(2)
( )6
, , , , ,
0 0 00 0 0
0 0
i i imt nt mn N N T
IN N T I I
X D D D Y Y d
XX
X
X
α γ λ × × ×
× × × × ×
⎡ ⎤= ⎣ ⎦
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
M M O M
L
6 June 2005 Roland-Holst and Yang Slide 18
OLS 3
( )( )1 2 1, , , I N N T I
ε ε ε′′ ′ ′
× × × ×= Lε
( )( ) 2
| 0
| N N T I
E X
E X Iσ′× × ×
=
=
ε
εε
Disturbances are given by
with
Now formulate the model as
and estimate with
Y X= +β ε
( ) ( )1OLS X X X Y−′ ′=)β
6 June 2005 Roland-Holst and Yang Slide 19
Seemingly Unrelated Regressions 1
In this case we use a technique called Feasible Generalized Least Squares (FGLS), with
Regressand: vector ( )iy consists of bilateral trade flows
of commodity 1,2,i I∈ L between region m (
1, ,m N= L ) and region ( 1, , 1, 1, , 1n i i N= − + +L L )
sorted by t ( 1, ,t T= L ).
( ) ( )( )( )( )
( ) ( ) ( ) ( ) ( ) ( ) ( )121 12 11 1 1 1 1
(1) (2) ( )
1
, , , , , , , , , ,
, , (where 1, , )
i i i i i i iT N N T N N N N T
I
N N T I
y y y y y y
Y i I
′
+ +
′′ ′ ′
× × × ×
=
= =
y
y y y
L L L L L
L L
6 June 2005 Roland-Holst and Yang Slide 20
SUR 2
Regressors: matrix ( )iX consists of dummy variables for home region m ,
foreign region n and time t :
1 for region 0 else
1 for region 0 else
1 for time 0 else
mD
nD
tD
α
γ
λ
⎧= ⎨⎩⎧
= ⎨⎩⎧
= ⎨⎩
( )
( ) ( )
( ) ( ) ( )
6
(1)
(2)
( )6
, , , , ,
0 0 00 0 0
0 0
i i imt nt mn N N T
IN N T I I
X D D D Y Y d
XX
X
X
α γ λ × × ×
× × × × ×
⎡ ⎤= ⎣ ⎦
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
M M O M
L
6 June 2005 Roland-Holst and Yang Slide 21
SUR 3
Disturbances in this case are given by
with
( )( ) ( ) ( )
( ) ( ) ( )
11 12 1
21 22 2
1 2
| 0
| N N T I N N T I
I I N N T N N T
I
I
I I II
E X
E X
I
σ σ σσ σ σ
σ σ σ
′× × × × × × ×
× × × × × ×
=
= Ω
= Σ ⊗
⎡ ⎤⎢ ⎥⎢ ⎥Σ =⎢ ⎥⎢ ⎥⎣ ⎦
L
L
M M O M
L
ε
εε
6 June 2005 Roland-Holst and Yang Slide 22
SUR 4
Now formulate the model as in OLS, i.e.
but estimate with FGLS as
where
Y X= +β ε
( ) ( ))( ) )( )
11 1
11 1
FGLS X X X Y
X I X X I Y
−− −
−− −
′ ′= Ω Ω
⎡ ⎤ ⎡ ⎤′ ′= Σ ⊗ Σ ⊗⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
)β
)
) ) )
) ) )
) ) )
11 12 1
21 22 2
1 2
I
I
I I II
σ σ σ
σ σ σ
σ σ σ
⎡ ⎤⎢ ⎥⎢ ⎥Σ = ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
L
L
M M O M
L
The least squares residuals OLSY X= −e)β can be used to
estimate consistently the elements of Σ with )
( ) , 1, ,i jij i j I
N N Tσ
′
= =× ×
e eL
6 June 2005 Roland-Holst and Yang Slide 23
Generalized Method of Moments 1
Using information of aggregate provincial/regional trade flows by commodity, we can add additional moment restrictions:
where IM denotes provincial/regional domestic import demand.
( ) ( )
1
(1)(1)
1
( )( )
1
( 1 , )N
i int
m
N
ntm
INInt
m
IM i I
IM
IM
⋅=
⋅=
⋅=
= =
⎡ ⎤∑ ⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦∑⎢ ⎥⎣ ⎦
∑y
y
y
L
M M
6 June 2005 Roland-Holst and Yang Slide 24
GMM 2
Regressand: The vector ( )iy consists of bilateral trade flows of commodity 1,2,i I∈ L between region
1, ,m N= L , and region 1, , 1, 1, , 1n i i N= − + +L L , sorted
by t ( 1, ,t T= L ).
( ) ( )( )( )( )
( ) ( ) ( ) ( ) ( ) ( ) ( )121 12 11 1 1 1 1
(1) (2) ( )
1
, , , , , , , , , ,
, , (where 1, , )
i i i i i i iT N N T N N N N T
I
N N T I
y y y y y y
Y i I
′
+ +
′′ ′ ′
× × × ×
=
= =
y
y y y
L L L L L
L L
6 June 2005 Roland-Holst and Yang Slide 25
GMM 3
Regressors: matrix ( )iX consists of dummy variables for home region m ,
foreign region n and time t :
1 for region 0 else
1 for region 0 else
1 for time 0 else
mD
nD
tD
α
γ
λ
⎧= ⎨⎩⎧
= ⎨⎩⎧
= ⎨⎩
( )
( ) ( )
( ) ( ) ( )
6
(1)
(2)
( )6
, , , , ,
0 0 00 0 0
0 0
i i imt nt mn N N T
IN N T I I
X D D D Y Y d
XX
X
X
α γ λ × × ×
× × × × ×
⎡ ⎤= ⎣ ⎦
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
M M O M
L
6 June 2005 Roland-Holst and Yang Slide 26
GMM 4
Disturbances in this case are given by
with
( )( ) ( ) ( )
( ) ( ) ( )
11 12 1
21 22 2
1 2
| 0
| N N T I N N T I
I I N N T N N T
I
I
I I II
E X
E X
I
σ σ σσ σ σ
σ σ σ
′× × × × × × ×
× × × × × ×
=
= Ω
= Σ ⊗
⎡ ⎤⎢ ⎥⎢ ⎥Σ =⎢ ⎥⎢ ⎥⎣ ⎦
L
L
M M O M
L
ε
εε
6 June 2005 Roland-Holst and Yang Slide 27
GMM 5
Again we formulate the model as in OLS, i.e.
but use the GMM estimator given by
Y X= +β ε
)
) )1 1( , | ) ( , | )arg min
N N T I N N T Ii ii i
GMMX Y X YW
N N T I N N T Iβ
ψ β ψ β′
× × × × × ×= =⋅ ⋅
⎛ ⎞ ⎛ ⎞∑ ∑= ⎜ ⎟ ⎜ ⎟× × × × × ×⎝ ⎠ ⎝ ⎠
)β
6 June 2005 Roland-Holst and Yang Slide 28
GMM 6
)
( )
)( ))( )
11 1 1
11 1
( ) (1)1 111
( ) ( )1 1
1
( , | )
N N T Ii i i iN N T I
N N T Ii iK i iN N T I
Ni iK I ntN Nm
NI IntN N
m
x y x
x y xX Y
IM
IM
β
βψ β
′× × ×=× × ×
′× × ×=× × ×
⋅+ × ⋅
=
⋅=
⎡ ⎤−∑⎢ ⎥⎢ ⎥⎢ ⎥
−⎢ ⎥∑⎢ ⎥= ⎢ ⎥∑ −⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
∑ −⎢ ⎥⎣ ⎦
y
y
M
M
)
) )( ))( ) )( )
1
1
Var ( , | )
( , | ) ( , | )
i
N N T Ii i i
W
X Y
X Y X Y
N N T I
ψ β
ψ β ψ β
−
⋅
′× × ×= ⋅ ⋅
= ∆
∆ =
⋅∑=
× × ×
where
and
6 June 2005 Roland-Holst and Yang Slide 29
Estimator Selection
• After generating estimates by all three methods, we can use a variety of criteria to choose between them.
• In traditional econometric analysis, one would use the goodness of fit measure, adjusted R2 as the selection criterion.
• For our primary objective is imputing missing bilateral trade flows, we would choose the estimator with the largest .
2R
6 June 2005 Roland-Holst and Yang Slide 30
References
• He, Jianwu, and Shantong Li (2004), “A Three-regional CGE Model for China,”presentation to the DRC, Beijing, November.
• He, Jianwu (2004), “A Social Accounting Matrix for China: 2000,” processed, DRC, Beijing, November.
• Mátyás, L. (1997), “Proper Econometric Specification of the Gravity Model,” The World Economy 20(3): 363--368.