Estimation of the Undervaluation of the Chinese Currency
by a Non-linear Model
Gene Hsin Chang
Department of Economics, the University of
Toledo, Toledo, USA
March 2007
©
Background
Senate Bill:
by Charles Schumer (D., N.Y.) and Lindsey
Graham (R., S.C.)
Currency manipulation by China.
If no RMB revaluation, imports from China can
be subject to 27.5% tariff.
Vote by July.
Background
House China Currency Act:
by Congressmen Duncan Hunter (R., Calif.) and Tim
Ryan (D., Ohio)
Currency manipulation as a "prohibited export subsidy"
by China, under Article VI of the GATT.
If no RMB revaluation, trigger an antidumping or
countervailing duty. Prohibition of importation of
Chinese defense products
US Trade with China
(million dollars)
250,000
200,000
Total
Deficit
150,000
100,000
50,000
19
85
19
87
19
89
19
91
19
93
19
95
19
97
19
99
20
01
20
03
20
05
0
Estimates of Undervaluation of
RMB
Jeffrey Frankel (2004): Using Rogoff model and
found that yuan is 42% undervalued
Lardy and Goldstein (2003): 15%-25%
undervalued. No formal model provided.
Gene Chang: Linear regression model: 19.2%
undervalued
Gene Chang and Shao (2004): linear model with
control of heteroskedasticity: 22.5%
undervalued.
Estimates of Undervaluation of
RMB
Zhang and Pan (2004): 15-22%
undervalued.
Steve Hanke and Michael Connoly (2004
WSJ): No undervaluation
Ronald McKinnon: No revaluation, at most
1%.
Robert Mundell: No need for revaluation
for RMB
Approaches to estimate Equilibrium
Value of Yuan
Determination in the short-run: Supply
and demand for the foreign exchanges
Estimating supply and demand for the
foreign exchanges, including trade
balance and current account balance.
Approaches to estimate Equilibrium
Value of Yuan
Determination in the long-run
Absolute purchasing power parity
Real Exchange Rate (RER)
RER = (E X PChina) / PU.S.
If absolute PPP holds, RER = 1
Data are available now for abs PPP
Approaches to estimate Equilibrium
Value of Yuan
Determination in the long-run
Purchasing power parity
E = PU.S. / PChina
Relative purchasing power parity
% depreciation in E
= inflation U.S. – inflation China
Problems with using relative PPP to
estimate equilibrium value of yuan
Real Exchange Rate of Countries
Estimation of Equilibrium Value
of Yuan
Why is RER greater than 1 for poor
countries?
The Balassa-Samuelson hypothesis
The Bhagwati-Kravis-Lipsey hypothesis
RER is a function of per capita income
level
Estimation of Equilibrium Value
of Yuan: Model Specification
Model with control of the income level:
RER = f (GDP per capita)
Data for RER
Linear or Rogoff log linear
(ln) RER = a + b X (ln) GDP per capita
Control heteroskedasticity
Estimation of Equilibrium Value
of Yuan: Simple OLS
Using the world sample to obtain the
estimates and the prediction equation
Intercept
Coefficients Standard
error
4.28039
0.15922
GDP p.c. -0.13386
0.01320
t -statistics
26.88387
-10.14495
Simple Linear Model: OLS
RER i  c  ( a  b GDPpci ) 1   i
The Rogoff Model
ln RER = a + b ln GDPpc + ε
Coefficients
a
b
0.68742
-0.38561
Sum of Squared Errors of the log RER
values: 11.105
Sum of Squared Errors of the true values*:
856.44
The Rogoff Specificatgion
RER i  c  ( a  b GDPpci ) 1   i
The New Non-linear Model
The new model
Non-linear regression equation
RER = c + (a + b GDPpc)-1 + ε
The New Non-linear Model
Regression results
Observations:160
Sum of squared errors: 299.0869
Estimated coefficients
a: 0.18903852
b: 0.023503552
c: 0.010
The New Non-linear Model
RMB Undervaluation Estimation
Non-linear Model
Year
GDP pc
2001
RER
actual
RER
predicted
Valuation
P-value**
1978
662
1248
1371
1502
1692
2394
2656
2876
2.00
3.26
4.13
4.15
4.44
4.83
4.28
4.09
4.90
4.59
4.53
4.47
4.38
4.09
3.99
3.91
59.2%
28.9%
8.9%
7.2%
-1.4%
-18.1%
-7.4%
-4.8%
0.084
1985
1986
1987
1991
1994
1995
1996
0.187
0.299
0.406
0.384
0.263
0.372
0.424
RMB Undervaluation Estimation
by Non-linear Model
Year
GDP pc
2001
RER
actual
RER
predicted
Valuation
P-value**
1998
3315
3506
3756
4020
4305
4647
4999
5462
4.27
4.41
4.46
4.53
4.59
4.55
4.32
3.96
3.76
3.69
3.62
3.54
3.46
3.36
3.27
3.16
-13.7%
-19.5%
-23.5%
-28.0%
-32.7%
-35.3%
-32.0%
-25.3%
0.437
1999
2000
2001
2002
2003
2004
2005*
0.409
0.357
0.319
0.304
0.278
0.286
0.315
Fittings of Different Models
1000
Sum of Square
Errors
800
600
400
200
0
Sum of Square
Errors
Rogoff
OLS
Non-linear
856.4
539.9
299.1
Comparison of various models
Year
OLS
Hetero
Rogoff
Non-linear
1978
1980
1981
1984
1986
1987
1990
1992
1993
52.3%
48.7%
42.3%
30.5%
20.7%
-0.7%
-2.5%
-16.3%
-32.2%
51.3%
44.9%
40.2%
33.8%
19.7%
-8.9%
-5.0%
-19.7%
-35.2%
6.2%
-2.8%
-16.2%
-45.6%
-70.2%
-118.8%
-126.2%
-162.4%
-201.7%
59.2%
55.8%
50.1%
38.7%
28.9%
8.9%
6.0%
-9.3%
-26.1%
Comparison of various models
Year
OLS
Hetero
Rogoff
Non-linear
1994
1995
1998
1999
2000
2001
2002
2003
2004
2005*
-21.9%
-9.1%
-11.3%
-15.8%
-18.2%
-21.0%
-23.8%
-24.4%
-19.6%
-11.5%
-24.3%
-12.6%
-8.9%
-14.3%
-18.4%
-20.1%
-23.2%
-22.5%
-19.2%
-181.0%
-153.6%
-162.4%
-173.9%
-180.2%
-187.5%
-194.6%
-196.0%
-184.4%
-164.6%
-18.1%
-7.4%
-13.7%
-19.5%
-23.5%
-28.0%
-32.7%
-35.3%
-32.0%
-25.3%
gyz
Re
-50.00%
-100.00%
pub
Bel lic
a
Co
ngo Camb rus
, De odi
a
m.
Re
p.
Ind
ia
Sou Ukrai
th A ne
fr
Vie ica
tn
B u am
lga
ria
C
Ban
h
gla ina
Phi desh
lipp
Ind ines
one
si a
Ru
Ita
ssi
an Thail ly
Fed
a
era nd
Ho
tion
ng
Ko Hung
a
ng,
Ch ry
i
Fin na
lan
Fra d
nc
Un
ited Swed e
en
Ki
Ko ngdom
rea
,R
ep
Jap .
a
Sau Kuw n
di A ait
rab
Ven
Zam ia
ez u
bia
Co ela, R
ngo
B
, Re
p.
Kyr
Under/over-valuation of currencies (2001)
by hetero-controlled linear model
100.00%
50.00%
0.00%
-150.00%
The theoretical justification for the
Rogoff-Frankel regression model
Why is the regression mean (predicted
line)) serves as the equilibrium exchange
rate?
Why does the error term (residual)
measure the magnitude of the under or
over valuation?
The theoretical justification
for the Rogoff-Frankel
regression model
• The Rogoff-Frankel regression model
RER i  f ( GDPpci )   i
Theoretical Justification
• Starting from a simple version
• n trading countries
• All are of the same economic size
and at the same development level
(same GDP per capita level).
Theoretical Justification
• Country i's trade balance (net exports) is
a function of its overvaluation or
undervaluation.
X i  a(RER i -RER*)
• where Xi is the next exports of country i.
RER* is the equilibrium real exchange rate.
RERi is the real exchange rate of country i,
and a is a positive constant.
Theoretical Justification
• If the country revalues/devalues its
currency back to the equilibrium
level thus, then its trade is in
balance Xi = 0.
• Conversely, if its trade is not
balanced, its currency is not at the
equilibrium level.
Theoretical Justification
• Globally, all trade deficits and
surpluses shall be cancelled out.
Summarize all countries trade, and
note that it must be globally
balanced: .
n
X
n
i
 a  (RER i  RER*)  0
i
 RER
i
i
i
 n RER*
Theoretical Justification
• The equilibrium real exchange rate
RER* is determined by
1
RER*=  RER i
n i
Theoretical Justification
• So, the equilibrium exchange rate is
the mean of the real exchange rates.
• Further, the difference between RERi
and RER*, , measures the under- or
overvaluation.
With different GDP sizes
• Let si be country i's share of the
global trade.
• The net exports volume of country i
will be affected by si.
• X  as (RER -RER*)
i
i
i
n
n
n
i
i
i
•  X i   asi (RER i  RER*)  a  si (RER i  RER*)  0
With different GDP sizes
•
n
 s RER
i
i
• Because
• RER* 
n
i
n
 RER* si
s

n
i
i
 1 , so,
 s RER
i
i
i
i
• Hence the equilibrium RER is trade share
weighted average of RERs of all trading
countries.
With the Balassa-Samuelson
effect
•
RER*=f ( GDPpc)
• there are m income-per-capita groups in the
world. There is only one country in each incomeper-capita group:. Without loss of generality, let
us assume that the income-per-capita level
follows the same order, where group 1 is the
poorest and group m is the richest. The nominal
exchange rates of all countries are at the
equilibrium levels. That is, all of them are trade
balanced:
X (e* )  0 j  1,..., m
j
j
With the Balassa-Samuelson
effect
• PjT the price of tradable goods of country j
• PjNT the price of the non tradable goods of
country j.
• k is the share of the tradable in the GDP.
•
T
NT
Pj  kPj  (1  k ) Pj
With the Balassa-Samuelson
effect
• The law of one price, or the principle of
purchasing power parity, only applies to the
tradable. The equilibrium exchange rate is:
•
•
e*j 
P
T
PjT
PT
the price of the tradables of the numeraire
country
*
numerair
e
PT
 T 1
P
With the Balassa-Samuelson
effect
• equilibrium real exchange rate of a
country at income group j is:
RER j * 
e*j P
Pj

e*j kP T  e*j (1  k ) P NT
kP  (1  k ) P
T
j
NT
j

ke*j P T  (1  k )e*j P NT
ke*j P T  (1  k ) PjNT
• Hence the equilibrium real exchange rate
RER* is a function of the price of nontradable of the country.
With the Balassa-Samuelson
effect
• The B-S effect implies the
nontradable price in poor countries is
lower,
• P NT  g (GDPpc )
j
j
• with
• g '(GDPpc)  0
With the Balassa-Samuelson
effect
• Hence,
RER j * 
ke*j P T  e*j (1  k ) P NT
ke*j P T  (1  k ) g (GDPpc j )
• RER * (GDPpc)  f (GDPpc)
• and f’ < 0
The Balassa-Samuelson effect
• Let country ij denote country i in income-percapita group j. Each group j has nj countries.
Drop the assumption that each country is in
equilibrium, but assume all of them are of the
same trade volume size, we have the net exports
of country ij be determined by its real exchange
rate against the equilibrium value:
•
X ij  a (RER ij -RER j *)
The Balassa-Samuelson effect
• Globally, all trade deficits and
surpluses shall be cancelled out.
Summarize all countries trade, and
note that it must be globally
balanced:
•
m
nj
 X
j
i
ij
0
The Balassa-Samuelson effect
•
m
nj
 X
ij
m
nj
j
i
 a  (RER ij  RER j *)  0
j
i
m
nj
m
nj
m
j
i
j
i
j
•  RER ij   RER* j  n j RER* j
The Balassa-Samuelson effect
• Then the following is true, the above equation is
true:
•
1
RER* j 
nj
nj
 RER
i
ij
j  1,..., m
• This shows that the equilibrium vales of the real
exchange rates is the means of the real exchange
rates of all countries in the same income groups.
The Balassa-Samuelson effect
• If each country ij adjust its RERij to
the mean of RERs of its income
group, then their trade is balanced
as indicates.
• The country's under- or
overvaluation currency can be
measured by the deviation from the
mean of RERs of its income group.
Trading across income groups
• Let Xi be any country, which can be
at different income level (but we
maintain the assumption that their
volume of trade is the same, or, its
net exports is only affected by its
under or overvaluation of exchange
rate but not the GDP size).
Trading across income groups
• X i  a(RER i - RER i *)
• RER*  E (RER i )  f ( ,GDPpc)
• where  is the parameter(s).
• X i  a(RER i - RER i *)  a[RER i - f ( ,GDPpci )]
Trading across income groups
• The condition of balance of the global
trade  X   a[RER - f ( ,GDPpc )]  0
n
n
i
i
•
n
 [RER
i
i
i
i
- f ( ,GDPpci )]  0
i
Trading across income groups
• Suppose we use the Rogoff-Frankel
model to regress RERi on GDPpc,
would the estimated RER be the
consistent estimate of , that satisfy
the condition of global trade balance?
Trading across income groups
• The regression model is
•
•
RER i  f ( ,GDPpci )   i
Suppose that the regression method is a nonlinear least square or maximum likelihood
estimation. Then it implies to minimize with
respect to the parameters:
• min

2
[RER

f
(

,GDPpc
)]

i
i
i
Trading across income groups
• If the specification is Chang 2006:
RER*  f ( ,GDPpc)  c  (a  b GDPpc)1
Trading across income groups
• The first order condition of nonlinear
least square regression is,
•


2
[RER

f
(

,GDPpc
)]

i
i
i

 2 {[RER i  f ( ,GDPpci )]
f ( ,GDPpc i )}

i
0
Trading across income groups
• The first-order-condition with
parameter c in the regression
estimation will lead to

f ( ,GDPpci )  1
c
Trading across income groups
• Then,
•

1 2
[RER

c

(
a

b
GDPpc
)
]

i
i
c i
2
1
{[RER

c

(
a

b
GDPpc
)
]

i
i
i
2
1
[RER

c

(
a

b
GDPpc
)
]1

i
i
i
0

f ( ,GDPpci )}
c
Trading across income groups
• By using MLE or nonlinear LS, the
estimates for parameters are
asymptotically consistent. The
estimated model servers as the
equilibrium values of the exchange
rates of trading countries, taking
account of the Balassa-Samuelson
effect.
Trading across income groups
• Our model is justified by three reasons:
• first, if a country adjusts its exchange rate to , its
trade is balanced as Xi = 0.
• Secondly, if all countries adjust their exchanges
to , each country is own trade balanced.
• Finally, by using Chang's model specification or
linear specification, the global trade balance
condition is always satisfied by the MLE estimates,
even if each country itself has trade surplus,
which is caused by an undervalued currency, or
has trade deficit, which is caused by an
overvalued currency.
Summary
The long run equilibrium value of RMB provides
the best information about the trend of the
valuation of RMB.
Absolute PPP with control of the BalassaSamuelson effect is the best approximation
available for the long-run equilibrium value of a
currency.
The suggested non-linear model provides better
fitting for the data than previous models.
Concluding Remarks
RMB is undervalued by 25.5% in 2005,
hence the revaluation pressure
continuously presents.
RMB has revalued substantially in real
term in 2005 by a nominal revaluation and
a higher inflation rate (10.46%) in the GDP
deflator.
Concluding Remarks
The magnitude of undervaluation will diminish in
near future due to: (1) revaluation of the nominal
exchange rate of RMB, and (2) a higher inflation
rate in China than that in U.S.
The undervaluation will intensify as China is
growing rapidly.
The net result depends on the relative
magnitudes of the two opposite forces. But
RMB revaluation represents the general trend,
which is in response to the market pressure.