This work is distributed as a Discussion Paper by the
STANFORD INSTITUTE FOR ECONOMIC POLICY RESEARCH
SIEPR Discussion Paper No. 00-25
A Purchasing PowerParity Paradox
Asaf ZussmanStanford University
January 2001
Stanford Institute for Economic Policy ResearchStanford University
Stanford, CA 94305(650) 725-1874
The Stanford Institute for Economic Policy Research at Stanford University supports research bearing oneconomic and public policy issues. The SIEPR Discussion Paper Series reports on research and policyanalysis conducted by researchers affiliated with the Institute. Working papers in this series reflect the viewsof the authors and not necessarily those of the Stanford Institute for Economic Policy Research or StanfordUniversity.
1
A Purchasing Power Parity Paradox*
Asaf Zussman
[January 2001]
Abstract
The Balassa-Samuelson hypothesis is considered the most important structural model oflong-run deviations from purchasing power parity (PPP). I present a simple model thatbuilds on the Balassa-Samuelson hypothesis and in which, paradoxically, PPP necessarilyholds in the long run. In this model real exchange rates converge because productivitylevels converge. The model’s predictions are tested empirically for 24 OECD countriesin the period 1950-92. Overall, the analysis supports the prediction that productivityconvergence is accompanied by real exchange rate convergence.
* I would like to thank William Baumol, Jeffrey Bergstrand, Menzie Chinn, CharlesEngel, Charles Jones, Michael Kumhof, Anne Krueger, Peyron Law, Ronald McKinnon,Kanda Naknoi, Assaf Razin and participants of the international economics “brown bag”seminar at Stanford University for their helpful suggestions and comments on previousversions of this paper.
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1 Introduction
This paper attempts to marry the purchasing power parity (PPP) and growth (in
particular “convergence club”) literatures. It raises and tests a very simple hypothesis:
that convergence in levels of per-capita income within a group of countries implies
convergence towards absolute PPP between them. The motivation for this exercise is
mainly empirical and it relies on two stylized facts. The first is that at any given point in
time rich countries tend to have higher same currency prices relative to poor ones. The
second stylized fact – the existence of convergence clubs - is that within specific groups
of countries and during specific periods of time, an inverse relationship can be found
between a country’s initial level of per-capita income and its rate of growth of per capita
income over time. Combining these two stylized facts, we can expect to find a third one:
within a convergence club there should be convergence towards absolute PPP.
At the heart of the paper’s analysis is the venerable Balassa-Samuelson (BS)
hypothesis. The BS hypothesis, advanced almost 40 years ago, is considered the most
important structural model of long-run deviations from PPP. The novelty of this paper’s
approach is that it shows that ultimately the BS hypothesis does not necessarily imply
that PPP does not hold in the long run. I present a model that augments the BS
framework with a technological diffusion mechanism to yield convergence in both per
capita incomes and in real exchange rates across countries over time.
Section 2 positions the paper within the existing literature and motivates the
theoretical and empirical analysis. Section 3 presents the theoretical model. Empirical
testing of the model’s predictions is carried out in section 4. Section 5 concludes.
3
2 The existing literature
PPP and structural determinants of the real exchange rate
PPP is a simple empirical proposition that, once converted into a common
currency, national price levels should be equal. The building block of PPP is the law of
one price, which says that goods market arbitrage will equate prices of traded goods
across countries. If all goods are traded and if weights of all goods in aggregate price
indices are identical across countries, aggregate price levels will be equated. Thus
according to PPP, the real exchange rate, which can be defined as the ratio of two
countries’ price levels, expressed in a common currency, should be equal to unity, for all
pairs of countries and at all times.
Since data on absolute price levels across countries is hard to obtain, empirical
testing has concentrated on relative PPP, which says that the real exchange rate should be
constant, or at least stationary, over time. In recent years there is a growing consensus in
the empirical literature that (relative) PPP tends to hold in the long run, i.e. that real
exchange rates are stationary with a very slow rate of mean reversion. In many cases
half-lives of deviations from relative PPP are estimated to be in the range of 3 to 5 years.1
On the other hand a well established stylized fact is that when all countries’ price
levels are translated to a common currency at prevailing nominal exchange rates, rich
countries tend to have higher price levels than poor ones.2 This inverse relationship
between a country’s relative per-capita income level and its real exchange rate obviously
1 The empirical PPP literature is very extensive. Some examples include Cheung and Lai (2000), Diebold
et. al. (1991), Edison et. al. (1997), Engel (1999, 2000), Engel et. al. 1996, Engel and Kim (1999),Frankel and Rose (1996), Jorion and Sweeney (1996), Koedijk et. al. (1998), Lothian (1997), Lothian(1998), Lothian and Taylor (1996), Oh (1996), Panos et. al. (1997), Papell (1997), Papell and Theodoridis(1998), Rogers and Jenkins (1995), Taylor (2001a, 2001b). It has to be emphasized that the consensus on
4
contradicts absolute PPP. A few theoretical explanations for this deviation from absolute
PPP have been offered, most of them focus on the distinction between tradable and non-
tradable goods.3 The most prominent among these explanations is the BS hypothesis.4 A
generalized formal model that captures the BS argument is presented in section 3. Here I
will concentrate on its basic features. Balassa and Samuelson argued that productivity
tends to be higher in rich countries than in poor ones and that this tendency is more
pronounced in the tradable than in the non-tradable goods sector. Assuming perfect
international capital mobility, wage equalization across sectors in each country and the
law of one price for tradable goods,5 this pattern of productivity differentials between
sectors and across countries will lead to higher non-tradable goods prices in rich
countries. Assuming identical shares of tradable and non-tradable goods in the overall
price indices across countries, this means that rich countries will tend to have higher price
levels and appreciated real exchange rates relative to poor ones. Moreover, even if the
productivity differential between two countries is identical in both sectors, the real
exchange rate in the rich one will still be appreciated if non-tradables are more labor
intensive than tradables. This is the Baumol-Bowen hypothesis that is also formally
analyzed in section 3.
PPP is not complete. For recent challenges see Engel (2000). For potential biases in the empiricalliterature see Taylor (2001b).
2 See for example Heston et. al. (1994), Rogoff (1996) and Summers and Heston (1991).3 See Bergsrand (1991).4 The original references are Balassa (1964) and Samuelson (1964). Discussions of the model can be
found in Asea and Corden (1994), Dornbusch (1987), Froot and Rogoff (1995), Rogoff (1996).5 The validity of the last assumption is contested in the empirical literature. Barriers to trade (tariff and
non-tariff), transportation costs, imperfect competition and other factors drive a wedge between prices oftraded goods in different countries. The first major attack on the empirical validity of the Law of OnePrice is Isard (1977). The literature on this issue was surveyed recently by Goldberg and Knetter (1997).The other assumptions underlying the BS argument could also, of course, be subject to criticism.
5
A second, related, theory that also predicts that rich countries will have
appreciated real exchange rates is due to Bhagwati (1984) and others. It focuses on
differences in endowments of capital and labor across countries. Rich countries tend to
have higher capital-labor ratios than poor countries (because of imperfect capital
mobility) and thus higher marginal product of labor and higher wages.6 Assuming again
that non-tradables (largely services) are labor-intensive relative to tradables (largely
commodities), non-tradables will tend to be cheaper in poor countries than in rich ones.
A third explanation is demand oriented.7 It suggests that, assuming non-
homothetic tastes, price levels are higher in countries with higher per capita income
because non-tradables are luxuries in consumption while tradables are necessities.
Countries with higher real per-capita income will exhibit, in equilibrium, stronger
demand for non-tradables relative to tradables, raising their relative price.8
The cross-section evidence on the relationship between relative incomes and real
exchange rates tends to raise the question: does it have a time-series counterpart? The
literature answers this question only indirectly, mainly by empirically testing variants of
the BS hypothesis. In the time-series context the BS proposition is that the real exchange
rate of country i relative to country j will tend to appreciate when i’s rate of growth of
productivity relative to j’s is faster in the tradable than in the non-tradable sector. The
results coming from this line of research lend some support to the BS hypothesis:
6 This argument assumes that factor endowment differences between rich and poor countries are
sufficiently great that factor-price equalization does not hold.7 See Bergstrand (1991, 1992).8 Of course, factors other than national income (including its supply and demand elements) influence real
exchange rates across countries. Some of the variables suggested in the literature as determinants of thereal exchange rate are the share of government expenditures (and specifically military expenditures) inGDP, terms of trade, foreign aid, relative abundance in mineral resources and others. See: Bergsrand(1992), Clague (1986), Connolly and Devreux (1992), Devreux and Connolly (1996), Edwards and van
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differential rates of growth of productivity across sectors and countries do, in certain
cases, influence real exchange rates behavior.9
The findings of the two lines of research presented above are to some extent
contradictory. Those of the “structural” literature say that we should expect real
exchange rates to follow productivity growth differentials and other real, structural,
factors. Thus long run trends of appreciation or depreciation of the real exchange rate are
possible. On the other hand the findings of the “non-structural” PPP literature, which
uses price and nominal exchange rate data only, say that we should not expect any long-
run trend in the real exchange rates. A natural environment for examining the validity of
the competing claims of the two schools is one in which different countries have very
different growth experiences over time. In such an environment one might expect real
exchange rates to have changed most dramatically over time. One such environment is a
convergence club.
Convergence clubs
Two main concepts of convergence appear in the growth literature. They are
termed β-convergence and σ-convergence. We say that there is absolute β-convergence
if poor economies tend to grow faster than rich ones. The concept of σ-convergence can
be defined as follows: a group of economies are converging in the sense of σ if the
dispersion of their real per-capita income levels tends to decrease over time. The two
Wijnbergen (1987), Kravis and Lipsey (1988), Neary (1988). Nonetheless, it is still the case that relativeper-capita national income is the best single explanatory variable of real exchange rates across countries.
9 There is a large body of empirical literature analyzing the behavior of real exchange rate over time usingthe differential productivity approach. In many cases this variable is augmented by other explanatoryvariables such as government expenditures and the terms of trade. See for example Amano and vanNorden (1995), Bahmani-Oskooee and Rhee (1996), Canzoneri et. al. (1999), Chinn (1997, 2000), Chinnand Johnston (1997), De Gregorio, Giovannini and Krueger (1994), De Gregorio, Giovannini and Wolf(1994), De Gregorio and Wolf (1994), Dutton and Strauss (1997), Edison and Klovland (1987), Engel
7
concepts are, of course, related: β-convergence is a necessary, but not a sufficient
condition for σ-convergence.10 Real exchange rate levels are heavily influenced by
monetary factors, especially in the short and the medium runs, and thus are prone to be
much more volatile than per-capita income levels. Because of this fact it would be much
harder for us to detect σ-convergence in our data. Therefore the focus of this paper will
be on β-convergence.
One of the key predictions of the neoclassical growth model is that the growth
rate of an economy will be positively related to the distance that separates it from its own
steady state. This is the concept known in the growth literature as conditional β-
convergence. To facilitate the distinction, the concept of β-convergence discussed above
is sometimes called absolute convergence. The conditional and the absolute convergence
hypotheses coincide only if all the economies have the same steady state. In order to test
the hypothesis of unconditional convergence we need a group of countries that are
expected to have the same steady state. A set of such countries would then tend to form a
“convergence club": a group of countries in which initially poor economies tend to grow
faster than the rich ones and thus catch-up or converge.
By testing for a negative relationship between average annual rates of growth and
initial levels of income, Baumol (1986) concluded that industrial countries appear to
belong to one convergence club, middle income countries to a separate, only moderately
converging club, and that low income countries actually diverged over time.11 Following
this analysis a large body of literature has tackled the empirical question of growth and
and Kim (1999), Ito (1997), Hsieh (1982), Marston (1987, 1990), Micossi and Milesi Ferretti (1994) andStrauss (1995, 1996, 1999) among others
10 For discussion of these concepts see Galor (1996) and Sala-i-Martin (1996).11 A similar early analysis of convergence is Abramovitz (1986).
8
convergence. The consensus that emerged in this literature suggested an absence of
global convergence to a single steady state. Yet the existence of a convergence club
among the top end of the income spectrum, specifically among OECD countries, is a
stylized fact of the empirical growth literature.12
12 See Barro and Sala-i-Martin (1992, 1995), and Sala-i-Martin (1996). For convergence among OECD
countries see Dowric and Nguyen (1989).
9
3 Model
This section presents a simple theoretical model that consists of three basic
elements: (1) the BS model in its cross-section dimension; (2) the BS model in its time-
series dimension; (3) a simple technological diffusion mechanism. The combination of
these three elements yields the result of simultaneous dual convergence: in levels of per
capita income and in real exchange rates. In other words under the assumptions of the
model absolute PPP necessarily holds in the long run.
I will assume a small open economy that takes the world real rate of interest as
given. Two goods are produced in the economy, traded and non-traded, according to the
following constant returns to scale Cobb-Douglas production functions:
1( ) ( )T T T Tt t t tY L Kα αθ −= (3.1)
1( ) ( )N N N Nt t t tY L Kβ βθ −= (3.2)
where , , ,i i i it t t tY L Kθ are output, productivity (or technology) level, labor input and capital
input at time t in sector i (i being T or N), while α and β are production function
parameters. Taking first order conditions with respect to capital and labor for each sector
yields the following equations:
(1 ) ( )T Tt t tR k αα θ −= − (3.3)
1( )T Tt t tW k ααθ −= (3.4)
(1 ) ( )N Nt t t tR P k ββ θ −= − (3.5)
1( )N Nt t t tW P k ββθ −= (3.6)
where tR and tW are the world real rate of interest and domestic wage rate, respectively,
itk is capital per worker in sector i and tP is the relative price of non-traded goods (i.e.
10
N Tt t tP P P= ). Solving for the relative price of non-traded goods we get the following
expression:
( ) ( ) ( ) ( )1
1 11 1
1
(1 )
(1 )
N T N Tt t t t t t tP R R
βα
β βα β βα αα α
ββ
α αθ θ γ θ θ
β β
− − −− − −
− = ≡
−
(3.7)
Note that if α=β then Tt
t Nt
Pθθ
= .
I will now define national price level in country i and time t, ,i tNPL , as a weighted
average of traded and non-traded goods prices. For simplicity I will assume that the
weights are constant and equal across time and countries.13 Therefore for every i and t
we have:
( ) ( )1
, , ,N T
i t i t i tNPL P Pδ δ−
= (3.8)
where ,N
i tP and ,T
i tP are the prices of the non-traded and traded goods, respectively. The
real exchange rate of country i relative to country j at time t is then defined as follows:
1
, , ,, ,
, , ,
N Tj t j t j t
i j t N Ti t i t i t
NPL P PQ
NPL P P
δ δ−
= = (3.9)
Assuming that purchasing power parity holds for traded goods, i.e. , ,T T
i t j tP P= ,
recalling the definition of the relative price of non-traded goods and using (3.7) we get:
( )( )
, ,
, ,
, ,
T Tj t i t
i j t N Nj t i t
Q
δβαθ θ
θ θ
=
(3.10)
13 This is the usual assumption made in the literature and is based on a Cobb-Douglas utility function with a
unit elasticity of substitution.
11
If j is a richer country than i, we would expect productivity levels in j to be higher
than in i. Moreover, we would expect the productivity ratio to be higher in the tradable
sector. According to the BS hypothesis this difference in productivity ratios is what
causes the real exchange rate of i to be depreciated relative to j. Formally: if
, , , ,T T N Nj t i t j t i tθ θ θ θ> and α=β then , , 1i j tQ > . On the other hand, if , , , , 1T T N N
j t i t j t i tθ θ θ θ= > and
α<β then we still get , , 1i j tQ > , as Baumol and Bowen have argued.
So far the discussion has concentrated on comparing price level across countries
at a given point in time. To see the time-series counterparts of these results we take time
derivatives of (3.10) to yield:
, , , ,, ,, , , ,
, , , , , ,
T NT Ni j t j t j ti t i t N T
i j t i j tT T N Ni j t j t i t j t i t
Qa a
Q
θ θθ θβ βδ δ
α θ θ θ θ α
= − − − = −
& & && &(3.11)
where , ,Ti j ta and , ,
Ni j ta are the differential productivity growth rates between countries i
and j in the tradable and the non-tradable sectors, respectively. Equation (3.11) says that,
assuming α=β, if the differential productivity growth is higher in the tradable sector than
in the non-tradable sector, country i will experience a real appreciation over time relative
to country j. If α<β, then even a balanced differential productivity growth will lead to
real appreciation.
The novel element in our theoretical framework comes from augmenting it by an
exogenous technological diffusion mechanism. 14 I will assume the following equations
for the rate of growth of productivity in the two sectors:
14 A similar specification of exogenous technological diffusion mechanism can be found in Dowrick and
Nguyen (1989) and Bernard and Jones (1996a, 1996b).
12
, ,
, ,
1T Ti t L tT
T Ti t i t
θ θφ
θ θ
= + −
&(3.12)
, ,
, ,
1N Ni t L tN
N Ni t i t
θ θφ
θ θ
= + −
&(3.13)
where Tφ and Nφ are positive productivity growth parameters, while ,TL tθ and ,
NL tθ are the
time t productivity levels in the economy with the highest productivity level (I will
assume that the same country has the productivity leadership in the two sectors).
Plugging (3.12) and (3.13) into (3.11) yields:
, , , ,
, , , ,
1 1T N
i L t L t L t
T Ni L t i t i t
Q
Q
θ θβδ
α θ θ
= − − −
&(3.14)
Thus if α=β country i will experience real appreciation relative to the productivity leader
as long as it is closer (in ratio terms) to the leader in non-tradable sector productivity than
in tradable sector productivity. When α<β country i will experience real appreciation
even if the productivity ratio between itself and the leader is the same across the two
sectors. If we make the simplifying assumption that non-tradable productivity is constant
and equal across time and countries, then real appreciation will result from tradable sector
productivity catch-up.
The model presented above implies that over time countries are catching up to the
leader, not only in terms of productivity, but also in terms of wages, capital-labor ratios,
per capita incomes and real exchange rates. Dual convergence is thus a result of
technological catch-up.
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4 Empirical analysis
Does the data exhibit the pattern of dual convergence predicted by the model? In
this section I will attempt to show that to a large extent the answer to this question is
positive. Within the group of OECD countries almost all had in 1950 a low level of per-
capita GDP and a depreciated real exchange rate relative to the U.S.; subsequently the
vast majority of these countries grew faster than the U.S. and experienced real exchange
rate appreciation. Thus convergence in per-capita GDP levels was accompanied by
convergence towards absolute PPP.
The database used in the empirical analysis is the Penn World Tables (PWT).
The PWT displays a set of national accounts economic time series. Its expenditure
entries are denominated in a common set of prices in a common currency so that real
quantity comparisons can be made, both between countries and over time. In its latest
version, which will be used in the paper, there are 152 countries and 43 years (1950-
1992). The analysis here focuses on 24 OECD countries.15
In this paper the variable relative per-capita GDP, Y, is defined as real GDP per-
capita, in current international prices, relative to the U.S. The variable y is the natural
logarithm of Y. The real exchange rate, Q, is defined as 100/P where P is the price level
of GDP (%) [(PPP of GDP relative to the U.S. dollar)/(exchange rate relative to the U.S.
dollar)]. The variable q is the natural logarithm of Q. For extended discussion of the
variables and the data set see Heston and Summers (1991). Figure 1 is a scatter diagram
15 The countries are Australia, Austria, Belgium, Canada, Denmark, Finland, France, West Germany,
Greece, Iceland, Ireland, Italy, Japan, Luxembourg, The Netherlands, New Zealand, Norway, Portugal,Spain, Sweden, Switzerland, Turkey, United Kingdom and the United States. These include the twentyoriginal (1961) members plus four members that joined the OECD by 1973. Current OECD membersthat joined only recently, namely the Czech republic, Hungary, Korea, Mexico and Poland, are excludedfrom the analysis.
14
of all (986) observations on relative per-capita GDP (Y) and real exchange rate (Q) for
the 23 countries during 1950-92.16 Visual inspection seems to indicate a strong negative
relationship between relative per-capita GDP and real exchange rate.
Figure 2 plots the time series of relative per-capita GDP and real exchange rate
for the twenty-three countries. For the majority of the countries a negative relationship
between the two variables is again apparent. A striking feature of the graphs displayed in
Figure 2 is that during the 1980s most countries experienced first a substantial real
depreciation and then a large real appreciation relative to the U.S. Dollar. This fact
points to the important influence of other variables, in this case mainly monetary, on the
behavior of real exchange rates.
The next stage in our analysis is more formal. We will perform a procedure to
test for unit roots in the individual time series, assuming that the data generating process
is unknown. The procedure, which is outlined in Enders (1995), has four steps. In the
first step we estimate the least restrictive model, which contains both a constant and a
time trend: 0 1 2 1
2
t t i t i t
p
i
y a y a t yγ β ε− − +
=
∆ = + + + ∆ +∑ . We then use the ττ statistic to test the
null hypothesis γ=0. If the null hypothesis of a unit root is rejected we conclude that the
series does not contain a unit root. Otherwise we go to step 2. In this step we have to
determine whether too many deterministic regressors were included in Step 1. We thus
test for the significance of the trend term under the null of a unit room (i.e. we use the
τβτ statistic to test the significance of a2). If the trend is not significant we proceed to
Step 3. Otherwise we retest for the presence of a unit root using the standardized normal
16 Greece has data only for the period 1950-91; Portugal for the period 1950-90.
15
distribution (z). If the null of a unit root is rejected we conclude that the series does not
contain a unit root. Otherwise we conclude that it does.
In Step 3 we run the regression without the trend. We test for the presence of a
unit root using the τµ statistic. If the null is rejected we conclude that the series does not
contain a unit root. Otherwise we test for the significance of the constant using the
ταµ statistic. If the drift is not significant proceed to Step 4. Otherwise we test for the
presence of a unit root using the standardized normal distribution. If the null hypothesis
of a unit root is rejected, we conclude that the series does not contain a unit root.
Otherwise we conclude that it does. In Step 4 we run the regression without the trend and
the constant. We use the τ statistic to test for the presence of a unit root. If the null
hypothesis of a unit root is rejected we conclude that the series does not contain a unit
root. Otherwise we conclude that it does. We first apply the procedure to the real
exchange rate series (q). Results are presented in Table 1.
Table 1: Unit-root tests for the real exchange rate (q)
Step 1 Step 2 Step 3 Step 4
Country τ−ττ−τ LL τ−βττ−βτ z τ−µτ−µ LL τ−αµτ−αµ ττ LL Conclusion
NEW ZEALAND -4.4321*** 1 Stationary
U.K. -4.3845*** 2 Stationary
IRELAND -4.3105*** 2 Stationary
GREECE -3.6051** 2 Stationary
FINLAND -3.5892** 4 Stationary
JAPAN -3.5608** 1 Stationary
TURKEY -3.2386* 1 Stationary
FRANCE -3.2234* 2 Stationary
SWEDEN -3.1929* 1 Stationary
AUSTRIA -3.1292 1 -3.0851** *** Stationary
ITALY -2.9839 1 -2.9585** *** Stationary
SWITZERLAND -3.0880 4 -2.9584** *** Stationary
SPAIN -3.0656 1 -2.7892* *** Stationary
NORWAY -3.0143 2 -2.7555* *** Stationary
LUXEMBOURG -3.1078 1 -2.7438* *** Stationary
DENMARK -2.9860 4 -2.6973* *** Stationary
W. GERMANY -2.9424 1 -2.6171* *** Stationary
16
ICELAND -3.1690 1 0.4000 -3.2124** 2 Stationary
PORTUGAL -3.0740 1 -1.1315 -2.8962* 1 Stationary
BELGIUM -3.1536 1 -2.1544 -2.2255 1 0.4874 -2.2501** 1 Stationary
CANADA -3.0316 1 1.9360 -2.2997 1 -0.9139 -2.1535** 2 Stationary
AUSTRALIA -2.1844 1 -1.1020 -0.8302 1 0.9310 -1.8966* 1 Stationary
NETHERLANDS -2.6186 4 -2.2517 -1.2677 4 -0.1333 -1.7186* 3 Stationary
Source: Penn World Tables and author’s calculations. Notes: *, **, *** denote, respectively, significance at the 10%, 5%, 1% levels;LL denotes lag-length. We allow for up to 4 lags. Critical values (for T=50) for the tests are given in Dickey and Fuller (1981) andHamilton (1994).
As Table 1 shows, application of the procedure to the real exchange rate (q) series
leads to rejection of the null hypothesis and to the conclusion that in all cases the series
does not contain a unit root. We next apply the procedure to the relative per-capita GDP
(y) series. Results are presented in Table 2:
Table 2: Unit-root tests for relative per-capita GDP (y)
Step 1 Step 2 Step 3 Step 4
Country τ−ττ−τ LL τ−βττ−βτ z τ−µτ−µ LL τ−αµτ−αµ ττ LL Conclusion
AUSTRALIA -5.7035*** 1 Stationary
AUSTRIA -4.1341** 3 Stationary
W. GERMANY -4.1203** 3 Stationary
TURKEY -4.0487** 0 Stationary
IRELAND -3.8857** 3 Stationary
NEW ZEALAND -3.6550** 3 Stationary
LUXEMBOURG -3.3227* 4 Stationary
PORTUGAL -3.0881 3 3.1383*** *** Stationary
ICELAND -2.9098 1 2.3914* *** Stationary
SWEDEN -2.3782 3 -0.3058 -3.6724*** 3 Stationary
SWITZERLAND -3.0708 3 0.3961 -3.4365** 3 Stationary
NETHERLANDS -2.1996 1 1.0270 -3.1880** 1 Stationary
FRANCE -1.7222 3 0.3859 -3.1847** 3 Stationary
GREECE -0.6919 3 -0.1274 -3.0158** 2 Stationary
JAPAN -1.5325 3 0.8810 -2.8249* 4 Stationary
ITALY -2.3157 1 1.6258 -2.7845* 3 Stationary
SPAIN -2.2982 3 1.2162 -2.6248* 3 Stationary
DENMARK -2.1189 1 1.1095 -2.2844* 2 -1.4791 -2.7521*** 2 Stationary
BELGIUM -2.4094 4 2.2897 -1.0967 3 -0.3732 -2.6218*** 1 Stationary
FINLAND -2.6067 3 2.2092 -2.2965 3 -1.4317 -2.5582** 4 Stationary
NORWAY -2.1077 4 1.8763 -1.3889 1 -0.7908 -2.2111** 0 Stationary
U.K. -2.5678 0 1.8063 -2.0299 1 -1.8242 -2.0775** 2 Stationary
CANADA -2.2826 2 2.2072 -1.0677 4 -0.2561 -1.7745* 3 Stationary
17
Source: Penn World Tables and author’s calculations. Notes: *, **, *** denote, respectively, significance at the 10%, 5%, 1% levels;LL denotes lag-length. We allow for up to 4 lags. Critical values (for T=50) for the tests are given in Dickey and Fuller (1981) andHamilton (1994).
As Table 2 shows, application of the procedure to the relative per-capita GDP
series leads to rejection of the null and to the conclusion that in all cases this series too
does not contain a unit root.
So far the formal analysis concentrated on the statistical properties of the series.
The more interesting questions, of course, concern the economics behind them. The first
issue that we have to address is the speed of reversion, which is captured in the parameter
γ. When the regression includes a constant only this parameter captures the speed of
mean reversion. In the context of the real exchange rate literature it measures the speed
of reversion to relative PPP. When the regression includes both a constant and a time
trend this parameter captures the speed of reversion to a trend. Table 3 contains data on
the average reversion parameter in the unit-root tests performed above for the twenty-
three individual time series.
Table 3: The reversion parameter, γγ
Real exchange rate – q Relative GDP per capita – yWith constant and
time trendWith constant
onlyWith constant and
time trendWith constant
onlyγ IHL γ IHL γ IHL γ IHL
Average -0.3369 1.6871 -0.1434 4.4783 -0.3239 1.7706 -0.1454 4.4122
Source: Penn World Tables and author’s calculations. IHL stands for “implied half
life”.
The results presented in the table are quite interesting. First we note the similarity
in the magnitude of the average γ coefficient for the regressions with a constant and a
trend: -0.3669 for the real exchange rate series and –0.3239 for the relative per-capita
GDP series. Second we note the similarity in the magnitude of the average γ for the
18
regressions with only a constant: -0.1434 for the real exchange rate series and –0.1454 for
the relative per-capita GDP series.
The most important observation in Table 3 is this: there is a large difference
between the average γ coefficients for the real exchange rate series between the
regressions with a constant and a time trend and those with only a constant.17 As
mentioned earlier, in essence the regression with only a constant tries to capture reversion
to relative PPP. The half-life of deviation from relative PPP implied by the value of the γ
coefficient is 4.48 years. This figure fits well with the “consensus” estimate of half-lives
of deviations from relative PPP discussed in Rogoff (1996): three to five years.18 The
fact that deviations are so persistent led Rogoff (1996) to coin the term “The Purchasing
Power Parity Puzzle”. But as shown above, when one includes a trend in the regressions
the implied half-life drops to only 1.69 years. A similar result was obtained by Taylor
(2001a) for a group of twenty countries in the post World War II years. The question
then is this: what explains the trend in the real exchange rate? The answer proposed here
is, of course, the process of dual convergence.
Table 4 provides evidence on the behavior on relative per-capita GDP and the real
exchange rate over time for all twenty-three countries. The column labeled Y0 shows the
initial, 1950, relative per-capita GDP. The next column shows the trend-change in the
natural logarithm of relative per-capita GDP, y, which was obtained by an OLS
regression of the natural logarithm of relative per-capita GDP on a constant and a linear
17 There is, of course, a corresponding difference in the relative per-capita GDP series figures, but here we
concentrate on the real exchange rate series.18 See also Obstfeld and Rogoff (2001)
19
time trend.19 The next two columns show the initial real exchange rate (Q0) and the
trend-change in the natural logarithm of the real exchange rate, q. The last two columns
attempt to estimate the long-run relationship between y and q. The penultimate column
shows the slope coefficient from an OLS regression of y on a constant and q.20 The last
column shows the intercept coefficient from that regression.
Table 4: Relative per-capita GDP and real exchange rate behavior
Regression of y on q and cCountry Y0 Trend in y Q0 Trend in q
Slope Intercept
Australia 0.8332 0.0019*** 1.5657 -0.0107*** -2.3911*** -0.3953***Austria 0.3251 0.0177*** 1.6450 -0.0192*** -0.9012*** -0.3546***Belgium 0.5071 0.0102*** 1.2718 -0.0098*** -1.0609*** -0.4086***Canada 0.7058 0.0090*** 1.0346 0.0026*** 0.2533*** 0.0134
Denmark 0.5915 0.0079*** 1.5610 -0.0212*** -1.8969*** -0.6040***Finland 0.3875 0.0147*** 1.2089 -0.0118*** -0.6994*** -0.3699***France 0.4573 0.0126*** 1.2560 -0.0071*** -0.4599*** -0.1596***
W. Germany 0.3776 0.0144*** 1.4196 -0.0165*** -0.9149*** -0.2875***Greece 0.1557 0.0242*** 0.9843 0.0035* 0.1123 0.4060***Iceland 0.3975 0.0189*** 0.7336 0.0028 0.0790 -0.2531***Ireland 0.3041 0.0130*** 1.6810 -0.0130*** -0.9489*** -0.6820***Italy 0.3152 0.0182*** 1.6431 -0.0145*** -0.7067*** -0.1897***Japan 0.1620 0.0382*** 2.2800 -0.0295*** -0.7066*** -0.3419***
Luxembourg 0.6707 0.0031*** 1.3875 -0.0126*** -0.6246* -0.0222Netherlands 0.5134 0.0092*** 1.8793 -0.0212*** -2.0049*** -0.6470***
New Zealand 0.8091 -0.0029*** 1.4255 -0.0031** 0.5403** 0.4590**Norway 0.4772 0.0130*** 1.4269 -0.0181*** -1.1460*** -0.5889***Portugal 0.1421 0.0231*** 1.6767 -0.0026 -0.1659** 0.3563**
Spain 0.2192 0.0192*** 1.8165 -0.0157*** -0.6253*** -0.1287Sweden 0.6597 0.0042*** 1.3963 -0.0148*** -1.9434*** -0.5215***
Switzerland 0.7635 0.0034*** 1.5161 -0.0261*** -1.9621*** -0.1364Turkey 0.1264 0.0076*** 1.4545 0.0163*** 0.9957*** 2.3695***U.K. 0.5889 0.0035*** 1.5635 -0.0104*** -2.2066*** -0.7195***
Average 0.4561 0.0124 1.4708 -0.0110 -0.8428 -0.1394
Source: Penn World Tables and author’s calculations. Notes: *, **, *** denote,
respectively, significance at the 10%, 5%, 1% levels in the relevant tests.
Did the OECD countries indeed form a convergence club? The data presented in
Table 4 shows that they did: all countries started out in 1950 with a lower per-capita GDP
than the U.S. and all, except for New Zealand, grew faster than the U.S. during the
following four decades. What happened to the real exchange rate of those countries
19 Natural logarithm of the original series is used in order to preserve consistency with the tests performed
above.
20
during this period? There is statistically strong evidence for the hypothesis of initial
depreciated real exchange rate and subsequent real appreciation in eighteen out of
twenty-three cases.21 The second to last column shows that in those eighteen cases there
is evidence for a long run inverse relationship between relative per-capita GDP and the
real exchange rate.22 The evidence presented in the last column lead to the conclusion
that although the real exchange rates are converging, they are not converging to unity as
implied by the model. If absolute PPP holds when relative per-capita GDP is unity, then
the intercept coefficient in our regression should be equal to zero. But in practice it is
significantly different from zero in nineteen cases. On average, the implied deviations
from absolute PPP are in the order of thirteen percent, i.e. if all countries had the same
per-capita GDP as the U.S. they would have, on average, prices that are thirteen percent
higher than those in the U.S. Nevertheless, Table 4 establishes that there a remarkable
amount of support for the dual β-convergence hypothesis.
20 The results from such regression are not spurious as Table 1 and 2 show that both variables are
stationary.21 Canada and Turkey started out with a depreciated real exchange rate but exhibited a statistically very
significant trend of real depreciation. Greece and Iceland started out with an appreciated currency butneither showed a very significant trend of either depreciation or appreciation. New Zealand, despite thefact that it had a lower per-capita GDP and a depreciated real exchange rate in 1950, showed astatistically very significant trend of growth divergence and real exchange rate convergence.
22 Statistical evidence against the dual convergence hypothesis is established in only three cases: Canada,New Zealand and Turkey.
21
5 Conclusion
The paper has shown that within a specific group of countries and over a specific
period of time, convergence in levels of per-capita GDP was accompanied to a large
extent by a convergence in real exchange rates. To the best of my knowledge this paper
is the first to raise and test this hypothesis of dual convergence. An important message
emerging form the exercise performed here is that real factors should be at the center of
real exchange rate analysis. Differences in technologies, endowments and tastes should
always be taken into account.
An interesting implication of the paper concerns the question of undervaluation
and overvaluation of currencies. The results presented above suggest that in order to
properly analyze the question of real undervaluation or overvaluation in the dynamic
sense we need to know the fundamental determinants of the steady state levels of per-
capita GDP of the countries involved and their current positions relative to these steady
states. This obviously makes the calculations extremely complicated.
The paper raises another interesting question. Is it possible to find a similar
relationship between convergence in levels of per-capita GDP and relative price levels
across other sets of countries and regions? Research in economic growth has shown that
a pattern of absolute convergence in levels of per-capita income characterizes earlier
historic periods (such as the late nineteenth century) and geographic and political units
(such as U.S. states). It would be interesting to examine whether in those cases, although
possibly somewhat different from the theoretical perspective, convergence in relative
prices can be detected.
22
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25
0
1
2
3
4
0.0 0.2 0.4 0.6 0.8 1.0
relative GDP per capita (US=1)
real
exc
hang
e ra
te (U
S=1
)Figure 1: Relative GDP per capita and real exchange rate
OECD, 1950-92
26
0.8
1.0
1.2
1.4
1.6
0.68
0.72
0.76
0.80
0.84
0.88
50 55 60 65 70 75 80 85 90
Australia
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.3
0.4
0.5
0.6
0.7
0.8
50 55 60 65 70 75 80 85 90
Austria
0.6
0.8
1.0
1.2
1.4
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
50 55 60 65 70 75 80 85 90
Belgium
0.8
0.9
1.0
1.1
1.2
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
50 55 60 65 70 75 80 85 90
Canada
0.6
0.8
1.0
1.2
1.4
1.6
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
50 55 60 65 70 75 80 85 90
Denmark
0.6
0.8
1.0
1.2
1.4
0.3
0.4
0.5
0.6
0.7
0.8
50 55 60 65 70 75 80 85 90
Finland
0.7
0.8
0.9
1.0
1.1
1.2
1.3
0.4
0.5
0.6
0.7
0.8
50 55 60 65 70 75 80 85 90
France
0.6
0.8
1.0
1.2
1.4
1.6
0.3
0.4
0.5
0.6
0.7
0.8
0.9
50 55 60 65 70 75 80 85 90
Germany
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.15
0.20
0.25
0.30
0.35
0.40
50 55 60 65 70 75 80 85 90
Greece
0.4
0.6
0.8
1.0
1.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
50 55 60 65 70 75 80 85 90
Iceland
0.8
1.0
1.2
1.4
1.6
1.8
0.25
0.30
0.35
0.40
0.45
0.50
0.55
50 55 60 65 70 75 80 85 90
Ireland
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.3
0.4
0.5
0.6
0.7
0.8
50 55 60 65 70 75 80 85 90
Italy
Figure 2: Relative GDP per capita and real exchange rate in OECD countries
0.5
1.0
1.5
2.0
2.5
0.0
0.2
0.4
0.6
0.8
1.0
50 55 60 65 70 75 80 85 90
J a p a n
0.6
0.8
1.0
1.2
1.4
1.6
0.6
0.7
0.8
0.9
1.0
50 55 60 65 70 75 80 85 90
Luxembourg
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
50 55 60 65 70 75 80 85 90
Q_133 Y_133
Netherlands
1.0
1.2
1.4
1.6
1.8
0.65
0.70
0.75
0.80
0.85
50 55 60 65 70 75 80 85 90
New Zealand
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.5
0.6
0.7
0.8
0.9
50 55 60 65 70 75 80 85 90
Norway
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
50 55 60 65 70 75 80 85 90
Portugal
0.8
1.2
1.6
2.0
2.4
0.2
0.3
0.4
0.5
0.6
50 55 60 65 70 75 80 85 90
Spain
0.6
0.8
1.0
1.2
1.4
1.6
0.65
0.70
0.75
0.80
0.85
0.90
50 55 60 65 70 75 80 85 90
Sweden
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.75
0.80
0.85
0.90
0.95
1.00
50 55 60 65 70 75 80 85 90
Switzerland
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.12
0.14
0.16
0.18
0.20
0.22
50 55 60 65 70 75 80 85 90
Turkey
0.8
1.0
1.2
1.4
1.6
0.56
0.60
0.64
0.68
0.72
0.76
50 55 60 65 70 75 80 85 90
U.K.