Purchasing Power Parity Theory

Published Date: 02 Nov 2017

Originated by Cassel (1918), Purchasing power parity (PPP) is considered as one of the

foundations of exchange rate behavior.

1. Definition

The theory is based on the simple idea of the law of one price which states that in the

presence of a competitive market, the absence of transport costs and other barriers to trade,

arbitrage will lead to the same goods having the same price in different markets. The law is

based on the idea of perfect goods arbitrage which occurs where economic agents figure out

the differences so as to provide a riskless profit.

Springing from this law, the PPP doctrine states that a relationship between exchange rate

and prices holds between pairs of countries. It comes in two forms: absolute PPP and

relative PPP.

The absolute PPP which relies strictly on the law of one price implies that the equilibrium

exchange rate between two national currencies equals the ratio between the domestic and

foreign price level. Algebraically, it can be expressed as:

S =

(1)

Where S is the exchange rate defined as domestic currency units per unit of foreign

currency, P and P* are the domestic price level and foreign price level respectively.

According to this version of the hypothesis, a fall in the domestic price level in comparison

to the foreign price level will lead to a proportional appreciation of the domestic currency

against the foreign currency.

The Relative PPP, the �weaker� variation, states that the exchange rate will adjust by the

amount of inflation differential between two economies. Algebraically it can be expressed

as:

%.S = %.P - %.P* (2)

Where: %.S is the percentage change in the exchange rate, %.P is the domestic inflation rate,

and %.P* is the foreign inflation rate. If the inflation rate in the home country is x% higher

than that in the foreign country, the exchange rate should be expected to depreciate by

approximately x%

Absolute PPP is argued to be unlikely to hold since the conditions needed are strict and

unrealistic. Isard (1977) says: �In reality the law of one price is flagrantly and

systematically violated by empirical data�. Meanwhile, the relative PPP can be expected to

hold even in the presence of distortions such as transport costs or trade impediment.

In addition, most empirical work has focused on the validity of PPP in the long-run than in

the short-run, which basically due to the greater volatility of the short-run exchange rate.

Dornbusch (1976) argues that in short run goods prices can be regarded as stable, while the

exchange rate is rapidly driven by news like announcements about interest rate changes or

other economic policies. For example, political uncertainty (the election of a Parti

Quebecois government in Quebec on 15 November 1976) and a substantial current account

deficit are two important causes for the depreciation of Canadian dollar by the end of

1970s. Meanwhile, PPP is based on only goods arbitrage, but says nothing about the role of

capital movements. Therefore, exchange rate deviations from PPP are substantial and

prolonged in the shot-run. Instead, PPP is supposed to describe the long-run behaviour of

exchange rates. The economic forces behind PPP will eventually equalize the purchasing

power of currencies. Furthermore, methodology used to calculate PPP in the short-run

makes the volatility readily. Whereas, it seems appropriate to use cointegration technique to

explain the concept of PPP as a long-run equilibrium. Long-run relationship in this sense

denotes the equilibrium to which a system converges over time, indicating that there is no

need for PPP to hold at every point in time. Instead, the PPP rate is thought to indicate a

target toward which the spot exchange rate is adjusted.

2. The limitations of PPP

Although considered as one of the foundations of exchange rate behavior, PPP is one of the

theories facing the heaviest criticism. In the following we discuss some main problems

making it difficult for the long-run PPP to hold in practice.

. Transport costs and trade impediments

According to Keith Pilbeam (1998), the absolute PPP is likely to not hold exactly due to the

existence of transportation cost and the distorting effects of protectionism. For instance, a

bundle of goods costs C$900 in Canada and $1000 in the US, the exchange rate is supposed

to be C$0.9/$1 under PPP. If the transport cost exists, say C$20, then the exchange rate will

fluctuate within C$0.7/$1 and C$1.1/$1.

. Imperfect competition

The key assumption of the PPP theory is that there is sufficient international competition to

keep the prices of a good equal no matter in any countries. Nonetheless, such competition is

not a case in reality. Different countries have been in different economic stages and

generally establish different sets of consumers. And with their price strategies, multinational

corporations obviously charge different prices in different countries. This argument can

partly explain why PPP is likely to perform better for a pair of industrial countries like

Canada and the US in our paper.

. Productivity differentials

Balassa (1964) and Samuelson (1964) argue that productivity differentials in the traded

sector between countries are one source causing deviation from PPP. They complain that

poor countries have lower price of non-tradables than rich countries because poor countries

have lower productivity only in traded sector than rich ones. Therefore, the aggregate price

indices which are set up by converting prices of similar baskets of both traded and non-

traded goods into a common currency are likely to be higher in rich countries than in poor

ones2.

2 see Balassa (1964) and Samuelson (1964) for more detail

. Statistical problems

The assumption of PPP that all goods are internationally traded is obviously unrealistic.

There is a kind of goods called nontraded goods, including services, properties.

Nonetheless, some authors argue this does not matter much in testing PPP because there is

a close relationship between two kinds of goods. Some nontradable goods serve as inputs

for tradable ones and vice versa. Also, under the PPP hypothesis, the exchange rate is

determined by comparing the price of identical bundles of goods in two countries.

However, different countries tend to put different weigh to various classes of goods and

services. CPIs in developing countries have higher weigh on basic consumption such as

food and clothing than that in developed countries, making it difficult for PPP to hold.

Bearing in mind these limitations of PPP, we proceed to the expectation of the performance

of PPP in the countries examined.

3. Expectations of the performance of PPP

In this part of the thesis, we will analyze many conditions of the countries examined in

order to make a prediction for the performance of long-run PPP among these countries.

Before analyzing, we give a brief note about the exchange rate characteristics of the

selected countries in the sample period. The Canadian Dollar (CAD) was floated since June

1970 while until 1976 Mexican Peso (MXN) was allowed to switch to the managed floating

exchange rates. Then the exchange rates have been determined largely on the basis of

demand and supply conditions in the exchange markets. However, the Bank of Canada and

the Bank of Mexico intervened when necessary to maintain orderly conditions in the

exchange markets. Whilst the Peso is always much weaker than the USD, The CAD is quite

strong against the USD. It was worth more than the USD for part of the 1970s. After two

series of downward pressures during the technological boom of the 1990s that was centered

in the US, its value has risen against the USD because of the continued strength of the

Canadian economy.

Two of the factors causing the poor performance of PPP in general or long-run PPP in

specific are transport costs and trade impediments. These factors partly explain for the

argument of Frankel (1981) that PPP performs better for countries that are geographically

close to one another and where trade linkages are high. In our case, it is reasonable to

expect PPP to hold between Canada, Mexico and the US. They are neighboured countries,

so the transport costs are no longer much matter to the performance of PPP hypothesis.

Furthermore, these North-America nations share the most comprehensive trading

relationship around the world. On January 1, 1994, the North American Free Trade

Agreement (NAFTA) between the United States, Canada, and Mexico entered into force.

Such agreements help to reduce trade impediments, making a good condition for PPP to

take place in the countries.

According to US Commercial Service, Canada and Mexico are two of the largest trading

partners of the US. Canada is the leading export market for 36 out of 50 U.S. States, and

ranked in the top three for another 10 States. On its turn, International Trade Administration

reports that Mexico-US trade has increased by over 225% since the NAFTA of 1994.

Meanwhile, IMF international statistics reports that the US is the largest trading partner of

both Canada and Mexico. Trading with the US accounts for about 73% of exports and 63%

of imports of Canada since 2009, while these numbers are 65% and 68% respectively for

Mexico.

Furthermore, previous studies support for the statement that high-inflation countries

provide good conditions for PPP to hold. Figure 1 presents Canadian inflation rate from

1977:I to 2010:IV. High inflation occurred in Canada during 1973 through 1979, but the

rate declines sharply since 1980�s. It has fluctuated around 2 percent from 1992 up to now.

On average, Canada is considered as a low-inflation country, with an average annual

inflation rate of 4.49%. Inversely, Mexico is a well-known high inflation country. Figure 2

presents Mexican inflation rate from 1977:I to 2010:IV. According to Bank of Mexico, the

average inflation rate in Mexico was 29.47% from 1977 until 2010. The rate reached an

historical high of 179.73 percent in February of 1988. Therefore, evidence of inflation

suggests PPP is likely to perform better for the case of Mexico-US than for the case of

Canada -US.

Nonetheless, the case of Canada and the US owns a condition which makes it easier for

long-run PPP to hold than the case of Mexico-US. In the previous part, we can see that the

two limitations of PPP, imperfect competition and productivity differentials, can be partly

overcame if we test PPP for two developed countries. Jayendu Patel (1990) supports for

this argument by stating that PPP is likely to hold only among developed relatively free-

market economies. The US is obviously the largest economy in the world, and Canada is in

the top of 10 world�s largest economies3. Since the two countries are ranked as developed

ones, there is not a large gap in income or living standard between them; they establish

similar sets of consumers. Therefore, the multinational corporations tend to charge same

level of price on the two countries.

On the other hand, Mexico is classified by the World Bank as an upper-middle-income

country. It is still considered as a developing country although by GDP it is ranked as the

thirteenth largest economy in the world in 20113. According to IMF, the Gross National

Income (per capita) of the US and Canada are about $33,000 and $21,000 respectively,

while that of Mexico is about only $5,000. Thus, an identical good often costs a lower price

in Mexico than in Canada and the US. All dresses belong to the 2011 summer collection of

Mango, for example, cost the same price for Canada and the US, but about 10% lower price

in Mexico.

Moreover, it is reasonable for the US to put same weighs with Canada, but different weighs

with Mexico to different classes of goods in constructing the price indices.

However, according to Someshwar Rao et al. (2004), although Canada and the US are both

ranked as developed countries, Canada�s labour productivity has grown slowly than the

US�s since 1995. In 2003, the US�s labour productivity was about 23 percent higher than

that of Canada�s. Therefore, when converting into a common currency, the prices of similar

baskets are still somehow higher in the US than in Canada, making difficulties for PPP to

hold. For the case of Mexico-US, there is still a quite large labor productivity differential

between two nations although the NAFTA agreement has helped to push up the technology

transfers, reducing the gap in productivity.

3 see World Development Indicators database, World Bank, 1 July 2011

In addition, both the Canadian and Mexican exchange markets are quite crowed. Stock

Exchange and TSX Venture Exchange of Canada are home to the largest number of

publicly traded companies of any exchange in North America. Likewise, the Mexican Stock

Exchange (Bolsa Mexicana de Valores) is the second largest stock exchange in Latin

America and the fourth largest in North America. Therefore, although the capital

movements are argued to affect the short-run PPP much more, it still makes the deviations

persistent and prolonged so as PPP cannot converge to the long-run equilibrium given the

important role of the capital market to these countries.

In short, both Mexico and Canada provide certain favorable backgrounds for PPP to hold in

comparison to other groups of countries. However, the previous studies have still

demonstrated mix findings as discussed in the next section.

III. LITERATURE REVIEW

So far, the validity of long-run PPP has remained an open question no matter which

econometric approaches are employed, which price indices are used or for which countries

PPP is tested.

At the beginning, PPP is often tested by traditional regression technique. Frankel (1981)

uses OLS to test PPP for the UK pound, German Mark and French Franc against the US

dollar and concludes that the hypothesis worked well in the 1920's, but not during the

1970's. Even he argues that PPP should not be considered as a theory of exchange rate

determination due to the fact that it specifies the relationship between endogenous variables

without providing the details about the process generating them.

On the other hand, using standard 2SLS and GLS Davutyan and Pippinger (1985) provide

evidence supporting for PPP during 1970's. These papers are excellent in their choice of

tested countries. The authors test PPP hypothesis for the group of developed countries

which have the approximately equal productivity. Moreover, it also figures out many

problems in testing PPP such as standard error or unequal weights constituting price levels

and complains such problems as the reasons for the Frankel�s argument of the collapse of

the theory.

However, Dean Corbae (1991) argues that in case exchange rates and prices are

nonstationary, standard regression may be biased towards rejection because of the serial

correlation.

After the introduction of cointegration and error-correction analysis, most recent studies

have adopted them in testing the PPP hypothesis in the long-run. This approach is said to be

more advanced than previous approaches in studying PPP since it deals with non-stationary

time series.

Those who employed OLS-based cointegrating technique of Engle and Granger (1987)

mostly reject PPP. Taylor (1988) conducts the Engle and Granger test for the long-run PPP

for five major exchange rates, including CAD/USD. The paper collected seasonally

adjusted data on relative prices and nominal exchange rate from 1973 through 1985 and

concluded that cointegrating relationship between exchange rate and relative prices does

not exist for any of the countries examined. Flynn and Boucher (1993), Mohsin (2004)

reject the hypothesis as well.

According to Muzafar Shah et al. (2006), nevertheless, the residual-based Engle-Granger

method tends to provides inconsistent results. Furthermore, they argue that Johansen's

multivariate framework would overcome some weaknesses from bivariate co-integration.

And often the Maximum-Likelihood based cointegration method of Johansen (1988) has

more support for the validity of PPP. Islam and Ahmed (1999) tested the PPP hypothesis

for Korean-US exchange rate and prices for the period from 1971 to 1996. The study

applied both the Engle-Granger method and the Johansen method. The paper provides

support for long-run PPP, and stronger support came from the Johansen method.

Furthermore, the paper also estimates the ECM and concludes that the exchange rate is a

stable function of the relative prices with a speed of adjustment of about 24% over a year.

Even those who used most recent developed techniques have provided mixed results.

Applying non-linear URTs, Cuestas (2009) rejected the hypothesis. Meanwhile, Telatar and

Hasanov (2009) who also use non-linear URTs for twelve CEE countries find evidence

supporting for it.

Turning to the researches for the case of North America, we also see mix findings about the

long-run PPP although there are only a few studies analyzing both the exchange rates of

Canada and Mexico against the US. According to Taylor (2002), PPP holds well for both

the cases of Canada-US and Mexico-US in the long-run over the 20th century. The paper

applies both the Johansen likelihood ratio JLR as Multivariate Test as well as ADF and DF-

GLS test as Univariate Tests. One of the outstanding points the author made is that he

collected data for a group of twenty countries over 100 years, a larger historical panel of

annual data than has ever been studied. He argues since PPP is likely to hold in the long

run, it is better to test the theory with long time dimension of the data. The findings are

supported by Wallace (2010) who reuses Taylor (2002) data set. The paper also claims the

important role of the instrument variables as reinforcement to the tests since they help to

eliminate nuisance parameters. The author concludes: �The ECM and ADL model, with or

without instrumental variables, and the traditional EG two-step approach provide some

support for the PPP hypothesis�. The ECM estimates that deviations move down in order to

adjust to long-run equilibrium with the speed of 21.7% and 58.9% respectively for Canada

and Mexico.

Nonetheless, Lopez at el (2005) argues that if Taylor (2002) had used an accurate lag

selection criterion, PPP just performs well for no more than 9 out of 16 cases. Specifically,

the authors fail to provide support for both Canada and Mexico.

The previous literature also provides evidence for the argument that PPP holds better for

the high-inflation countries. Mahdavi and Zhou (1994) apply the Johansen framework to

analyze PPP in a sample of less-developed countries (LDCs) using quarterly data for

1973Q2 onwards. They conclude that PPP holds more frequently among high inflation

countries, including Mexico. This finding is supported by Su Zhou (1997) who examines

the long-run PPP for four high-inflation countries, including Mexico. The co-integration

tests in this paper are conducted with the correction of the finite sample bias and the

adjustment for trend breaks. Like the previous, the paper concludes that: �The results are

consistent with the argument that, during the recent floating exchange-rate period, PPP

holds well, at least in a weak form, in high-inflation countries where the general price level

movement overshadows the factors causing deviations from PPP.�

On the other hand, Holmes (2002) testing PPP for a sample of thirty LDCs over the period

1973-2001 finds evidence against long-run PPP for the case of Mexico-US.

Turning to the case of Canada-US, there are numerous researches about this pair of

industrial countries. Johnson (1990) applying both Eagle-granger cointegration techniques

and ECM framework finds supportive evidence for PPP as a long-run equilibrium

relationship for the case of Canada-US. Furthermore, the study concludes that estimates of

the ECM depend on exchange rate regimes. If exchange rates are fixed, adjustment towards

PPP occurs mainly through the adjustment of the domestic price level. If exchange rates are

flexible, then both the domestic price level and the level of the exchange rate can do the

adjustment to reach the long-run PPP equilibrium.

Investing the validity of long-run PPP between Canada and the US in the 1980s and 1990s,

Beiling Yan (2002) generally rejected the theory. This paper is very professional at

Commodity Groups Classification. The paper finds some support only from homogeneous

goods within the tradables.

Yan (2002)�s findings raise a notice that it should be careful to distinguish between

different commodity groups as well as which price index should be used as the proxy for

the price level when testing PPP. On one hand, some authors argue the WPI is more

favorable to PPP than CPI. Su Zhou (1997) states: �That PPP often holds better for the WPI

pairs than the CPI pairs could be explained by the fact that the CPI does not include

exported goods and thus is weighted more toward nontraded goods than is the WPI.�

According to McNown and Wallace (1989), cointegration between the exchange rate and

the WPIs occurs in two out of four high-inflation countries, but the relationship between the

exchange rate and the CPIs does not exist in any of the four cases. Kim (1990) also

supports for this argument. On the other hand, some authors argue that PPP should be

applicable to CPI since such general price index can represent the whole mass of

commodities in the economy. Johnson(1990) finds evidence supporting for long-run PPP

between Canada-US exchange rate and CPI�s.

Bearing in mind the advantages of co-integration technique as well as the facts that the

conditions needed for PPP to hold in short-run are strict and unrealistic, the main purpose

of our paper is testing the validity of PPP as a long-run relationship using co-integration

methods. Furthermore, we will give more detail about the ECM interpretation. For the first

time, our paper will focus on the three countries: The US, Canada, and Mexico in an

attempt to check the predictions that PPP holds better for high-inflation countries (Mexico-

US) and a pair of developed countries (Canada-US). Finally, we also test the theory using

both CPI and WPI for comparison.

In the next section we discuss the analytical model, the methodology as well as the sample

of data used to test the validity of long-run PPP.

IV. METHODOLOGY AND DATA

1. Econometric methodology

The long-run PPP implies the following relationship between the nominal exchange rate

and the price levels:

st = a0 + a1pt + a2pt* + .t (3)

Where st, pt, pt* are the logarithms of the exchange rate, domestic price level and foreign

price level respectively. .t is the disturbance term.

In the cointegrating context, the proposition that PPP holds in the long run implies that the

three variables st, pt and pt* are cointegrated. The first requirement for a cointegration

relationship is that three variables are integrated of the same order.

1.1. Tests for unit root

To determine if the nominal exchange rate and domestic/foreign price level are integrated

of the same order, we apply the augmented Dicky-Fuller (ADF) test for a unit root. The

general form of ADF test is:

.yt = � + .yt-1 + dt + S

+ .t (4)

Where .yt is the first difference of the variable yt, � is the drift term, t stands for the trend

term, m is the number of required lags so as to achieve non autocorrelation of the error

term, and .t represents the error term. The null hypothesis of the test is that the series has a

unit root.

Lag length is one important part of the ADF test. Enders (1948) claims that too few lags

may cause the estimates incorrect, while using too many lags for argumentation lowers the

performance of the test. Therefore, to ensure the power of the test we apply the general-to-

specific approach presented in Schwert (1987) to choose the most appropriate number of

lags. We start to run the test with a long lag length, then gradually decrease the lags which

are shown insignificant by the t or F values. Finally, we have to make sure the residuals are

white noise once the tentative lag length has been chosen.

If the variables are found to have a unit root at the same level or to be integrated of same

order, we will apply two tests, the Engle-Granger and Johansen, for co-integration which

represents long-run equilibrium relationship of non-stationary variables.

1.2. Tests for co-integration

Following the Engle-Granger (1987), we first estimate the cointegrating regression

(equation 3) by the standard regression method OLS. Then the residuals from the regression

will be tested by the ADF test for a unit root. If the residuals have no unit root or are

stationary, the variables are co-integrated and vice versa.

Following the Johansen (1988) approach, 5 Information Criterions: LR, FPE, AIC, HQIC

and SBIC are first applied to specify the appropriate lag length of the VAR system in order

to make sure the residuals uncorrelated. Within the Johansen's maximum likelihood

procedure, the matrix notation of the Vector error correction model is specified as follow:

.Xt = A0 + .Xt-1 + A1.Xt-1 + A2.Xt-2 + ... + Ap.Xt-p + Et (5)

Where Xt is a (nx1) vector of I(1) processes, A0 is the (nx1) vector of intercepts, Ai is the

matrix of coefficients, Et is the vector of error term. And . is the matrix of parameters such

as at least one element is non-zero. Johansen test is a test for the rank of matrix.. Denote

rank (.) = r. Johansen (1995) suggests a tests statistic to determine the cointegration rank

known as the trace statistic:

trace(r0/k) = -TS

^

) (6)

Where

^ are the estimated eigenvalues .1 > .2 > .3 > � > .k and r0 ranges from 0 to k-1

depending on the stage in the sequence. This is the relevant test statistic for the null

hypothesis r < r0 against the alternative r > r0 + 1.

If r = 0, we have no co-integration. If 0 < r < n, then we have r co-integration vectors. On

the other hand, if r = n, all series in vector Xt are stationary.

As long as the variables are found to be co-integrated, they share a common trend even

though they are individually non-stationary. Thus, one can lead to the conclusion that PPP

relationship holds in the long-run.

1.3. The Error-Correction Model

If the variables are found to be co-integrated, there must exist an associated error-correction

model (ECM) which provides the short-run dynamics or how the system converges to the

long-run equilibrium. Generally, an ECM for 3 variables can be expressed as:

.st = a10 + Sas(j)CEj+ Sa11(i) .st-i + Sa12(i) .pt-i + Sa13(i) .p*t-i + est (7)

Where CEj are the error correction terms and are the residuals from the cointegrating

regression equations. If this term is larger than zero, yt in the previous period overshoots the

equilibrium and yt will fall unless yt-1 = � + �xt-1. . denotes the first differential. a11(i),

a12(i) and a13(i) are the coefficients representing the short-run dynamics of .st with respect

to .pt-1, .p*t-1 and .st-1, and eyt is a white noise process.

as is the speed-of-adjustment parameter. Larger as is, greater is the response of st to the

previous period�s deviation from the long-run equilibrium and vice versa. For an ECM to

exist at least one of the speed-of-adjustment parameters must be different from zero.

2. Data

As discussed in the literature review, we follow Taylor (2002) who argues that empirical

tests of long-run relationship require considerable amounts of data over a long period4. Our

paper tests the hypothesis for a sample of quarterly data of thirty-four years.

4 Frankel (1986) and Kim (1990) also support this argument

The data examined are quarterly series taken from IMF�s International Financial Statistics

covering the floating period from 1977:I to 2010:IV. The exchange rate series include

nominal Canada-US exchange rate (CAD/USD) and Mexico-US exchange rate

(MXN/USD). Finally, both the WPI and CPI are used as the proxy for the price level in

order to ascertain if the choice of price index matters. The data used are described in the

table 1and graphs 1.

Table 1: Price Indices Summary Statistics

Sample period: 1977:I to 2010:IV

Variable

Consumer price

Wholesale Price

Exchange Rate

CAD

MXN

USD

CAD

MXN

USD

CAD/USD

MXN/USD

Maximum

109.808

126.047

112.282

111.717

130.304

127.361

1.593

14.332

Minimum

21.744

0.052

22.486

24.059

0.142

40.344

0.968

0.023

Mean

70.939

44.123

69.068

75.164

44.093

78.026

1.243

5.475

Std.

Deviation

25.849

43.680

26.194

23.608

43.907

19.120

0.162

4.702

The first 6 columns summarize the price indices of Canada (CAD), Mexico (MXN) and the

US (USD). As we can see, the greatest deviations are in the Mexico prices, indicating

Mexico is the most inflationary country. Furthermore, Canadian dollar and US dollar are

quite similar, indicating the similar purchasing power of the two Dollars. Finally, the

exchange rate columns show that the more stable currency is the Canadian Dollar.

Given this set of data, we proceed to the empirical results.

V. EMPERICAL RESULTS

1. Graphical evidence

Before conducting cointegrating tests, we give graphical evidence to present first

diagrammatically if the PPP hypothesis holds among the selected countries.

Graph 2 plots the actual exchange rates and PPP rates for the countries examined. The

figure shows significant divergences of the exchange rate from that suggested by PPP.

Graph 2(a) shows the prolonged divergence of PPP from the real exchange rate of Canadian

Dollar-US Dollar when the CPI is the proxy. Between mid 1979 and early 1981 there was a

dramatic depreciation of the Canadian Dollar while PPP would have predicted an

appreciation. Thereafter, the Canadian Dollar has a brief period of undervaluation in

relation to PPP. After mid 1986, PPP provided the contrast predictions to the movements of

the actual exchange rate. For example, between mid 1986 and last 1989 while Canadian

Dollar appreciated, PPP would have shown a slight depreciation.

On the other hand, although the PPP rates which are computed by the WPI indicate

prolonged overvaluation of the Canadian Dollar in the whole period examined, the PPP

performs well in predicting the movements of the actual exchange rate since exchange rates

generally move in the same direction with PPP rates. Furthermore, it appears that the

magnitude of the divergence has been getting small and small.

In short, WPI�s do a better job at tracking the Canadian Dollar- US Dollar parity than the

CPI�s.

Graphs 2(c) and 2(d) tell us the performance of PPP for the case of Mexico-US. Different

from the previous cases, the PPP rates made up from WPI and CPI behave similarly. The

choice of the price indices does not matter. Both cases show that the Mexican Peso has

been undervalued in relation to PPP in the whole period, but the PPP is useful in predicting

the movement direction of the exchange rate.

It is noticeable in all plots, especially for the case of Canada-US, that although the

exchange rate is frequently far from PPP it has a propensity to come back towards the PPP

rates over the longer term. Therefore, PPP may be useful to determine the long-run

exchange rate.

In the next part, we present co-integrating tests and ECM estimation to give econometric

evidence for the existence of the long-run PPP.

2. Econometric results

2.1. Unit root tests

The results of ADF tests are reported in tables 2. Almost previous studies run the ADF tests

only with the trend and without trend specifications, but nothing about the constant or the

drift term. Our paper runs the F-test for the need of not only trend but also the constant.

We then choose the most appropriate specifications for the ADF tests and only report the

ADF test statistics for these specifications.

ADF tests reveal that the null hypothesis of a unit root cannot be rejected for all variables in

their levels but rejected in their first differences. These variables are thus found to be non-

stationary in their levels (or integrated of order one, I(1)). The results allow us to proceed to

cointegrating tests.

2.2 . Cointegration tests

The results of the cointegrating Eagle- Granger tests are presented in the table 3. Two cases

are considered.

First we test whether there is a cointegrating relationship between exchange rates and CPIs.

Due to the fact that all the variables are non-stationary, the estimated coefficients are

invalid; therefore, we have to test the unit roots of the residuals. The ADF test statistics of

the residuals for the cases of Canada-US and Mexico-US are -1.119 and -2.377

respectively. They are both smaller than the critical value at 5% significant level (-3.785) in

absolute value; the null hypothesis of a unit root cannot be rejected. Therefore, one can

conclude that the cointegrating relationship does not exist or the long-run PPP does not

hold in case CPI is used as the proxy for the price level.

Even when WPI is employed, the residuals are still non-stationary. The deviations from

PPP have no tendency to converge to a long-run equilibrium path. Our paper provides

evidence consistent with Taylor (1988), Flynn and Boucher (1993), and Mohsin (2004)

who also apply Engle-Granger method and reject the hypothesis, but contrary to the

conclusions reached in some other studies such as Johnson (1990) and Kim (1990) which

support for the long-run PPP.

In the followings we analyze the results of Johansen co-integration tests as shown in the

table 4. In contrast to Eagle-grange tests, the Johansen tests show evidence supporting for

the long-run PPP relationship for two pairs of countries no matter CPI or WPI are in use,

but with different numbers of cointegrating vectors.

The exchange rate and the CPIs of Canada and the US share 1 cointegrating vector while

there are 2 vectors for the case of Mexico-US. On the other hand, there are 2 cointegrating

vectors exist for the case of Canada-US and only 1 vector for the case of Mexico-US when

WPI is employed. However, no matter how many cointegration vectors are found, the

Johansen tests are supportive for the validity of long-run PPP. This result is against Lopez

at el (2005), but consistent with almost previous studies such as Mahdavi and Zhou (1994),

Su Zho (1997), Islam and Ahmed (1999) or Taylor (2002) and provides more evidence for

the argument of Muzafar Shah et al. (2006) that Johansen test will give stronger support for

the long-run PPP relationship than the Eagle-Granger method.

Furthermore, findings from the cointegrating tests provide the evidence that both the CPI

and WPI bring about similar results for the existence of long-run PPP relationship between

the exchange rate and the price levels. Therefore, one should keep suspect eyes on the

argument of McNown and Wallace (1989) or Kim (1990) about the advantage of WPIs

over CPIs in testing PPP.

2.3. The Error-Correction Model

Tables 5-6 represent the results of the ECM estimation. Table 5 shows how the system

converges to the long-run equilibrium implied by the speed-of-adjustment parameters. The

condition that at least one speed-of-adjustment parameter is different from zero is satisfied

in all cases. Therefore, the ECMs exist and PPP holds in the long-run in all cases.

For the pair of Canada and the US, there exists 1 error correction term (CE) when CPI is

employed. The p-values of the speed-of-adjustment parameters a11, a21, a31 are equal to

0.294, 0.000 and 0.000 respectively, so only the speed-of-adjustment parameters in

equations of .p and .p* are significant. Therefore, most of the adjustment to reach the

long-run equilibrium path is done by the two price levels. The magnitude and the sign of

the parameters are almost the same (-0.0055). Intuitively, if there are depreciations or

appreciations in the exchange rate in previous period, the US CPI and the Canadian CPI

will play almost equally important roles in adjusting the exchange rate to fall back again to

the equilibrium with a slow speed of 0.55%.

On the other hand, the exchange rate and WPIs of the 2 nations are cointegrated through 2

vectors. For the first CE, only the speed-of-adjustment parameter in the equation of .p* is

significant, meaning that the US WPI plays the most important role in adjusting the

exchange rate. Furthermore, the parameter is equal to -0.035, meaning that deviations will

move down with the speed of 3.5%. Otherwise, deviations in the second CE move down to

eliminate disequilibrium with faster speed of 5.3% mainly through the Canadian WPI.

For the case of Mexico-US, there are two CE as CPI is the proxy. In the first CE,

adjustment to reach the long-run PPP equilibrium path is done by the exchange rate and

Mexico CPI. However, the signs of the parameters are opposite, indicating opposite

movement directions of the convergence. The absolute value of the parameter in the

exchange rate equation is 24% which is much larger than that of 5.7% in the Mexico CPI

equation. Therefore, one can lead to the conclusion that the deviations made up through the

exchange rate donate those through the Mexico price in the convergence process. On the

other hand, in the second CE, all speed-of-adjustment parameters are significant.

Deviations move down with speed of 5.9% through Mexico CPI and 1.2% through US CPI,

but move up with much faster speed of 24% by the lagged exchange rate.

The exchange rate and the WPIs of Mexico and the US share only 1 cointegrating vector.

Through Mexico WPI, deviations will move up with a speed of 13% while they will move

down with a slower speed of only 3.3% through the US WPI.

In short, for the case of Canada-US, both the ECMs with CPI and WPI indicate deviations

move down mainly through the two price levels with a low average speed of 2.4% towards

the long-run equilibrium. On the other hand, while the ECM with CPI shows that all 3

variables can make deviations towards equilibrium, the other with WPI indicates only the 2

price levels can do in case of Mexico-US. The average upward speed is 14.2% and the

average downward speed is 8.6%.

In comparison with previous findings, our results show some difference in detail.

Johson(1990) concludes the domestic price level and the level of the exchange rate can do

the adjustment for the case of Canada-US while our paper shows the two price levels. Also,

the speed of adjustment in our paper is much lower than in Wallace (2010).

Another interest finding is about the interaction between the variables as presented in table

6. The p-values of the estimated coefficients in the .s equation are 0.398, 0.294, 0.755,

0.917 and 0.056 respectively. They are all larger than the critical value, so they are all

insignificant, indicating no variables have impact on the future value of the exchange rate

in case CPI is used as the price levels of Canada and the US. However, when the WPI is

employed, one-period past difference of exchange rate has effects on predicting the future

values of the exchange rate.

For the case of Mexico-US with CPI, only lag 1 and lag 3 of the exchange rate first

difference are significant, thereby having impact on the exchange rate future values.

Meanwhile, lag 1, 3, 6 of the exchange rate first difference and lag 4, 5 of the Mexico WPI

do impact in case WPI is employed.

In summary, while the Eagle-Granger cointegrating test rejects the long-run PPP, the

Johansen and the ECM are supportive for it. According to Duasa (2004), Johansen�s

approach has several advantages over the more traditional Eagle-Granger procedure. Unlike

the Eagle-Granger test, the Johansen test can work in the multivariate framework and

enables one to determine the number of cointegrating relations. Furthermore, the maximum

likelihood Johansen does not depend on arbitrary normalization rules, whereas results of

the OLS-based Eagle-Granger depend on the normalization implicit in the choice of the

regress and in the cointegrating regression. Given these advantages, our paper follows the

results of the latter ones5. Therefore, one can lead to the conclusion that long-run PPP holds

among these countries as expected in the previous section. Our results are in line with

Johnson (1990), Mahdavi and Zhou (1994), Su Zhou (1997), Islam and Ahmed (1999),

Taylor (2002) or Wallace (2010).

5 Harris and Sollis (20003) discusses more detail about the problems of Eagle-Granger method.

Table 2: Unit root Tests in nominal exchange rates and price indices

F-test statistic ADF test statistic

Trend Constant Level First differences

Exchange rate

CAD/USD 2.60 1.83 -0.904*** -9.801***

(1) (1)

MXN/USD 2.36 3.88 -1.350*** 0.029***

(3) (3)

Price index

CAD - CPI 10.35 11.76 0.143* 0.000*

(3) (1)

MXN � CPI 3.97 5.67 0.058** 0.002**

(5) (5)

USD � CPI 12.80 14.56 0.471* 0.000*

(3) (1)

CAD � WPI 12.76 14.23 0.058** 0.006**

(5) (5)

MXN � WPI 1.71 5.46 0.361** 0.000**

(1) (1)

USD � WPI 5.21 7.24 0.472** 0.000**

(1) (1)

Note: F-test for trend: the null hypothesis: there is no trend. The critical value at 5% is 6.49.

F-test for the constant: the null hypothesis: there is no constant. The critical value at 5% is 4.71.

ADF test statistic: *, ** and *** denote the specifications with both trend and the constant, specifications with only the

constant and the ones without both trend and the constant respectively. The null hypothesis of the ADF and PP test is that:

the series has a unit root. The critical value for the first 2 specifications at 5% significance level is 0.05 while that for the

latter is -2.888. The lag length is chosen using the general-to-specific approach and reported in parentheses.

Table 3: The cointegrating Eagle- Granger tests

Period: 1977:I - 2010:IV

CPI WPI

Canada-US

st = - 0.122+ 0.798pt � 0.720p*t st = 0.553+ 1.282pt -1.362p*t

(0.327) (0.001) (0.003) (0.000) (0.000) (0.000)

R2 = 0.0969 F-statistic = 7.13 R2 = 0.788 F-statistic =246.79

Unit Root Test in the Residuals Unit Root Test in the Residuals

ADF test statistic -1.119 [-3.78] - 0 lag ADF test statistic -2.985 [-3.785] � 0 lag

Mexico-US

st = 4.371 + 1.044pt � 1.46p*t st = -0.788+ 0.983pt � 0.251p*t

(0.703) (0.023) (0.177) (0.453) (0.000) (0.328)

R2 = 0.993 F-statistic = 10092.41 R2 = 0.976 F-statistic = 2713.45

Unit Root Test in the Residuals Unit Root Test in the Residuals

ADF test statistic -2.377 [-3.785] - 2 lags ADF test statistic -2.021 [-3.785] � 1 lag

Note: t-statistics in parentheses and critical values at the 5% S.L. for the ADF tests in [ ].

Table 4: The cointegrating Johansen tests

Period: 1977:I - 2010:IV

r Eigenvalue Trace statistic 5% critical value

Series: ln(CAD/USD), ln(Canadian CPI), ln(US CPI) Maximum lag in VAR = 1

None - 145.3597 29.68

At most 1 0.6458 1.8564 * 15.41

At most 2 0.01344 0.0300 3.76

Series: ln(CAD/USD), ln(Canadian WPI), ln(US WPI) Maximum lag in VAR = 1

None - 78.8852 29.68

At most 1 0.34719 21.3113 15.41

At most 2 0.14410 0.3058* 3.76

Series: ln(MXN/USD), ln(Mexican CPI), ln(US CPI) Maximum lag in VAR = 4

None - 58.0980 29.68

At most 1 0.20109 28.4638 15.41

At most 2 0.17471 3.1179* 3.76

Series: ln(MXN/USD), ln(Mexican WPI), ln(US WPI) Maximum lag in VAR = 6

None - 38.4430 29.68

At most 1 0.16394 15.1666* 15.41

At most 2 0.09129 2.7224 3.76

Note: r is the number of cointegration vectors under the null hypothesis. The appropriate lag length is based on 5

information criteria: LR, FPE, AIC, HQIC and SBIC. The stars denote the rank of the matrix . where the trace statistics

are smaller than the critical values at 5% significant level.

Table 5: The Speed of adjustment

Equation Speed of adjustment

CPI WPI

Canada-US Mexico-US Canada-US Mexico-US

.s CE1(a11) � 0.0096 - 0.244 -0.034 0.083

(0.109) (0.021) (0.547) (0.055)

CE2(a21) 0.242 0.005

(0.023) (0.947)

.p CE1(a21) � 0.0056 0.058 0.018 0.130

(0.000) (0.032) (0.365) (0.004)

CE2(a22) - 0.059 -0.053

(0.031) (0.033)

.p* CE1(a31) � 0.0055 0.013 -0.035 -0.033

(0.000) (0.064) (0.006) (0.000)

CE2(a23) - 0.012 0.022

(0.048) (0.550)

Note: .s, .p and .p* are respectively the equations of the first difference of exchange rate, domestic price level and the

foreign price level in the ECM estimation. CEs denote the cointegrating vectors. The p-values are in parentheses and the

critical value at 5% significant level is 0.05

Table 6: the Error Correction Model

Canada-US -CPI

Estimates of regression

.s = -0.003 � 0.0020CE1 � 0.169.pt-1 + 0.054.p*t-1 + 0.164.st-1

(0.398) (0.294) (0.755) (0.917) (0.056)

R2 = 0.155

Normalized Cointegrating Vector

CE1= st-1 + 21.671 � 9.581pt-1 + 5.029p*t-1

(0.059) (0.313)

P > Chi2 = 0.000

Mexico-US-CPI

Estimates of regression

.s = -0.002 - 0.244CE1+ 0.242CE2 + 0.433.pt-1 � 0.318.pt-2 + 0.169.pt-3 + 0.009.pt-4

(0.991) (0.021) (0.023) (0.229) (0.412) (0.658) (0.976)

- 2.033.p*t-1 + 2.153.p*t-2 � 1.842.p*t-3 � 0.984.p*t-4

(0.212) (0.196) (0.272) (0.535)

+ 0.329.st-1 + 0.053 .st-2 + 0.474.st-3 + 0.016.st-4

(0.008) (0.684) (0.000) (0.900)

R2 = 0.4669

Normalized Cointegrating Vector

CE1= st-1 + 48.209 - 9.445p*t-1

(0.002)

P > Chi2 = 0.0016

CE2= pt-1 + 45.005 � 8.283p*t-1

(0.005)

P > Chi2 = 0.0049

Canada-US - WPI

Estimates of regression

.s = -0.003 � 0.023CE1 + 0.009CE2 � 0.208.pt-1 + 0.158.p*t-1 + 0.222.st-1

(0.343) (0.654) (0.881) (0.490) (0.467) (0.027)

R2 = 0.052

Normalized Cointegrating Vector

CE1= st-1 + 3.269 � 0.658p*t-1

(0.002)

P > Chi2 = 0.0021

CE2= pt-1 + 2.503 + 0.465p*t-1

(0.002)

P > Chi2 = 0.0017

Mexico-US-WPI � 6 lags

Estimates of regression

.s = 0.004+ 0.083CE1 +0.083.pt-1� 0.103.pt-2 +0.443.pt-3 �0.095.pt-4 + 0.111.pt-5 �0.194.pt-6

(0.782) (0.055) (0.358) (0.646) (0.399) (0.000) (0.044) (0.677)

+ 0.044.p*t-1 + 0.479.p*t-2 + 0.348.p*t-3 � 0.116.p*t-4 � 0.108.p*t-5 + 0.237.p*t-6

(0.925) (0.355) (0.518) (0.830) (0.835) (0.610)

+ 0.232.st-1 � 0.103 .st-2 + 0.443.st-3 � 0.095.st-4 +0.112.st-5 � 0.194.st-6

(0.030) (0.353) (0.000) (0.359) (0.254) (0.048)

R2 = 0.052

Normalized Cointegrating Vector

CE1= st-1 + 1.956 + 1.067pt-1 � 0.985p*t-1

(0.000) (0.007)

P > Chi2 = 0.0000

Note: In the Estimates of regression, the p-values related to t-statistics are in parentheses. The Normalized Cointegrating

Vector expresses how the variables are cointegrated. The P > Chi2 denotes the p-value associated with the F-test for the

significance of the cointegrating vectors; they are all smaller than 0.05, the critical value at 5% significant level. Thus, the

ECMs are all meaning.