The Purchasing Power Parity (PPP) is

undoubtedly one of the most tested theory and an important approach to the

theory of exchange rate determination. The law of one price is at the core of

the PPP condition, it states that:

(1)

Where is the price level in the domestic country

relating to the good or service number , is the foreign price level relating to the

good or service number and the nominal spot exchange rate. Equation 1 essentially

says that the same good should have the same price in different countries if

the prices are expressed in the same currency. The rational for the law of one

price condition to hold is that arbitrage will take place if price differences

for perfect substitutes are observed. We assume that equation 1 holds with

respect to all goods and services consumed in the two countries.

By assuming that the weights used in

compiling the respective price indices are identical, then a general equation is

derived where equation 1 applies equally to the general price levels and not to

individual goods or services:

(2)

Equation 2 is known as the absolute

PPP, it states that the general level of prices, when converted to a common

currency, will be the same in every country. The PPP in its absolute form is

the application of the law of one price to the aggregate level.

Solving for S:

(3)

The nominal exchange rate between

two currencies is, according to the PPP, equal to the ratio of aggregate price

levels between the two countries. Consequently, the purchasing power of a unit

of one currency would be the same in both economies.

But for this equation to hold, three

conditions need to be met:

The goods of each country must be freely tradable on the

international market.The price index for each of the two countries must be comprised of

the same basket of goods.All of the prices need to be indexed to the same year.

In its weaker form, called the

relative PPP, equation 2 is reformulated in terms of rate of growth by taking

the logs of the values and their derivatives:

(4)

Equation 4 says that the home

country’s inflation rate, , is equal to

the sum of the foregin inflation rate and the rate of currency depreciation, .

It can be rearranged to:

(5)

Thus, equation 5 shows the

relationship between inflation rate and depreciation: the higher inflation, the

lower the value of money. In a two-country context, the more rapidly prices

rise in the home country relative to the foreign country, the more rapidly the

domestic currency loses value relative to the foreign country; that is the more

rapidly its exchange rate depreciates. So the relative PPP sees exchange rates

as a compensation for inflation differentials.

Distinguishing the absolute PPP from

its relative form has important consequences when it comes to test the PPP

hypothesis: using the relative PPP avoids the difficulties of choosing a base

for the price index.

The extent to which the PPP condition

may hold can be immediately objected by the presence, for instance, of

transaction costs – arising from taxes, tariffs, transportation costs…- that

would inevitably make the PPP equation impossible to hold. If for instance

transportation costs were to be observed, then equation 1 would not hold since

transportation costs would need to be added. Another theory that directly explains

deviation from the PPP is the presence of non-tradeable goods (Balassa, 1964;

Samuelson, 1964). Because no arbitrage opportunity can arise with the presence

of non-traded goods in different countries.

Also, in theory, it appears that the

PPP is essentially a long-term hypothesis: in the short term, it does not make

much sense as prices are sticky while nominal exchange rates can jump up and

down. For example, a temporary money supply expansion will temporarily

depreciate the country’s exchange rate leaving both the domestic and foreign

price levels unchanged. Therefore, the real exchange rate will temporarily

depreciate, decreasing the domestic price level with respect to the foreign

price level, both expressed in the same currency: this means that in the short

run PPP will not hold. This reasoning is confirmed empirically by Kim and Lima

(2010): one needs to differentiate short-term from long-term when testing the

PPP since prices cannot adjust in the short term. Deviations from the PPP in

the short-run are also confirmed by the observed short-term high volatility of

real exchange rates and their high level of persistence (Rogoff, 1996).

For the PPP to hold, a constant real

exchange rate much be observed, which is not the case in reality. This is why

most empirical studies focus on the long-term PPP. It allows for short-run

deviations of the PPP arising from the above reasons. But while the invalidity

of the short-term PPP is commonly admitted, the debate is more vivid when it

comes to admit whether the PPP, in some forms, holds in the long-run. The empirical

evidences found on the long-term PPP depend very much on the methodology

chosen.

The early empirical testings of the

PPP were usually based on estimates of equations of the form:

(6)

where is the error term, and where the absolute PPP

would depict et .

Frenkel (1978) does find convincing

estimates of and on high inflation countries, suggesting that the

PPP explains exchange rates in the long-run. But the estimates are supportive

of the PPP only in hyperinflationary economies, in countries facing normal

inflation, the PPP is rejected. Also, Frenkel does not test for stationarity: this

critic founds the core of the next empirical tests of the PPP.

To counter this caveat, empirical

tests started to concentrate on whether the exchange rate itself is stationary,

thus validating the long-run PPP, or whether it follows a unit root process;

i.e. the exchange rate follow a random walk pattern. Fleissig and Straaus (2000)

are unable to reject the stationarity hypothesis for six price indices

differing in their traded and non-traded components. They thus validate the

long-term PPP and the Balassa-Samuelson effect since stationarity is higher for

price indices with largely proportion of traded goods.

The literature also looked for

evidence of co-integration between prices and real exchange rates. This kind of

regression thus tests whether there is a mean reverting pattern depicted by

exchange rates, hence testing the long-run PPP. Taylor (1988) for example

rejects the unit root hypothesis for five major exchange rates: the PPP

hypothesis does not fit the fact, even in the long-run. Even when taking into

consideration transaction costs and measurement errors, he fails at finding a

long-run proportionality between exchange rates and relative prices. Enders

(1988) uses both ARIMA and cointegration and finds mixed evidence of PPP: depending

on the countries tested, he either rejects the stationarity hypothesis or is

unable to reject it.

It seems that the difference in data

used to assess the PPP has an important impact on the results. Some argue that

data sample should be very large to allow mean-reversion. Frankel (1986) is

able, by using 116 year of data on the US-UK exchange rate, to reject the

random walk hypothesis and to find a statistically significant speed of

adjustment to the PPP: 5 years. The PPP may have to be tested over very long

periods. A similar long-period test is conducted by Lothian and Taylor (1996).

They use two centuries annual data for dollar-sterling and franc-sterling real

exchange rates and find that they are both significantly mean-reverting. It

appears however that the mean-reversion is a relatively slow process: 3 years

for franc-sterling and 6 years for dollar-sterling.

Overall, the PPP is both

theoretically and empirically believed not to hold in the short-run since

prices are sticky. Many real world factors also prevent the PPP from holding: The

PPP cannot hold when obstacles to free trade, such as tariffs or transaction

costs, possibly in combination with monopolistic limits to free competition,

prevent the law of one price for one or many goods to hold. Also, the PPP

cannot hold if different basket of goods are used across countries to measure and . The presence

of non-tradable goods also prevent the PPP from holding. All in all, obstacles

to the PPP are such that the relationship never really holds: even Frenkel’s

estimates (1978) are not perfectly validating the PPP. The empirical results on

the long-run PPP are mixed, the main obstacles to consensus on the matter are

more of an econometrical nature, notably on the data used. The poor performance

of the PPP empirically does not necessarily mean that the PPP is worthless, it

is a useful approximation of the ongoing relationship between price levels and

exchange rates.