The Law of One Price (LOP) is the whole basis and building block of any formation of PPP. In its most intuitive form, it states that the price of the same good in two different countries must be the same once converted into a common currency. This is also known as its absolute form and requires that:
Where is the price of good denoted in the currency of the domestic currency at time , is the same good at time in terms of the foreign currency, and is the nominal exchange rate, which is the domestic price of one unit of the foreign currency. Evidently, the LOP only holds in a very specific environment of both frictionless international arbitrage, and for goods with perfect substitution. In such an environment prices and exchange rates adjust instantaneously in order to sustain a real exchange rate of unity for any good:
Where denotes the real exchange rate between the two countries. In its weaker relative form, it only requires:
Clearly, the absolute version implies the relative, but not vice versa. Unlike the absolute which requires the real exchange rate of the goods to be in parity, the relative only require that it is constant over time. If the LOP were to hold for each individual good, it must be that it then holds for a basket of goods. Even if deviations from the LOP occur, it may be that they cancel out when aggregated. In the original Casselian view of PPP this basket refers to the general price level of a country, which is measured by the Consumer Price Index (CPI) as a weighted sum of prices. In its absolute form, the PPP hypothesis requires:
Or if we were to take its logarithmic form:
Where and are the logs of the price levels for the domestic and foreign countries respectively. In its absolute form, PPP requires that nominal exchange rates be equal to the price differentials of two identical baskets. However, one glaring problem is that the composition of these baskets and their weightings can very immensely between countries, despite being conceptually similar.
Heterogeneous baskets are less of a concern if we were to believe that price impulses effect all goods in the baskets homogenously. This is, however, unlikely to be the case. There have been efforts to construct an international standardised CPI. The most notable is the “International Comparison Programme” (ICP) by Summers and Heston (1991), which reports absolute PPP estimates using a constructed common basket of goods. However, this is less valuable empirically due to its infrequency and availability for only a few select countries. Sarno and Taylor (2002) argue that the degree of extrapolation used by the ICP makes these estimates partially artificial, and thus less reliable.
There are also other indices related problems, especially in the context of a time series analysis. One is how to address changes in weights, or the introduction and/or removal of goods. In order to account for possible constant differentials between price levels much of the literature focuses on examining PPP in its relative form:
In its relative form, PPP requires that changes in the growth of exchange rates are offset by changes in the growth of price level differences. In other words, whereas absolute PPP is concerned with price level differentials, relative PPP focusses on inflation differentials of the two countries. A second problem is that CPI portrays prices that are relative to its base year value, and test using it only examine deviations of exchange rates and price level differentials from said year. Therefore, unless we are to assume that PPP held over some base period, there is no way to estimate its deviation from the absolute PPP condition. As a result, any empirical work using price indices can only test for relative, and not absolute, PPP. (Crownover et al., 1996)
Relative PPP is the focus of the majority of empirical work done in this field. This is due to the use of price indices as its logarithmic changes roughly equates to its inflation.
1.2 Empirics of PPP
PPP is an almost century old concept. Despite its deceptively simple formulation it has remained a challenge to conclusively identify its existence. In order to appreciate its complexity, it is important to review it empirical history. With each new iteration of econometric techniques there is a wave of research as we are better able to model the dynamics of PPP.
Stage 1: Simple PPP
Early works of PPP testing, through the 1970s, attempted to model Cassel’s (1922) view that PPP was a tendency for exchange rates to revert to PPP that was subject to short run deviations. However, at this point in time there was no theoretical or statistical tools to distinguish between short and long run real effects. Unable to fully model the dynamics, this first stage short run modelling of absolute PPP was usually characterised as:
Where is an error term, or alternatively:
In this formulation PPP is thought to exist under two restrictions:
1. Symmetry Condition:
2. Proportionality Condition:
Both the symmetry and proportionality are required by the strong form of PPP, whereas its weak form poses no restriction. Whereas this specification tests for absolute PPP, testing for relative PPP was done by running the model in first differences.
In a widely cited study, Frenkel (1978) finds strong evidence of PPP in countries with high inflation, with coefficients close to plus and minus unity. However, most stage-one tests done on non-high inflation countries yield strong rejections of PPP. These early empirics are largely seen as flawed for two main reasons. Firstly, Frenkel (1978), much like others at the time, neglected to investigate the nature of the error term. In the case that prices and exchange rates are nonstationary and the resulting residuals are also nonstationary, this regression would be spurious and thus invalid. In the case that the residuals are stationary, it would mean that exchange rates and prices are cointegrated and exhibit a long run relationship. However, test statistics are also invalid due to bias in the standard errors (Engle and Granger, 1987). Secondly, there may exist endogeneity between exchange rates and prices. Krugman (1987) attempts to address this by using instrumented variables (IV) and ordinary least squares (OLS), and finds that the absolute value of coefficients are closer to unity than its non-instrumented counterpart, but still rejects PPP.
Stage 2: Testing for Unit Roots
Recall from equation (2), in logarithmic form the real exchange rate is characterised as:
In the failures to find PPP in stage one tests, attention then shifted to modelling PPP as a long run phenomenon by testing for possible stationarity of the real exchange rate. This was most commonly done through testing for unit roots in the real rate using variants of the Augmented Dickey Fuller (ADF) test:
Where is lag operator for the the lag, and is some white noise disturbance. Testing the null hypothesis that is equivalent for testing for unit roots, where failure to reject the null implies that the process is not a mean reverting process. Early stage two empirical work include, inter alia, Roll (1979), Adler and Lehmann (1983), Huizinga (1987), and Meese and Rogoff (1988). Other less commonly used tests include nonparametric variance ratio tests of the real rate (Cochrane, 1988) and fractional integration (Diebold et al., 1991).
Although stage two tests incorporate some notion of a long run relationship, they have largely failed to find evidence of PPP, especially when restricting observations to post-Bretton Woods data (Meese and Rogoff, 1988; Mark, 1990; Edison and Pauls 1993). The collapse of the gold standard and the Bretton Woods era in the 1970s have been used as the de facto period for the beginning of floating regimes for most of the developed world. These tests, as will be discussed below, suffered from a lack of predictive power, which was then only exacerbated by the highly volatile nature of the floating regime. However, it the evidence is more mixed when testing for fixed or formally stabilised currencies. For example, Chowdhury and Sdogati (1993) finds strong evidence to reject nonstationarity of real exchange rates between several European currencies against the Deustche mark during the time the European Monetary System (EMS) was in place (1979-90). However, they could not find the same when currencies were set against the U.S. Dollar.
Stage 3: Cointegration
Cointegration, as pioneered by Engle and Granger (1987), seemed like a natural candidate to modelling PPP, and has become an increasingly popular method to do so. Cointegration implies that these two processes will move in such a way that they hold a long run equilibrium regardless of any short run deviation. Cointegration offered a new breath of hope to model Cassel’s (1822) idea of PPP, over half century later. Cointegration requires that for any two nonstationary processes that are integrated to the same order they share a linear combination, which itself is integrated to an order below its sum. In the case of PPP testing we want exchange rates and price differentials to both be I(1), and for there to be some cointegrating parameter that results in a linear I(0) process. Cointegration is “at least a necessary condition for [two variables] to have a stable long-run (linear) relationship” (Taylor, 1988). As a result, if the two variables were in fact cointegrated, this would be a taken as evidence that real exchange rates are in fact stationary and thus mean reverting. However, in the case that the two are not cointegrated, we are left with a spurious relationship.
A large difference is between cointegration and unit root tests is that they do not impose conditions for symmetry and proportionality, neither are they possible to test due to the bias that exists in the standard errors. Cointegration tests are either bivariate, where only the restriction of symmetry is placed, or trivariate, where there are no restrictions. Reasons as to why these conditions may include systematic differences in basket composition, trade barriers and others have been investigated by, inter alios, Taylor (1988), Fisher and Park (1991), and Cheung and Lai (1993). There are a number of ways to addressing this issue, one of which is by using Johansen’s (1988 and 1991) maximum likelihood estimator, which tests for the presence of multiple cointegrating vectors simultaneously, and allows for the direct testing of said conditions.
Cointegration inspired yet another wave of research. Early work failed to find evidence of mean reversion for the recently floating regimes (Taylor, 1988 and Mark, 1990). However, more recent work has found evidence of the long run PPP hypothesis in the major industrialised economies (Corbae and Ouliaris 1988; Kim, 1990; and Cheung and Lai, 1993). This flood of research proved to reveal several insights into PPP. Firstly, it appears that cointegration, and thus mean reversion of real rates, is less likely to be found in floating currency pairs; a finding which supports the insight gained in stage two testing. Secondly, cointegration is more likely to be found when using the Wholesale Price Index (WPI) rather than the CPI. This is due to the WPI having relatively less nontraded components. Finally, rejections of cointegration occurred more often when using bivariate tests then when using trivariate ones. This would imply that residuals appear to be more stationary when we relax the conditions of symmetry and proportionality. However, despite these findings the results for and can be incredibly varied, and at times nonsensical. For example, Cheung and Lai find coefficient estimates that range from 1.03 to 25.4 for CPIs and 0.3 to 11.4 for WPIs, which far from the theorised value of unity.
One main issue that has plagued these tests is that of low power. These studies, especially the earlier ones, were done during the 1980s, which meant that they had only roughly 15 years of floating data. Sarno and Taylor (2002) demonstrates this problem with a simple Monte Carlo experiment using findings of a yearly mean reversion 11 percent for pound sterling-U.S. dollar currency pair by Lothian and Taylor (1996). They find that with only 15 years of data the chances of correctly rejecting the null hypothesis of unit roots falls in between 4.8 and 7.4 percent. Even with the use of a century’s worth of data, as did Lothian and Taylor (1996), the chances of correct rejection increases to between 12.5 to 78 percent given a 95 percent confidence interval. With the floating period only currently spanning almost 50 years, there is a clear shortage of data. Furthermore, Shiller and Perron (1985) provides Monte Carlo evidence that to increase the probability of correct rejections, it is not enough just to increase the frequency of sampling. As PPP is seen to now be most valid as a long run preposition, it requires longer spans of data. In order to address this issue, studies have taken one of two approaches.
In response to the low power of tests that was forewarned by Frankel (1986), several studies began to use long-horizon studies. In order to reconcile apparent volatility of exchange rates and possible, albeit slow, reversion rates researchers extended their samples to cover much longer periods, which also inevitably covered multiple exchange rate regimes.