Purchasing Power Parity: A Unit Root, Cointegration and VAR Analysis in Emerging and Advanced Countries

The purpose of this study is to investigate the validity of the absolute version of the purchasing power parity (PPP) of a sample of four advanced and four emerging countries covering the period from 1993 to 2014. To examine the existence of PPP we apply the Augmented Dickey-Fuller, DF-GLS and KPSS tests for non-stationarity, and the Johansen procedure for cointegration between exchange rates and consumer price indices. The impulse response function presents a graphical view which is consistent with impressions from the statistics of stationarity tests. We also employ the variance decomposition method to analyze the movements in the exchange rates and the price indices that are caused by their own shocks, and shocks caused by other variables. With respect to half-life estimates, the results from a shock to the real exchange rate range from 9,76 to 77,39 months. Overall, unit root tests show that absolute PPP may hold, but this depends on the country and the selected method. In contrast, the Johansen approach does not support the existence of PPP in any country.


Introduction
The prevalence of the free exchange rates regime has undoubtedly contributed to the growth of international financial markets, but also to a radical increase in the volatility of exchange rates, undermining the national economic policies and the level of future investments. In this regard, a thorough understanding of the causes and the implications of exchange rate uncertainty is of fundamental importance for the efficiency of money and capital markets, as numerous historical events have illustrated.
The academic literature has devoted a considerable amount of time examining the relationship between exchange rates and fundamental macroeconomic variables, focusing either on the sources of the causal relationship or to an implied level determined by an equilibrium theoretical framework. For instance, Rogoff (1996) and Taylor and Taylor (2004) have debated extensively about the validity of the Purchasing Power Parity.
In this paper we restrict our attention to the existence and the degree of validation of the absolute Purchasing Power Parity (PPP) hypothesis. The motivation for this topic comes from several sources. First, in order to evaluate the validity of previous research findings we employ a sample spanning a longer and more recent time period. Second, we select two groups of countries: emerging and advanced. The first group consists of Mexico, China, Hungary and South Africa and the second one of Germany, Japan, the United Kingdom and Canada. The particular set comprises countries with heterogeneous economic characteristics and covers almost all continents. Finally, we perform a wider range of econometric techniques to gain a more complete picture, which will be discussed in detail in the following sections.
According to the PPP hypothesis, the exchange rate will adjust to equalize the price levels between countries. In detail, the formula of the absolute PPP is: where e is the nominal exchange rate measured in units of domestic currency per unit of foreign currency, P is the domestic price level and P * is the foreign price level. Note also that the logarithmic transformation of (1) has the form: where s, p, p* are the logarithms of e, P, P*. PPP is often used as a benchmark model which provides a baseline forecast of future exchange rates. Therefore, it can be viewed as a measure of deviations from the equilibrium level. An algebraic manipulation of equation (2) gives: Equation (3) is essentially a logarithmic transformation of the real exchange rate which is defined as the nominal exchange rate multiplied by the ratio of the price levels. Thus, we can directly base our inferences regarding the validity of the absolute PPP by focusing on the time series properties of the real exchange rates.
The statistical tools applied in this study contain unit root tests (i.e., ADF, DF-GLS and KPSS).
A complementary approach (half-life) estimates the number of years required to correct 50 percent of deviations from PPP levels resulting from a unit shock response in the levels of the series of the real rates. Furthermore, we perform a long-run relationship test (Johansen approach) and impulse response functions (IRF) coupled with variance decomposition in order to observe the effects of a shock in the exchange rate and price indices and to analyze the variance of the variables which are contained in the VAR model. The remainder of the paper is organized as follows. Section 2 reviews previous literature findings. Section 3 describes the sample, methodology and a discussion of the empirical results.
Finally, section 4 concludes the paper.

Literature Review
A wide range of approaches that test the existence of the PPP hypothesis can be categorized in two parts, initially testing the stationarity of the real exchange rate, and secondly, determining the cointegration relationship among the nominal exchange rate and the relative prices. Results are dependent on the performed methods, the characteristics of the sample, the number of the observations and the macroeconomic variables.
The test results for PPP by McDonald (1993) are encouraging and provide empirical evidence for the existence of purchasing power parity in the long-run. He found evidence that PPP does not hold in its strong form for Canada, France, Germany, Japan and the United Kingdom.
Cointegration tests did not support the existence of a long-run equilibrium relationship between the consumer price ratio and nominal exchange rate with US dollar as base currency for any of the five countries.
Another study of Alba and Papell (2005)

Empirical Data
To assess our hypothesis, we use monthly data of eight exchange rates and monthly consumer price indices from OECD. All bilateral exchange rates have a common denominator, the US dollar. The time period under investigation is from August 1993 to August 2014.

Empirical Analysis and Results
In the first part of the empirical analysis, we employ two unit root tests (ADF and DF-GLS) and one stationarity test (KPSS) in order to investigate the existence of unit roots and determine the order of integration of the real exchange rates. If there is a unit root, we are unable to reject the hypothesis that PPP does not hold. Then, we apply the Johansen test to investigate the longrun relationship between nominal exchange rate and consumer price index of each country.
Finally, we use a VAR model in order to extract firstly, the impulse response function to predict the movements of the variables, which are included in the PPP, due to shocks and secondly, variance decomposition to evaluate how shocks reverberate through a system.

Unit Root Test
Before we proceed to test whether there is a cointegrating relationship between exchange rates and price levels, it is essential to test whether the real exchange rate series are non-stationary.
Therefore, we perform unit root tests by including both a constant and a trend term 7 .
Following standard practice, we include lags, thus, in our case we use the Schwarz Information Criteria (SIC) in order to estimate the appropriate number of lags before proceeding to identify the probable order of stationarity. The results of the tests on the levels and the first differences are presented in table 1.
7 Augmented-Dickey Fuller with intercept and trend: q t = β 1 + β 2 t + γq t−1 + Σ i=1 p β ι Δq t−1 + ε t , namely the series are non-stationary in the levels, but they are stationary in the first differences, that is, they are integrated of order one.

Half-Lives
After formulating the unit root tests, it is important to calculate each country's half-life. The half-life of deviations from purchasing power parity (PPP) is used as a measure of persistence to quantify the degree of mean reversion in real exchange rates. It is defined as the number of years that it takes for deviations from PPP to subside permanently below 50 percent in response to a shock in the levels of the series. This particular notion of the degree of mean reversion in 8 The null hypothesis is that the real exchange rate has a unit root. 9 The null hypothesis is that the real exchange rate has a unit root. 10 The null hypothesis is that the real exchange rate is stationary. where γ is the coefficient of the lagged value in the regression model of the ADF test. The half-lives from the ADF regressions 11 for all samples are also presented in

Cointegration Tests using Johansen Procedure
To gain further insight into the issue, we examine the long-run equilibrium relation between the nominal exchange rate and the two price indices using the Johansen multivariate cointegration technique 12 . A necessary condition in order to investigate the cointegrating relationship between consumer price indices and exchange rates is that these variables must be integrated of order one. However, Japan, the United Kingdom, Canada, China and Hungary do not satisfy this requirement, thus, we examine the remaining countries for long-run relationship.
The issue of finding the appropriate lag length is very important, so we use the most common procedure by estimating the VAR model, inspecting the value of the Schwarz Information Criterion (SIC) and choosing the model that minimizes the SIC as the one with the optimal lag length. The Johansen cointegration test, then, has been conducted and the outcomes of the maximum eigenvalue and trace statistics are reported in We then proceed to estimate the vector error correction model (VECM) in order to analyze how short-run divergence, if any, was corrected so as to capture how rapidly long-run PPP is attained. Also, it should be noted that a necessary and sufficient condition for PPP to hold is that the logarithm of the exchange rate between countries and the logarithm of the price levels should be cointegrated with cointegrating vector [1 -1 1]. As we can see from table 3, the pvalue of Mexico is smaller than the five percent level of significance, so it does not satisfy the restrictions for the validity of PPP. As a consequence, the Johansen multivariate cointegration test concludes that the absolute PPP does not hold in any country.

Impulse Response Function from Domestic Price to Exchange Rate and to U.S. Price
The findings from the cointegration analysis are also reinforced with an alternative analysis, the impulse response function. The rationale beyond IRF is that it traces out the responsiveness of the dependent variables in the VAR model to shocks to the error term of each equation. For this purpose, we give a shock to the residuals of the VAR equations, to see how it affects the whole VAR model.
where Et is nominal exchange rate defined in local currency units per foreign currency unit, CPIDOM,t is the domestic consumer price index, CPIUSA,t is the foreign (USA) consumer price index and ui,t are the residuals of each equation.
In the following charts ( Africa. Finally, the impulse response functions indicate that the adjustment process of the US price level is not completed within these twenty months due to the various shocks to all of the variables.

Variance Decomposition
The second standard tool to analyze the properties of the estimated structural VAR is the variance decomposition. Variance decomposition offers a slightly different method to examine VAR system dynamics. It gives the proportion of the movement in the dependent variables that are due to their own shocks, versus shocks to the other variables. To further illustrate this point, a shock to one of the variables will directly affect its own, but it will also be transmitted to all of the other variables in the system through the dynamic structure of the VAR. Figure  contrast with the increasing percentage that is due to the price level of each country, while the economy of Canada constitutes a special case in which the proportion that is attributed to its price level is greater than the proportion that is explained by the price level of the U.S.A..

Conclusion
The purpose of this paper is to examine the long-run validity of the absolute PPP in eight developing and developed countries. Our discussion is concentrated on five broadly defined topics. Initially, we examine the stationarity of the real exchange rates using ADF, DF-GLS and KPSS tests and estimate the half-life of each country in comparison with the findings of previous studies. In addition, we conduct a cointegration test, using the Johansen approach and finally we extract the impulse response function and the variance decomposition from the VAR.
Our empirical results indicate that the real exchange rates of the examined economies are all non-stationary except for Mexico in ADF test and the United Kingdom in KPSS test. The findings of the half-life estimates are striking and vary in comparison with the results of previous studies. More specifically, Mexico exhibits the highest speed of mean reversion of the real exchange rate. With respect to the cointegration test between exchange rate and price levels, we found no evidence supporting a long-run relationship between these variables.
An alternative way to assess the adjustment of the exchange rate and the price levels to an unexpected temporary shock is the impulse response function because it provides useful insight to interplay with PPP components. Our findings are in line with the existing literature except for the economy of Mexico which demonstrates an inconsistent picture with PPP predictions.
Another notable finding is in the variance decomposition section where the variance of the U.S.A. price level is less explained by itself and more by the price level of Canada after the fourth month, which could be attributed to the fact that these two countries share the world's largest and most comprehensive trading relationship.