Purchasing Power Parity Between Zambia And South Africa

Published Date: 02 Nov 2017

Exchange rates are a very important aspect of the day to day running of every economy and their determination has engulfed much of the empirical literature in Macroeconomics. Since the demise of the Bretton Woods agreement in the early 1970s and the consequent adoption of flexible exchange rates, there has been great concern into the determinants of a country’s exchange rate.

This paper is aimed at testing the validity of one of the heavily debated theories in international macroeconomics, the Purchasing Power Parity (PPP). The study will test the PPP using two sub-Saharan Africa countries; Zambia and South Africa from January 1992 to November 2009. This is the period in which most African countries adopted floating exchange rate systems. In terms of exchange rate management, Zambia adopted a freely floating exchange rate in 1992 as part of the economic reforms being implemented at that time. From 1992, the Zambian kwacha has experienced high levels of volatility against its major trading currencies.

The PPP draws it empirical popularity from the work of the Swedish economist Karl Gustav Cassel in the 1920s. The PPP as a model of exchange rate determination is based on the assumption of the law of one price; that the price of a commodity should be the same in one country as in another once converted to the same currency. In this respect, the PPP relates the exchange rate to the ratio of prices of countries. According to its specification, a country with a higher inflation will have a depreciating exchange rate and vice-versa.

In this study we subject the PPP to Zambian and South African data using various econometric techniques from Ordinary Least Squares specification, Mean Reversion Tests and Finally Cointegration Tests. The rest of the paper is organised as follows; section 2 will explain the data used, source and detailed description of the estimation techniques adopted. Section 3 will report the empirical results and brief discussion of the results. Finally, section 4 will conclude the study.

Data and Methods

There is a wealth of methods in the literature of international finance that has been used to test the purchasing power parity. This study adopts some of the most commonly used methods. The first method will involve the use of Ordinary Least Squares (OLS). Most studies have applied this method hence it will allow this study to compare its results with other studies. Further, the use of OLS in analysis gives the opportunity to test for the stationarity of the data since its time series. The second method will involve testing for mean reversion of the real exchange rate. The final part of the analysis will involve a Cointegration test of the series. These are discussed in more detail below.

2.1 Model Specifications

As discussed earlier, the PPP theory is premised on the law of one price. According to the law of one price, assuming perfect competition in trade (i.e., no tariffs, no trade barrier, no quotas or legal impediments to trade), a good from one country should sell for the same price in another country once the price is converted into a common currency. Thus, given any good j,

………….. (1)

Where, is the domestic price of good j, foreign price of good j and the domestic nominal exchange rate defined as domestic currency units per unit of foreign currency.

The law of one price in this sense is based on the most restrictive assumption; Perfect Competition. In reality, however, there are barriers to trade; transportation costs as well as other trade costs that make it impossible for the two prices to converge even when converted to a common currency.

Drawing from the law of one price, the absolute version of PPP as a model of exchange rate determination hypothesises that in the same currency, a bundle of similar good will attract the same price in any country. Thus, inverting equation (1) above, we have

……………… (2)

Where, , are the price indices of similar bundles of goods in the home and foreign country and the domestic nominal exchange rate as defined above. This version of the PPP is less likely to hold due to the same reasons raised against the law of one price. Further, the existence of non-tradable goods jeopardises arbitrage investment opportunities (Sarno et al, 2002).

To counter the short comings of the absolute version of the PPP, an alternative version called Relative Purchasing Power Parity is used. This version looks at the change in the exchange rate, rather than the absolute value. It states that the percentage change in the exchange rate should equal the difference between change in the growth rate between the domestic and foreign price levels. Formerly,

……………. (3)

Where gives the change in the exchange rate, represent the change in the domestic and foreign piece levels respectively. What the above specification entails is that if the domestic prices increase faster than foreign prices, then the exchange rate will have to depreciate to maintain PPP.

The empirical models specified to test the above two versions are:

………… (4) For the absolute version of PPP

and

………. (5) For the relative version of PPP.

Where all the variables are defined in logs with being the actual exchange rate between the Zambian Kwacha and the South African Rand, and are the Zambian and South African consumer price indices respectively and is a white noise error term~

In both specifications, the PPP holds only if, and otherwise fails it to hold.

A further specification is employed in this study. This relies on testing the mean reversion of the real exchange rate. According to Mkenda (2006), due to the fact that exchange rates fluctuate more than prices, the only realistic way to test for PPP is to examine its long-run behaviour. An exchange rate fluctuating and reverting towards a constant level mean entails PPP holds. Defining the real exchange rate as;

……………. (6)

Testing for PPP with the above specification is testing for mean reversion of the real exchange rate. What this test entails ideally is testing whether or not the real exchange rate is stationary, that in the long run, it tends to converge to a constant mean. Formerly, we specify the following equation:

…….. (7)

Where is the white noise error term~. The hypothesis tested here is that the real exchange rate has a unit root, . If the real exchange rate has a unit root, then it is not stationary, hence the PPP fails to hold.

Further, in log form, equation (6) cab be specified as

…………… (8)

What equation (8) above means is that the real exchange rate may be viewed as a measure of deviations from PPP (Sarno et al, 2002).

The final test will involve conducting a Cointegration test between the nominal exchange rate and the ratio of the prices from the two countries (see Johansen, 1985, 1995 for a detailed description of the test).

2.2 Data

The data for this analysis was collected from three sources; these are the IMF’s International Financial Statistics, the Reserve Bank of South Africa and the Central Bank of Zambia. Just like other studies on this subject, the series required are the nominal exchange rate, the domestic consumer price index and the foreign consumer price index. The data covers the period of floating exchange rates from January 1992 to November 2009 obtained monthly. This gives a high frequency monthly data.

2.2.1 Price Indices

This study uses the Zambian and South African Consumer Price Indices (CPI) to represent domestic and foreign price levels respectively. A consumer price index represents the cost of a representative bundle of goods overtime relative to some arbitrary base year (Tshipinare, 2006). However, we note the biases the use of CPIs as price indices creates due to the existence of a substantial component of non–tradables such as housing, and other domestic services. We stick to the CPIs, however, for two reasons: Firstly, data collection and reporting in most developing countries is still problematic. While we were able to find data on Industrial Production(Producer Price Index) for South Africa, that wasn’t the case for Zambia. Secondly, in as much as The CPI may have non-tradables, it can be argued that changes in the prices of non-traded goods does affect the price of traded goods through their indirect impacts on wage demands and cost of living (Tshipinare, 2006). The graphs below show the CPIs for South Africa and Zambia over the period of study.

Figure 2.1. South Africa and Zambia Consumer Price Indices (Jan 1992 to November 2009)

Source: IMF International Financial Statistics

Beginning from January 1992 to about July 2004, the Zambian CPI was below that of South Africa. However, the gap between the two indices has not been extremely wide after 2004 compared to the way it was before 2004.

2.2.2 Exchange Rates

The exchange rate is just the price of one currency in denominated in another. It can either be real (adjusted for price differentials) or nominal. The domestic exchange rate in thus study is the Zambian kwacha per unit of South African Rand (foreign currency).

The graph below displays the nominal and real exchange rate of the Zambian Kwacha against the South African Rand for the period January 1992 to November 2009. The Zambian Kwacha is arguably a commodity currency, whose value is largely determined by the price of Copper, Zambia’s major export. Consequently much of the movements and volatility of the Kwacha is due to the movements in the copper price (Bova, 2009). In the initial stages of general floating, the Kwacha was relatively stable but later begun to experience high levels of volatility. This was during the period of souring copper prices. However, we note a period of appreciation between 2005 and 2006; this too could be attributed to the copper price boom.

Figure 2.2. Zambian Kwacha/ South African Rand (Jan 1992 to November 2009)

Source: Bank of Zambia, Statistics Fortnightly

From about July 2006, the Kwacha has been unstable for a various number of reasons among them the decreased copper exports as the copper price fell. Further, the country has had two elections between 2006 and 2008, leading to a bit of instability during the election periods, hence souring the exchange rate.

Empirical Analysis

3.1 The Basics

A closer look at the time series variables outlined in the previous sections yields some important results. The graphical display of all the series needed for this study clearly shows that they all have a time trend in them. The two price indices clearly show an increasing positive trend. The same is the case for the exchange rate though it tends to fluctuate more than the price indices. We can thus naively say that the series are non-stationary. Stationarity in time series is an important concept and in order to avoid spurious regressions, it’s important that we make all the variables of interest stationary.

In brief, a time series, say is said to be covariance stationary if it has a constant mean and none of its autocovariances are time dependent, i.e.,

Since non stationary series yield biased and spurious results, Granger and Newbold (1974) suggest that one way to get round this problem is to perform regression analysis based on the changes of the variables rather than in level form hence solves the problem. First differencing of the variables thus solves the problem of non-stationarity (see appendix for graphs).

Besides graphical inspection, we formerly tested for stationarity in both levels and first difference using the Dickey Fuller Unit Root test. The Dickey Fuller test is based on the premise of testing for a unit root in the coefficient of the lagged dependent variable. The null hypothesis is that the coefficient on the lagged dependent variable is one, thus the variables exhibit the properties of a non-stationary random walk (see Hamilton 1994). The results of the Dickey Fuller test are reported below:

Augmented Dickey Fuller (ADF) Unit Root Test-5% Level

Variable

ADF

Critical Value

Decision

In Levels

 

Exchange Rate

-1.675

-2.882

 Non-Stationary

Zambia CPI

8.533

-2.882

 Non-Stationary

South Africa CPI

1.860

-2.882

 Non-Stationary

First Difference

 

Exchange Rate

-12.71

-2.882

 Stationary (1(1))

Zambia CPI

-6.741

-2.882

 Stationary (I(1))

South Africa CPI

-9.312

-2.882

 Stationary (I(1))

Note: The decision rule is to reject the null of a unit root if the ADF is less than the critical value at the chosen level of significance.

From the regression results, we could not reject the null hypothesis of a unit root in all the series, and thus concluded that they were non stationary in levels. However, when transformed by first differencing, the null of a unit root is rejected at the 5% level and the series become stationary. Thus, all the series are integrated of order one, I (1), (see appendix for transformed series). The order of integration is the number of times a non-stationary series must be differenced to make it stationary (Fagereng, 2007).We now report the regression results

3.2 Absolute PPP Results

Results for equation (4) are shown below

Dependent Variable is exchange rate ()

Coefficient

Standard Error

T-statistic

Significance

Constant ()

6.43421

0.0185042

44.75

0.000

0.65977

0.0147426

347.72

0.000

R squared

0.90390

Adjusted R squared

0.90340

F-statistic

2002.81

0.000

Sample Size(N)

215

Degrees of Freedom

214

Clearly, none of the estimated coefficients satisfy the condition for PPP. The constant and slope coefficient terms are significantly different from zero and one respectively, hence the PPP fails to hold. These results are supported by the graphical representation of the actual exchange rate plotted against the exchange rate as predicted by PPP below. It is clear and evident that the two exchange rates drift apart, hence the PPP does not approximate the actual exchange rate at all, thus it fails to hold.

Figure 3.1 Actual Vs Absolute PPP Exchange Rate Series, K/Rand, January 1992 to November 2009

From January 1992 to about April 2000, we note that was PPP closely approximated the real exchange rate, and then drifted way apart after 2000. The interesting bit of it is that that is during the time that the South African CPI was higher than that of Zambia. We notice a scenario where closer the CPIs, the more the deviation from PPP is the exchange rate

3.3 Relative PPP Results

Results for equation (5) are shown below

Dependent Variable is exchange rate ()

Coefficient

Standard Error

T-statistic

Significance

Constant ()

0. 0027386

0.0054343

0.50

0.615

0.6381073

0.1743879

3.66

0.000

R squared

0.0594

Adjusted R squared

0.0550

F-statistic

13.39

0.0003

Sample Size(N)

215

Degrees of Freedom

214

Just as in the absolute version case, the relative version of PPP also performs pretty badly. While the constant term meets the requirement of being zero, the slope coefficient is significantly different from one, thus the PPP fails to hold in totality.

Figure 3.2 Monthly Percent Changes in the Kwacha-Rand (K/ZAR) Exchange Rate and Monthly Inflation Differentials

As we would expect, the graph above shows just how the differences in inflation between the two countries fails to adequately predict the changes in the exchange rate. On this basis, we conclude that the relative version of the PPP does not hold. Notice also that the fluctuations in the exchange rate are more pronounced that in the price differentials. This gives the basis for the mean reversion test below.

3.4 Mean Reversion Test of PPP Results

Drawing on Sarno et al (2002) and specifying the exchange rate as earlier defined, then the real exchange rate is the nominal exchange rate adjusted for the relative national price level differences. With PPP holding, we would expect the real exchange rate to be constant, thus movements in the exchange rate represent deviations from PPP. Results for equation 7 are shown below;

Mean Reversion Test Results

Dependent Variable is Real exchange rate ()

Coefficient

Standard Error

T-statistic

Significance

Constant

14.725

13.559630

1.09

0.279

-0.1948

0.0143479

-1.36

0.176

R squared

0.0086

Adjusted R squared

0.0039

F-statistic

1.84

0.1760

Sample Size(N)

215

Degrees of Freedom

214

NB: This test should not be confused with the ADF test conducted earlier, though they both test for unit root.

This test of PPP relied on the real exchange rate between the Zambian Kwacha and the South African Rand. The null hypothesis was that the coefficient on the lagged value of the real exchange rate is equal to zero, thus the real exchange rate has a unit, hence non stationary. Failure to reject the null implies that the PPP fails to hold as the real exchange rate does not revert to a constant mean nor does it converge to a long run equilibrium level and thus follows and random walk. The result in the table above indicates that the real exchange rate indeed has a unit root, hence non stationary. As can be seen from the reported statistic, we fail to reject the null that = 0, hence the real exchange rate does not exhibit mean reversion. The graph below clearly supports the non stationarity result of the test.

Figure 3.4 Real Exchange Rate of the Z/ZAR, January 1992 to November 2009

If PPP held, the real exchange rate was expected to be constant. The above graph shows a downward tend, hence non–stationary. On this basis, it is clear that the real exchange rate has no tendency to converge to on a long run equilibrium level, thus PPP fails.

3.5 COINTEGRATION TESTS

Originally developed by Engle and Granger (1987), Cointegration provides a basis for testing the PPP. In as much as the PPP may fail to hold in the short run, it is possible that there could exist a long run equilibrium.

In brief, Cointegration tests are concerned with the existence of a long run stable relationship between or among a set of variables. Individually, a set of time series variables may be non-stationary but a linear combination of them could be stationary and thus integrated of order zero, I (0). A number of tests have been proposed to conduct Cointegration tests, in this study, we employ the Engel and Granger (1987) tests for Cointegration.

The starting point of a Cointegration test is to ensure that all the series are integrated of the same order. From the preceding analysis, we established that all the series are integrated of order one; hence we proceed to conduct a Cointegration test.

The procedure for this test involves testing for stationarity of the residuals obtained from equation specification (4). If the two series are cointegrated, then there should be no pattern in the residuals. Thus ideally, this is a test for a unit root in the residuals. The results are shown below;

Unit Root Test on the Residuals for Cointegration Test

Variable

ADF Test Statistic

1% CV

5% CV

10% CV

Decision

Residual

-1.966

-3.472

-2.882

-2.572

Non-Stationary

Note: The decision rule is to reject the null of a unit root if the ADF is less than the critical value at the chosen level of significance

Clearly we fail to reject the null of a unit root in the residuals. We should expect the residuals to exhibit some pattern hence the series are not cointegrated. Below is a graph of the residuals.

Figure 3.5 Residuals from Equation (4)

As expected from the ADF test on the residuals, there is a pattern in the residuals from the regression equation estimated. This is a clear indication of non stationarity in the residuals. Thus, based on the Engle Granger test, we fail to find any long run stable relationship between the exchange rate and the price levels. The PPP fails to hold.

Conclusion

This study sought to test the validity of the Purchasing Power Parity between two Zambia and South Africa employing various econometric methods. In line with earlier and current studies in the area, we used the nominal exchange rate of the Zambian Kwacha alongside the Consumer price indices for Zambia and South Africa.

As with most studies, the series were non stationary in levels, hence the estimation was conducted in first differences. The results from the absolute and relative PPP estimations rejected the PPP. The estimated coefficients clearly could not meet the required criterion to support the PPP theory. Further, this study also went on to test the PPP using the mean reversion test as done in studies such as Rogoff (1996). Here too, we failed to reject the null hypothesis of a random walk of the real exchange rate, it is non stationary, and thus does not support the PPP.

A final test of the PPP was conducted using a cointegrated analysis. This test failed to establish the existence of a long run stable relationship between the exchange rate and the price indices. It further supports the results from other studies that indeed the PPP does not hold.

Therefore, as with most studies in this area, this study too fails to support the PPP when using consumer price indices. The major limitation of using consumer price indices is that they contain a large component of non tradable goods which do not enter the international market to influence the exchange rate. Further, the existence of imperfect competition in international trade; tariffs and other non tariff barriers; transportation costs, and heterogeneity of consumer indices invalidates the very foundation of PPP. Aside that, we also suspect that the Ballssa -Samuelson Effect could be at play in the two countries. Further, we are also of the view that long run adjustment to PPP would take quite some time to be established, hence despite having a large number of monthly observations, we only relied on 18 years worth of data. Further studies in the area could be done using price indices of tradable goods spanning over a long period of time especially in regions where trade imperfections have been reduced.

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