Economic theories provide indications about the speed of reaching a given long-run effect even though it's silent on the processes of adjustment to equilibria. A very good measure of this speed of adjustment is the half-lives. "The half-life indicates the number of year for deviations from PPP to subside permanently below one-half in response to unit shock." (Murray and Papell 2005).
The general consensus from extensive literature indicates that even though real exchange rates may converge to parity in the long-run, it takes time to get there and the process is slow. "The PPP puzzle arises because the estimated real exchange rate persistence is construed to be excessive in reference to theories where differences in prices are sustained by limits to arbitrage or nominal rigidities." (Imbs, Mumtaz et al. 2005). (Rogoff 1996) explains that the PPP puzzle is the difficulty in explaining why the very high short-term volatilities in the real exchange rate have slow mean reversions. The general result from research is that the PPP deviations dampen out at a slow rate of 15% and thus, the half life for it to dampen out is 3 - 5 years.
In our analysis, however, we find that the PPP puzzles strongly persist considering that there is no mean reversion in the long-run using the generalized impulse response. However, from the vector error correction model result, there exists some dampening out of deviations from the PPP to its long-run equilibrium. Section 2 is the literature reviews of the PPP hypothesis, the puzzle, as well as research works. The next section is the empirical work carried out by us. The last section is the conclusion from our findings.
2.0 Literature Review
The law of one price (LOP) states that the price of a good in a home country should be equal to that of foreign countries when converted to a common currency. This is the building block behind purchasing power parity (PPP). Over the years, several works have been done on PPP, which indicated that, in reality, this does not hold, except for few exceptions. This, according to earlier empirical works, is due to tariffs, transportation costs and non-tariffs barriers. The concept was treated empirically for the first time by Gustav Cassel (1918), even though it's been earlier discussed by John Stuart Mill, Viscount Goschen, Alfred Marshal and Ludwid von Mises. Of course, this empirical works have the shortcomings of data inconsistency and limitations.
This deviation from the LOP are remarkable volatile across broad range of goods, especially highly traded similar goods. This is the empirical result from the work of Peter Isard (1977), who found that these deviations are persistent and reflect nominal exchange rate movements. Rogers and Jenkins (1995) also discovered that not only is the relative price differential for similar goods volatile across borders but also persistent. This was also the findings of Engel (1993). Kenneth Froot, Michael Kim and Rogoff (1995) found consistency in the volatility of international prices of goods over the centuries.
Given the failure of the LOP, tests carried out on the aggregate price indices reject PPP as a short-run relationship. Such of those tests are Krugman (1978) and Frenkel (1978). Drawing back on Dornbusch's (1976) overshooting model of nominal and real exchange rate volatility, the failure in the short-run PPP can be attributed to the stickiness of nominal prices and changes in the real exchange rate in the short-run. They found it hard to see any convergence towards PPP in the long run using the cointegration methods on modern floating rate data. Obstefd and Rogoff (1995) pointed out that if sticky prices could account for the short-run real effects on monetary shocks, they may also be responsible for the long-run effect on fixed exchange rate.
Frankel (1986) argued that if the deviations from PPP dampen out extremely slowly, then tests should be based on long-horizon data sets. It is also believed that to enhance the unit root power of these tests, there should be an expansion in the range of countries being considered.
Murray and Papell (ibid) used Andrews' (1993) exactly median unbiased estimation methods with the results for dollar-sterling rates as increasing from the 5.78 to 6.5 years. In critic of the least square approach used by Lothian and Taylor, they explained that it was downward median biased. Even though the result for the franc-sterling rates was below Rogoff's consensus, shocks were discussed to decay only at a rate of 11 - 30% per year.
Rogoff proposes a 15% dying out of deviations from PPP per year for glacial rates and that there is a mean reversion within 3-5 years. Murray and Papell (2004) noted the results of Lothian and Taylor (1996; 2000) on dollar-sterling exchange rates as 10% dying out rates, which was slower than Rogoff's. The half-life deviations from PPP took 5.78 years and 2.73 years for the dollar-sterling and franc-sterling exchange rates respectively, showing a longer period particularly for the dollar-sterling exchange rates mean reversion as against Rogoff's 3-5 years. They also pointed out that there were underestimations of the half-lives of PPP deviations and that they overestimate the speed of mean reversion. When we allow for serial correlation, the errors of the proposed mean reversions are revealed. The arguments against blindly calculating half-lives brought about the need for better inspection of impulse response functions.
The impulse response as defined by Pesaran and Shin (1998), "measures the time profile of the effect of shock at a point in time on the expected future values of variables in a dynamic system." The three issues pointed out were:
The nature of the shock that hit the country at time t.
The state of the economy at the time before t (before the shock occurs, i.e. t-1)
The types of shocks expected to hit the economy in the future (t+1,..., t+n).
They introduced an alternative approach to the use of orthogonalized impulse response in solving for the shortcomings of not being invariant of the ordering of the variables in the VAR. The proposed approach is the generalized impulse response analysis.
3.0 Methodology and Empirical Analysis
We use the PPP model
PPP = P - F - E
where P is the log of UK wholesale price, F is the log of weighted index of foreign prices and E is the log of UK effective exchange rates, to analyse the PPP puzzle from the data file G7PPP.fit containing the quarterly observations from the period 1972Q1 to 1992Q3. We also make seasonal adjustments with quarterly dummies.
Our analysis follows the method put forward by Pesaran and Shin (1996) on the time profile of the effect of shocks on cointegrating relations of a VAR model for the PPP hypothesis with expectation that the impact of shocks will dampen out taking the economy to its long-run equilibrium. We also use the impulse response function analysis. We also employed Johansen's (1988, 1991) maximum likelihood (ML) method in measuring the persistence profiles based on Gaussian vector error correction (VEC) model.
We used the lag order of 2 for the vector autoregressive (VAR) model based on the Akaike Information and Swarz Bayesian selection criteria, and the test for cointegrating relations using the Johansen's log likelihood ratio.
Table 1 - Number of Cointegrating Relations
LR Maximal
LR Trace
Null
Statistic
95% Critical Value
90% Critical Value
Statistic
95% Critical Value
90% Critical Value
r = 0
26.9832
25.4200
23.1000
45.7604
42.3400
39.3400
r = 1
13.0981
19.2200
17.1800
18.7772
25.7700
23.0800
r = 2
5.6791
12.3900
10.5500
5.6791
12.3900
10.5500
From Table 1, the test results from both maximal and trace reveal cointegration at both 5% and 10% significance levels. Where the null r=1 and r=2, we do not reject the cointegration because 13.1 and 5.7 are less than 17.2 and 10.6 respectively for LR maximal. Also for LR trace, 18.8 and 5.7 are less than 23.1 and 10.6 respectively. These are at a 90% critical value for both maximal and trace.
From Appendix 2, we have an exact identification for one restriction and we denote the cointegrating vector of the PPP as shown below:
β =
HE : β =
The result for the exactly identified ML estimators is has follows
β = , lE = 728.8501
Results for the test for restrictions are shown in Table 4 in the Appendix. The test for co-trending hypothesis shows that we do not reject the null at a p-value of 0.712 and a LR statistic of 0.136. The PPP hypothesis alongside the co-trending is also tested. The value of the LR statistic for testing the PPP is 10.72 with a p-value of 0.015. Therefore, we strongly reject the hypothesis. We could assume that the effect of the UIP hypothesis not tested for explains the rejection, as in Pesaran and Shin (1996).
We estimate the following dynamic vector error correction (VEC) model
Δzt = c0 + d1S1t + d2S2t + d3S3t + α1ξ'zt-1 + Γ1Δzt-1 + εt
where α1 is the vector of error correction. In the table below, the figures in the bracket for the coefficient column are the standard errors, while the others are the p-values at 95% confidence level. The error correction equation for Δpt and Δft show statistical significance. However, the possibility of serial correlation and heteroscedasticity is evident in Δpt, while heteroscedasticity can be traced in Δft. The possible explanation for this could be the effect of changes in the real oil price, particularly, the serial correlation. As for the Δet, the diagnostic test fails in significance. Also the variation in exchange rates is not well explained for (3%). However, domestic price equations have a 74% explanation for price variations.
Table 2 - Estimates of the Error Correction coefficients and Diagnostic Statistics
Equation
α1
T-ratio for α1
Ṝ2
Χ2SC(4)
Χ2FF(1)
Χ2N(2)
Χ2HE(1)
Δpt
-0.0320
(.0088)
-3.6403
(0.001)
0.7442
13.3227
(0.010)
0.0145
(0.904)
154.3071
(0.0000)
9.3239
(0.002)
Δft
-0.0239
(0.0087)
-2.7518
(0.007)
0.5389
3.9726
(0.410)
0.6600
(0.417)
73.3938
(0.0000)
6.7815
(0.009)
Δet
-0.0098
(0.0418)
-.23449
(0.815)
0.03
1.1425
(0.887)
1.2079
(0.272)
0.3992
(0.819)
1.1735
(0.279)
The error correction term for the cointegrating relation explaining long-run price movement as a significant but small negative impact on current price changes, suggesting an equilibrating but slow adjustment process for UK prices in response to changes in exchange rate and foreign prices. However, when examined in an impulse response function with respect to the generalized impulse response function as proposed by Pesaran and Shin (1998), we noticed that the PPP puzzle still holds. We find that in our generalized impulse response analysis, the PPP relations do not converge to equilibrium, bringing more uncertainty as to the hypothesis. Although the cointegrating relations previously converge to zero in the vector error correcting model, it does not hold in the generalized impulse response analysis.
Figure 1 - Generalized Impulse Response(s) to one S.E. shock in the equation for f
From Figure 1, we noticed that foreign price equation have a non-zero permanent impact on the level of other I(1) variables. It has the largest impact on UK domestic prices, rising over 10% in the first year and a negative effect of over 10% in the first year as well.
4.0 Conclusion
The PPP puzzle posited by Rogoff for a mean reversion to the long run equilibrium over a 3 - 5 year period with a 15% rate of dying out of deviations from PPP per year for glacial rates may even be more serious than shown by the research of Lothian and Taylor as noted by Murray and Papell. Our analysis of the cointegrating relation of the PPP with the vector error correction model and the generalized impulse analysis reveal an equilibrating but slow adjustment process for UK prices in response to changes in exchange rate and foreign prices as well as a non-zero permanent impact on the UK domestic prices, totally away from the PPP hypothesis.
The PPP puzzle in this analysis, is not explained with reference to the UIP hypothesis as shown in the works of Pesaran and Shin (ibid) to accommodate the interest rate effect of the rate of mean reversion of the PPP to its long run equilibrium. Also the empirical work is not free of the possibility of serial correlation resulting from the non inclusion of the effects of variables such as the real oil prices shocks as show in economic and econometric literature. However, we have persuading evidence that the PPP puzzle still remains for academic research both on an economic and econometric frontier.
