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Chapter 4: Testing for PPP using SADC real exchange rates
Introduction
Recent literature recognizes the need to assess nonlinearities in the adjustment dynamics of real exchange rates. The main reason for this paradigm shift is that, due to lower power in small sample sizes, the standard Dickey-Fuller-type tests do not provide a solid foundation for an inference that reduces the probability of committing a Type 2 error. This has been the case even when stronger versions of Dickey-Fuller tests – such as the one suggested by Elliot, Rotenberg and Stock (1996) – were used.
The focus on nonlinearities has been reinforced by Taylor, Peel and Sarno (2001) who provide strong evidence that four major real bilateral dollar exchange rates were characterized by nonlinear mean-reversion. One influential study that has also corroborated nonlinear mean-reversion is by Michael, Nobay and Peel (1997). In the nonlinear models, an equilibrium level of the real exchange rate in the regime in which the log-level of the real exchange rate is close to a random walk becomes increasingly mean reverting as the absolute size of the deviation from equilibrium increases. This is consistent with the recent theoretical literature on the nature of real exchange rate dynamics in the presence of transaction costs (See Sercu, Uppal and van Hulle (1995)).
This chapter presents hypothesis testing in respect of joint tests of nonlinearity and stationarity associated with the seminal contribution by Kapetanios, Shin and Snell (2003), henceforth denoted KSS. It also presents the results of ADF tests and Bayesian unit root tests at conventional levels.
In addition to non-ESTAR alternative unit root testing, the chapter provides a background description of the Bayesian unit-root testing framework.
The ESTAR testing framework
In the context of nonlinearities, the testing framework is the smooth transition autoregressive modelling. In particular, we focus on the exponential version of the model, which is often used when the economic agents can have arbitrage opportunities by facing some deviation from the long-run equilibrium. In such a case, the unit root regime becomes an inner regime, and the mean-reverting regime becomes the outer regime.
In equation (4.1), yt −d is a transition variable, making this ESTAR model a self-exciting one. The delay parameter d is an integer, which can be fixed or be determined endogenously by means of a supremum LM test in the spirit of Norman (2006a). In equation (4.2), F( yt −d ) represents the exponential transition function. We note that the extreme values of the transition function are 0 and 1. So, for F () > 0 and F() < 1, the model exhibits a smooth regime-switching behaviour. The parameter determines the smoothness of the transition from one regime to another. We note that as yt → ±∞ , then the transition function F() → 0 . In addition, as γ → 0 orγ → ∞ , then F(st ,γ , c) = 0 .
Bayesian unit root testing
Over and above the ESTAR testing, for comparative purposes we employ Bayesian unit root tests in conjunction with the ADF tests. Bayesian unit root testing was introduced by Zellner and Siow (1980) but was popularised by Koop (1992) and Ahking (1997, 2004), among others. Under the null hypothesis, a times series is an autoregressive model of order p with a linear time trend. In short, hypothesis testing takes the following form:
where t denotes a linear deterministic time trend; and ε jt , j = 2,3,4 represents a serially uncorrelated error process with zero mean and constant variance. H1 represents the null model, hypothesising a trend-stationary auto-regressive process of order p. The first alternative against the null, hypothesises a stationary AR( p) process, while the second alternative hypothesises an AR( p) process with a unit root. It is to be noted that H 2 and H3 are special cases of the null hypothesis, with linear restrictions imposed on the null model. The trend-stationary hypothesis is included because it is the leading alternative to unit-root non-stationarity in macroeconomic time series. According to Ahking (2004) the stationary alternative is included to appraise the extent to which the Bayesian test can distinguish between nonstationary series and a stationary one with a high degree of persistence, as is frequently encountered in time series econometrics.
We compare the four hypotheses, based on both prior and sample information, by calculating the posterior odds ratios:
note that the posterior odds ratio gives the ratio of the probabilities of the two hypotheses holding given the sample data. On the assumption that all three hypotheses have equal prior probability, then the posterior odds ratio becomes
Empirical evidence
SADC dollar-based real exchanges were chosen on the basis of adequate data availability. We used the International Financial Statistics database of the International Monetary Fund. Real exchange rates were derived from the relative form of the purchasing power parity hypothesis, namely
where ln Yt is the logarithm of a real exchange rate (domestic price of foreign currency) at time t ; ln Pt and ln Pt are the logarithms of foreign and domestic price levels, respectively. The United States consumer price index (CPI) inflation is the all-item CPI inflation and the foreign CPI inflation rates are the general CPI inflation rates of the chosen countries. Sample periods varied according to data availability in respect of CPI inflation and nominal exchange rate series.
The results of Bayesian unit root testing
According to the results appearing in Table 2, nonstationarity hypothesis receives small posterior probability relative to other hypotheses. In this setting, the Bayesian results strongly support the hypothesis that all the real exchange rates are trend-stationary autoregressive processes.
It is necessary to point out that the Bayesian unit root test results are sharply at odds with the ADF results in that the hypothesis of a unit root does not receive significant posterior probability in all cases. Instead the results seem to support the hypothesis of trend-stationarity for all cases. That been said, Ahking (2004) found that that the Bayesian test used in this paper could not distinguish between a trend-stationary autoregressive model from a stationary autoregressive one, especially when the time trend effect was relatively small, and the time series was highly persistent. The latter author found that the bias was in favour of finding a trend-stationary model. Thus, the results should be treated with caution.
Results from nonlinear tests of nonstationarity
In the context of nonlinear analysis, we used partial autocorrelation function to determine the optimal lags. This approach is recommended by Granger and Terasvirta (1993) and Terasvirta (1994). The usage of PACF over that of information criteria represents an effort to avoid possible bias when choosing lag length. The delay parameter was fixed at 1. In general, the appropriate choice of the delay parameter is the one associated with the highest test statistic. However, fixing the delay parameter is generally of little consequence since economic intuition would suggest that smaller values of the delay parameter were to be preferred.
We employed an auxiliary regression appearing in (4.5) or (4.6). In this setting, the null and alternative hypotheses are of the form:
H0:δ =0,
H1:δ <0.
Failing to reject the null implies that the real exchange rate should be treated as nonstationary. By contrast, the rejection of the null hypothesis in favour of the alternative implies that the exchange rate is mean-reverting and nonlinear.
KSS provide the simulated asymptotic critical values for the nonlinear unit root tests. For both the nonlinear Dickey-Fuller test and nonlinear ADF tests, the 1, 5 and 10 per cent critical values for the demeaned series are –3.48, -2.93 and –2.66 respectively, whereas for the demeaned and detrended series are, in the same order, –3.93, –3.40 and –3.13.
In the context of linear analysis, our dataset has log-levels of real exchange rates and demeaned series. In the case of non-demeaned data, we apply the ADF test. When testing for unit roots, we allow for a constant but no deterministic time trends in the test regression.
The above being said, the pitfalls of the tests should be noted. As was noted by Hall (1994) and Ng and Perron (1995), the ADF tests suffer from low power when the lag length is too small. In some cases, lag selection alone may be responsible for the difference in rejections rates.
Table 3 summarises key inferences that can be made from the above estimation. The results from KSS nonlinear unit root and linear ADF tests are based on the demeaned series and suggest that the null hypothesis of nonstationarity should be rejected at 1 per cent significance level for 4 real exchange rates (Mauritius, South Africa, Swaziland, and Tanzania) out of 10 country exchange rates under study. This suggests that a linear specification for these countries would be inappropriate. In addition, these real exchange rates are mean-reverting but in a nonlinear fashion.
At 1 per cent significance level, all the series were nonstationary. However at 10 per cent significance level the real exchange rates of 6 countries were stationary: Mozambique, Madagascar, Mauritius, South Africa, Swaziland, and Tanzania were found to be stationary.
Chapter 1: Background to the Thesis
1.1 Background to the Thesis.
1.2 A brief overview of the foreign exchange markets in the SADC
1.3 The evolution of South Africa’s foreign exchange system.
1.4 Objectives of the Thesis
1.5 Resolving the mean reversion version of the PPP puzzle
1.6 Resolving the half-life version of the PPP puzzle
1.7 Resolving the exchange rate determination puzzle
1.8 Resolving the exchange rate disconnect puzzle
1.9 Contributions of the Thesis to economic literature
1.10 The structure of the Thesis
2 Chapter 2: Introducing the puzzles
2.1 Introduction.
2.2 The PPP puzzle: mean-reversion
2.3 The half-life version of the PPP puzzle
2.4 The exchange rate disconnect puzzle
2.5 The exchange rate determination puzzle
2.6 Conclusion
3 Chapter 3: Recent theoretical and empirical developments
3.1 Introduction
3.2 Recent developments: PPP mean-reversion puzzle
3.3 Threshold and STAR approaches to the PPP puzzle
3.4 Recent developments: Half-lives
3.5 Testing for long memory in respect of the PPP puzzle
3.6 Recent developments: Exchange rate disconnect puzzle
3.7 A survey of GE models in respect of the disconnect puzzle
3.8 Critical assessment of the models and conclusions
3.9 Recent developments: Exchange rate determination puzzle
4 Chapter 4: Testing for PPP using SADC real exchange rates
4.1 Introduction
4.2 The ESTAR testing framework
4.3 Bayesian unit root testing
4.4 Empirical evidence
4.5 Conclusions.
5 Chapter 5: Half-life deviations from PPP in the SADC
5.1 Introduction
5.2 Rossi (2005a) methodology for general AR( p) processes
5.3 Empirical results
5.4 Conclusions
6 Chapter 6: Tests of long memory regarding the PPP puzzle
6.1 Introduction
6.2 Concept of fractional integration
6.3 Testing methodology
6.4 Results
6.5 Conclusions
7 Chapter 7: Microstructure approach to the determination puzzle
7.1 Introduction
7.2 The basic model
7.3 Hybrid regression models
7.4 Econometric issues and data analysis
7.5 Empirical evidence
7.6 Conclusions
8 Chapter 8: Conclusions and implications
8.1 Background summary
8.2 Conclusions
8.3 Issues for future research
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