Purchasing Power Parity and the Irish Experience: Unit Roots and Cointegration Tests for Two Industrial Countries

Fountas, Stilianos; Wu, Jyh-lin Article Purchasing Power Parity and the Irish Experience: Unit Roots and Cointegration Tests for Two Industrial Countries Kredit und Kapital Provided in Cooperation with: Duncker & Humblot, Berlin Suggested Citation: Fountas, Stilianos; Wu, Jyh-lin (1995) : Purchasing Power Parity and the Irish Experience: Unit Roots and Cointegration Tests for Two Industrial Countries, Kredit und Kapital, ISSN 0023-4591, Duncker & Humblot, Berlin, Vol. 28, Iss. 2, pp. 201-215, https://doi.org/10.3790/ccm.28.2.201 This Version is available at: https://hdl.handle.net/10419/293298 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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Introduction The Purchasing Power Parity (PPP) relationship has been in the forefront of modern international finance. This relation has been tested empirically by many economists and the evidence on its relevance has been mixed. It is generally accepted by applied and theoretical economists that the parity does not hold in the short run. However, there is no consensus on the empirical evidence on the long run parity. Modern time-series econometrics that include unit root tests and cointegration techniques have been used extensively to test for the long run parity.1 Unit root tests have tried to determine whether the real exchange rate is a random walk (i.e., a nonstationary series). If the real exchange rate follows a random walk, there would be no tendency to return to its long run value (i.e., the deviations from its long run value would be permanent). Dickey-Fuller tests have been used to test the null hypothesis of unit roots. However, this type of classical statistical tests has been criticized recently by statisticians and economists. This criticism is based on the fact that classical tests cannot discriminate between large autoregressive coefficients and unit roots. In other words, classical tests have low power. In response to the above-mentioned widely-accepted criticism of unit root tests, Kwiatkowski, Phillips, Schmidt and Shin (hereafter referred to as KPSS, 1992) have suggested the use of tests for stationarity along with tests for unit roots. Then, if one finds that the null of stationarity is rejected and the null of unit root cannot be rejected, the series is considered to be nonstationary. KPSS (1992), after testing for both null hypoi Examples include Abuaf-Jorion (1989), Corbae-Ouliaris [(1988), (1990)], Cheung-Lai (1993), Coughlin-Koedijk (1990), Enders (1988), Johnson (1991), Mark (1990), Pippenger (1993), Taylor (1988), Thorn (1989) and Wright (1993). OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.28.2.201 | Generated on 2023-01-16 13:05:58 202 Stilianos Fountas and Jyh-lin Wu theses of unit root and stationarity, conclude that for many of the US time-series examined by Nelson and Plosser (1982) the existence of unit roots is in doubt. Therefore, they argue in favour of a combination of tests (eg., DF and KPSS) to test for the stationarity of a time series. We have decided to combine the KPSS test with the Augmented DickeyFuller (ADF) test. This paper contributes to the empirical literature on PPP in three ways: first, we use a relatively new test, the classical KPSS test to test for the null of stationarity of the real exchange rate. Second, using monthly data for the 1981.1 - 1992.4 period, we find strong evidence that PPP holds, even with monthly data, for the UK and Germany based on Johansen's cointegration test.2 These results imply that Ireland's competitive position against an ERM and a non-ERM country has been maintained following the country's entry to the ERM. PPP against Germany, in particular, implies that Ireland has linked its inflation rate to the low German rate justifying the country's membership in the EMS. Third, the estimation of the Error-Correction Mechanism (ECM) representations shows that adjustment towards the long-run PPP takes place through changes in the Irish price level rather than the foreign price level. These results are, in general, consistent with the small-open economy (SOE) version of PPP where the price level of the small economy adjusts to restore the PPP relation. The ECM evidence also shows that PPP against the UK has been maintained through changes in the (sterling/pound) nominal exchange rate. In a recent paper, Wright (1993) tests for PPP for the Irish pound/sterling and Irish pound/DM exchange rates using monthly data for the 1981.1 - 1992.6 period. Wright (1993) using Johansen's cointegration approach, i.e., a different technique from that applied by Thom (1989) and Callan and Fitzgerald (1989), finds a similar result: PPP using both Irish/UK and Irish/German data is rejected. However, the author finds that PPP holds if the cointegrating vector is expanded to include the domestic and foreign interest rate. Our paper differs from Wright (1993) for four main reasons: first, we concentrate only on price levels and the nominal exchange rate and use two different testing procedures. Second, following Cheung-Lai (1993) and Pippenger (1993), we provide a different interpretation to the Johansen's test results on the proportionality hypothesis in the cointegrating vector and, hence, are able to accept PPP 2 These results contrast sharply with those by Thom (1989) who could not provide supportive evidence for the PPP against the US and UK using Engle-Granger cointegration techniques and monthly data for the 1980 - 1987 period. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.28.2.201 | Generated on 2023-01-16 13:05:58 Purchasing Power Parity and the Irish Experience 203 provided the deviation from the exact PPP relationship may be due to measurement errors in prices. Third, to deal with the low power of unit root tests, we make use of tests for both a unit root null and a stationarity null (using the recently developed KPSS test) when testing for the integration properties of the real exchange rate, the nominal exchange rate, and the price levels. Finally, using the ECM regressions, we determine the relative importance of domestic prices, foreign prices, and the nominal exchange rate in establishing the long-run PPP relationship. The remaining of the paper is structured as follows: section II describes the ADF and KPSS tests, Johansen's procedure and discusses the methodology. Section III provides the empirical results for the real exchange rate using the combination of KPSS and the ADF tests. Section IV presents the cointegration results using Johansen's approach and the error-correction estimations. Finally, section V summarizes the main conclusions. II. Testing for PPP: the Empirical Methodology To test for PPP we employ two testing procedures. First, we use two classical tests, ADF and KPSS. The first tests for the null of a unit root and the second for the null of stationarity in the real exchange rate. Second, we apply Johansen's procedure to test for a long-run relation among the nominal exchange rate and the price levels. 1. PPP and Statistical Tests Let R = S + P - P*, be the real exchange rate, where S, P, and P* are the logs of the nominal exchange rate, domestic price level, and foreign price level respectively. Provided that S, P, and P* have unit roots, the following definitions of PPP can be considered: (i) PPP exists if R is stationary. (ii) PPP exists if (S, P, P*) is cointegrated. (i) is stronger than (ii) since it implies (ii) plus the restriction that the cointegrating vector is (1, 1, -1). PPP tests of type (i) include Corbae and Ouliaris (1988), Whitt (1992), and tests of type (ii) include Cheung and Lai (1993), Enders (1988), and Wright (1993). In this paper, we employ statistical tests to test for both PPP definitions (i) and (ii). Even if R is not stationary, cointegration among price levels and the nominal exchange rate would be consistent with PPP provided the deviation from OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.28.2.201 | Generated on 2023-01-16 13:05:58 204 Stilianos Fountas and Jyh-lin Wu the "exact" PPP relation is due to measurement errors in price indices as explained below. 2. ADF and KPSS In this section we combine two classical tests, ADF and KPSS, in order to test for the stationarity of the real exchange rate. According to the ADF test, the following regression is run p (1) Rt = P 0 + PrRt1 + 7 * + + i = l and the null hypothesis pT = 1 is tested using a Student's t statistic denoted rT. We chose p = 4 since it is the minimum number of lags necessary for white noise residuals. The normalized bias test, when the errors are serially correlated, is given by the statistic cT(pT-1), where c = 1/(1 ~/32 -03 -^4). Its critical values are given in Fuller (1976, p. 371).3 Dickey, Bell, and Miller (1986) in their survey of unit root tests mention that the powers of the two tests, rT and normalized bias, are the same. The inclusion of a linear time trend in equation (1) above requires some justification. Economic theory suggests that there might be a linear trend in the PPP relation. In other words, the equilibrium value of the real exchange rate may vary over time. The argument (Balassa (1964), Samuelson (1964)) is as follows: an increase in the factor productivity of tradable goods, as a country grows more rapidly than another, causes a movement of capital and labour from the nontradable goods sector to the tradable goods sector. The reduction in the supply of nontradables leads to an increase in their relative price. Assuming that the prices of tradables are not very sensitive to domestic conditions, the domestic-foreign price level ratio and, therefore, the real exchange rate will increase. This is more likely to happen the greater the weight of nontradables in the price index being used. This argument shows that the real exchange rate of high-growth countries should be appreciating. A finding that the unit root null cannot be rejected should not necessarily imply that the series is nonstationary since as KPSS (1992), among others, have argued, unit root tests cannot discriminate against close alternatives. Therefore, KPSS (1992) and Fisher and Park (1991) have 3 Fuller (p. 374, (1976)) mentions that the asymptotic distribution of this statistic is the same as that of the unadjusted T(pT - 1) statistic. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.28.2.201 | Generated on 2023-01-16 13:05:58 Purchasing Power Parity and the Irish Experience 205 suggested the use of a test for stationarity where the null hypothesis is that the series is stationary. The KPSS test is described below. Suppose the series yt consists of a deterministic trend, a random walk, and a stationary error: (2) yt = ft + rt + et where rt is a random walk, i.e., (3) rt = rt-1 + ut where ut is iid (0, The initial value of rt, r0, is assumed fixed and, therefore, represents the constant of the equation. The null hypothesis of stationarity is that = 0. Then, y is regressed on a constant and a time trend and the residuals are denoted by et. The partial sum process of the residuals is t (4) ST = t=l, 2,..., T ¿ = i To derive the asymptotic distribution of the test statistic, assuming that the errors et are serially correlated, KPSS (1992) define first the "longrun variance": (5) = Mm T-1 E(S2 T) Then, assuming that f ^0, i.e., that the null hypothesis is trend stationarity, KPSS derive the asymptotic test statistic fjT: (6) rK = T-2X>?/s2(Z) t = l where s2(l) is an estimator of the "long-run variance" (given by equation (10) in KPSS (1992)), and Z, the lag truncation parameter, is the number of lags used to calculate the variance s2(l). Asymptotic critical values for fjT are provided in Table 1 in KPSS (1992). 3. Johansen's Cointegration Analysis An alternative way to test for the PPP theory is to determine whether there is a long-run relationship among the nominal exchange rate and the price levels in the two countries. In other words, if the three series OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.28.2.201 | Generated on 2023-01-16 13:05:58 206 Stilianos Fountas and Jyh-lin Wu are integrated of order 1, a cointegration test can be run to examine the possibility of a long-run equilibrium relation among these series. Engle and Granger (1987) were the first to introduce a procedure for a cointegration test. However, their approach does not allow for the determination of the number of cointegrating vectors (CIV) and for the testing of certain hypotheses on the cointegration parameters. Johansen and Juselius (1990) have provided maximum likelihood tests for the number of CIV in two cases: first, a model that allows for deterministic trends in the integrated variables, and second, a model that does not allow for deterministic trends. We choose the first type of models. Johansen's cointegration approach applies the reduced rank regression. Assume a n-dimensional vector Xt — (a?it, ..., xnt)'. Regress AXt and Xt-k-i on a constant and the lagged differences of AXt (up to k lags) and derive the residuals uu and u2t respectively. Denote the product moment matrices of the residuals as T Sii = T~l Z^itttjt» t = 1 where i, j = 1,2, and T is the sample size. Then, the equation (7) \XS22-S2iS^S12\ = 0 is solved for the eigenvalues A. Johansen and Juselius (1990) specify two likelihood ratio (LR) test statistics to test for the number of cointegrating vectors. First, the likelihood ratio test statistic for the hypothesis of at most r cointegrating vectors against a general alternative, also called trace statistic, is: n -21nQr = ~T ln(l-A0 i = r+ 1 where A* are the n-r smallest estimated eigenvalues derived from equation (7). The second LR statistic for the null of exactly r cointegrating vectors against the alternative of r + 1 vectors is the maximum eigenvalue statistic: -21nQr,r+1 = -Tln(l-Ar+i) Critical values for the above test statistics are tabulated in Johansen and Juselius (1990, p. 208 - 209). The second test is more powerful since the alternative hypothesis is an equality. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.28.2.201 | Generated on 2023-01-16 13:05:58 Purchasing Power Parity and the Irish Experience 207 Once one finds that two or more variables are cointegrated, restrictions on the estimated cointegrating vector parameters can be tested using a likelihood ratio test also suggested by Johansen and Juselius (1990). The test statistic for the hypothesis of n - s restrictions on all CIV is: -2 In Qn_s = T ln{ (1 - Ai) / (1 - A4) } i = 1 where s is the number of independent cointegrating parameters, r is the number of CIV established through the use of the trace and maximum eigenvalue statistics, and Aj and A $ are the estimated eigenvalues from the restricted and unrestricted models respectively. Under the null, this statistic follows ax2 distribution with r(n - s) degrees of freedom. III. Unit Root Tests for the Irish Real Exchange Rate 1. Data We use monthly data for the 1981.1 - 1992.4 period. Due to unavailability of monthly series of CPI for Ireland, we use the PPI (or WPI) series of the three countries in all regressions. The exchange rates are end-ofperiod rates in the Dublin market obtained from the Quarterly Bulletin of the Central Bank of Ireland. Price indexes come from the OECD Main Economic Indicators. 2. ADF and KPSS Unit Root Tests of the Real Exchange Rate In order to test for a unit root in the real exchange rate according to the ADF test, the regression given by equation (1) is run for p = 4 and the Student's t value on the estimate of p is determined. The values of this statistic, rr, are given in Table 1. Alternatively, one may use the normalized bias statistic cT(pT — 1). According to table 1, based on the rT and normalized bias statistics, the unit root null on the real exchange rate can be rejected at 5% significance level only for the UK. The values of the KPSS statistic, 77r, in a model for the real exchange rate with a constant and a deterministic trend included are given in Table 1. We report results for three different values of I, the lag truncation parameter that is used to estimate the long-run variance and, hence, derive the asymptotic critical values. KPSS (1992) argue that the best choice is I = 8, since this achieves an optimal trade off between large size 14 Kredit und Kapital 2/1995 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.28.2.201 | Generated on 2023-01-16 13:05:58 208 Stilianos Fountas and Jyh-lin Wu Table 1 Unit root tests on the real exchange rate Monthly data 1981.1 - 1992.4 Country KPSS Tt CT (pT - 1) (1 = 4) (i = 8) (Z = 12) UK - 3.57* - 22.95* 0.380* 0.259* 0.215* Germany -2.44 - 14.74 0.133 0.085 0.069 Notes: rT refers to tests of H0: pT = 1 in the regression Rt = /3o + pTRt-I + 71 + = l Pi&Rt-i + stThe critical value at 5% is - 3.45. (Table 8.5.2 in Fuller, 1976). cT(pT-l) is the normalized bias test for the same null hypothesis, c equals 1/(1-/3i-/?2-/33-/?4). The 5% critical value is approximately -20.9. KPSS (1992) assumes that the null is stationarity in the real exchange rate. The critical value for KPSS at the 5% level is 0.146. I = 4, 8, or 12 specifies the number of lags in the long-run variance. * indicates rejection of null at 5%. distortions and low power. Based on the results of Table 1, the null of stationarity cannot be rejected for Germany at the 5 % significance level. The results of Table 1 imply that, based on the combination of ADFKPSS we cannot conclude on the existence of a unit root in the real exchange rate for the UK and Germany. This provides the motivation for the use of Johansen's cointegration tests as an alternative approach to test for PPP which does not impose any a priori restrictions on the cointegrating vector. IV. Johansen's Cointegration Tests First, the individual series, i.e., the nominal exchange rate and the domestic and foreign price levels, are subjected to the ADF and KPSS tests in order to determine their integration properties. The results are given in Tables 2 and 3. Table 2 lists the results for the price level. Based on the combination of KPSS and normalized bias statistic (or rT), we conclude that the price levels have a unit root in all countries since we reject the stationarity null hypothesis and cannot reject the unit root null. Table 3 includes the results for the nominal exchange rates. According to both Tt and normalized bias we conclude that all nominal exchange rates have a unit root.4 KPSS implies that the null of stationarity is 4 The only exception being the UK under the normalized bias statistic. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.28.2.201 | Generated on 2023-01-16 13:05:58 Purchasing Power Parity and the Irish Experience 215 Zusammenfassung Kaufkraftparität und die irische Erfahrung: Einheitswurzelund Kointegrationstests bei zwei Industrieländern In diesem Beitrag werden Einheitswurzel-/stationäre Prozeßund Kointegrationstests für die Prüfung der Kaufkraftparität Irlands im Vergleich mit zwei Industrieländern verwendet. Auf der Grundlage von monatlich erhobenen Daten für den Zeitraum seit 1981 zeigen wir, daß die Kaufkraftparitätsbeziehungen Bestand haben, sofern die Ablehnung der Hypothesen von Symmetrie und Verhältnismäßigkeit auf Meßfehler bei den Preisindizes zurückgeführt werden kann. Insbesondere implizieren die Kaufkraftparität im Vergleich mit Deutschland und die dadurch bewirkte Bindung der irischen Inflationsan die deutsche Inflationsrate, daß die Mitgliedschaft Irlands beim EWS gerechtfertigt ist. Es wird bewiesen, daß in Übereinstimmung mit der Kaufkraftparitätstheorie einer kleinen offenen Volkswirtschaft die Anpassung an die Kaufkraftparitätsbeziehungen in erster Linie durch Veränderungen im Inlandspreisniveau erfolgt. Résumé La parité du pouvoir d'achat et l'expérience irlandaise: unit roots et tests de cointégration pour deux pays industrialisés Ce travail utilise l'unit root et des tests de cointégration pour examiner la parité du pouvoir d'achat entre l'Irlande et deux pays industrialisés. Sur base de données mensuelles de la période postérieure à 1981, il est montré que, en supposant le rejet des hypothèses de symétrie et de proportionalité, les rapports de la parité du pouvoir d'achat peuvent être attribués à des erreurs de mesure dans les indices de prix. En particulier, la parité du pouvoir d'achat face à l'Allemagne et le lien entre les taux d'inflation irlandais et allemand implique que l'adhésion de l'Irlande au SME est justifiée. Il est aussi montré que, suivant la théorie de la parité du pouvoir d'achat en petite économie ouverte, ce sont principalement les changements dans le niveau des prix nationaux qui permettent à la parité du pouvoir d'achat de se réajuster. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.28.2.201 | Generated on 2023-01-16 13:05:58