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The long-run behaviour of the real exchange rate: evidence from colonial Pennsylvania
Economics Letters 74 (2001) 25–30 www.elsevier.com / locate / econbase
The long-run behaviour of the real exchange rate: evidence from colonial Pennsylvania Taufiq Choudhry a , *, Kul B. Luintel b a
b
School of Management, University of Southampton, Southampton SO17 1 BJ, UK Department of Economics & Finance, Brunel University, Uxbridge UB8 3 PH, UK Received 28 January 2001; accepted 6 June 2001
Abstract This paper empirically investigates the time series properties of real exchange rate between Great Britain and colonial Pennsylvania during 1727–1775. Results provide ample evidence of mean-reversion in the real exchange rate. 2001 Elsevier Science B.V. All rights reserved. Keywords: Exchange rate; Fractionally-integrated; ARFIMA; Mean-reversion; Variance ratio JEL classification: F40; F41
1. Introduction This paper empirically investigates the time series properties of real exchange rate between Great Britain and colonial Pennsylvania during 1727–1775. Between 1727 and 1775, colonial Pennsylvania was heavily involved in trade with Britain and other regions of the world (Bezanson et al., 1935). Trade mostly involved agricultural and durable goods. The nominal exchange rate between the Pennsylvania pound and the British pound sterling varied considerably during the stated period due to changes in the demand for and supply of Pennsylvania currency and the political situations in America and Europe. Three different real exchange rate series are used based on three different price indices: a general price, wheat price, and bread price. Wheat was traded between Pennsylvania and Britain while bread was not (Bezanson et al., 1935). This paper is different from other PPP studies appealing in the following aspect; (i) no other study has gone that far back in history to investigate the real exchange
* Corresponding author. Tel.: 144-2380-593-887; fax: 144-2380-593-844. E-mail addresses: [email protected] (T. Choudhry), [email protected] (K.B. Luintel). 0165-1765 / 01 / $ – see front matter PII: S0165-1765( 01 )00532-8
2001 Elsevier Science B.V. All rights reserved.
T. Choudhry, K.B. Luintel / Economics Letters 74 (2001) 25 – 30
26
rate,1 (ii) our data consist of a pure floating rate period which does not mix different exchange rate regimes and (iii) the paper investigates both a tradeable good (wheat) and a non-tradeable good (bread). The real exchange rate (e t ) in this paper is defined as the nominal exchange rate deflated by a ratio of foreign and domestic price levels. In logarithmic form e t 5 s t 1 p t* 2 pt
(1)
where e t is the log of real exchange rate, s t is the log of the nominal spot exchange rate defined in local currency units per foreign currency unit, and pt and p *t are the log of domestic and foreign price level, respectively.2 We treat Great Britain as the foreign country and Pennsylvania as the domestic country. Since the real exchange rate can also be viewed as the deviation from purchasing power parity (PPP), a nonstationary real exchange rate implies that PPP does not hold. The empirical investigation is conducted by means of two different tests, a fractional-integrated test (the GPH test) and the variance ratio test.
2. Methodology
2.1. Variance ratio test By determining the long-run response of a time series to a shock, one can obtain an estimate of the degree of persistence. The variance ratio test of Cochrane (1988) provides a method of measuring the degree of persistence in time series. If the natural logarithm of a time series Yt is a pure random then the variance of its k-difference grows linearly with the difference k. Thus, in the case of a random walk, the variance of the k-difference is k times the variance of the first difference: var(Yt 2 Yt2k ) 5 k 3 var(Yt 2 Yt 21 ).
(2)
Hence using Eq. (2) the variance ratio (VR) test is defined as VR 5 (1 /k) 3 var(Yt 2 Yt 2k ) /(Yt 2 Yt21 ).
(3)
A VR of about unity or higher indicates the presence of a stochastic trend. If the series is trend stationary (mean-reverting), VR approaches zero as k approaches infinity. According to Campbell and Mankiw (1987) there is a downward bias in VR; this implies that the measurement of persistence should be corrected by multiplying by (T /T 2 k), where T is the number of observations. Lo and MacKinlay (1988) provide a homoskedasticity and heteroskedasticity consistent test (Z-statistics) to
1 Lothian and Taylor (1996) study the long-run behavior of real exchange rate between the United States, the United Kingdom and France from 1791 to 1990. 2 The nominal exchange rate, and the price indices of Pennsylvania are taken from Historical Statistics of the United States Colonial Times to 1957 and Bezanson et al. (1935), while the British price indices are obtained from Mitchell and Dean (1962).
T. Choudhry, K.B. Luintel / Economics Letters 74 (2001) 25 – 30
27
test the null hypothesis of a random walk in the variance ratio test. The Z-statistics are also applied in this paper.
2.2. Fractional-integration test A flexible and parsimonious way to model short term and long term behaviour of time series is by means of an autoregressive fractionally integrated moving average (ARFIMA) model. An autoregressive fractionally integrated moving average (ARFIMA) process can be postulated as:
F (L)(1 2 L)d Yt 5 Q (L) ´t , ´t | (0, s 2 )
(4)
where d is the long memory parameter (difference operator), and F (L) and Q (L) are the usual autoregressive and moving average polynomials in the lag operator with roots outside the unit circle. An integer value of d 5 0 gives rise to the standard ARMA processes whereas d 5 1 implies the unit-root nonstationary process. For any process Yt | (d) and for d , 1 implies that the process is mean-reverting; however, covariance of the process would be infinite unless d , 0.5. Geweke and Porter-Hudak (1983, henceforth GPH) suggest the semi-parametric estimator of d based on the following OLS-based estimating equation: ln (I(w j )) 5 b0 1 b1 ln (4 sin 2 (w j / 2)) 1 dt ; j 5 1, . . . , n
(5)
with b1 5 2 d, where I(w j ) is the periodogram of a series at frequency w j , and w j 5 2pj /T ( j 5 1, . . . , T 2 1). The number of low frequency ordinates (n) used in this test is n 5 T m , where T is the number of observations. The value of d can be used to test the null hypothesis of a unit root. To check the sensitivity of results to the choice of the sample size of the spectral regression the GPH test is conducted with two different values of m, 0.55 and 0.60.
3. Empirical results Table 1 presents the results from the variance ratio test and the Z-statistics. For all series the maximum length of the window size (k) applied is 10. For all three series the value of VR stays one when k ranges from 1 to 3. Increasing the length of k beyond 3 reduces the size of the VR below unity. The Z-statistics indicate that VR is significantly less than unity when k is greater than 3. When k increases the VR falls, which indicates all three exchange rates are trend stationary. Table 2 shows results from the GPH test. The estimates of d employing both values of m (i.e. 0.55 and 0.60) along with their respective t-ratio based on the asymptotic standard error are shown. All point estimates of d are below unity which indicates real exchange rates to be fractionally integrated and mean-reverting. However, using the two-sided test the null d 5 1 (i.e. unit root) against the alternative d ± 1 is rejected at the 5% level or above in all cases except the real exchange rate based on the general price index. The estimated value of d for wheat and bread falls between 0 and 20.5 which suggests that they are not only mean reverting but also covariance stationary. Thus, the GPH test indicates that the bread- and wheat-based real exchange rates are fractionally integrated with intermediate memory but the real exchange rate based on the general price index contains a unit root. The two tests provide conflicting results only in the case of general price index real exchange rate.
T. Choudhry, K.B. Luintel / Economics Letters 74 (2001) 25 – 30
28 Table 1 Variance ratio test k
General price
Wheat price
Bread price
1 2
1.021 1.013 (0.125) 0.820 (21.434) 0.706** (22.084) 0.705* (21.974) 0.390*** (23.522)
1.021 1.162 (1.554) 0.911 (20.709) 0.616** (22.724) 0.395*** (24.046) 0.279*** (24.163)
1.021 1.0793 (0.760) 0.802 (21.576) 0.652** (22.464) 0.524*** (23.183) 0.294*** (24.077)
3 4 5 10
Z-statistics in parentheses. ***, ** and * significantly different from one at 1, 5 and 10% level, respectively.
Though the potential power advantage of the GPH test over the standard unit root tests is quite relevant, it needs further investigation in view of our sample size and its low frequency. To clarify this issue further we report the correlogram of each of these exchange rates. Fig. 1 presents the sample autocorrelation function of the three exchange rate series along with 5% level confidence bands. In all three cases, the autocorrelation becomes insignificant quite early and stays insignificant. A visual inspection of the autocorrelations indicates all real exchange rates are stationary.
4. Conclusion and implications This paper investigates the time series properties of the real exchange rates between British sterling and the Pennsylvania currency during the American colonial period, 1727–1775. The empirical investigation is conducted using a fractional-integration test (the GPH test) and the variance ratio test. Three different real exchange rates based on three different price indices are applied; a general price index, wheat price index and bread price index. Wheat was heavily traded between colonial Pennsylvania and Britain, while bread was not. Our results indicate a few important points. Firstly, we confirm mean-reverting real exchange rates Table 2 GPH test
m Real exchange rate
0.55 0.60 The value of d and t-statistics
General price
0.654 (20.983) 20.082*** (23.077) 20.143*** (23.250)
Wheat price Bread price
0.713 (20.935) 0.021*** (23.261) 0.253** (22.485)
*** and ** imply rejection of the null hypothesis (d 5 1) of the two-sided test at 1 and 5% level, respectively.
T. Choudhry, K.B. Luintel / Economics Letters 74 (2001) 25 – 30
29
Fig. 1. Autocorrelation function of real exchange rates.
during a very early floating period. Given that most studies find non-stationary real exchange rate during the recent float, our results may imply that mean-reversion in real exchange rate may be time period specific. Secondly, we find mean-reversion using both a traded (wheat) and non-traded (bread) goods price. Given the debate regarding the use of the proper price index in real exchange rate, we find real exchange rates to be stationary using both traded and non-traded goods prices.
Acknowledgements We thank an anonymous referee for helpful comments. The remaining errors and omissions are our responsibility alone.
References Bezanson, A., Gray, R., Hussey, M., 1935. Prices in Colonial Pennsylvania. University of Pennsylvania Press, Philadelphia. Campbell, J., Mankiw, N.G., 1987. Permanent and transitory components in macroeconomics fluctuations. American Economic Review Papers and Proceedings 77, 111–117. Cochrane, J., 1988. How big is the random walk in GNP? Journal of Political Economy 96, 893–920.
30
T. Choudhry, K.B. Luintel / Economics Letters 74 (2001) 25 – 30
Geweke, J., Porter-Hudak, S., 1983. The estimation and application of long memory time series models. Journal of Time Series Analysis 4, 221–238. Lo, A., MacKinlay, A.C., 1988. Stock market prices do not follow a random walk: evidence from a new specification test. Review of Financial Studies 1, 41–67. Lothian, J., Taylor, M., 1996. Real exchange rate behaviour: the recent float from the perspective of the past two centuries. Journal of Political Economy 104, 488–509. Mitchell, B., Dean, P., 1962. Abstract Of British Historical Statistics. Cambridge University Press, London.
The long-run behaviour of the real exchange rate: evidence from colonial Pennsylvania Taufiq Choudhry a , *, Kul B. Luintel b a
b
School of Management, University of Southampton, Southampton SO17 1 BJ, UK Department of Economics & Finance, Brunel University, Uxbridge UB8 3 PH, UK Received 28 January 2001; accepted 6 June 2001
Abstract This paper empirically investigates the time series properties of real exchange rate between Great Britain and colonial Pennsylvania during 1727–1775. Results provide ample evidence of mean-reversion in the real exchange rate. 2001 Elsevier Science B.V. All rights reserved. Keywords: Exchange rate; Fractionally-integrated; ARFIMA; Mean-reversion; Variance ratio JEL classification: F40; F41
1. Introduction This paper empirically investigates the time series properties of real exchange rate between Great Britain and colonial Pennsylvania during 1727–1775. Between 1727 and 1775, colonial Pennsylvania was heavily involved in trade with Britain and other regions of the world (Bezanson et al., 1935). Trade mostly involved agricultural and durable goods. The nominal exchange rate between the Pennsylvania pound and the British pound sterling varied considerably during the stated period due to changes in the demand for and supply of Pennsylvania currency and the political situations in America and Europe. Three different real exchange rate series are used based on three different price indices: a general price, wheat price, and bread price. Wheat was traded between Pennsylvania and Britain while bread was not (Bezanson et al., 1935). This paper is different from other PPP studies appealing in the following aspect; (i) no other study has gone that far back in history to investigate the real exchange
* Corresponding author. Tel.: 144-2380-593-887; fax: 144-2380-593-844. E-mail addresses: [email protected] (T. Choudhry), [email protected] (K.B. Luintel). 0165-1765 / 01 / $ – see front matter PII: S0165-1765( 01 )00532-8
2001 Elsevier Science B.V. All rights reserved.
T. Choudhry, K.B. Luintel / Economics Letters 74 (2001) 25 – 30
26
rate,1 (ii) our data consist of a pure floating rate period which does not mix different exchange rate regimes and (iii) the paper investigates both a tradeable good (wheat) and a non-tradeable good (bread). The real exchange rate (e t ) in this paper is defined as the nominal exchange rate deflated by a ratio of foreign and domestic price levels. In logarithmic form e t 5 s t 1 p t* 2 pt
(1)
where e t is the log of real exchange rate, s t is the log of the nominal spot exchange rate defined in local currency units per foreign currency unit, and pt and p *t are the log of domestic and foreign price level, respectively.2 We treat Great Britain as the foreign country and Pennsylvania as the domestic country. Since the real exchange rate can also be viewed as the deviation from purchasing power parity (PPP), a nonstationary real exchange rate implies that PPP does not hold. The empirical investigation is conducted by means of two different tests, a fractional-integrated test (the GPH test) and the variance ratio test.
2. Methodology
2.1. Variance ratio test By determining the long-run response of a time series to a shock, one can obtain an estimate of the degree of persistence. The variance ratio test of Cochrane (1988) provides a method of measuring the degree of persistence in time series. If the natural logarithm of a time series Yt is a pure random then the variance of its k-difference grows linearly with the difference k. Thus, in the case of a random walk, the variance of the k-difference is k times the variance of the first difference: var(Yt 2 Yt2k ) 5 k 3 var(Yt 2 Yt 21 ).
(2)
Hence using Eq. (2) the variance ratio (VR) test is defined as VR 5 (1 /k) 3 var(Yt 2 Yt 2k ) /(Yt 2 Yt21 ).
(3)
A VR of about unity or higher indicates the presence of a stochastic trend. If the series is trend stationary (mean-reverting), VR approaches zero as k approaches infinity. According to Campbell and Mankiw (1987) there is a downward bias in VR; this implies that the measurement of persistence should be corrected by multiplying by (T /T 2 k), where T is the number of observations. Lo and MacKinlay (1988) provide a homoskedasticity and heteroskedasticity consistent test (Z-statistics) to
1 Lothian and Taylor (1996) study the long-run behavior of real exchange rate between the United States, the United Kingdom and France from 1791 to 1990. 2 The nominal exchange rate, and the price indices of Pennsylvania are taken from Historical Statistics of the United States Colonial Times to 1957 and Bezanson et al. (1935), while the British price indices are obtained from Mitchell and Dean (1962).
T. Choudhry, K.B. Luintel / Economics Letters 74 (2001) 25 – 30
27
test the null hypothesis of a random walk in the variance ratio test. The Z-statistics are also applied in this paper.
2.2. Fractional-integration test A flexible and parsimonious way to model short term and long term behaviour of time series is by means of an autoregressive fractionally integrated moving average (ARFIMA) model. An autoregressive fractionally integrated moving average (ARFIMA) process can be postulated as:
F (L)(1 2 L)d Yt 5 Q (L) ´t , ´t | (0, s 2 )
(4)
where d is the long memory parameter (difference operator), and F (L) and Q (L) are the usual autoregressive and moving average polynomials in the lag operator with roots outside the unit circle. An integer value of d 5 0 gives rise to the standard ARMA processes whereas d 5 1 implies the unit-root nonstationary process. For any process Yt | (d) and for d , 1 implies that the process is mean-reverting; however, covariance of the process would be infinite unless d , 0.5. Geweke and Porter-Hudak (1983, henceforth GPH) suggest the semi-parametric estimator of d based on the following OLS-based estimating equation: ln (I(w j )) 5 b0 1 b1 ln (4 sin 2 (w j / 2)) 1 dt ; j 5 1, . . . , n
(5)
with b1 5 2 d, where I(w j ) is the periodogram of a series at frequency w j , and w j 5 2pj /T ( j 5 1, . . . , T 2 1). The number of low frequency ordinates (n) used in this test is n 5 T m , where T is the number of observations. The value of d can be used to test the null hypothesis of a unit root. To check the sensitivity of results to the choice of the sample size of the spectral regression the GPH test is conducted with two different values of m, 0.55 and 0.60.
3. Empirical results Table 1 presents the results from the variance ratio test and the Z-statistics. For all series the maximum length of the window size (k) applied is 10. For all three series the value of VR stays one when k ranges from 1 to 3. Increasing the length of k beyond 3 reduces the size of the VR below unity. The Z-statistics indicate that VR is significantly less than unity when k is greater than 3. When k increases the VR falls, which indicates all three exchange rates are trend stationary. Table 2 shows results from the GPH test. The estimates of d employing both values of m (i.e. 0.55 and 0.60) along with their respective t-ratio based on the asymptotic standard error are shown. All point estimates of d are below unity which indicates real exchange rates to be fractionally integrated and mean-reverting. However, using the two-sided test the null d 5 1 (i.e. unit root) against the alternative d ± 1 is rejected at the 5% level or above in all cases except the real exchange rate based on the general price index. The estimated value of d for wheat and bread falls between 0 and 20.5 which suggests that they are not only mean reverting but also covariance stationary. Thus, the GPH test indicates that the bread- and wheat-based real exchange rates are fractionally integrated with intermediate memory but the real exchange rate based on the general price index contains a unit root. The two tests provide conflicting results only in the case of general price index real exchange rate.
T. Choudhry, K.B. Luintel / Economics Letters 74 (2001) 25 – 30
28 Table 1 Variance ratio test k
General price
Wheat price
Bread price
1 2
1.021 1.013 (0.125) 0.820 (21.434) 0.706** (22.084) 0.705* (21.974) 0.390*** (23.522)
1.021 1.162 (1.554) 0.911 (20.709) 0.616** (22.724) 0.395*** (24.046) 0.279*** (24.163)
1.021 1.0793 (0.760) 0.802 (21.576) 0.652** (22.464) 0.524*** (23.183) 0.294*** (24.077)
3 4 5 10
Z-statistics in parentheses. ***, ** and * significantly different from one at 1, 5 and 10% level, respectively.
Though the potential power advantage of the GPH test over the standard unit root tests is quite relevant, it needs further investigation in view of our sample size and its low frequency. To clarify this issue further we report the correlogram of each of these exchange rates. Fig. 1 presents the sample autocorrelation function of the three exchange rate series along with 5% level confidence bands. In all three cases, the autocorrelation becomes insignificant quite early and stays insignificant. A visual inspection of the autocorrelations indicates all real exchange rates are stationary.
4. Conclusion and implications This paper investigates the time series properties of the real exchange rates between British sterling and the Pennsylvania currency during the American colonial period, 1727–1775. The empirical investigation is conducted using a fractional-integration test (the GPH test) and the variance ratio test. Three different real exchange rates based on three different price indices are applied; a general price index, wheat price index and bread price index. Wheat was heavily traded between colonial Pennsylvania and Britain, while bread was not. Our results indicate a few important points. Firstly, we confirm mean-reverting real exchange rates Table 2 GPH test
m Real exchange rate
0.55 0.60 The value of d and t-statistics
General price
0.654 (20.983) 20.082*** (23.077) 20.143*** (23.250)
Wheat price Bread price
0.713 (20.935) 0.021*** (23.261) 0.253** (22.485)
*** and ** imply rejection of the null hypothesis (d 5 1) of the two-sided test at 1 and 5% level, respectively.
T. Choudhry, K.B. Luintel / Economics Letters 74 (2001) 25 – 30
29
Fig. 1. Autocorrelation function of real exchange rates.
during a very early floating period. Given that most studies find non-stationary real exchange rate during the recent float, our results may imply that mean-reversion in real exchange rate may be time period specific. Secondly, we find mean-reversion using both a traded (wheat) and non-traded (bread) goods price. Given the debate regarding the use of the proper price index in real exchange rate, we find real exchange rates to be stationary using both traded and non-traded goods prices.
Acknowledgements We thank an anonymous referee for helpful comments. The remaining errors and omissions are our responsibility alone.
References Bezanson, A., Gray, R., Hussey, M., 1935. Prices in Colonial Pennsylvania. University of Pennsylvania Press, Philadelphia. Campbell, J., Mankiw, N.G., 1987. Permanent and transitory components in macroeconomics fluctuations. American Economic Review Papers and Proceedings 77, 111–117. Cochrane, J., 1988. How big is the random walk in GNP? Journal of Political Economy 96, 893–920.
30
T. Choudhry, K.B. Luintel / Economics Letters 74 (2001) 25 – 30
Geweke, J., Porter-Hudak, S., 1983. The estimation and application of long memory time series models. Journal of Time Series Analysis 4, 221–238. Lo, A., MacKinlay, A.C., 1988. Stock market prices do not follow a random walk: evidence from a new specification test. Review of Financial Studies 1, 41–67. Lothian, J., Taylor, M., 1996. Real exchange rate behaviour: the recent float from the perspective of the past two centuries. Journal of Political Economy 104, 488–509. Mitchell, B., Dean, P., 1962. Abstract Of British Historical Statistics. Cambridge University Press, London.