STAT421 (Final Report for term project)
Ryu, Deockhyun
Nov. 27, 2001
Does a real exchange rate have a mean reverting property?
1. Introduction
In international economics, the Purchasing Power Parity (hereafter PPP) is a well-known theory. It states that the real exchange rate (nominal exchange rate adjusted for differences in national price level) tends toward PPP in the very long run. Let be the logarithm of exchange rate of a country and be logarithms of price level of base country and the country, respectively. Then the real exchange rate is
Whether PPP holds in the long run or not is equivalent to whether real exchange rate is stationary or not. More specifically we equate this hypothesis to whether or not the real exchange rate is a random walk. Thus in the literature of economics, the test of PPP is same as the test of stationarity for real exchange rate.
Consider the following simple AR(1) model.
where is white noise with mean 0 and finite variance.
If we reject the null hypothesis of , then we can say that is stationary and has a property of mean reversion. Thus, we can say the PPP theory holds in the long run.
In this paper, we want to examine this property with the real exchange rate of British Pound via US dollar.
2. Description about data
As defined in the above, we need three time series, a country’s nominal exchange rate, price level (Consumer Price Index, CPI), and CPI for base country (here, USA). The data were taken from International Financial Statistics (IFS) CD-ROM from International Monetary Fund (IMF). The period we consider is 1973:1-1998:12. Thus the length is 312. After the Bretton Woods system collapsed in 1973, the exchange rate regime shifted from fixed rate system to floating rate system. Thus, it is interesting to test PPP theory over those periods. We consider the monthly data and annual data. For annual data, we take this from Lothian and Taylor (1996). The annual data is running from 1791 to 1990.
3. Preliminary analyses
We begin by examining a few basic plots for each of the data sets; time series plot, estimated ACF, PACF, and the histogram of the time series.
First, below are the basic plots of monthly exchange rate. The exponential decay of ACF and a basically zero PACF after the first lag seems to indicate AR(1) process. But, from the time series plot, we guess this is not a stationary because the mean is not same over the sub period. Also the PACF at lag 1 is almost 1 as well as the ACF. Thus, we need to stationary version of this data and from figure 2 we know that first difference is stationary.
Below (figure 3, 4 ) are basic plots of annual data. It shows similar pattern with monthly data. It seems non-stationary. Thus we need also first differencing.
Figure 1. The basic plot of monthly data
From the figure 2, we know that first differencing make the series as stationary. In glance, the differenced series seems MA(2) from ACF or AR(16) from PACF.
Figure 2. The basic plot of first difference of monthly data
Figure 3. The basic plot of annual data
Figure 4. The basic plot of first difference of annual data
From the preliminary analyses of data, it seems that the PPP does not hold for the real exchange rate of British pound in monthly and annual data set.
The next step is to confirm this result by statistical method. We will estimate the coefficients of AR(1) model for those two series. Thus we can tell more correctly whether the PPP theory holds or not.
4. Estimation and Unit root test
4.1 Estimation
We estimate the AR(1) coefficient to evaluate the PPP hypothesis. We consider AR(1) model with and without constant term. Consider the following models;
Model 1: rerm(t)=a+b*rerm(t-1)+w(t)
Model 2: rerm(t)= bb*rerm(t-1)+v(t)
Model 3: rery(t)=c+d*rery(t-1)+e(t)
Model 4: rery(t)= dd*rery(t-1)+d(t)
Where rerm and rery denote monthly and yearly real exchange rates, respectively. And w(t), v(t), e(t) and d(t) are white noise.
The table1 shows the results of estimation. We estimate these models by Gauss. For the more complete regression results, please see the appendix 2. From the table 1, it seems that every AR(1) coefficients are close to 1, i.e., the real exchange rate seems to follow random walk irrespective of its frequency. But we should be cautious of this. As stated in next section, the judgement will be deferred after the unit root test.
< Table 1. Regression results for real exchange rate >
Variable / Coefficient / Standard error / t-RatioModel 1 / Constant
AR(1)variable / -0.00679512
0.985912 / 0.00444419
0.00573372 / -1.52899
171.950
Model 2 / -
AR(1)variable / 0.993931 / 0.00232135 / 428.170
Model 3 / Constant
AR(1)variable / -0.161493
0.897915 / 0.0508652
0.0319532 / -3.17492
28.1009
Model 4 / -
AR(1)variable / 0.998878 / 0.00319828 / 312.317
4.2 Unit root test
It is well known that the use of data characterized by unit roots has the potential to lead to serious errors in inferences. Thus, instead of using conventional t-statistics, Dickey-Fuller propose an appropriate set of critical values for testing the hypothesis that AR(1) coefficient equals 1 in AR(1) regression when there truly is a unit root by Monte Carlo methods. The hypothesis may be carried out with a conventional t test, but with a revised set of critical values. A few of the values from the Dickey-Fuller tables are reproduced in Table 2.
For example, consider the model 1
rerm(t)= -0.00679512+0.985912*rerm(t-1)
(0.00444) (0.00573372)
Standard errors are in parenthesis. We have a null hypothesis that the AR(1) coefficient equals 1, i.e. H: b=1. By simple calculation, we get the conventional t
t=(0.985912-1)/0.00573372=-2.00
Based on the conventional critical point of –1.96, we would reject the hypothesis of a unit root. But the value from table 2 for 311 observations would be roughly –3.12 (for 5% size test). So the hypothesis of a unit root is decidedly not rejected. That means the real exchange rate shows random walk with drift and the PPP hypothesis does not hold. The annual data is same. Consider model 4. By doing same t test, we can get –0.35 of t-statistics (=(0.998878-1)/0.003198). It clearly shows that the annual model also cannot reject the null of random walk.
< Table 2. Critical Values for the Dickey-Fuller Test >
Sample Size
25 / 50 / 100 /AR model
0.010.025
0.0975
0.99 / -2.66
-2.26
1.70
2.16 / -2.62
-2.25
1.66
2.08 / -2.60
-2.24
1.64
2.03 / -2.58
-2.23
1.62
2.00
AR model w/constant
0.01
0.025
0.0975
0.99 / -3.75
-3.33
0.34
0.72 / -3.58
-3.22
0.29
0.66 / -3.51
-3.17
0.26
0.63 / -3.33
-3.12
0.23
0.60
5. Conclusion
We studied the dynamic process of real exchange rate by simple AR(1) process. As expected in preliminary data analyses, we could know that the real exchange rate is not stationary but first order difference stationary. And we confirm the first glance judgement by conducting Dickey –Fuller Unit root test. From the unit root test, we can say the real exchange rate shows random walk not mean-reverting process. This implies the PPP theory does not hold over the British pound via. American dollar.
But we have several drawbacks.
First, the models that we choose are all linear. There are many articles arguing nonlinearity or nonlinear adjustment of real exchange rate. The potential nonlinear models used in recent literature are Threshold Auto-Regression (TAR), Exponetianl Transition Auto-Regression (ESTAR) and Logistic Transition Auto-Regression (LSTAR).
Second, we did not consider the serial correlation in our model. Then, we conduct the different unit root test, for example, Augmented Dickey-Fuller (ADF) test.
Finally, the frequency of data we consider is too low. The problem is that we cannot find the higher frequency price data though we can have very high frequency exchange rate data.
References
Shumway, Robert H. and David S.Stoffer (2000), Time Series Analysis and
its Application. New York:Spring-Verlag
Greene, William (2000), Econometric Analysis, 4th Edition. Prentice Hall
< Appendix 1: S-plus code >
################################################################
# Data retrieving
################################################################
rerm_scan("d:\\dhryu\\stat421\\reruk.txt")
rery_scan("d:\\dhryu\\stat421\\uk_lt.txt")
rerm<-rts(rerm, start=c(1973,1), end=c(1998,12), units="months", frequency=12)
rery<-rts(rery, start=1791, end=1990, units="years", frequency=1)
nm<-length(rerm)
ny<-length(rery)
drerm<-diff(rerm)
drery<-diff(rery)
ndm<-length(drerm)
ndy<-length(drery)
ndm
ndy
################################################################
# Basic time series plot
################################################################
prettyplot<-function(x,y)
{
graphseet()
par(mfrow=c(x,y))
}
prettyplot(2,2)
ts.plot(rerm)
title(" The Monthly Real Exchange Rate \n 1973:1-1998:12")
ts.plot(drerm)
title(" The Plot of first difference: monthly")
ts.plot(rery)
title(" The Yearly Real Exchange Rate \n 1791-1990")
ts.plot(drery)
title(" The Plot of first difference: yearly")
# Some basic plots of the data
prettyplot(2,2)
ts.plot(rerm)
title("The Monthly Real Exchange rate")
hist(rerm)
title("Histogram")
acf(rerm)
acf(rerm, type="partial")
prettyplot(2,2)
ts.plot(rery)
title("The Yearly Real Exchange rate")
hist(rery)
title("Histogram")
acf(rery)
acf(rery, type="partial")
prettyplot(2,2)
ts.plot(drerm)
title("First Difference of monthly data")
hist(drerm)
title("Histogram")
acf(drerm)
acf(drerm, type="partial")
prettyplot(2,2)
ts.plot(drery)
title("First Difference of yearly data")
hist(drery)
title("Histogram")
acf(drery)
acf(drery, type="partial")
################################################################
# Model estimation by linear AR(1) model: gauss code
################################################################
new;
format 8,6;
output file=stat1.out reset;
load data[312,1] = reruk.txt;
n=rows(data);
k=cols(data);
y=data[2:312,1];
y=y[.,1];
xx=data[1:311,1];
x=xx[.,1];
x=ones(n-1,1)~x;
k=cols(x);
@ (a)-1 Basic Regression:calculate OLS estimates & Summary of statistis @
b = inv(x'x)*x'y;
df = n-k;
e = y-x*b;
ssr = e'e;
s2 = (e'e)/df;
ybar = sumc(y)/n;
r2 = 1-(e'e)/((y-ybar)'(y-ybar));
ar2 = 1-((n-1)/df)*(1-r2);
/* Preliminary calculation for LR,LM statistics */
ix = inv(x'x);
iix = x*inv(x'x)*x';
@ (a)-2 Calculate covariance matrix, standard errors, and t-ratios @
c = s2*inv(x'x);
se = sqrt(diag(c));
tr = b./se;
pv = 2*cdftc(abs(tr),df);
/* print result */
"degree of freedom = " df ;
"number of observation = " n;
"standard error of regression = " sqrt(s2);
"Sum of squared residuals = " ssr;
"R-squared = " r2 ;
"Adjusted R-squared = " ar2;
t = "Coeff" ~ "Estimate" ~ "S.E." ~ "t-ratio" ; Print $t;
format 8,6;
d = Seqa(1,1,k)~b~se~tr; Print d;
"";
"Covariance Matrix";
t = "b1" ~"b2" ; print $t;
"";
c[1,1];
c[2,1:2];
end;
< Appendix 2: Regression results>
Model 1:
degree of freedom = 310.000
number of observation = 312.000
standard error of regression = 0.0316625
Sum of squared residuals = 0.310779
R-squared = 0.989625
Adjusted R-squared = 0.989591
Coeff Estimate S.E. t-ratio
1.00000 -0.006795 0.00444419 -1.52899
2.00000 0.985912 0.00573372 171.950
Covariance Matrix
b1 b2
1.97509e-05
2.33098e-05 3.28756e-05
Model 2:
degree of freedom = 311.000
number of observation = 312.000
standard error of regression = 0.0317305
Sum of squared residuals = 0.313122
R-squared = 0.989546
Adjusted R-squared = 0.989546
Coeff Estimate S.E. t-ratio
1.00000 0.993931 0.00232135 428.170
Covariance Matrix
b1
5.38866e-06
Model 3:
degree of freedom = 187.000
number of observation = 189.000
standard error of regression = 0.0681804
Sum of squared residuals = 0.869282
R-squared = 0.809087
Adjusted R-squared = 0.808066
Coeff Estimate S.E. t-ratio
1.00000 -0.161493 0.0508652 -3.17492
2.00000 0.897915 0.0319532 28.1009
Covariance Matrix
b1 b2
0.00258727
0.00161752 0.00102101
Model 4:
degree of freedom = 188.000
number of observation = 189.000
standard error of regression = 0.0698075
Sum of squared residuals = 0.916140
R-squared = 0.798796
Adjusted R-squared = 0.798796
Coeff Estimate S.E. t-ratio
1.00000 0.998878 0.00319828 312.317
Covariance Matrix
b1
1.02290e-05
1