4. Long-Run Analysis (Year 1870~2003)
4.1. The CHEER Model of Exchange Rate Determination
4.1.1. Unit root tests
+1
The CHEER Model of Exchange Rate Determination
Analysis of Chapter 3 is based on the monthly data in the post floating period spanning 31 years (from 1973 to 2004); this chapter employs longer data span, which is annual data spanning from year 1870 to 2003. This data set contains the exchange rate, the CPI index and the long-term interest rates for U.S and Japan33.
4.1.1. Unit root tests
The unit root tests for PPP and UIP are performed in the first step to determine whether they hold. Here we also assume perfect foresight, i.e., in UIP hypothesis. As shown in equation (9) and (15), the unit root tests can be conducted by testing the null hypothesis that
( 1)
t t t
E s+ =s
γ =0. The test statistics are abbreviated as Table 9.
Table 9: Unit root tests
Test DF PP ADF34
γ in PPP equation -3.3485 -3.2633 -1.61463
γ in UIP equation -8.1145 -8.3654 -3.19689
The 5 percent critical value for the null that γ =0 is -2.883 and all the
33 Long-term interest rates are U.S 10 year government bond yield and Japanese 7 year government bond yield. The maturities are different because no other long-run rate is available.
34 The lags in ADF equations are 8.
41
absolute value of the statistics reported in Table 9 exceed this except the ADF statistics for PPP, which is -1.61463. As Campbell and Perron (1991) pointed out, nonrejection of the unit root hypothesis may be due to the existence of structural breaks. From year 1870 to 2003, at least two structural breaks occurred. As Figure 8 shows, one is year 1945, in which the yen depreciated more than 200 percent (from 4.29 to 15). The other notable break is year 1970, in which the yen began to appreciate sharply due to the oil shock35.
Considering the existence of two structural breaks, we should detrend the real exchange rate before performing the unit root tests. First, we regress:
0 1 1 1970 2 1985
qt =a +a t+µ DL +µ DL +∧qt
where and are level dummies representing structural breaks in year 1970 and 1985, respectively;
1970
DL DL1985
µ1 and µ2 are coefficients of DL1970 and DL1985, respectively;
t represents time and a0, a1 are parameteres.
Then, we perform the unit root test on the detrended real exchange rate, qt
∧
The estimated t-statistics for γ =0 is -3.1515, exceeding the 5 percent critical value of -2.883. Therefore, we conclude that the null hypothesis of unit root can be rejected and both PPP and UIP hold in the long run. The graph of nominal and real exchange rate, changes in real exchange rate and interest rate
42 35 Year 1985 may be considered as another break for the advent of the Plaza Agreement. It is not necessary to deal with it here, however, since we have taken year 1970 as a break and we can view year 1985 as a point in the appreciation trend of yen.
differential are depicted in Figure 8,9,10 and 11.
Figure 8
Nominal and Real Exchange Rate (Yen/Dollar)
1870 1881 1892 1903 1914 1925 1936 1947 1958 1969 1980 1991 2002 0
100 200 300 400 500 600 700
NOMINAL REAL
Source: DataStream and the author’s calculation
Figure 9
Nominal and Real Exchange Rates for Yen (in logs)
1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 -1
0 1 2 3 4 5 6 7
nominal rea;
Source: DataStream and the author’s calculation
43
Figure 10
Changes In Real Exchange Rate of Yen
1871 1883 1895 1907 1919 1931 1943 1955 1967 1979 1991 2003 -1.5
-1.0 -0.5 0.0 0.5 1.0 1.5
Source: DataStream and the author’s calculation
Figure 11
Nominal Interest DIfferential(Japan-U.S) in log value
1870 1882 1894 1906 1918 1930 1942 1954 1966 1978 1990 2002 -0.06
-0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10
Source: DataStream
44
4.1.2.
p
The nominal exchange rate determination model
The PPP and UIP can be written in the following equations:
*
Regress equation (25) and (26) and we obtain:
5.748 1.017( *)
The t-statistics are shown in the parenthesis and the null hypothesis of coefficients of (pt−p*t and (it−it*) equal to unity cannot be rejected
t
36.
By substituting (25) into equation (26), we can obtain the following equation:
* *
1 ( ) ( )
t t t t t
s+ = p −p + −i i +ε (27) where ε η ηt = p+ i.
Equation (27) is the determination of nominal exchange rate in the long run.
It can be interpreted that the future nominal exchange rate is determined by the current price and interest rate differentials. In this equation, three variables are concerned,st+1, (pt −pt*) and (it −it*). According to equation (27), we can set 36 The F-statistics of the null that the coefficient equal to 1 is tested to be 1.3576 and 0.09183 in equation (25) and (26) respectively, which is less than the 1 percent critical value 6.63.
* *
The lag length in system (28) is determined using AIC and SBC criteria, whose value is the smallest when 3 lags are used. Therefore, we set .
4.2. Engle-Granger Cointegration Test and ECM
We also adopt Engle-Granger approach to test the cointegration relationship among the three variables concerned in equation (27):s , and (it−it*). Pretests of the order of their integration indicate that they are all I(1) processes.
The next step entails regressing on the following equation to get the long-run equilibrium relationship.
(29) where εt/ is the estimated error. The result of the regression is reported in Table 10.
Table 10: Regression Results in the Long Run
coefficient significance
α 5.7580 0.0433 133.001 0.000
βp 1.0519 0.012 87.8757 0.000
βi 10.8902 1.0525 10.3468 0.000
Just the same as equation (22) in Section 3.3.1, we regress the following equation to know the t-statistics of α1.
46
/ / respectively, and both exceeding the critical values for the null of no cointegration at 5 percent level (-3.17 with lags and –3.37 without lags respectively). Moreover, by changing the regression order, i.e., changing the left side variable in equation (29) do not lead to different results. The t-values of the estimated errors both exceed the critical value in the Engle-Granger no cointegration table. Therefore, we can reject the null of no cointegration.
After the cointegration test, we then estimate the ECM, which should have the form as the following if we substitute εt/−1 with the real error termεt−1,
, and e white-noise disturbances which may be correlated with each other; and
e1t e2t 3t =
α , β are parameters.
System (31) is estimated to be:
/
` The value in the parenthesis below each estimated coefficient is the corresponding t-value. Careful examination of system (32) would reveal that if
/ 1
εt− is positive, i.e., if is larger than the long-run equilibrium level, then the ECM will make it to decrease while
1
st+
(p−p*)t and ( to increase, making it possible to restoring to the equilibrium. Therefore, different from the short run ECM, the long-run ECM is stable.
*
i i− )t
Although the ECM is stable in the long run, the coefficient of εt/−1 in equation (32-1) is not significant, implying the exchange rate does not respond to the long-run deviation. What makes this coefficient be insignificant? We expect to answer this question by performing the Granger causality tests to reveal the intrinsic relationship between the variables in system (28).
4.3. Granger Causality and Johansen Methodology
Granger causality test betweenst+1,(pt− pt*), (it−it*) are helpful to further reveal the roles each variable plays in the long-run nominal exchange rate determination. The joint F-statistics and the significance levels of system (28) are reported in Table 11.
Table 11: Granger Causality Tests for the Long Run
variable st+1 (pt− pt*) (it −it*)
1
st+ 89.91 0.000 12.87 0.000 0.042 0.989 (pt−pt*) 16.91 0.000 154.13 0.000 1.482 0.223 (it−it*) 2.300 0.081 1.696 0.172 59.404 0.000
48
Table 11 indicates that at 5 percent significance level, st+1 and (pt −pt*) both Granger-cause themselves and each other; (it −it*) Granger-causes only itself and is not Granger-caused by any of the other two. These results imply that the interest rate differential is not an endogenous variable in the system of exchange rate determination.
(it−it*)
)
The exogenous property of interest rate differential raises a serious question of system (28), i.e., is it appropriate to include (it −it* in the VAR system?
Because the likely ratio test can determine whether a variable should be included, we perform the likely ratio test of(it−it*). Begin with PPP equation (25), suppose that should not be included, the likely ratio statistics
4 with significance level , less than the 5 percent critical value of
. Therefore, the null hypothesis of exclusion of ( can not be rejected.
0.54812587
18.307 it−it*
The Granger Causality tests and likely ratio tests both questions the hypothesis of CHEER in the long run. Because CHEER approach is often tested with the cointegration method, we proceed to use Johansen methodology, the more powerful cointegration test, to reinvestigate the cointetgration relationship.
As stated in Section 3.3.2., Johansen methodology circumvents the small sample size distortion problem and it can detect the multiple cointegration vectors.
Further, the unnecessary step-by-step approach in Johansen methodology avoids any potential enlarged error from the previous step.
According to the CHEER model of equation (27), we perform the Johansen
49 37 The test equation is similar with equation (15). Here the number of restrictions is 10 (lag 0,1,2,3,4 in each equation) and the unrestricted model contains 12 coefficients.
test to detect the cointegration between the exchange rate, price and interest rate differentials. The Johansen test results are summarized in Table 12.
Table 12: Johansen Cointegration Test
Eigenv L-max Trace H0: r p-r L-max95 Trace95
0.1286 17.89 27.22 0 3 21.07 31.52
0.0510 6.80 9.33 1 2 14.90 17.95
0.0192 2.52 2.52 2 1 8.18 8.18
Table 12 indicates that the null hypothesis of no cointegration vector among
1,
st+ (p− p*)tand can not be rejected either by the L-max or Trace statistics. Therefore, the CHEER approach fails in the long run and the exchange rate determination model is only represented by PPP.
(i−i*)t
)
The fact of CHEER hypothesis in the long run is a surprise because we have found cointegration in the short run and many economic hypotheses tends to hold better in the long run. We can explain the puzzle from the history of financial markets. Financial markets did not develop well until the past twenty years and its role in the exchange rate determination is not apparent if we view it in a very long data span. If we test CHEER hypothesis using the recent data, the interest rate differential tends to become more significant because of its notable size. In short, the 130 years is too long and the effect of financial market in exchange rate determination in recent years is “diluted”.
The forecast error decomposition also suggests that the price differential (pt−pt* is the dominant factor in the determination of st+1. In the 5-lag ahead
forecast horizon, (pt −p*t) explains about 10 percent of the error of st+1, while
50
(it−it*) explains about 0.1 percent (See Appendix 5).
Figure 12 depict the impulse response functions. The responses of st+1 resembles that of (i i− *)t in the short run; however, it resembles (p−p*)in the long run and the price differential affects it more in magnitude.
Figure 12-1
Plot of Responses To s
0 2 4 6 8 10 12 14 16 18 20 22
-0.4 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8
s dp di
Figure 12-2
Plot of Responses To dp
0 2 4 6 8 10 12 14 16 18 20 22
-0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50
s dp di
51
Figure 12-3
Plot of Responses To di
0 2 4 6 8 10 12 14 16 18 20 22
-0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6 0.8 1.0
s dp di
Note: s = st+1,dp=(pt −pt*)anddi=(it −it*).
Source: DataStream and the author’s calculation
52
5. Conclusion
The major conclusion of this paper is that the validity of the exchange rate model combing PPP and UIP strongly depends on the data span. Either failure of PPP or UIP will lead to no cointegration between the future exchange rate, price and interest rate differentials without appropriate modification of the cointegration variables. The cointegration relationship does not necessarily exist, however, even when both PPP and UIP hold. Further, in the VAR model of exchange rate determination, the roles of price and interest rate differentials play in the exchange rate determination are remarkably different between the short run and the long run.
In the short-run analysis during the floating period with monthly data, PPP fails and UIP holds. The variables in the CHEER approach are differenced to improve the test power of the cointegration analysis. With this modification, cointegration is found between the future exchange rate, price and interest rate differentials. The likely ratio test indicates that the interest rate differential is an endogenous variable and it explains more of the movement of the future exchange rate. The Granger causality tests indicates that although both the price and interest differentials Granger-cause the future nominal exchange rate, some effects of the price differential is transmitted by the interest differential, making the latter explain more forecast variance of the exchange rate. The coefficients in ECM are significant; however, because of the price rigidity, the ECM is not stable in the short run.
In the long-run analysis using yearly data of longer than one century, both UIP and PPP hold. The interest rate differential becomes an exogenous variable
53
in that it only Granger-cause itself and is not Granger-caused by any of the other two. Further, the likely ratio tests suggest that the interest rate differential does not belong to the exchange rate determination model, and therefore it should not be included in the VAR system. The price differential is the dominant factor in the future exchange rate determination and it is also strongly affected by the exchange rate. The interest rate differential only explains a negligible portion of the forecast error of the exchange rate. Although the Engle-Granger methodology yields cointegration between the future exchange rate, price and interest rate differentials, the ECM is of poor quality because the coefficient of exchange rate is not significant, implying the exchange rate does not respond to the long-run deviation. Therefore, the Johansen methodology is employed to reinvestigate the relationship since it is more powerful. The Johansen test rejects cointegration. Therefore, the CHEER hypothesis does not work well in the long run.
The validity of CHEER in the short run and its failure in the long run looks like a puzzle because many economic theories tend to hold better in the long run.
This puzzle can be explained by the development of financial markets. The world financial markets have not been fully developed until recently. As Reinhart and Rogoff (2002) pointed out, the freely floating regime for yen did not begin until the end of 1977, before which the yen was either managed to float or pegged.
Because the basic idea of the CHEER approach is to jointly consider the goods and financial markets to determine the exchange rate, it is not a surprise that cointegration is easy to find if we use the recent data, in which the financial markets have developed and become large in comparison to the goods market.
54
For the same reason, supportive evidence of CHEER becomes difficult to find when we use the long-run historical data, in which the world financial markets are not fully developed. In this paper, the short-run and long-run data span the past 30 years and the 130 years, respectively. The effect of financial markets in the long run is “diluted” by the 100 years (from 1870 to 1970s) of the long-run data. Therefore, we expect that the CHEER approach will be supported with new future data.
The findings of this paper have important policy implications for monetary authorities. In the short run, it is more effective to control the interest rate to reduce the exchange rate volatility because the interest rate differential explains more of the forecast decomposition. In the long run, however, the price level becomes a dominant factor in the determination of the exchange rate. Therefore, the control of money supply should be given high priority because the interest rate differential becomes exogenous and it only explains a negligible portion of the long-run exchange rate movement.
55
Appendix
.
Appendix 1: Program Description
Software used:
I have used the following software to test the hypotheses, estimate the equations and depict the graphs.
1. Regression Analysis of Time Series (RATS) of version 5.11, Estima Company (2002).
2. Cointegration Analysis of Time Series (CATS) of version 1.03; Estima Company (2002).
Software Manual:
I have consulted the following three books to write and run the programs.
1. “RATS User’s Guide” (version 5, 2002), Estima Company. 2. “Rats Reference Manual” (version 5, 2002), Estima Company.
3. “RATS Handbook For Econometric Time Series” (1996), Water Enders, John Wiley & Sons, Inc.
56
Appendix 2: Data Source and Description
Data Source:
Short Run( Jan.1973〜 Jun. 2004, monthly): DataStream and the author’s calculation (data on the first day of each month)
Long Run(Year 1870〜 Year 2003,annually): Global Financial Data, Incorporation.
Directory of “Sample Data Access” on http://www.globalfindata.com/ (Sections of
“Commodity Prices”, “Exchange Rates”, and “Interest Rates” of U.S. and Japan).
Data Description:
S: yen/dollar exchange rate (New York market buying rates for the short run;
close rates on the last day of each year for the long run).
IJ: Japanese nominal interest rate level (euro rates in London market for the short run; 7-year government bond rate for the long run);
IU: U.S. nominal interest rate level (euro rates in London market for the short run; 10-year government bond rate for the long run);
PU: U.S. consumer price index (CPI);
PJ: Japanese consumer price index;
INFU: U.S. inflation level (calculated from “PU”);
INFJ: Japanese inflation level (calculated from “PJ”);
RIU: U.S real interest level (calculated from “IU” and “INFU”);
RIJ: Japanese real interest level (calculated from “IJ” and “INFJ”).
57
Appendix 3: Data Summary
Appendix 3.1: Short Run
Series Obs Mean Std Error Minimum Maximum
S 377 176.567533 67.484020 83.690000 305.67000 IU 353 0.070099 0.036694 0.010313 0.193750 IJ 311 0.037805 0.032370 -0.005000 0.144375 PU 376 219.346888 78.871551 78.681454 347.01170 PJ 376 155.058245 30.585180 67.300000 186.60000 INFJ 376 0.034114 0.050785 -0.016000 0.251000 INFU 376 0.049125 0.032070 0.011000 0.148000 RIU 351 0.024031 0.025832 -0.044500 0.090375 RIJ 309 0.022462 0.020216 -0.042250 0.069125
Appendix 3.2: Long Run
Series Obs Mean Std Error Minimum Maximum
S 134 107.617443 140.804934 0.902700 360.0000 PJ 134 23.806538 35.651962 0.018500 101.3000 PU 134 40.087280 49.057558 6.762000 184.3000 IJ 134 0.058510 0.021692 0.005400 0.148200 IU 134 0.045743 0.023343 0.016700 0.139800
58
Appendix 4: Short-Run Forecast Error
Appendix 4.1:
5
Step Ahead Forecast Error.
Appendix 4.2: Forecast Error Decomposition
Decomposition of Variance for Series ( )
Step Std Error ( ) ( )
Decomposition of Variance for Series ( )
Step Std Error ( ) ( )
Appendix 5: Long-Run Forecast Error Decomposition
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