The effect of the modification and the increased test power

It is worthy noting that the supportive evidence of CHEER approach is found according to equation (19), the first-order differenced model. In most of the recent CHEER papers, however, cointegration is directly searched between s_t+1,

*

p t and ( . Here we will show that this modification is necessary because we can not find cointegration if using the same method as the recent papers.

*

t t

i −i

) )

Now we proceed to use Johansen methodology, the more powerful cointegration test, to investigate the cointetgration relationship betweens_t+1,

(p_t − p*_t) and (i_t −i_t^*). Johansen methodology circumvents the small sample size

28 Table 5 is excerpted from Enders (1995).

31

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. The Johansen test results are summarized in Table 6²⁹.

Table 6-1: Summary of the Johansen test betweens_t, (p_t − p^*_t) and (i_t −i_t^*)

Eigenv L-max Trace H0: r p-r _{L-max95 Trace95}

0.0679 20.83 39.77 0 3 21.07 31.52

0.0483 14.66 18.95 1 2 14.90 17.95

0.0144 4.29 4.29 2 1 8.18 8.18

Table 6-2: Summary of the Johansen test betweens_t+1, (p_t −p_t^*) and (i_t −i_t^*)

Eigenv L-max Trace H0: r p-r _{L-max95 Trace95}

0.0617 18.85 40.37 0 3 21.07 31.52

0.0562 17.11 21.52 1 2 14.90 17.95

0.0148 4.40 4.40 2 1 8.18 8.18

Table 6-3: Summary of the Johansen test between∆s_t+1,∆(p_t −p^*_t)and ∆ −(i_t i_t^*)

Eigenv L-max Trace H0: r p-r _{L-max95 Trace95}

0.1487 47.51 87.52 0 3 21.07 31.52

0.0817 25.14 40.01 1 2 14.90 17.95

0.0492 14.88 14.88 2 1 8.18 8.18

Table 6-1 is the result of Johansen methodology based on the cointegration

29 Source of the L-Max and L-Trace statistics: Enders (1996)

32

analysis between the contemporaneous exchange rate, price and interest rate differentials, which is the method adopted by most of the other researchers.

Table 6-2 gives the cointegration analysis between the expected future exchange rate, price and interest rate differentials while Table 6-3 is the result of the modified model, i.e., the differenced model of equation (19). Comparing the results in the three sub-tables, we conclude that the effect of this modification is significant. Only Table 6-3, i.e., the sub-table with the modified model can yield the cointegration relationship³⁰.

3.3.3.  

e1t

The error correction model (ECM)

According to Granger representation theorem, cointegration and error correction are equivalent representations, i.e. cointegration implies error correction and vice versa. The next stage involves the error correction model for the VAR system (20), which should have the form of:

1 1 1 1

30 In Table 6-1 and Table 6-2, L-Max and L-Trace contradicts each other in that only the latter rejects cointegration. We should pin down the number of cointegration vectors and therefore we conclude that there is no cointegration.

31

βp and β_i are the cointegrationg vectors given by equation (21).

33

estimation of deviation from long-run equilibrium in period(t−1), as proposed by Engle and Granger, it is possible to use the saved residuals {ε_t^/₋₁}

1 1

obtained from the regression of equation (21) as an instrument for the expression . Based on this substitution, the error-correction model (ECM) is estimated to be:

Φ

The value in the parenthesis under each equation shows t-value and the significance level. In equation (24.1), the error-correction term is of highly significance level. This suggests that the future change of exchange rate is strongly determined by the long-run equilibrium. Further, the sign of the error-correction terms ε_t−1 in equation (24.2) is not consistent with the theory, making (24.2) looks more like error-amplifying rather than error-correction. The significance level, however, is very low. This indicates that the differenced price level is rather rigid and does not respond to the deviation from the long-run equilibrium level much.

3.3.4. Causality test and innovation accounting

In VAR analysis, we say that does not Granger-cause q if the lagged values of do not appear in the equation for , that is, the current and lagged do not help to predict the future value of . The null hypothesis that does not Granger-cause can be tested by doing a joint F-test: regress on both the lagged and the lagged and see the significance of the coefficient of lagged .

q1t

The following part of this section investigates the Granger causality relationship among the three variables in the exchange rate determination model: ∆s_t+1 ,

(p_t p_t*)

∆ − and ∆ −(i_t i_t^* while performing the innovation accounting.

∆ )

Based on the VAR system (20), we first conduct the Granger causality test. The F-test and the corresponding significance level are reported in Table 7.

Table 7: Summary of Granger-Causality Tests

variable st₊1 ∆(p_t −p^*_t) ∆ −(i_t i_t^*)

1

st₊

∆ 21.57 0.000 0.024 0.976 0.722 0.487

4.121 0.017 7.184 0.001 5.234 0.006 (i_t i_t*)

∆ − 1.741 0.177 0.361 0.697 2.622 0.074

(p_t p*_t)

∆ −

Table 7 shows that ∆s_t+1 Granger-causes only itself; ∆ −(i_t i_t^*) also roughly Granger-causes only itself, while ∆(p_t− p_t^*) Granger-causes all the three variables.

To further identify the different roles ∆(pt −p^*t) and ∆ −(it it^*) play in the model of exchange rate determination, we can decompose the forecast error variance. The forecast error variance decomposition tells us the proportion of the movements in a sequence due to its own shocks versus shocks to the other variables.

We use the variance matrix in the VAR system (20) to obtain 1-step ahead through 24-step ahead forecast errors. Appendix 4 shows the first five impulses, together with the variance decomposition. To measure all responses in terms of standard deviations, we depict the following impulse response functions.

35

Figure 6-1

Plot of Responses To ds

0 2 4 6 8 10 12 14 16 18 20 22

-0.2 0.0 0.2 0.4 0.6 0.8 1.0

ds ddp ddi

Figure 6-2

Plot of Responses To ddp

0 2 4 6 8 10 12 14 16 18 20 22

-0.25 0.00 0.25 0.50 0.75 1.00

ds ddp ddi

36

Figure 6-3 Source: DataStream and the author’s calculation

From the Granger causality test and innovation accounting analysis, we can see that the price differential itself can not explain all the movements of the future nominal exchange rate. ∆(pt−pt^*)

)

st

explains only 0.011 percent of the movement of while explains 0.34 percent in the 5-lag (five month) ahead horizon. The differenced interest rate differential serves as a “channel” in the sense that it does not Granger cause

1

st₊

∆ ∆ −(i_t i_t^*

+1

∆ directly but it explains more of the movement of the exchange rate. Some of the effects of ∆(p_t−p_t^*)on ∆s_t+1are conveyed by this “channel”: ∆(p_t−p_t^*) affects ∆ −(i_t i_t^*) and then affects the nominal exchange rate movement.

(i_t *

∆ −i_t)

The exchange rate determination model is derived from the economic theories and the differenced variables make it a little difficult to grasp the real effects since differencing tends to smooth the various shocks. Moving away the

37

difference in equation (19) to set up a VAR system containings_t+1, (p_t− p_t^*) and and plotting the impulse response function will give a more apparent impression, as shown in Figure 7. The movement of

(i_t−i_t*)

Figure 7-3

Plot of Responses To di

0 2 4 6 8 10 12 14 16 18 20 22

-0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25

s dp di

Note: s = s_t+1,dp=(p_t − p_t^*) and di=(i_t −i_t^*)

Source: DataStream and the author’s calculation

Moreover, considering the interest rate differential is small in value and differencing it may cause it to appear white noise³², its effect tends to be underestimated in model (19). Granger causality test betweens_t+1, (p_t−p_t^*) and

yields a different result, as is shown in Table 8.

(i_t−i_t*)

Table 8: Summary of Granger Causality Tests

variable st₊1 (p_t −p^*_t) (i_t−i_t^*)

1

st₊ 6599 0.000 2.286 0.103 7.41 0.001

(p_t −p*_t) 3.45 0.033 61750 0.000 5.86 0.003 (i_t−i_t*) 4.832 0.009 1.423 0.243 1229 0.000

32 The mean of ∆ −(it it^*) is0.000201^.

39

The value in italic forms shows the significance level. As indicated from the Table 8, in the 5 percent significance level, the price differential (p_t −p_t^*)

(i_t i_t*) Granger-cause both s_t+1 and interest rate differential (i_t −i_t^*) ; − Granger-causes and it explains 9.715 percent of the 12-lag ahead forecast error of , leaving

1

st₊ 1

st₊ (p_t−p_t^*) only account for 0.772 percent in the same period.

Therefore, the interest differential seems more essential in the determination of exchange rate movement in the short run.

40

4. Long-Run Analysis (Year 1870~2003)

4.1.

+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 D_L1985

µ1 and µ₂ are coefficients of D_L1970 and D_L1985, 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 (p_t−p^*_t and (i_t−i_t^*) 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 η_t = _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,s_t+1, (p_t −p_t^*) and (i_t −i_t^*). 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 (i_t−i_t^*). 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 ε_t^/ 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 ε_t^/₋₁ with the real error termε_t−1,

, and e white-noise disturbances which may be correlated with each other; and

e1t e_{2t 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 ε_t^/₋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 betweens_t+1,(p_t− p_t^*), (i_t−i_t^*) 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 (p_t− p_t^*) (i_t −i_t^*)

1

st₊ 89.91 0.000 12.87 0.000 0.042 0.989 (p_t−p_t*) 16.91 0.000 154.13 0.000 1.482 0.223 (i_t−i_t*) 2.300 0.081 1.696 0.172 59.404 0.000

48

Table 11 indicates that at 5 percent significance level, s_t+1 and (p_t −p_t^*) both Granger-cause themselves and each other; (i_t −i_t^*) 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.

(i_t−i_t*)

)

The exogenous property of interest rate differential raises a serious question of system (28), i.e., is it appropriate to include (i_t −i_t^* 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 i_t−i_t^*

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 (p_t−p_t* is the dominant factor in the determination of s_t+1. In the 5-lag ahead

forecast horizon, (p_t −p^*_t) explains about 10 percent of the error of s_t+1, while

50

(i_t−i_t*) explains about 0.1 percent (See Appendix 5).

Figure 12 depict the impulse response functions. The responses of s_t+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 = s_t+1,dp=(p_t −p_t^*)anddi=(i_t −i_t^*).

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

This puzzle can be explained by the development of financial markets. The world financial markets have not been fully developed until recently. As

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