3. Short-Run Analysis (Jan. 1973~Jun. 2004)
3.1. Pretests of PPP and UIP
3.1.1.
t
Unit root process and Dickey-Fuller test
If PPP holds between U.S. and Japan, the shocks to yen’s real exchange rate,q , should be temporary in that will be mean-reversion. Specifically is called stationary if it exhibits mean reversion and has a finite variance. A unit root process is not stationary. Consider the simple AR (1) process:
t qt qt in the autoregressive polynomial (1−ρL) lies outside the unit circle.
Ifρ =1, then qt becomes a random walk process,∆ =qt εt, and
23 Considering a growing importance of international financial markets, I use the euro rates in this section.
24 For any variable ,qt L qk t =qt− .
18
assume q1 =0. Thus,
Therefore, is no longer stationary because the variance depends on t. We should be careful about the nonstationary variables because there might be what Granger and Newbold (1974) called a spurious regression. A spurious regression has significant t-statistics and a high
qt
R2, but the process is without any economic meaning. Therefore pretesting for nonstationarity the variables in a regression equation is of extremely importance. There are many kinds of unit-root tests and we first introduce the Dicker-Fuller (DF) test since it is the most straightforward and it is also the foundation of many other more complicated tests.
contains a unit root. The test involves estimating equation (9) using OLS in order to obtain the estimated value of
t
γ and the associated standard error. By comparing the resulting t-statistic with the appropriate value reported in the DF tables25, we can determine whether to accept or reject the null hypothesis γ =0.
3.1.2.
Nonstationary process and PPP test
After introducing the basic property of unit root and its test method, we proceed to conduct some unit root tests on the yen/dollar real exchange rate.
t
19 25 DF table contains three sub-tables for three different regression equations, respectively.
Besides equation (6), the other two are: ∆ =qt α γ0+ qt−1+ε and∆ =qt α γ0+ qt−1+α1t+εt.
Some tests used in the following sections will be more complicated and powerful than the DF test.
3.1.2-1 The DF test
We add a constant to the right side of equation (2) and regress the nominal exchange rate st on a constant and the price differential(pt−p*t), The restriction is binding; however, the F-statistics value 188.3298 of equation (11) exceeds the 5 percent critical value of the 3.84 in the statistical table. Let denotes the real exchange rate and we depict the graph of the nominal and real exchange rates and price differential as Figure 1.
*
Nominal and Real Exchange Rate in logs (Yen/Dollar)
1973 1975 1977 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 4.4
Figure 1-B
Price Differential in logs (JP-U.S.)
1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 -0.7
Source: DataStream and the author’s calculation
We test for the unit root according to equation (9) ∆ =qt γqt−1+ε to obtain:γ =-2.19143. Comparing the value of γ with the DF table implies that we can not reject the null of unit root process inqt, because the absolute value of γ is smaller than that of the 10 percent critical value , not to mention the 5 and 1 percent critical values .
(-2.571) (-2.87 and -3.44)
3.1.2-2 The Augmented DF test
Not all time-series processes can be well represented by the first-order autoregressive process∆ =qt α γ0+ qt−1+α1t+εt. When we test unit roots in higher order equations such as the pth order− autoregressive process, the DF test equation should be modified as:
Equation (11) is called the augmented Dicker-Fuller (ADF) test and the coefficient of interest isγ ; ifγ =0, the equation is entirely in first differences and so has a unit root. The ADF statistics table is the same as DF test.
Here we select to test the sequence of Japanese real exchange rate . The test result is:
4
p= qt
0.013159
γ = − witht−value −1.9910. Since the is smaller than that of the corresponding 10 percent value
t−value
( 2.57)− , the ADF test suggests that we can not reject the null hypothesis thatγ =0.
3.1.2-3 The Phillips-Perron Tests
The distribution theory supporting the Dicker-Fuller tests assumes that the errors are statistically independent and have a constant variance. The changes of the real exchange rate are depicted as Figure 2, which strongly indicates that the sequence is serially correlated in that positive (negative) deviation persists for a rather period. Therefore, we should take some reservation about the power of the DF test.
qt
Figure 2
Changes in Real Exchange Rate (Yen/Dollar)
1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003
-0.125
Source: DataStream and the author’s calculation
22
Phillips and Perron (1988) developed a test procedure that allows for the distribution of errors. The Phillips-Perron test (PP test) statistics are modifications of the DF t-statistics and the critical values are precisely the same.
The conclusion based on the PP test is the same as DF test; the γ statistics value is tested to be -2.3588, absolute value lesser than the 10 percent critical value( 2− .57).
From above we know that all kinds of unit root tests yield the same conclusion:
we can not reject the null hypothesis of unit root in the process of real exchange rate qt. Therefore, PPP does not seem to hold in the floating period.
Although PPP fails in this case, we can investigate further by testing relative PPP indicated by equation (3):∆ = ∆ − ∆st pt pt*. Thus, unit root test on the
“relative” real exchange rate is necessary since it becomes the error term in equation (3).
qt
∆
*
t t t
q s p
∆ = ∆ − ∆ + ∆pt (13) The test results can be summarized as Table 1 with γ denoting the coefficient of the first difference of ∆qt−1 in unit root tests.
Table 1: Summary of the Unit Root Tests for ∆qt
Test DF ADF PP
t-statistics of γ -13.8278 -8.5079 -13.7641
Compare the t-statistics or γ with the DF test table to know that all exceed the 1 percent critical value(- . Therefore relative PPP seems to hold in the floating exchange rate period.
3.44)
23
3.1.3. Stationary process and UIP test
Recall the UIP theory E st( t+1)− = −st it it* and we know that to test UIP involves the estimation of future nominal exchange ratest+1. In this paper we assume the perfect foresight, i.e. there is no expectation error: . Under this assumption, UIP becomes: Then we use the following regression to test UIP:
1
The test result can be summarized as the Table 2. Since all the absolute values of t-statistics are greater than that of the 1 percent critical value (-3.453), we can reject the null of unit root. Therefore we conclude that the error term in UIP, εtis stationary and UIP holds in the floating period.
Table 2: Summary of the Unit Root Test for UIP
Test DF ADF PP
t-statistic of γ -7.4504 -4.6556 -7.3324
Figure 3, 4 and 5 depict the nominal and real interest differential and the
26 ∆st+1 and ∆it are both tested to be I(1) processes.
24
relative CPI changes between U.S and Japan in the floating period.
Figure 3
Nominal Interest DIfferential(Japan-U.S)
1978 1981 1984 1987 1990 1993 1996 1999 2002 -0.12
-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04
Source: DataStream
Figure 4
Real Interest Differential(Japan-U.S)
1978 1981 1984 1987 1990 1993 1996 1999 2002 -0.075
-0.050 -0.025 0.000 0.025 0.050 0.075 0.100 0.125
Source: DataStream and the author’s calculation
25
Figure 5
CPI Index Change Differential(Japan-U.S)
1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 -4
Source: DataStream and the author’s calculation