Longer Data, Less “CHEER”—Case Study of Yen-Dollar Exchange Rate

This paper compares CHEER approach in both short-run (since 1973) and long-run (since 1870) with the yen-dollar exchange rate. The most important result is that CHEER is valid only in the period when the international capital market is developed enough. Historical data will render the interest rate parity insignificant and thus CHEER will fail. Also, the paper demonstrates that when either PPP or UIP fails, modification of the cointegration variables im-proves the power of the CHEER test.

Since then, the CHEER approach has become popular in the study of exchange rates. Extensively studied, the conclusion for CHEER is still mixed: some researchers find supportive evidence while others cannot. 1 Considering the different exchange rates, dissimilar empirical methods and varying data span in CHEER tests, mixed results may not be unusual. Still, it is important to summarize the key characteristics of the current research. Careful review of these papers raises at least two questions. The first question is why the cointegration relationship is often investigated between prices, interest rates and the contemporaneous, not the expected future exchange rate? Substituting the current exchange rate for the future rate is not in consistency with the CHEER approach because UIP hypothesis describes the relationship between interest rates and expected future exchange rate, not the current rate. Therefore, literally speaking, PPP and UIP are not combined correctly in the papers where the contemporaneous exchange rate is put to test.
IF PPP and UIP are the underlying theoretical framework for CHEER, then we have the second question: why PPP and UIP are not explicitly tested before checking CHEER? Or, equivalently, can the non-rejection of no cointegration be ascribed to the failure of PPP or UIP? This paper shows that either a failure of PPP or UIP does result in non-rejection of no-cointegration. The reason is that the linear sum of error terms in PPP and UIP will not be stationary if exactly one of them fails to hold. 2 In this case, it is impossible to find evidence supporting the CHEER approach. Modification of the variables under study, however, may increase the possibility of finding cointegration. Thus, an appropriate step in the investigation of whether PPP and UIP hold is essential to improve the power of cointegration analysis.
This paper aims at demonstrating that ignorance of the two above mentioned questions may be the reasons of mixed evidence for CHEER using the Yen/Dollar exchange rate. For the first question, perfect foresight is assumed to circumvent the lack of expectation in UIP testing. The result reveals that this simple modification is not trivial: cointegration among prices, interest rates and the exchange rate would not exist without adding expectations. 1 For positive results, see [5], [6], [7], etc. For the negative results, see [8], [9], [10] [11], and [12], etc. 2 Here it is important to note that when both PPP and UIP fail, it becomes possible to find evidence of supporting CHEER. The reason is that the sum of the two nonstationary residuals in PPP and UIP may be stationary because of the interaction of goods market and financial market. In [13], for example, although both PPP and UIP fail, cointegration relationship among prices, interest rates and exchange rate with dollar still exists. The remainder of this paper is structured as follows. Sections 2 and 3 analyze the short-run and long-run, respectively. Section 4 summarizes this paper.

PPP and UIP
Let t P and t P * denote the price levels for the home and foreign country respectively, and t S represents the nominal exchange rate (foreign price of domestic currency). PPP can be expressed as t t t P S P * = . By changing to lower-case letters to denote the natural logs, it can be rewritten as: Traditionally, Equation (1) is referred as the absolute PPP and the relative PPP is its first order difference: Tests for PPP refers to the investigation of time series properties of the real Uncovered Interest Parity (UIP) states that one unit of currency should have the same return whether invested in the domestic or the foreign markets at equilibrium. Let t I and t I * denote the domestic and foreign interest rates, respectively, and ( ) is stationary or not. This paper studies the yen/dollar exchange rate for three reasons. First, Japan and the US are both large trading countries and their economy has a substantial weight in the world. Second, the yen/dollar exchange rate is among the few main currencies that have historical data, which serves the purpose well. Third, the studies on yen/dollar exchange rate abound, making it easy to compare. The short-run data spans from January 1973 to November 2012, taken on the first day of each month from "DataStream". 3 Figure 1 presents the nominal, real exchange rates and changes in the real exchange rates in the short-run. Figure 2 plots the price and interest rate differentials. September 1985 is tested to be a structural break following the procedures in [14], which is widely believed to be the consequence of the Plaza . Price and interest rate difference in logs. 3 The specific time-series data consist of the following. 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); I jp : Japanese nominal interest rate level (euro rates in London market for the short run; 7-year government bond rate for the long run); I us : U.S. nominal interest rate level (euro rates in London market for the short run; 10-year government bond rate for the long run); P us : U.S. consumer price index (CPI); P jp : Japanese CPI; inf us : U.S. inflation level (calculated from "PU"); inf jp : Japanese inflation level (calculated from "PJ"); ri us : U.S real interest level (calculated from "IU" and "INFU"); ri jp : Japanese real interest level (calculated from "IJ" and "INFJ").
Accord. In the ADF test, the t-value is −2.2870, smaller in absolute terms than the 5-percent critical value −2.57. Similarly, unit root tests for UIP are performed, and the results are summarized in the following. Table 1 indicates that absolute PPP fails, while relative PPP and UIP hold in the short-run.

Exchange Rate Determination
The success of relative PPP may lead someone to believe that the price differential is enough to explain the movement of the nominal exchange rate. This section, however, argues that we should discard this optimistic idea. Assume that only the price differential between Japan and U.S. determines the Japanese nominal exchange rate, then we can write out this as the following th p order bivariate vector autoregressive (VAR) system in its standard form: where 1t e and 2t e are white-noise disturbances.
Equations (7a)  Therefore the interest differential is essential in the determination of the exchange rate in the short-run.
The above preliminary tests suggest that CHEER may be more appropriate to forecast the exchange rate in the short-run. Recall that the relative PPP and UIP are: and substituting Equation (8a) into Equation (9), if a country is suffering higher inflation or sharp interest rate increasing, its exchange rate will depreciate. It is worth noting that, compared with normal CHEER approach, which usually searches cointegration directly between t s , t t p p * − and t t i i * − , Equation (10) has two modifications. One is that it is of first-order difference, and the other is that it involves expectations, thought the expectations here are assumed to be perfect. Here we will show that the two modifications are necessary because we cannot find cointegration with either modification absent. To see this, consider the following three models: Model 1 is the most often seen practice in most papers, which investigate the relationship between the exchange rate, price and interest rate differentials. Model 2 adds expectation, which comes from the UIP hypothesis. Model 3 is further modified by adding a first-order difference, which is based on the empirical tests of PPP and UIP. The results of the cointegration tests are summarized in Table 2. Comparing the results in Table 2, we conclude that the effect of the two modifications is significant. Only model 3, i.e., the model with expectation and first-order difference can yield the cointegration relationship. 5

The VAR Analysis
Model 3 implies that the short-run model can be presented by a structural VAR system ( ) ( ) Based on the VAR system (11a), (11b) and (11c), the Granger causality test can be performed. The F-test and the corresponding significance level are reported in Table 3. Table 3 shows that It is worth noting that 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 difference in (10) to set up a VAR system containing Moreover, considering the interest rate differential is small in value and differencing it may cause it to appear white noise, its effect tends to be underestimated in (10). 6 Granger causality test between  Table 4.  0.54 percent in the same period. Therefore, the interest differential seems more essential in the determination of exchange rate movement in the short run.

Long-Run Analysis
The long-run data from the year 1870 to 2012, consisting of the exchange rate, the CPI index and the long-term interest rates for the US and Japan. 7 Figure 3 depicts the exchange rates and Figure 4 shows the price and interest rate differentials. [14] tests reveal that from 1870 to 2012, two structural breaks occurred. One is the year 1945, in which the yen depreciated more than 200 percent (from 4.29 to 15). The other notable break is the year 1970, in which the yen began to appreciate sharply due to the oil shock. We next proceed to test PPP and UIP, the same as the analysis for the short run. ADFs test of ˆt q and uip t  yield the statistics of −2.93 and −3.30, respectively. Both statistics exceed the 5-percent critical value of −2.88. Therefore, both PPP and UIP hold in the long-run. Substitute t s in Equation (1) into (5), and assuming perfect foresight, the long-run exchange rate model can be written as:

(
) ( ) s p p i i ξ * * + = − + − + (12) where t ξ is the sum of errors from PPP and UIP. The Johansen test results of Equation (12) are summarized in Table 5. Table 5 indicates that the null hypothesis of no cointegration vector among i i * − cannot be rejected either by the Maxλ or Trace statistics. Therefore, the CHEER approach fails in the long run.
To further understand the internal mechanism, a VAR system consisting of  ( ) ( ) terest rates in the long run. 11 The forecast error decomposition also suggests that the price differential ( )