Introduction
Exchange rate fluctuations affect economic decisions in a field of business activities like import and export, risk management. Therefore, evaluation of exchange rate behavior pattern seems important and both researchers and investors are interested about it. In this article, we will examine some characteristics of exchange rate for 3 countries: France, Belgium and Germany during Post-Bretton-Woods period (1973-1990). Firstly, summary statistics of these time series data are analyzed to capture basic data characteristics. Second, unit root test is performed by augmented Dickey-Fuller (ADF) approach to examine stationarity of time series data. Then, I applied Johansen approach to test whether long-run purchasing power parity (PPP) holds for bilateral exchange rates across time. Next, appropriate models are built and then used to forecast fluctuation of exchange rate in the future. Final part concludes.
Data
Two categories of data are provided here. The first one includes bilateral exchange rates among three countries: France, Belgium and Germany. The second one contains CPI of each currency. It is noted that these monthly data were downloaded from WRDS and stats.oecd.org and the time ranges from 1973 to 1990. In addition, the data has been split to two parts; the first one excluded last 10 observations whereas the second one is complete and used for purpose of forecasting. By taking logarithm of each series, discrepancy of measurement is eliminated. Besides, monthly bilateral exchange rate return is specified by taking first difference of logarithm exchange rate series and corresponding denotations are given below.
Table Denotations of variable
LB/F: logarithm of exchange rate for France against Belgium.
LB/G: logarithm of exchange rate for Germany against Belgium.
LF/G: logarithm of exchange rate for Germany against France.
LFRCPI: logarithm of price level for France.
LBECPI: logarithm of price level for Belgium.
LGECPI: logarithm of price level for Germany.

To identify distribution of each series, various descriptive statistics were calculated and reported in table 2. As can be seen from it, skewness and excess kurtosis that indicate deviation from normal distribution (with both skewness and excess kurtosis of 0) are observed for each series. More specifically, higher kurtosis like for DLB/F and DLB/G means that future return will either be extremely large or small, which is also known as fat tail distribution. Results of normality test confirmed this finding. Standard deviation, which measures deviation from average mean, is selected to quantify volatility level to some extent. By comparison of standard deviation of LFRCPI and LGECPI, one could conclude that price level in France fluctuated more than Germany and thus we could think that inflation level is not stable in France. Moreover, if we compare mean and standard deviation together for exchange rate series, it is not surprising to find that statement of ‘higher volatility tends to follow higher return’ is valid for DLB/G and DLF/G whereas fails for DLB/F, which might need further investigation.
Besides, last three columns calculated real exchange rate, which are supposed to be stationary if long-run PPP hold. However, from figure 1, it is obvious that this hypothesis fails for B/G, while nothing can be claimed for the rest. The equation for real exchange rate is written as, (1)
Where E (R) is the nominal (real) exchange rate in domestic currency per unit of foreign currency in period t; is the domestic price level and is the foreign price level.
Table Summary statistics

Figure Trajectories of real exchange rate

Unit root test
Recall, that one series is said to be integrated with order zero (I (0)) if it is stationary and I (1) if it becomes stationary after differencing once and so on. Generally, stationarity is an important feature that must be satisfied before it is ready to forecast trend in finance.