Daily data for the High, Low, Open and Close prices for the US Dollar/German Mark (USDDEM), US Dollar/Japanese Yen (USDJPY) and Pound Sterling/US Dollar (GBPUSD) were used. The period studied was from August 1986 to August 1996 for USDDEM and USDJPY, yielding 2770 observations for each series. For GBPUSD only data from August 1989 to August 1986 was available, producing 1890 observations. The data was log-transformed and checked for outliers and missing observations. Obvious outliers were removed and missing observations in the series were closed by linear interpolation. The data set was obtained from Knight Ridder Financial Ltd.
The currency pairs selected for this study comprise the three most actively traded currencies. The average daily turnover for these three currency pairs in April 1995 comprised more than half (50.4%) of the total foreign exchange market turnover11. According to the Bank of International Settlements, the daily average of foreign exchange market turnover in April 1995, net of local and cross-border interdealer double counting, totalled for all currency pairs to 1136.9 billions US dollar. Trading in USDDEM had a percentage share of 22.3%, USDJPY of 21.3% and GBPUSD of 6.8%.
Since non-stationarity is a pre-condition of cointegration, the daily data for the three exchange rate series on High, Low, Open and Close prices was tested for unit roots using an Augmented Dickey Fuller (ADF) and Phillips Perron tests. USDJPY and GBPUSD appeared to be I(1) since the null hypothesis of a unit root could not be rejected for either of the eight series at the 5% level, and the first differences proved to be stationary. The results of the ADF test are presented in the appendix, in Tables 1 to 6.
However, USDDEM appeared to be borderline stationary under the conventional ADF test and the Phillips Perron test. When considering heteroskedasticity-consistent standard errors in the ADF test, the null hypothesis for a unit root could not be rejected for USDDEM12. All three currencies therefore proved to be I(1) and integrated of the same order and thus satisfy an important precondition of cointegration. Clearly, a cointegration analysis is only meaningful if the time series properties of the variables under consideration have different time series properties.
Since High, Low and Close prices are drawn from the same time series, one might argue that these would share the same (indeed identical) time series properties and hence finding a cointegrating vector is an expected feature. It is, however, exactly this point that presents the motivation of our analysis: High, Low and Close prices of the same time series have different time series properties. Looking at daily exchange rate data, then according to Takens’ delay time concept (Takens 1981), the Open series represents an embedding of the Close series since Open and Close are always recorded at the same points in time; i.e. the opening and closing time of the corresponding market13.
Low and High prices, however, carry different information - they do not coincide with a certain times of the day but with the extreme prices of the trading day. Since Low and High prices do not occur at specific time of the day, neither Low nor High series can therefore be expressed as an embedding of the Close series. Thus, High, Low and Close should not have the same dynamic characteristics and it should be possible to treat them as three different discrete time series.
One way to illustrate the different dynamics of the High, Low and Close is to look at the difference between the autocorrelation function (ACF) of the differences of these series. If High, Low and Close had the same time series properties, the difference between the series should follow a white noise process: the ACF between the differences of High and Close, High and Close and Close and Low should therefore decay immediately to zero.
However, this is not the case as Figure 1, which represents the plot of the ACF of the difference between the High and the Close prices for the first 500 lags of GBPUSD, demonstrates. The slow decay of the ACF is significant and hence the difference between the three series are structural and not random in nature. It takes roughly around 100 lags for both series before the ACF becomes zero for the first time14.
Another indication for the different time series properties can be derived from the results of the Augmented Dickey Fuller unit root test for the different time series. The results of the ADF test, presented in the appendix, show that different lag lengths have to be included in order to yield zero correlation in the residuals. The different lag lengths for High, Low and Close and the same lag lengths of Close and Open series reflect the different degree of serial correlation in the residuals and hence support the different time series properties hypothesis15.
11 Bank of International Settlements (BIS), 1996, table F-4.
12 The borderline nonstationarity of USDDEM could be due to heteroskedasticity in the residuals and/or due to additive outliers. Since heteroskedasticity in the residuals of an otherwise properly specified linear model leads to inconsistent covariance matrix estimates, and thus faulty statistical inference (White, 1980), and since the presence of additive outliers move standard inference towards stationarity (Hoeck, 1995), the unit root hypothesis could be wrongly rejected. An easy remedy for these problems is the combination of the OLS-based ADF with heteroskedasticity-consistent standard errors (robusterrors), a procedure proposed by Lucas (1995) in order to yield more robust standard DF t-statistics and also to provide protection against the distortive effect of additive outliers.12 The inclusion of heteroskedasticity consistent standard errors yields different critical values for the DF test statistics, however, since these values converge with increasing sample size towards the standard DF test statistics, these values did not prove critical for our sample. An alternative approach is to use maximum likelihood estimators which have, in general, a higher robustness than OLS estimators; see Lucas(1995), Hoeck(1995).
13 Takens theorem states (Takens 1981) that if x(t) is a discrete representation of a continuous time series, by applying delay time co-ordinates, a different discrete time series yt = g(xt) can be constructed that has the same dynamic characteristics as x(t). yt m = (yt, yt+m,...,yt+m-1) is then called an m-embedding of x(t).
14 The ACF for the differences between the High, Low and Close series show similar results for USDDEM and USDJPY and are therefore not presented here in order to save space.
15 The ACF of the difference between High and Low (figure 3) shows a particularly slow decay and thus reflects a high degree of structural relationship between these two series. This, however, is not surprising since the absolute difference between High and Low denotes a simple measurement of daily volatility and a slowly decaying ACF therefore indicates the presence of autoregressive conditional heteroskedasticity (ARCH), a feature of daily exchange rates widely accepted within the foreign exchange market microstructure theory. See, for example, Baillie and Bollerslev (1989) and the references therein.
Prof. Ronald MacDonald, Prof. Norbert Fiess
Next: Structural Econometric Modelling
Summary: Index