Analysis and forecast of the daily and intraday exchange rates with VAR and gls
Following Engle and Bollerslev, we studied the volatility transfers between different financial markets during the Asian crisis of 1998-1999. Our PhD thesis at the University of Basel, Switzerland (Zahnd, 2002) was published in 2002 and included mainly analysis examples for weekly exchange rates and daily stock exchange indexes. In the present work, we go beyond GARCH models and we try to model volatility by using the high and low prices of the currency for a given period (daily, hourly, 5mn data).
Models for the daily exchange rates
We use the high, low and close prices for 2 time series: EUR/USD and GBP/USD over a period of 6 to 9 years (more than 1400 daily observations). We have then a total of 6 times series and we look for cointegration vectors for these series. We use several cointegration vectors and each of them is statistically significant in at least one VAR-VEC equation (in the vector error correction form).
We include dummy variables for the different days of the week. We also multiply the regressors by these dummy variables, to study the differential impact of each day of the week on each series. We include only 3 lagged values in the VAR-VEC model, because this model is for daily values. We study the autocorrelations of the residuals and of the square residuals of the model with Portmanteau tests with Q-statistics of Chi-Square. We also study a model where a moving average of order two or three is applied to our original time series.
We apply the same procedure for two other series: USD/CH and USD/JPY. We compare both models with a neural network model using 20 or 25 explanatory variables.
Models for the hourly exchange rates
We use the same approach. We look for a cointegration relation in the period of interest. Sometimes, less than 3000 observations must be used, otherwise we could not find any cointegration relations. We use a high number of lagged values of high, low and close prices. We project hourly forecasts over 24 hours.
Use of GLS
The Gauss-Markov theorem says that the OLS estimators have the smallest variance among the unbiased estimators. The Gauss-Markov theorem depends on the assumption that the errors are uncorrelated with constant variance. In presence of autocorrelations and time varying variance, we use GLS.