The Angular Distribution of Asset Returns in Delay Space

We develop and apply a set of hypothesis tests with which to study changes in the angular distribution of points in delay space. Crack and Ledoit (1996) plotted daily stock returns against themselves with one day's lag. (This might be described as a plot in \delay space"). The graph shows these points collected along several rays from the origin. They correctly attribute this \compass rose" pattern to discreteness in the data. Asset prices move in discrete ticks. Our testing procedures allow one to test for changes in Crack and Ledoit's compass rose pattern. Our case study gives an example of such a change in distribution being caused by a change in regime. We plot the number of points along a given ray of the compass rose against the angle of that ray. This creates a \theta histogram" which describes the angular distribution of the points in delay space. We compare this distribution to a standard theta histogram created by a simple bootstrap procedure. The 2 test is then performed in order to estimate quantitatively the consistency of the actual data with the standard theta histogram. Extensions of this technique are discussed. We apply our technique to an important episode of Russian monetary history. In the late nineteenth century, the \credit ruble" was a oating currency unlinked to precious metals. Generally, the nance ministry actively intervened to in uence the ruble exchange rate. The one exception was during Nicolai Bunge's tenure as nance minister. Bunge's successor, Ivan Vyshnegradsky, was an unusually vigorous interventionist. The shift in regime from Bunge the non-interventionist to Vyshnegradsky the interventionist produced a marked change in the behavior of the ruble exchange rate. The angular distribution in delay space of the ruble's exchange rate against the German mark shifted dramatically under Vyshnegradsky. Hypothesis tests support the view that Vyshnegradsky's activism caused a disproportionate number of points of the compass rose to accumulate on the main diagonals in delay space. The theory of \Big Players" (Koppl and Yeager 1996) helps to explain why. Our results are consistent with those of Broussard and Koppl (1996) who use a GARCH(1,1) model.