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. The graph shows these points collected along several rays from the origin. They correctly attribute this ``compass rose'' pattern to discreteness in the data. 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. We apply our technique to an important episode of Russian monetary history. Generally, the finance ministry actively intervened to influence the ruble exchange rate. The one exception was during Nicolai Bunge's tenure as finance minister. Bunge's successor, Ivan Vyshnegradsky, was an unusually vigorous interventionist. 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.