In Chapter 1, I develop a test for the martingale hypothesis using the fact that a continuous martingale is time-deformed browninan motion, where the deforming process is quadratic variation of the martingale. Sampling a martingale at equal increases in quadratic variation and taking first differences, we may obtain variables that are identically, independently distributed as normal. I propose tests that involve thus transforming a sample and testing for the martinagale hypothesis by using the distance between the empirical distribution of the transformed variables and the standard normal distribution using Kolmogorov-Smirnov and Cramer-von Mises statistics. Asymptotics in this setting involve sampling the process more frequently over a given sampling horizon in addition to sampling over longer horizons. Simulations show that the size and power performance of the tests is rather satisfactory. The test is employed on a variety of financial futures to test for a risk premium.
In Chapter 2, I develop a method of estimating drift terms in diffusion models that involves transforming the discretely observed sample and performing a least squares regression. This method allows for the estimation of a parametrically specified drift term when the diffusion term is unspecified. The procedure involves sampling data at equal increases in estimated quadratic variation and taking first differences. The transformed variables can be written as the sum of two quantities, a function of the drift parameters and, a regression error that forms a martingale difference sequence and is homoscedastic in the limit when the sampling interval is small. Thus the procedure relies on asymptotics of a small sampling interval as well as a large sample. Simulations show that the procedure outperforms ordinary least squares for nonstationary processes.
In Chapter 3, I revisit the testing problem studied in Chapter 1 and consider a test based on second moments instead of distrbutional distance. Simulations show that for mean-reverting processes, the test is more powerful than the tests of Chapter 1. Further, I present a time-change result for discrete martingales, and show that discrete martingales sampled at equal increases in quadratic variation and differenced are homoscedastic to an error.